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Lecture Notes
Financial Mathematics 1 & 2
Gerald Trutnau
Department of Mathematical Sciences
Seoul National University
Version: December 1, 2015
Non-Corrected version
This text is a summary of the lecture
Financial Mathematics 1 & 2 held atSeoul National University, Spring and Fall Term 2015
Please email all misprints and mistakes to
1
The main source for this lecture is
- Lamberton, Damien; Lapeyre, Bernard: Introduction to stochastic calculus applied tofinance. Second edition. Chapman & Hall/CRC Financial Mathematics Series, Boca Ra-ton, FL, 2008.
Here are some additional references:
1. Baxter, Martin, Andrew Rennie: Financial Calculus: An Introduction to DerivativePricing, Cambridge University Press, 1996.
2. Bjork, Tamas: Arbitrage Theory in Continuous Time, Oxford University Press.
3. Elliott, Robert J.; Kopp, P. Ekkehard: Mathematics of financial markets. Secondedition. Springer Finance. Springer-Verlag, New York, 2005.
4. Hull, John: Options, Futures, and Other Derivatives, 6th ed., Prentice Hall, 2006.
5. Karatzas, Ioannis and Shreve, Steven: Methods of mathematical finance - Springer,1998.
6. Shreve, Steven E.: Stochastic Calculus for finance I, II, Springer, 2004.
7. Wilmott, Paul; Dewynne, Jeff; Howison, Sam: Option Pricing: Mathematical ModelsAnd Computation, Oxford Financial Press; 1994.
2
1 Introduction to options
An option gives its holder the right (but not the obligation) to buy or sell a certainamount of financial asset, by a certain date for a certain strike price.
The writer (=seller) of the option has to specify:
(i) type of option
call : option to buy
put : option to sell
(ii) underlying asset (stock, bond, currency, etc. ...)
(iii) the amount of underlying asset
(iv) the expiration date (=maturity)American option: can be exercised at any time before maturity
European option: can be exercised only at maturity
(v) the exercise price (=strike price = price at which the transaction is done, if theoption is exercised)
The price of an option is also called premium.
In an organized market, the premium is quoted by the market. Otherwise, the problem isto price the option, but even if the option is traded on a market, it can be interesting todetect possible abnormalities in the market.
Example 1.1 European call option
Ct = its price at time t < TT = expiration dateK = exercise priceSt = stock price at time t
3
At time T , the buyer of the European call option has the right to buy the stock for K.Thus the value of the call at maturity is
(ST −K)+ = max(0, ST −K) =
0, if ST ≤ K
(”holder does not exercise the option”)
ST −K, if ST > K
(”holder buys the stock for K and sells it back on
the market for ST”)
Two questions arise:
(i) How much should the buyer pay for the option (at time 0) ?
”C0 = premium =?”
(”Problem of pricing the option”)
(ii) The writer of the call, who earns the premium initially, must be able to deliver astock at price K at time T . How should the writer generate an amount of (ST −K)+
at time T ?
(”Problem of hedging the option”)
We can only answer under additional assumptions:
basic assumption: absence of arbitrage opportunity in the market, i.e.
”there is no riskless profit available in the market”
A rigorous mathematical description will be given in chapter 2. Here as an example con-sider:
Arbitrage and put/call parity:
Let Ct (resp. Pt) be the value at time t of an European call (resp. put) on the samestock with value St at time t ∈ [0, T ], and
T = maturity of both, put and call
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K = exercise price of both, put and call
At time T
CT = (ST −K)+
PT = (K − ST )+
ThusCT − PT = (ST −K)+ − (K − ST )+ = ST −K (1)
Assume additionally: it is possible to borrow or to invest money throughout [0, T ] atconstant rate r.The last is called continuous compounding. Thus a capital X deposited in the bank (ormoney market account) at time t accumulates to
X · limn→∞
(1 +
r
n
)n(T−t)
︸ ︷︷ ︸”periodic compounding withn periods during time T−t”
= X · er(T−t)
a cash sum K at time T can be generated by depositing Ke−r(T−t) at time t.
Claim: under absence of arbitrage opportunity, the call/put prices on the stockS must satisfy (1) with appropriate discounting of K for all t < T , i.e.
Ct − Pt = St −Ke−r(T−t), ∀t < T.
(”put/call parity”)Assume, for instance
Ct − Pt > St −Ke−r(T−t) for some t < T. (2)
Then, a writer could do the following at time t:
He buys a share of stock and a put, and sells a call.
”net value of the operation” Ct − Pt − St =: Vt
If Vt ≥ 0: he invests Vt at rate r until T .
If Vt < 0: he borrows Vt at rate r until T .
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Two outcomes are possible:
1)ST > K︸ ︷︷ ︸
put notexercised !
: K − ST︸ ︷︷ ︸”he has to pay the
exercised call”
+ (Ct − Pt − St)er(T−t)︸ ︷︷ ︸”sum he gets/pays for
investing/borrowing at interest rate r”
+ ST︸︷︷︸”he sells his
share of stock”
>(2)
0
2)ST ≤ K︸ ︷︷ ︸
call notexercised !
: (K − ST )︸ ︷︷ ︸”he exercises
his put”
+ (Ct − Pt − St)er(T−t)︸ ︷︷ ︸”same, he clears his bank
account again”
+ ST︸︷︷︸”he sells his shareof stock again”
>(2)
0
In both cases he made a profit without using any initial capital.
(”arbitrage strategy”)
Black-Scholes model and its extensions:
- no arbitrage arguments lead to many interesting equations, but are not suffi-cient for deriving pricing formulas
- Black-Scholes (1973) were the first to suggest a model whereby one can derive anexplicit price for a European call that pays no dividend. According to their model,the writer of the option can hedge himself perfectly.
The Black-Scholes model and its extensions are based on stochastic calculus, and in par-ticular on Ito’s formula.
2 Discrete time models
In this chapter we present the main ideas of option theory in discrete-time models.
2.1 Discrete-time formalism
2.1.1 Assets
Let (Ω,F, P ) be a finite probability space, i.e. Ω = ω1, . . . , ωN0, N0 ∈ N, P(Ω) the powerset of Ω, i.e. the set of all subsets of Ω. Consider an increasing sequence of σ-algebras inΩ:
∅,Ω = F0 ⊂ F1 ⊂ · · · ⊂ FN = P(Ω) = F
N = ”time horizon” (often the maturity of the options)
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Fn = ”information available at time n”
We assume P (ω) > 0 for all ω ∈ Ω. For any time n = 0, . . . , N , we consider (d + 1)positive Fn-measurable random variables S0
n, S1n, . . . , S
dn (in this chapter all r.v.’s are
real-valued), which are financial assets
Sn := ( S0n︸︷︷︸
riskless
asset
, S1n, . . . , S
dn︸ ︷︷ ︸
risky assets
) ”vector of stock prices at time n”
We set S00 = 1 and
βn :=1
S0n
, n = 0, . . . , N ”discount factor from period n to 0”
Example 2.1 If the return of the riskless asset (e.g. bond) is constant over one periodand equal to r, then
S0n = S0
n−1 + S0n−1 · r = S0
n−1(1 + r) = · · · = S00︸︷︷︸
=1
(1 + r)n = (1 + r)n.
Thus βn = 1(1+r)n
. So, if the amount of βn is invested in the riskless asset S00 at time 0,
then the amount of 1 (e.g. one dollar) will be available at time n.
2.1.2 Strategies
A trading strategy is a (discrete) stochastic process
φ = (φn)n=0,...,N = ((φ0n, φ
1n, . . . , φ
dn))n=0,...,N
in Rd+1, where for each i = 0, ..., d and n = 0, . . . , N
φin = number of shares of asset i held in the portfolio at time n.
φ is assumed to be predictable, i.e.
∀i = 0, . . . , d
φi0 ∈ F0
φin ∈ Fn−1, if n ≥ 1
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Illustration :
at time 0: S0 = (1, S10 , ..., S
d0) and the positions φ0
0, φ10, ..., φ
d0 ∈ F0 are given
at time n− 1, n ≥ 1: select the positions φ0n, φ
1n, ..., φ
dn ∈ Fn−1
(Sn−1 is known) (based on information at time n− 1)
”the positions of the portfolio at time n are decided with respect to the informationavailable at time n− 1”
The value of the portfolio at time n is the scalar product
Vn(φ) = φn · Sn =d∑i=0
φinSin n = 0, . . . , N.
The discounted value is
Vn(φ) = βn(φn · Sn) = φn · Sn,
with βn =1
S0n
and Sn︸︷︷︸”vector of
discounted prices”
= (1, βnS1n, . . . , βnS
dn), n = 0, . . . , N.
By considering discounted prices, the price of the non-risky asset becomes the monetaryunit (a so called numeraire). A strategy is called self financing, if
φn+1 · Sn = φn · Sn ∀n = 0, . . . , N − 1 (3)
This means that the positions in the portfolio are changed without changing the totalvalue of the portfolio (”portfolio is only redistributed”).
Remark 2.2 (3) is equivalent to
Vn+1(φ)− Vn(φ)︸ ︷︷ ︸ = φn+1 · Sn+1 − φn · Sn = φn+1 (Sn+1 − Sn)︸ ︷︷ ︸ n = 0, . . . , N − 1. (4)
”net gain from period n to n+ 1” only due to ”price moves”
Proposition 2.3 Equivalent are
(i) The strategy φ is self-financing.
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(ii) For any n = 1, . . . N
Vn(φ) = V0(φ) +n∑j=1
φj ·∆Sj (∆Sj := Sj − Sj−1)
(iii) For any n = 1, . . . , N
Vn(φ) = V0(φ) +n∑j=1
φj ·∆Sj (∆Sj := Sj − Sj−1)
Proof (i) ⇐⇒ (ii) is (4) summed up. (i) ⇐⇒ (iii) holds because:
(3) ⇐⇒ φn · Sn = φn+1 · Sn ⇐⇒ Vn+1(φ)− Vn(φ) = φn+1(Sn+1− Sn) n = 0, . . . N − 1.
The proposition shows that if one follows a self-financing strategy, then the discountedvalue of the portfolio (hence its value) is completely defined by the initial wealth and thestrategy ((φ1
n, . . . , φdn))n=0,...,N (this is only true because ∆S0
j = 0). More precisely, thefollowing holds.
Proposition 2.4 For any predictable process ((φ1n, . . . , φ
dn))n=0,...,N and any F0-measurable
random variable V0, there exists a unique predictable process (φ0n)n=0,...,N such that the
strategy φ = (φ0, φ1, . . . , φd) is self-financing and such that V0(φ) = V0.
Proof Suppose φ0 = (φ0n)n=0,...,N such as in the statement of the proposition exists. Then
V0(φ) = φ00 +
d∑i=1
φi0 Si0 = V0, hence
φ00 = V0 −
d∑i=1
φi0 Si0, (5)
and for n ≥ 1
(∗)
Vn(φ) = φ0
n +∑d
i=1 φin S
in (by definition)
Vn(φ) = V0(φ) +∑n
j=1
∑di=1 φ
ij (Sij − Sij−1) (by Proposition 2.3 (iii))
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By (∗)
φ0n = V0(φ)︸ ︷︷ ︸
=V0
+n−1∑j=1
d∑i=1
φij (Sij − Sij−1) +d∑i=1
φin (−Sin−1) (6)
Therefore, if we define φ0 through (5), and (6), then φ0 satisfies all the desired properties.
2.1.3 Admissible strategies and arbitrage
The quantities φin were not supposed to be positive, in fact, they were allowed to haveany value.
φ0n < 0 means that we have borrowed the amount |φ0
n| in the riskless asset.
For i ≥ 1 : φin < 0 means that we are ”short” a number φin of asset i.
In this model borrowing and short-selling are allowed, but we shall restrict the class ofstrategies by the following definition.
Definition 2.5 A strategy φ is admissible if it is self-financing and if Vn(φ)(ω) ≥ 0 forall ω ∈ Ω, n = 0, . . . , N (Thus the investor must be able to pay back his debts at anytime).
Definition 2.6 An arbitrage strategy is an admissible strategy φ with zero initial valueand non-zero final value, i.e. V0(φ)(ω) = 0 for all ω ∈ Ω, but P (VN(φ) > 0) > 0.
2.2 Martingales and arbitrage opportunities
2.2.1 Martingales and martingale transforms
As before let (Ω,F, P ) be finite probability space with F = P(Ω), and P (ω) > 0 ∀ω ∈ Ω,(Fn)n=0,...N be a filtration on F, i.e. Fi ⊂ F are sub-σ-algebras and
F0 ⊂ F1 ⊂ · · · ⊂ FN ⊂ F.
In this section 2.2.1 we do not assume F0 = ∅,Ω,FN = F.We say that (Xn)n=0,...,N is adapted (to (Fn)n=0,...N) if Xn ∈ Fn ∀n = 0, . . . , N .
Definition 2.7 An adapted sequence (Mn)n=0,...,N (automatically integrable since |Ω| <∞ and real-valued !) of real-valued r.v.’s is
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- a martingale, if E[Mn+1 | Fn] = Mn n = 0, . . . , N − 1
- a supermartingale, if E[Mn+1 | Fn] ≤Mn n = 0, . . . , N − 1
- a submartingale, if E[Mn+1 | Fn] ≥Mn n = 0, . . . , N − 1
Remark 2.8 (i) Definition 2.7 can be generalized to multidimensions : e.g. a sequenceof Rd-valued r.v.’s (Mn)n=0,...,N is a martingale, if each component is a real-valuedmartingale.
(ii) Let i = 1, . . . , d. If the price of asset i at time n (Sin)n=0,...,N is a martingale, then
E[(Sin+1 − Sin)2] ≤ minS∈Fn
E[(Sin+1 − S)2] n = 0, . . . , N − 1
”using information Fn, Sin is the best guess for Sin+1 in the mean square error”
(All Fn-measurable r.v.’s are in L2(Ω,Fn, P ), since |Ω| <∞ and since they are supposedto be real-valued).
Definition 2.9 A sequence (Hn)n=0,...N of r.v.’s is called predictable, if H0 ∈ F0, andHn ∈ Fn−1 for all n ≥ 1.
Proposition 2.10 Let (Mn)n=0,...,N be a martingale, (Hn)n=0,...,N be predictable, bothw.r.t. the filtration (Fn)n=0,...,N . Then
Xn := H0M0 +n∑k=1
Hk (Mk −Mk−1)︸ ︷︷ ︸=:∆Mk
n = 0, . . . , N
is a martingale w.r.t. (Fn).
Remark: Xn in 2.10 is a discrete version of the stochastic integral∫ t
0
Hs dMs
that we will consider later.
Proof (of 2.10) (Xn) is adapted. For n ≥ 0
E[Xn+1 −Xn | Fn] = E[Hn+1(Mn+1 −Mn) | Fn]
= Hn+1 E[Mn+1 −Mn | Fn]︸ ︷︷ ︸=0 (Mn Martingale)
(since Hn+1 ∈ Fn)
= 0
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thus (Xn) is a martingale.
Proposition 2.11 Let (Mn)n=0,...,N be an adapted sequence of r.v.’s. Equivalent are :
(i) (Mn) is a martingale.
(ii) For any predictable sequence (Hn), we have
E[N∑k=1
Hk (Mk −Mk−1)︸ ︷︷ ︸=∆Mk
] = 0 (7)
Proof Let (Mn) be a martingale, and (Hn) any predictable process. Then by 2.10 (sincea martingale has constant expectation)
E[H0M0 +N∑k=1
Hk(Mk −Mk−1)] = E[H0M0]
thus (7) holds. Conversely, suppose that (Mn) is adapted, and that (ii) holds. Let A ∈ Fjand j ∈ 0, . . . , N − 1 be fixed but arbitrary. Then (ii) holds in particular for
Hn = Hj,An :=
0 if n ∈ 0, . . . , N − 1 − j + 11A if n = j + 1
and so (7) impliesE[Mj+1 1A] = E[Mj 1A].
Since A ∈ Fj, and j ∈ 0, . . . , N − 1 were arbitrary, it follows that
E[Mj+1 | Fj] = Mj j = 0, . . . , N − 1
and the result follows.
2.2.2 Viable financial markets
We suppose again the assumptions of 2.1.1 to hold. A probability measure P ∗ on (Ω,F)is called equivalent to P , if P ∗(ω) > 0 for all ω ∈ Ω.Notation: P ∗ ≈ P .
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Definition 2.12 A market is viable, if there is no arbitrage opportunity.
Theorem 2.13 (Fundamental Theorem of Asset Pricing) The market is viable, ifand only if there exists a probability measure P ∗ equivalent to P such that the discountedprices of assets (Sin)n=0,...,N are P ∗-martingales for i = 1, . . . , d.
Proof Assume: ∃P ∗ ≈ P , such that (Sin)n=0,...,N are P ∗-martingales. For any self-financingstrategy φ = (φn)
Vn(φ) =2.3
V0(φ) +n∑j=1
φj ·∆Sj
=d∑i=0
[φi0 S
i0 +
n∑j=1
φij ·∆Sij
].︸ ︷︷ ︸
P ∗−martingales ∀i by assumption and 2.10
Therefore (Vn(φ))n=0,...,N is a P ∗-martingale and so
E∗[Vn(φ)] = E∗[V0(φ)︸ ︷︷ ︸=V0(φ)
], n = 0, . . . , N . (E∗ = ”expectation w.r.t. P ∗”)
If φ is admissible and V0(φ) = 0, it follows:
• VN(φ)(ω) ≥ 0 for all ω (by definition, since φ is admissible)
• E∗[VN(φ)] = 0 (since V0(φ) = 0)
But0 = E∗[VN(φ)] =
∑ω∈Ω
VN(φ)(ω)P ∗(ω)
implies VN(φ)(ω) = 0 for all ω since P ∗(ω) > 0 for all ω ∈ Ω. Therefore VN(φ)(ω) = 0 forall ω and so an arbitrage strategy cannot exist.
The converse is a little tricky. Thus assume the market is viable. Want to show : ∃P ∗ ≈ Psuch that (Sin)n is a P ∗-martingale ∀i = 1, . . . , d. Define
Γ := X r.v. | X ≥ 0, and P (X > 0) > 0.
Note: Γ is a convex set, i.e. t ∈ [0, 1], X, Y ∈ Γ ⇒ X + t(Y −X) ∈ Γ ∀t ∈ [0, 1].Now:
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the market is viable ⇐⇒ ∀φ admissible with V0(φ) = 0, we have VN(φ) /∈ Γ.
For any predictable process((φ1
n, . . . , φdn))n=0,...,N
define
Gn(φ) : =n∑j=1
(φ1j ∆S1
j + · · ·+ φdj ∆Sdj ).
Note that
Gn(φ) =2.3(iii)
2.4
Vn(φ0, φ1, . . . , φd) (∗)
for uniquely determined predictable φ0, such that (φ0, φ1, . . . , φd) is self-financing andV0(φ0, . . . , φd) = 0.
Lemma 2.14 Suppose the market is viable. Then for any predictable process (φ1, . . . , φd)we have GN(φ) /∈ Γ.
Proof (of Lemma 2.14) Assume to the contrary GN(φ) ∈ Γ.
(a) If Gn(φ) ≥ 0, for all n = 1, ..., N , then add φ0 to (φ1, . . . , φd) as in (∗). It follows, that(φ0, φ1 . . . , φd) is admissible, V0(φ0, φ1 . . . , φd) = G0(φ) = 0, but VN(φ0, φ1 . . . , φd) =GN(φ) ∈ Γ. Thus (φ0, φ1 . . . , φd) is an arbitrage strategy and the market is notviable. This is a contradiction.
(b) Suppose not all Gn(φ) ≥ 0. Then
n := supk ≥ 1 | P (Gk(φ) < 0) > 0 ≤ N−1 (since GN(φ) ∈ Γ, hence GN(φ) ≥ 0).
From the definition of n, we get
P (Gn(φ) < 0) > 0 and ∀j ∈ n+ 1, ..., N we have Gj(φ) ≥ 0.
Define
ψj :=
0, if j ∈ 1, ..., n1Gn(φ)<0 φj, if j ∈ n+ 1, ..., N.
Then ψ is predictable, because Gn(φ) < 0 ∈ Fn ⊂ Fj−1 if j ∈ n + 1, ..., N andφj ∈ Fj−1. Moreover
Gj(ψ) =
0 if j ∈ 1, ..., n1Gn(φ)<0( Gj(φ)︸ ︷︷ ︸
≥0, since j≥n+1
− Gn(φ)), if j ∈ n+ 1, ..., N.
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Thus Gj(ψ) ≥ 0 for any j = 1, . . . , N and GN(ψ) > 0 on the set Gn(φ) < 0, whichhas strictly positive probability. It follows GN(ψ) ∈ Γ. Therefore by (a) applied toψ instead of φ we get again the contradiction that the market is not viable.
We have seen that the assumption GN(φ) ∈ Γ leads to a contradiction in all cases. It cantherefore not hold and so GN(φ) /∈ Γ.
To continue the proof of Theorem 2.13, we let
V := GN(φ) | φ predictable process in Rd ⊂ RΩ = RN0 ,
”subspace of RN0”.
Since the market is viable by assumption, Lemma 2.14 implies V ∩ Γ = ∅. Let
K := X ∈ Γ |∑ω∈Ω
X(ω) = 1
Clearly K is convex, compact, 0 /∈ K, and V ∩K = ∅. Now, we need another lemma.
Lemma 2.15 Let K ⊂ RN0 be a compact and convex set, 0 /∈ K, and let V be a subspaceof RN0. If V ∩K = ∅, then there exists a linear map ξ : RN0 −→ R, with
(1) ξ(x) > 0 ∀x ∈ K
(2) ξ(x) = 0 ∀x ∈ V
(Note that (1) and (2) imply V ⊂ ξ−1(0) (hyperplane) and ξ−1(0) ∩K = ∅).
We will prove Lemma 2.15 later below. By Lemma 2.15 there exists (λ(ω))ω∈Ω, such that
(i)∑
ω∈Ω λ(ω)X(ω) > 0 for all X ∈ K (because any linear map ξ : RN0 −→ R has the
form ξ(z1, . . . , zN0) =∑N0
i=1 λizi for some (λ1, ..., λN0) ∈ RN0)
(ii)∑
ω∈Ω λ(ω) GN(φ)(ω) = 0 for all φ = (φ1, . . . , φd) predictable.
Choosing X(ω) = 0 for all but one ω in (i), implies that λ(ω) > 0. Thus λ(ω) > 0 for allω ∈ Ω. Therefore
P ∗(ω) :=λ(ω)∑ω∈Ω λ(ω)
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defines a probability measure on Ω with P ∗ ≈ P . We have
E∗[N∑j=1
φj ∆Sj] =∑ω∈Ω
GN(φ)(ω)P ∗(ω) =1∑
ω∈Ω λ(ω)
∑ω∈Ω
λ(ω)GN(φ)(ω) =(ii)
0
for any predictable process (φ1, . . . , φd). Therefore, choosing all but one coordinate φi of(φ1, . . . , φd) to be 0, we obtain for any predictable process (φin)n=0,...,N in R and for anyi = 1, . . . d, that
E∗[N∑j=1
φij ∆Sij] = 0.
Now Proposition 2.11 implies that ((Sin)n=0,...,N , P∗) is a martingale for any i.
It remains to show Lemma 2.15. For the proof of Lemma 2.15, we need:
Lemma 2.16 Let C ⊂ RN0 be closed, convex, and 0 /∈ C. Then there is α > 0 and alinear map ξ : RN0 −→ R with
ξ(x) ≥ α ∀x ∈ C
Proof (of Lemma 2.16) Let Bλ(0) := x ∈ RN0 | ‖x‖ ≤ λ, ‖x‖ = (x, x)1/2 = Euclideannorm. Then we can find some λ > 0 with Bλ(0) ∩ C 6= ∅. Since Bλ(0) ∩ C is compact,and x 7→ ‖x‖ is continuous, there exists a unique x0 ∈ Bλ(0) ∩ C, with
infx∈Bλ(0)∩C
‖x‖ = ‖x0‖ = dist(0, Bλ(0) ∩ C) = ”orthogonal projection of 0 on C”.
If x ∈ Bλ(0)c ∩ C, then ‖x‖ > λ ≥ ‖x0‖, thus
‖x‖ ≥ ‖x0‖ ∀x ∈ C.
In particular, since C is convex, for any x ∈ C
‖x0 + t(x− x0)‖2 ≥ ‖x0‖2, ∀t ∈ [0, 1].
Thus
(x0, x− x0) ≥ − t2‖x− x0‖2 0 (as t 0),
and soξ(x) := (x0, x) ≥ ‖x0‖2︸ ︷︷ ︸
6=0since 0/∈C
:= α > 0, ∀x ∈ C.
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Proof (of Lemma 2.15) Set
C = K − V := x ∈ RN0 | x = y − z; y ∈ K, z ∈ V .
Then C is convex, closed, and 0 /∈ C. It follows hence from Lemma 2.16 that there existsξ : RN0 −→ R linear and α > 0 such that ξ(x) ≥ α for all x ∈ C. This implies
ξ(x) = ξ(y)− ξ(z) ≥ α ∀y ∈ K ∀z ∈ V.
If we choose z = 0, we obtain ξ(y) ≥ α > 0 for all y ∈ K and replacing z by λz; λ ∈ R,we obtain ξ(y)− λξ(z) ≥ α for all λ ∈ R, hence ξ(z) = 0 for all z ∈ V .
2.3 Complete markets and option pricing
2.3.1 Complete markets
A European option can be characterized by its payoff h at maturity, which is a non-negative FN -mb r.v. For instance, for the asset S1, h = (S1
N −K)+ in case of a call withstrike price K, and h = (K − S1
N)+ in case of a put with strike price K. More generally,we define:
Definition 2.17 (i) A contingent claim (with non-negative payoff) is a non-negativereal-valued FN -measurable random variable h.
(ii) The contingent claim defined by h is called attainable, if there exists an admissiblestrategy φ worth h at time N , i.e. VN(φ) = h.
Remark 2.18 For contingent claims in a viable market we have:
∃ self-financing φ with VN(φ) = h =⇒ h is attainable.
Proof Since the market is viable by 2.13 ∃P ∗ ≈ P such that (Sin)n=0,...,N , i = 1, . . . , d areP ∗-martingales. Thus the martingale transform (Vn(φ))n=0,...,N is also a P ∗-martingale by2.10. It follows
Vn(φ) = E∗[ VN(φ)︸ ︷︷ ︸= 1
S0N
h≥0
| Fn] ≥ 0 ∀n ≤ N.
Thus Vn(φ) = S0nVn(φ) ≥ 0 for any n ≤ N , and so φ is admissible.
17
Definition 2.19 The market is complete, if every contingent claim is attainable.
Remark 2.20 The assumption of completeness of a market is restrictive and not as usefulas the no-arbitrage assumption. It is, however, useful for pricing and hedging contingentclaims (see also model of Cox-Ross-Rubinstein below).
Theorem 2.21 Let the market be viable. Then it is complete, if and only if there existsa unique measure P ∗ ≈ P under which the discounted prices (Sin)n=0,...,N , 1 ≤ i ≤ d, areP ∗-martingales. (Here the probability P ∗ will become the computing tool whereby one canderive pricing formulas and hedging strategies).
Proof ”⇒ ” Assume the market is viable and complete: Then
∀h ≥ 0 FN -mb ∃ admissible strategy φ with VN(φ) = h,
and furtherh
S0N
=︸︷︷︸completeness
VN(φ) =︸︷︷︸self-financing
V0(φ) +N∑j=1
φj ·∆Sj. (8)
Since the market is viable: ∃P 1 ≈ P under which the (Sin)n=0,...,N are martingales. Suppose∃P 2 with the same properties. Then by (8) and 2.10
Ei[h
S0N
]= Ei[VN(φ)] = Ei[V0(φ)] = V0(φ), i = 1, 2.
The last equality holds since V0(φ) ∈ F0 = ∅,Ω. Thus
E1
[h
S0N
]= E2
[h
S0N
].
Choose h = 1A S0N , A ∈ FN = F ⇒ P 1 = P 2, and uniqueness is proved.
”⇐ ” Let the market be viable but incomplete: Then
∃h ≥ 0 FN -mb r.v., that is not attainable.
Note: h 6= 0 otherwise h would be attainable. Define
V := Y r.v. | Y = u0 +N∑n=1
φn ·∆Sn, u0 ∈ F0, (φn)n=1,...,N predictable n = 1, . . . , N.
18
We have h /∈ V , because otherwise by Proposition 2.4 and Remark 2.18, h would beattainable. Thus V ⊂⊂ X | X r.v. ∼= RN0 is a true linear subspace. Let P ∗ ≈ P be anequivalent martingale measure. Then, there exists a r.v. X 6= 0 with E∗[XY ] = 0 for allY ∈ V (exercise). Define
P ∗∗(ω) :=
(1 +
X(ω)
2 supω∈Ω |X(ω)|
)P ∗(ω).
Since E∗[X] = E∗[X · 1Ω︸︷︷︸∈V
] = 0, we get P ∗∗(Ω) = 1. Clearly, P ≈ P ∗∗ and P ∗∗ 6= P ∗.
Moreover for any predictable process (φn)n=1,...,N = ((φ1n, . . . , φ
dn))n=1,...,N we get
E∗∗[
N∑n=1
φn ·∆Sn
]= 0. (exercise)
Thus by 2.11 (Sin)n=0,...,N , i = 1, . . . , d is a P ∗∗-martingale and so there is more than oneequivalent martingale measure.
2.3.2 Pricing and hedging contingent claims in complete markets
Assume the market is viable and complete and let P ∗ be unique equivalent martingalemeasure (also called risk-neutral measure). Let h be a contingent claim and φ an
admissible strategy with VN(φ) = h. We know from section 2.3.1, that(Vn(φ)
)n=0,...,N
is
a P ∗-martingale, and that
V0(φ) = E∗[VN(φ)] = E∗[h
S0N
],
as well as
Vn(φ) = S0nVn(φ) = S0
n E∗[h
S0N
∣∣∣ Fn] n = 0, . . . , N. (9)
Thus, at any time n = 0, . . . , N , the value of the admissible strategy that generates h is
given by S0n E∗[hS0N
∣∣∣ Fn] (”the value of the option at time n”). Therefore, if the investor
sells the option for the price of
E∗[h
S0N
]at time n = 0, and then follows the admissible strategy φ he can exactly generate thenecessary payoff h at time n = N (”the investor is perfectly hedged”). In particular,
E∗[hS0N
]is the fair price of the option at time n = 0.
19
Remark 2.22 In a complete market the calculaton of the fair option price only requiresthe knowledge of the risk-neutral measure P ∗ (not P ). We could have started with only
(Ω,F) and (Fn)n=0,...,N .
If these are specified, we do not need to find the true probability P (e.g. by empiricalobservations, statistics) in order to price the option!
2.3.3 Introduction to American options
Assume the market is viable and complete, and let P ∗ be the unique equivalent martingalemeasure (risk-neutral measure). An American option can be exercised at any time n =0, . . . , N . Thus it should be defined by an adapted sequence of non-negative r.v.’s
Zn ≥ 0, Zn ∈ Fn, n = 0, . . . , N.
Example 2.23
American call on S1 with strike price K payoff at time n is Zn = (S1n −K)+
American put on S1 with strike price K payoff at time n is Zn = (K − S1n)+.
In order to determine fair price for the option with possible payoff Zn proceed backwards
price at time N : UN = ZN
price at time N − 1 (if the option is not directly exercised (at time N − 1), thewriter must deliver ZN at time N , so (at time N − 1) he needs the amount that isnecessary to deliver ZN at time N):
UN−1 = max
ZN−1︸ ︷︷ ︸”option directly
exercised”
, S0N−1 E
∗[ZNS0N
∣∣∣ FN−1
]︸ ︷︷ ︸
value of the option at time N − 1 thatgenerates ZN at time N” see (9)
By induction for n = 1, . . . , N :
Un−1 = max
(Zn−1, S
0n−1 E
∗[UnS0n
∣∣∣ Fn−1
]).
20
Assume a constant return at rate r for riskless asset, i.e.
S0n = (1 + r)n.
Then
Un−1 = max
(Zn−1,
1
1 + rE∗[Un
∣∣∣ Fn−1
]),
and so for the discounted price of the American option Un := UnS0n
, it holds
Un−1 = max( Zn−1︸︷︷︸:=
Zn−1
S0n−1
,E∗[Un | Fn−1]).
Proposition 2.24 (Un)n=0,...,N is a P ∗-supermartingale. It is the smallest P ∗-supermartingalethat dominates (Zn)n=0,...,N .
Proof As max of adapted r.v.’s Un is adapted, and
Un−1 = E∗[Un−1 | Fn−1] = E∗[max(Zn−1,E∗[Un | Fn−1]) | Fn−1]
≥ E∗[E∗[Un | Fn−1] | Fn−1] = E∗[Un | Fn−1],
thus the first claim holds. Let (Tn)n=0,...,N be another P ∗-supermartingale that dominates(Zn)n=0,...,N . By backward induction :
TN ≥ ZN = UN
Assume Tn ≥ Un n ≤ N , then
Tn−1 ≥ E∗[Tn | Fn−1] ≥ E∗[Un | Fn−1],
henceTn−1 ≥ max(Zn−1,E∗[Un | Fn−1]) = Un−1.
2.4 Cox-Ross-Rubinstein (CRR) model
The CRR model is the discrete version of the Black-Scholes model. In this model Sn =(S0
n, S1n), n = 0, . . . , N , with riskless asset
S0n = (1 + r)n, for some r > −1,
21
and the risky asset is supposed to satisfy
S1n+1 =
S1n(1 + a) or
S1n(1 + b) where − 1 < a < b,
for n = 0, ..., N − 1, with S10 = c > 0 for a given constant c. The possible states of
Tn :=S1n
S1n−1
∈ 1 + a, 1 + b, n = 1, . . . , N,
are then given by
Ω = ω = (x1, . . . , xN) | xi ∈ 1 + a, 1 + b, i = 1, . . . , N.
Each N -tuple in ω = (x1, . . . , xN) ∈ Ω represents the successive values of the ratio Tn(ω),n = 1, . . . , N , on Ω, i.e. Tn(ω) = xn for any n.
Definition: Let n ∈ N and Xi : Ω→ R, i = 1, ..., n, be maps. Then
σ(X1, ..., Xn)
is defined to be the smallest σ-algebra A in Ω for which the maps Xi are A/B(R)-mb,i = 1, ..., n. σ(X1, ..., Xn) is called the σ-algebra generated by (the maps) X1, ..., Xn.
Consider the filtration
F0 := ∅,Ω, Fn := σ(T1, . . . , Tn), n = 1, . . . , N,
on F := P(Ω). We have (exercise):
Fn = σ(S10 , . . . , S
1n), S1
n = S10
n∏k=1
Tk, n = 0, . . . , N.
It holdsP ((x1, . . . , xN)) = P (T1 = x1, . . . , Tn = xn).
(In particular, knowing P is equivalent to knowing the joint distribution of T1, . . . , Tn, seelater). We suppose that P (ω) > 0 for all ω ∈ Ω.
22
1. The discounted price (Sn) is a martingale under a probability measure P on (Ω,F), ifand only if E[Tn+1 | Fn] = 1+r, n = 0, . . . , N−1, where E denote the expectation w.r.t. P
Proof Clearly (Sn) is adapted. Thus
(Sn)P -martingale ⇐⇒ (S1n)P -martingale
⇐⇒ E[S1n+1 | Fn] = S1
n
⇐⇒ E[ S1
n+1
S1n
| Fn]
= 1 (1
S1n
∈ Fn and S1n > 0)
⇐⇒ E[Tn+1 | Fn] = 1 + r (S1n+1
S1n
=S1n+1
S1n
· 1
1 + r= Tn+1 ·
1
1 + r)
2. If the market is arbitrage-free, then r ∈ (a, b).
Proof If there is no arbitrage opportunity, then ∃P ∗ ≈ P , and (Sn) is a P ∗-martingale.By 1. applied with P = P ∗
E∗[Tn+1 | Fn] = 1 + r,
hence
E∗[Tn+1 | Fn] = 1 + r = (1 + a)P ∗(Tn+1 = 1 + a) + (1 + b)P ∗(Tn+1 = 1 + b)
= 1 + aP ∗(Tn+1 = 1 + a) + b P ∗(Tn+1 = 1 + b) ∈ (1 + a, 1 + b).
3. By 2., if r /∈ (a, b), then there are arbitrage strategies.
E.g. if r ≤ a, then at time n = 0: borrow the amount of S10 in the riskless asset to
finance a share of stock S10 , i.e.
φ0 = (−S10 , 1), and keep the strategy up to time N , i.e.
φn = (−S10 , 1), n = 0, . . . , N , (i.e. at time N : pay back the loan and sell risky asset).
Then φ = (φn)n is predictable, self-financing, and for all n = 0, . . . , N
Vn(φ) = φn · Sn = S1n − S1
0(1 + r)n ≥︸︷︷︸(r≤a)
S10
(n∏k=1
Tk − (1 + a)n
)︸ ︷︷ ︸
≥0
≥ 0.
23
Thus φ is admissible, V0(φ) = 0, and
P (VN(φ) > 0) ≥ P (N∏k=1
Tk − (1 + a)N > 0)
= 1− P ((1 + a, . . . , 1 + a))︸ ︷︷ ︸∈(0,1)
> 0.
So φ is an arbitrage strategy.If r ≥ b, we can make a riskless profit by short-selling the risky asset, e.g.
φn = (S10 ,−1), n = 0, . . . , N is an arbitrage strategy (exercise).
4. From now on suppose: r ∈ (a, b). Set p = b−rb−a ∈ (0, 1). Then
(S1n) is a P -martingale ⇐⇒
(Tn)n=1,...,N is independent and
identically distributed (i.i.d) with
P (T1 = 1 + a) = p = 1− P (T1 = 1 + b)
(10)
Proof ”⇐ ”:
E[Tn+1 | Fn] =︸︷︷︸Tn+1 independent
of σ(T1,...,Tn)=Fn
E[Tn+1] =︸︷︷︸id
n=1,...,N−1
(1 + a)p+ (1 + b)(1− p)
= 1 + ap+ b(1− p) = 1 + (a− b)p+ b
= 1 + r − b+ b = 1 + r.
Thus (S1n) is a P -martingale by 1.
”⇒ ”: If E[Tn+1 | Fn] = 1 + r n = 0, . . . , N − 1, then
1 + r = (1 + a) E[1Tn+1=1+a | Fn] + (1 + b) E[1Tn+1=1+b | Fn]︸ ︷︷ ︸=1−E[1Tn+1=1+a | Fn]
.
Thus, we obtain E[1Tn+1=1+a | Fn] = p = 1−E[1Tn+1=1+b | Fn], n = 0, . . . , N − 1. Byinduction we get (exercise)
P (T1 = x1, . . . , Tn = xn) =n∏i=1
pi,
24
where pi = p, if xi = 1 + a and pi = 1 − p, if xi = 1 + b. Thus T1, . . . , TN is i.i.d, andP (T1 = 1 + a) = p as desired.
If the l.h.s. or the r.h.s. of (10) holds, then (S1
n) is a P -martingale, and P is uniquelydetermined. Thus the market is viable (free of arbitrage) and complete, and P = P ∗. Wewill assume this from now on.
5. Let Cn (resp. Pn) the value at time n = 0, ..., N , of a European call (resp. put) on ashare of the stock S1 with strike price K and maturity N .
(a) The put/call parity equation holds
Cn − Pn = S1n −K(1 + r)n−N , n = 0, ..., N.
Proof By (9) we know that the value (fair price) of these options at any time n isgiven by
Cn = S0n E∗
[(S1
N −K)+
S0N
∣∣∣ Fn] , (resp. Pn = S0n E∗
[(K − S1
N)+
S0N
∣∣∣ Fn])Then with S0
n = (1 + r)n we obtain
Cn − Pn = (1 + r)n−N E∗[(S1N −K)+ − (K − S1
N)+ | Fn]
= (1 + r)n−N E∗[S1N −K | Fn]
= (1 + r)n E∗[S1N | Fn]−K(1 + r)n−N
= S1n −K(1 + r)n−N (since (S1
n)n=0,...,N is a martingale)
as desired.
(b) Derive an explicit formula for the price of a call. Show: we can write Cn = c(n, S1n),
where c is a function that depends on K, a, b, r, and N .
Proof For n = 0, ..., N , write
S1N = S1
n
N∏i=n+1
Ti.
25
Then S1n is Fn-mb and
∏Ni=n+1 Ti is independent of Fn = σ(T1, . . . , Tn). Hence, using
Proposition 4.12 from the probabilistic part of the lecture with Φ(x, y) = (xy−K)+,we obtain
Cn = (1 + r)n−N E∗[(S1n
N∏i=n+1
Ti −K)+
︸ ︷︷ ︸=Φ(S1
n,∏Ni=n+1 Ti)
| Fn] = (1 + r)n−Nϕn(S1n),
where
ϕn(x) = E∗[Φ(x,N∏
i=n+1
Ti)] = E∗[(x ·N∏
i=n+1
Ti −K)+]
=N−n∑j=0
(N − nj
)pj(1− p)N−n−j
(x · (1 + a)j(1 + b)N−n−j −K
).
Thus c(n, x) = (1 + r)n−Nϕn(x).
(c) c(n, x) satisfies the recursive equations
c(n, x) =p c(n+ 1, x(1 + a)) + (1− p)c(n+ 1, x(1 + b))
1 + r, n = 0, . . . , N − 1.
Proof We have
ϕn(x) = E∗[(x ·N∏
i=n+1
Ti −K)+]
= E∗[
(x ·N∏
i=n+1
Ti −K)+(1Tn+1=1+a + 1Tn+1=1+b)
]
= E∗(x(1 + a)
N∏i=n+2
Ti −K
)+
1Tn+1=1+a
+E∗
(x(1 + b)N∏
i=n+2
Ti −K
)+
1Tn+1=1+b
=︸︷︷︸
Tn+1 indep.
of Tn+2,...,TN
pϕn+1(x(1 + a)) + (1− p)ϕn+1(x(1 + b)).
26
Here the last equality follows as in the proof of Propodition 4.6. of the Probabilisticbackground.
6. If (Hn) is the replicating (=hedging) strategy of a call, then H1n = ∆(n, S1
n−1) at timen = 1, ..., N , where ∆ is expressed in terms of c.
Proof We have
Vn(H) = Hn · Sn = H0n(1 + r)n +H1
nS1n =
(9)Cn =
5.c(n, S1
n).
Since S1n = S1
n−1(1 + a) or S1n = S1
n−1(1 + b), we obtain
H0n(1 + r)n +H1
nS1n−1(1 + a) = c(n, S1
n−1(1 + a)) (∗)H0n(1 + r)n +H1
nS1n−1(1 + b) = c(n, S1
n−1(1 + b)) (∗∗)
(∗∗)− (∗) H1n =
c(n, S1n−1(1 + b))− c(n, S1
n−1(1 + a))
S1n−1(b− a)
,
thus
∆(n, x) =c(n, x(1 + b))− c(n, x(1 + a))
x(b− a).
7. Study the asymptotics in the N -period model as N → ∞. In particular derive theasymptotics of put and call prices.
For this, let T > 0. Divide the (continuous time) interval [0, T ] in N subintervals oflength T
Nand let R > 0 be the instantaneous interest rate (continuous compounding), i.e.
eRT = limN→∞
1 +RT
N︸︷︷︸=:rN
N
.
Let σ > 0 be given (we will see later: σ “=” limN→∞ var(
log(S1N))).
Let −1 < aN < bN , and aN , bN be given as unique solutions to
log
(1 + aN1 + rN
)= −σ
√T
N, log
(1 + bN1 + rN
)= σ
√T
N.
27
(a) Suppose YN = XN1 + · · ·+XN
N with
XNi , 1 ≤ i ≤ N i.i.d., XN
i ∈
−σ√T
N, σ
√T
N
, E[XN
1 ] = µN , and NµN →N→∞
µ ∈ R.
For the characteristic function P YN of P YN we have for any ξ ∈ R:
P YN (ξ) = E[exp(iξYN)] =indep.
N∏j=1
E[exp(iξXNj )] =
i.d.E[exp(iξXN
1 )]N
= E[1 + iξXN1 −
ξ2
2(XN
1 )2 + rest︸︷︷︸=o( 1
N )
]N
=
(1 +
iξNµN − ξ2
2σ2T +N · o
(1N
)N
)N
−→N→∞
exp(iξµ− ξ2
2σ2T ) = E[exp(iξY )], Y ∼ N(µ, σ2T ).
Thus the characteristic functions P YN converge pointwise to the characteristic functionof N(µ, σ2T ) which is obviously continuous in ξ = 0. It follows from Levy’s continuitytheorem that P YN −→
N→∞N(µ, σ2T ) weakly.
(b) Calculate the asymptotics of the fair put/call prices.
For given N ≥ 1 the (fair) put price at time n = 0 is:
P(N)0 = (1 + rN)−N E∗[(K − S1
0
N∏j=1
Tj)+]
= E∗[(K(1 + rN)−N − S1
0
N∏j=1
(Tj
1 + rN
))+],
= E∗[(K(1 + rN)−N − S1
0eYN)+]
with
YN :=N∑j=1
log( Tj
1 + rN
).
28
Assuming Tj ∈ 1 + aN , 1 + bN, we have that
XNj := log
(Tj
1 + rN
)∈
−σ√T
N, σ
√T
N
, 1 ≤ j ≤ N.
Furthermore (XNj )1≤j≤N is i.i.d. under P ∗ (cf. 4.) and
E∗[XNj ] =
(−σ√T
N
)p+
(σ
√T
N
)(1− p) = (1− 2p)σ
√T
N,
where p = bN−rNbN−aN
. It follows
(1− 2p) =2− eσ
√TN − e−σ
√TN
eσ√
TN − e−σ
√TN
,
and
N · E∗[XNj ] −→
N→∞µ = −σ
2T
2.
Thus (YN)N≥1, (XNj )1≤j≤N , satisfy the conditions of 7.(a). Define
ψ(y) := (K e−RT − S10 e
y)+ ∈ Cb(R) (for call not bounded!)
Using that x 7→ (x− const)+ is Lipschitz continuous with Lipschitz constant one, we get∣∣∣P (N)0 − E∗[ψ(YN)]
∣∣∣ =∣∣∣E∗[(K(1 + rN)−N − S1
0 exp(YN))+ −
(K e−RT − S1
0 exp(YN))+
]∣∣∣
≤ K∣∣∣(1 + rN)−N − e−RT
∣∣∣ −→N→∞
0,
and since P ∗YN −→ N(µ, σ2T ) weakly and ψ ∈ Cb(R), we obtain
limN→∞
P(N)0 = lim
N→∞E∗[ψ(YN)] =
∫Rψ(y)N(µ, σ2T )(dy) =
1√2πσ2T
∫ ∞−∞
ψ(y) e−(y−µ)2
2σ2T dy.
The first zero of the decreasing function ψ is given by
Z0 = log
(Ke−RT
S10
)= log
(K
S10
)−RT.
29
Thus
limN→∞
P(N)0 =
1√2πσ2T
∫ Z0
−∞ψ(y) e−
(y−µ)2
2σ2T dy
=1√
2πσ2T
∫ Z0
−∞e−
(y−µ)2
2σ2T dy ·Ke−RT − 1√2πσ2T
∫ Z0
−∞ey−
(y−µ)2
2σ2T dy · S10 .
In the first integral of the last expression we make the substitution z = y−µσ√T
. Noting that
y − (y − µ)2
2σ2T=−4µy − (y − µ)2
2σ2T=
(y + µ)2
2σ2T,
we make the substitution z = y+µ
σ√T
in the second integral. By this, we obtain
limN→∞
P(N)0 =
1√2π
∫ Z0−µσ√T
−∞e−
z2
2 dz ·Ke−RT − 1√2π
∫ Z0+µ
σ√T
−∞e−
z2
2 dz · S10
= Ke−RT Φ
log(KS10
)−RT + σ2T
2
σ√T
− S10 Φ
log(KS10
)−RT − σ2T
2
σ√T
where Φ(x) = 1√
2π
∫ x−∞ e
− z2
2 dz. Therefore the asymptotic put price is
limN→∞
P(N)0 = E∗[ψ(Y )] = Ke−RT Φ(−d2)− S1
0 Φ(−d1)
with
d1 =− log
(KS10
)+RT + σ2T
2
σ√T
, d2 = d1 − σ√T .
The asymptotic call price now follows from the put/call parity:
limN→∞
C(N)0 = lim
N→∞
(P
(N)0 + S
(N)0︸︷︷︸
≡S10=c
−K(1 + rN)−N︸ ︷︷ ︸→Ke−RT
)= Ke−RT
(Φ(−d2)− 1︸ ︷︷ ︸∫−d2−∞ −
∫∞−∞=−
∫∞−d2
=−∫ d2−∞ by symmetry
)− S1
0
(Φ(−d1)− 1︸ ︷︷ ︸
=−∫ d1−∞
)
= S10Φ(d1)−Ke−RTΦ(d2).
Remark 2.25 The only non-observable parameter is σ! Its interpretation as variancesuggests it should be estimated by statistical methods.
30
3 Brownian motion and Stochastic Differential Equa-
tions (SDEs)
3.1 General notions
Definition 3.1 Let (Ω,A, P ) be a probability space, and (E,E) a measurable space. Astochastic process is a family of r.v.s (Xt)t≥0 from (Ω,A, P ) with values in (E,E) (E iscalled state space). Thus
∀t ≥ 0 Xt : (Ω,A) −→ (E,E) measurable.
∀ω ∈ Ω the map X.(ω) : [0,∞) −→ E
t 7→ Xt(ω),
is called a sample path (or a trajectory).
Remark 3.2 - t stands for time
- a stochastic process can be considered as map
X := X.(·) : R+ × Ω −→ E.
We always assume that X is B(R+)⊗A︸ ︷︷ ︸product σ−algebra
\E measurable.
- we will mainly work with stochastic processes on a finite time interval [0, T ].
Definition 3.3 Let (Ω,A, P ) be a probability space. A filtration on A is an increasingfamily (Ft)t≥0 of sub-σ-algebras of A, i.e.
Fs ⊂ Ft ⊂ A ∀s ≤ t.
A process (Xt)t≥0 is adapted to (Ft)t≥0, if Xt is Ft-mb ∀t ≥ 0, i.e. Xt ∈ A := ω ∈Ω | Xt(ω) ∈ A ∈ Ft ∀A ∈ E, t ≥ 0.
For any t ≥ 0, Ft is interpreted as information (knowledge) at time t. To express that(Ft)t>0 is a filtration on (Ω,A, P ) we shortly write (Ω,A, (Ft)t>0, P ).
Remark 3.4 We will only work with filtrations (Ft)t≥0 such that F0 contains all P -zerosets of A, i.e.
A′ ⊂ A ∈ A and P (A) = 0⇒ A′ ∈ F0. (11)
31
A filtration as in Remark 3.4 can always be constructed from an ordinary one. Indeed, let
NP := A′ ⊂ Ω | A′ ⊂ A ∈ A, P (A) = 0,
andAP := A ∪ A′ | A ∈ A, A′ ∈ NP,
then one can show that AP is a σ-algebra on Ω, and that P extends to AP by the formulaP (A∪A′) = P (A), if A ∈ A, A′ ∈ NP (exercise). (Ω,AP , P ) is called the completion of Aw.r.t. P . Then let
Ft := σ(Ft ∪NP ) t ≥ 0.
Then (Ft)t≥0 is a filtration on AP that satisfies (11) with A replaced by AP .A useful consequence of (11) is the following:
X = Y P -a.s. and Y Ft-mb⇒ X Ft-mb (exercise)
Gt := σ(Xs; s ≤ t) is called the filtration generated by (Xt)t≥0. If (Xt)t≥0 is adapted tosome filtration (Ft)t≥0, then Gt ⊂ Ft,∀t ≥ 0 (since Xs ∈ Ft for any s ≤ t and (Gt)t≥0 isthe smallest filtration for which this is the case). In general (σ(Xs; s ≤ t))t≥0 does notsatisfy (11), but by the above consideration the completed filtration Gt := σ(σ(Xs; s ≤t)∪NP ), t ≥ 0, satisfies (11). This (Gt)t≥0 is called the natural filtration of (Xt). Obviously(Xt)t≥0 is also adapted to (Gt)t≥0.
Definition 3.5 A stopping time w.r.t. the filtration (Ft)t≥0 is a map τ : Ω→ [0,∞] suchthat
τ ≤ t ∈ Ft, ∀t ≥ 0.
In particular τ is A-measurable. The σ-algebra associated with τ is
Fτ := A ∈ A | A ∩ τ ≤ t ∈ Ft ∀t ≥ 0
It is called the σ-algebra of the ”τ -past” or of the ”events prior to τ”. Fτ is interpreted asinformation available before τ .
For technical reasons it is also convenient to consider right-continuous filtrations (Ft)t≥0,i.e. filtrations (Ft)t≥0 that satisfy Ft = Ft+ := ∩ε>0Ft+ε for any t ≥ 0. By considering theright-continuous filtration (Ft+)t≥0 instead of (Ft)t≥0, we may often assume that (Ft)t≥0
is right-continuous.(Ft)t>0 is said to satisfy the “usual conditions” if it is right-continuous, (Ω,A, P ) acomplete probability space, and N ∈ A such that P (N) = 0 implies N ∈ F0. From nowon we assume whenever necessary the usual conditions.
32
Proposition 3.6 (Let (Ft)t≥0 satisfy the usual conditions)
(1) If S is a stopping time, S is FS-mb.
(2) Let S be a stopping time, P (S < ∞) = 1, (Xt)t≥0 be continuous (i.e. P (t 7→Xt is continuous) = 1), and adapted to (Ft)t≥0. Then XS is FS-mb, where
(XS)(ω) :=
XS(ω)(ω), if S(ω) <∞,0 otherwise.
(3) If S, T are stopping times with S ≤ T P -a.s, then FS ⊂ FT .
(4) If S, T are stopping times, then S ∧ T := inf(S, T ) is a stopping time. In particular∀t ≥ 0, t fixed S ∧ t is a stopping time.
(5) τ : Ω→ [0,∞] is a stopping time, if and only if it is a weak stopping time, i.e.
τ < t ∈ Ft, ∀t ≥ 0.
Proof Exercise, cf. e.g. lecture SDEs.
3.2 Brownian motion
Definition 3.7 A real-valued stochastic process (Bt)t≥0 is called a (standard) Brownianmotion (BM for short, or BM starting in zero), if :
(1) P (ω ∈ Ω | B0(ω) = 0) = 1 (”start in zero”, ”standard”).
(2) ∀0 ≤ s < t, the r.v. Bt−Bs is normally distributed with mean 0 and variance t− s,i.e. for any b ∈ R
P (Bt −Bs ≤ b) =1√
2π(t− s)
∫ b
−∞e−
x2
2(t−s) dx.
(3) (Bt)t≥0 has independent increments, i.e. for any 0 ≤ t1 < · · · < tn, the r.v.’s
Bt1 , Bt2 −Bt1 , . . . , Btn −Btn−1
are independent.
33
(4) Almost all sample paths of (Bt)t≥0 are continuous, i.e.
P (ω | t 7→ Bt(ω) is continuous) = 1.
Remark 3.8 The construction, i.e. existence of a BM is not trivial to show (see forinstance lecture SDEs). Here, we accept the existence of a BM as in 3.7.
Definition 3.9 Let (Gt)t≥0 be a filtration on (Ω,A, P ). An (Gt)-adapted process (Bt)t≥0
is called a (Gt)t≥0-BM (or BM w.r.t. (Gt)t≥0), if
(1) (Bt)t≥0 is a BM.
(2) ∀s ≤ t: Bt −Bs is independent of Gs.
Remark 3.10 Any BM (Bt)t≥0 is an σ(Bs; s ≤ t)-BM. One can also check that any BM(Bt)t≥0 is a BM w.r.t. its natural filtration.
3.3 Continuous-time martingales
Definition 3.11 Let (Ω, (Ft)t≥0,A, P ) be a filtered probability space. An (Ft)-adaptedfamily (Mt)t≥0 of integrable r.v.’s (i.e. E[|Mt|] <∞ for any t ≥ 0) is a
- a martingale, if Ms = E[Mt | Fs] for all s ≤ t.
- a supermartingale, if Ms ≥ E[Mt | Fs] for all s ≤ t.
- a submartingale, if Ms ≤ E[Mt | Fs] for all s ≤ t.
Proposition 3.12 If (Xt)t≥0 is a standard (Ft)-BM, then:
(1) (Xt)t≥0 is an (Ft)-martingale.
(2) (X2t − t)t≥0 is an (Ft)-martingale.
(3) (eσXt−σ2
2t)t≥0 is an (Ft)-martingale for any σ ∈ R.
Proof
(1) One easily checks E[|Xt|] <∞ for any t ≥ 0 and ∀s ≤ t
E[Xt | Fs]−Xs =Xs∈Fs
E[Xt −Xs | Fs] =3.11(2)
E[Xt −Xs] =3.7(2)
0.
34
(2) We have E[X2t ] <∞ for any t ≥ 0 and for any s ≤ t
E[X2t −X2
s | Fs] = E[(Xt −Xs)2 + 2Xs(Xt −Xs) | Fs]
= E[(Xt −Xs)2 | Fs] + 2Xs E[(Xt −Xs) | Fs]︸ ︷︷ ︸
=0 by (1)
=3.11(2)
E[(Xt −Xs)2] =
3.7(2)t− s,
thus using X2s ∈ Fs, we obtain E[X2
s | Fs]− s = E[X2t − t | Fs] for any s ≤ t.
(3) is left as an exercise.
The (sub)martingale property remains true for bounded (!) stopping times.
Theorem 3.13 (optional sampling theorem) Let (Mt)t≥0 be a continuous martingalew.r.t. (Ft)t≥0. If τ1, τ2 are two bounded stopping times with τ1 ≤ τ2, then Mτ2 is integrableand
E[Mτ2 | Fτ1 ] = Mτ1 P -a.s.
(The above equation holds with ” ≥ ”, if (Mt)t≥0 is a submartingale).
Remark 3.14 Theorem 3.13 implies:
∀ bounded stopping times τ, Mτ is integrable, and E[Mτ ] = E[M0].
(In fact: If we preassume continuity and adaptedness of (Mt)t≥0 as in 3.13, this is alsoequivalent to the statement in 3.13).
Proposition 3.15 Let (Xt)t≥0 be an (Ft)-BM. Define
Ta := infs ≥ 0 | Xs = a, a ∈ R (inf ∅ := +∞).
Then Ta is a stopping time, finite P -a.s. and its distribution is characterized by its Laplacetransform
E[e−λTa ] = e−√
2λ|a|, λ ≥ 0.
Proof By 3.4 we may assume that t 7→ Xt(ω) is continuous ∀ω ∈ Ω. Let first a > 0.Then
Ta ≤ t = sups≤t
Xs ≥ a =⋂
ε>0,ε∈Q
sups≤t
Xs > a− ε =⋂
ε>0,ε∈Q
⋃s∈Qs≤t
Xs > a− ε ∈ Ft.
35
Thus Ta is a stopping time. Hence Ta ∧ n is a bounded stopping time ∀n ≥ 1, and so
E[eσXTa∧n−σ2
2Ta∧n] =
3.12(3)+3.141 ∀n ≥ 0, σ ∈ R. (12)
From now on, we let σ > 0. Then, we have 0 ≤ eσXTa∧n−σ2
2Ta∧n ≤ eσa P -a.s. ∀n ≥
1 (integrable majorant) and
limn→∞
1Ta<∞eσXTa∧n−
σ2
2Ta∧n = 1Ta<∞e
σXTa−σ2
2Ta =
XTa=a1Ta<∞e
σa−σ2
2Ta
and since Xt ≤ a on Ta =∞, for σ > 0
limn→∞
1Ta=∞eσXTa∧n−
σ2
2Ta∧n = lim
n→∞1Ta=∞e
σXn−σ2
2n = 0.
Thus for (any) σ > 0
1 =(12)
limn→∞
E[eσXTa∧n−σ2
2Ta∧n] =
LebesgueE[ lim
n→∞eσXTa∧n−
σ2
2Ta∧n] = E[1Ta<∞e
σa−σ2
2Ta ]
This implies
E[1Ta<∞e−σ
2
2Ta ] = e−σa (13)
Letting σ 0, we obtain again using Lebesgue E[1Ta<∞] = 1, hence
P (Ta <∞) = 1,
and so, by (13)
E[e−σ2
2Ta ] = e−σa.
If a < 0, then since Wt := −Xt is also a (standard) (Ft)-BM, and
Ta = infs ≥ 0 | Xs = a = infs ≥ 0 | Ws = −a︸︷︷︸>0
,
we conclude as before E[e−σ2
2Ta ] = e−σ(−a). Now, we let λ := σ2
2, then the assertion follows
for any λ > 0. The assertion for λ = 0 is trivial.
Theorem 3.16 (Doob’s inequality) Let (Mt)t∈[0,T ] be a continuous martingale. Then
E
[supt∈[0.T ]
|Mt|2]≤ 4E[M2
T ]
Proof See textbooks on probability theory or martingale theory.
36
3.4 Stochastic integral and Ito claculus
Let (Wt)t≥0 be a standard (Ft)-BM on a filtered probability space (Ω,A, (Ft)t≥0, P ). Fora certain class (that we will precise later) of adapted process (Ht)t≥0 we want to definethe stochastic integral ∫ t
0
Hs dWs, t ≥ 0 (14)
From the next theorem and the following remark, we can see that it is not possible todefine (14) in the Lebesgue-Stieltjes sense.
Theorem 3.17 Let τn = 0 = tn0 < tn1 < · · · < tnN(τn) <∞, tni ∈ R, n ≥ 1, be a sequence
of partitions of [0,∞) with
|τn|︸︷︷︸”mesh of τn”
:= sup1≤i≤N(τn)
|tni − tni−1| −→n→∞
0, tnN(τn) −→n→∞
∞.
Define
Snt :=∑tni∈τn
tni+1≤t
(Wti+1−Wti)
2.
Then Snt −→n→∞
t in L2(Ω,A, P ). If additionally τn ⊂ τn+1, then
P (Snt −→n→∞
t ∀t ≥ 0) = 1.
Proof L2-convergence: We have
E[Snt ] =∑tni∈τn
tni+1≤t
E[(Wti+1−Wti)
2] =∑tni∈τn
tni+1≤t
(tni+1 − tni ) −→n→∞
t,
and
var(Snt ) =∑tni∈τn
tni+1≤t
var((Wtni+1
−Wtni)2)
=∑tni∈τn
tni+1≤t
(E[(Wtni+1
−Wtni)4]︸ ︷︷ ︸
=3(tni+1−tni )2 (exercise)
−(tni+1 − tni )2)
= 2∑tni∈τn
tni+1≤t
(tni+1 − tni )2 −→n→∞
0.
Thus Snt − E[Snt ] −→n→∞
0 in L2, hence also Snt − t −→n→∞
in L2.
P -a.s. convergence: Proof only for τn = dyadic partition, i.e.
tni =i
2n, i = 0, . . . , n · 2n = N(τn), n ≥ 1.
37
Then
var(Snt ) = 2
i+12n≤t∑
i=0
(1
2n
)2
≤ 2k·2n−1∑i=0
(1
2n
)2
=2k
2n(for t ≤ k).
Thus using Chebyshev’s inequality, i.e.
P (|Snt − E[Snt ]| ≥ ε) ≤ 1
ε2var(Snt ) ∀ε > 0,
we get∞∑n=1
P (|Snt − E[Snt ]| ≥ ε) ≤ 2k
ε2<∞ ∀ε > 0
Thus (Snt − E[Snt ])n≥1 converges fast in probability to zero. Consequently (cf. lectureTIM1, Lemma 7.7), (Snt − E[Snt ])n≥1 converges P -a.s. to zero. Since limn→∞ E[Snt ] = t bythe above, we have hence shown that
P ( limn→∞
Snt = t) = 1 for any t ≥ 0.
This means: ∀t ≥ 0 ∃Ωt with P (Ωt) = 1 and
limn→∞
Snt (ω) = t ∀ω ∈ Ωt (dependence on t !).
Let Q+2 := m · 2−n ; m,n ∈ N ∪ 0 be the positive dyadic rationals and
Ω :=⋂t∈Q+
2
Ωt.
Then P (Ω) = 1, since P ((⋂t∈Q2
Ωt)c) = P (
⋃t∈Q2
Ωct) ≤
∑t∈Q2
P (Ωct) = 0 and
Snt (ω)n→∞−→ t ω ∈ Ω and t ∈ Q+
2 .
Let t ∈ [0,∞) \Q+2 . Then ∃(t+k )k≥1, (t
−k )k≥1 ⊂ Q+
2 , with t+k t, t−k t as k ∞ and forall ω ∈ Ω, n, k ≥ 1, we get
Snt−k
(ω) ≤ Snt (ω) ≤ Snt+k
(ω).
Thust−k = lim
n→∞Snt−k
(ω) ≤ lim infn→∞
Snt (ω) ≤ lim supn→∞
Snt (ω) ≤ limn→∞
Snt+k
(ω) = t+k
for any k ≥ 1. Letting k →∞ we get: limn→∞ Snt (ω) = t for all ω ∈ Ω, t ≥ 0 (”Sandwich
argument”).
38
Remark 3.18 Theorem 3.17 shows that P -a.e. path of BM is of infinite variation. Indeed,suppose (Wt(ω))t≥0 is of finite variation, then∑
tni∈τn
tni+1≤t
(Wtni+1(ω)−Wtni
(ω))2 ≤ maxtni∈τn
tni+1≤t
|Wtni+1(ω)−Wtni
(ω)|
︸ ︷︷ ︸<ε, for |τn|<δ
by uniform continuity on [0,t]
∑tni∈τn
tni+1≤t
|Wtni+1(ω)−Wtni
(ω)|
︸ ︷︷ ︸<∞ by assumption
⇒ 〈W 〉t(ω) := limn→∞
Snt (ω) = 0, ∀t ≥ 0.
But by Theorem 3.17 this only possible for ω in some P -zero set, i.e. ω /∈ Ω. Therefore,as we already said, we can not define (14) pathwise as an Lebesgue-Stieltjes integral.However, the quadratic variation of (Wt)t≥0 along (τn) is a well behaved function (positive,continuous and increasing). This fact can be exploited in order to develop a new approachfor the definition of (14).
3.4.1 Construction of the stochastic integral
Let (Wt)t≥0 be a standard (Ft)-BM on
(Ω,A, (Ft)t≥0, P ).
Fix T > 0.
Definition 3.19 (Ht)t∈[0,T ] is called a simple process, if
Ht(ω) =
p∑i=1
φi(ω)1(ti−1,ti](t),
where 0 = t0 < t1 < · · · < tp = T , and φi, i = 1, ..., p, is Fti−1-measurable and bounded.
The stochastic integral of H is defined for all t ∈ [0, T ] by
I(H)t :=
p∑i=1
φi(Wti∧t −Wti−1∧t)
⇒ (I(H)t)t∈[0,T ] is continuous, (Ft)-adapted and (square) integrable.
We have
I(H)t =k∑i=1
φi(Wti −Wti−1) + φk+1(Wt −Wtk), if t ∈ (tk, tk+1]. (15)
39
One writes ∫ t
0
Hs dWs := I(H)t, t ∈ [0, T ].
Proposition 3.20 If (Ht)t∈[0,T ] is a simple process, then
(1)(∫ t
0
Hs dWs
)t∈[0,T ]
is a continuous (Ft)-martingale.
(2) E[( ∫ t
0
Hs dWs
)2]
= E[ ∫ t
0
H2s ds
], t ∈ [0, T ].
(3) E[
supt≤T
∣∣ ∫ t
0
Hs dWs
∣∣2] ≤ 4E[ ∫ T
0
H2s ds
].
Proof
(1) Let s < t. Include s, t in 0 = t0 < · · · < tp = T .
0 = t0 < · · · < tp = T, p ∈ p, p+ 1, p+ 2.
For n ≤ p we can use (15), and write
Mn :=
∫ tn
0
Hu dWu =n∑i=1
φi(Wti −Wti−1)
with φi ∈ Gi−1 := Fti−1, i = 1, ..., p.
Further Xj := Wtj , j = 0, ..., p is a (Gj)-martingale since (Wt)t≥0 is an (Ft)-martingale. Thus (Mn) is a martingale transform (see Proposition 2.10). Then∃j < l ≤ p with s = tj, t = tl, and so
E[ ∫ t
0
Hu dWu | Fs]
= E[Ml | Gj] = Mj =
∫ t
0
Hu dWu.
We already know that( ∫ t
0Hu dWu
)t≥0
is continuous adapted and integrable, thus
(1) follows.
(2) Using the notations of (1), there exists n ≤ p, with t = tn. Then
E[M2n] =
n∑i,j=1
E[φi φj(Xi −Xi−1)(Xj −Xj−1)]. (16)
40
If i < j, then
E[φi φj(Xi −Xi−1)(Xj −Xj−1)]
= E[E[φi φj(Xi −Xi−1)(Xj −Xj−1) | Gj−1]
]= E
[φi φj(Xi −Xi−1)E[(Xj −Xj−1) | Gj−1]︸ ︷︷ ︸
=0
]= 0, (17)
and for j > i we get the same. For i = j, we have by independence of the Brownianincrements
E[(φi)2(Xi −Xi−1)2] = E
[E[(φi)
2(Xi −Xi−1)2 | Gi−1]]
= E[(φi)
2 E[(Xi −Xi−1)2 | Gi−1]︸ ︷︷ ︸=E[(Wti
−Wti−1)2]=ti−ti−1
]. (18)
Since t = tn for some n ∈ 0, . . . , p, we obtain
E[( ∫ t
0
Hs dWs
)2]
= E[M2n] =
(16),(17),(18)
E[n∑i=1
(φi)2(ti − ti−1)] = E
[ ∫ t
0
H2s ds
].
(3) Since( ∫ t
0Hs dWs
)t∈[0,T ]
is a continuous martingale, (3) is just Doob’s inequality
together with (2).
Remark 3.21 Define∫ T
t
Hs dWs :=
∫ T
0
Hs dWs −∫ t
0
Hs dWs, t ≤ T.
If A ∈ Ft, then s 7→ 1A1s>tHs is still simple process, since
∀s ≤ T 1A 1s>t︸ ︷︷ ︸1(t,T ](s)
Hs =
p∑i=1
φi1A1(ti−1,ti]∩(t,T ](s)
Here
φi1A ∈
Fti−1
if t ≤ ti−1
Ft if t ∈ (ti−1, ti)
Ft if t ≥ ti
and (ti−1, ti] ∩ (t, T ] =
(ti−1, ti] if t ≤ ti−1
(t, ti] if t ∈ (ti−1, ti)
∅ if t ≥ ti.
41
Further, one can easily see from the definition of stochastic integral that∫ T
0
1A1s>tHs dWs = 1A
∫ T
t
Hs dWs. (19)
Define
H :=
(Ht)t∈[0,T ] | H is (Ft)-adapted and E[ ∫ T
0
H2s ds
]<∞
.
Obviously H contains all simple processes.
Proposition 3.22 Let (Wt)t≥0 be an (Ft)-BM. There exists a unique linear map
J : H −→Mc = continuous (Ft)-martingales on [0, T ],
H 7→ (J(H)t)t∈[0,T ]
such that:
(1) If (Ht)t∈[0,T ] is a simple process, then
P (I(H)t = J(H)t ∀t ∈ [0, T ]) = 1.
(2) If t ∈ [0, T ], then
E[J(H)2t ] = E
[ ∫ t
0
H2s ds
]for any H ∈ H.
The uniqueness of J is in the following sense: Let J ′ be another map with the sameproperties as J , then
P (J(H)t = J ′(H)t ∀t ∈ [0, T ]) = 1 for any H ∈ H.
Define: ∫ t
0
Hs dWs := J(H)t for t ≥ 0, H ∈ H. (”stochastic integral”)
Proposition 3.23 Let H ∈ H. Then:
(1)
E[
supt∈[0,T ]
∣∣ ∫ t
0
Hs dWs
∣∣2] ≤ 4E[ ∫ T
0
H2s ds
](20)
42
(2) If τ ≤ T is an (Ft)-stopping time, then
P(∫ τ
0
Hs dWs =
∫ T
0
1s≤τHs dWs
)= 1. (21)
Proof of 3.22 (and 3.23). We shall accept: H ∈ H⇒ ∃(Hn)n∈N simple processes with
E[ ∫ T
0
(Hs −Hns )2 ds
]−→n→∞
0.
For a sketch of proof of this fact see e.g. [Karatzas/Shreve, Brownian motion and stochasticcalculus (2nd edition 1991), 2.5 Problem, p.135]. For H ∈ H and (Hn)n∈N like above, and∀p ≥ 1
E[
supt∈[0,T ]
∣∣∣I(Hn+p)t − I(Hn)t
∣∣∣2] ≤ 4E[ ∫ T
0
(Hn+ps −Hn
s )2 ds]
(22)
⇒ ∃ subsequence (nk) with
E[
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣2] 12 ≤ 1
2k
⇒ E[ ∞∑k=1
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣]≤
∞∑k=1
E[
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣2] 12 ≤ 1 <∞
⇒ P( ∞∑k=1
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣ <∞) = 1
⇒ ∃J(H)t := limk→∞
I(Hnk)t uniformly on [0, T ] P -a.s.
In particular: (J(H)t)t∈[0,T ] is continuous and adapted. Taking the limit in (22) along thesubsequence (n+ p nk), we obtain with Fatou’s lemma
E[
supt∈[0,T ]
∣∣∣J(H)t − I(Hn)t
∣∣∣2] ≤ 4E[ ∫ T
0
(Hs −Hns )2 ds
]. (23)
(23) implies: J(H)t does not depend on the choice of approximating sequence (Hn)n∈Nand 3.22 (1) holds. Moreover: (23) ⇒ I(Hn)t → J(H)t in L2 for any t ∈ [0, T ]. ThenJ(H) is a martingale, since for any s ≤ t P -a.s.
E[J(H)t | Fs] = E[ limk→∞
I(Hnk)t | Fs] = limk→∞
E[I(Hnk)t | Fs] = limk→∞
I(Hnk)s = J(H)s.
43
3.22 (2) holds, because
E[J(H)2t ] =
(23)limn→∞
E[I(Hn)2t ] =
3.20limn→∞
E[ ∫ t
0
(Hns )2 ds
]= E
[ ∫ t
0
H2s ds
].
(20) holds, because
E[
supt∈[0,T ]
|J(H)t|2]
= limk
E[
supt∈[0,T ]
∣∣∣ I(Hnk)t
∣∣∣2] ≤ limk
4E[ ∫ T
0
(Hnks )2 ds
]= 4E
[ ∫ T
0
(Hs)2 ds]
and the uniqueness follows, since for any n ≥ 1
E[ supt∈[0,T ]
|J(H)t − J ′(H)t|2]
≤ 2(E[ sup
t∈[0,T ]
∣∣∣J(H)t − I(Hn)t
∣∣∣2] + E[ supt∈[0,T ]
∣∣∣I(Hn)t − J ′(H)t
∣∣∣2])
≤ 16E[ ∫ T
0
(Hs −Hns )2 ds
]−→n→∞
0.
Finally, we want to show (21): Start with stopping times of the form
τ =n∑i=1
ti1Ai , 0 < t1 < · · · < tn = T, Ai ∈ Fti , pairwise disjoint , 1 ≤ i ≤ n, P (∪ni=1Ai) = 1.
τ is indeed a stopping time by Proposition 3.6(5), since
1s>τ = 1s>∑ni=1 ti1Ai =
n∑i=1
1s>ti1Ai︸ ︷︷ ︸=
0 if s ≤ ti
1Ai ∈ Fti ⊂ Fsif s > ti.
44
Hence 1s>τHs ∈ H and also 1s>ti1AiHs ∈ H if H ∈ H, i = 1, ..., n. Then∫ T
0
1s>τHs dWs =n∑i=1
∫ T
0
1s>ti1AiHs dWs
= limk→∞
n∑i=1
∫ T
0
1s>ti1AiHnks dWs
=(19)
limk→∞
n∑i=1
1Ai
∫ T
ti
Hnks dWs
=n∑i=1
1Ai
∫ T
ti
Hs dWs =
∫ T
τ
Hs dWs,
and so ∫ T
0
1s≤τHs dWs =
∫ τ
0
Hs dWs.
For general stopping time τ ≤ T we have that
τn := T1τ=T +2n−1∑k=0
(k + 1)T
2n1τ∈[ kT
2n,(k+1)T
2n) ( τ as n→∞)
is of the previous form and by continuity∫ τn
0
Hs dWs −→n→∞
∫ τ
0
Hs dWs P -a.s.
Further
E[∣∣∣ ∫ T
0
1s≤τnHs dWs −∫ T
0
1s≤τHs dWs
∣∣∣2] =3.22(2)
E[ ∫ T
0
1τ<s≤τnH2s ds
]−→n→∞
0
by Lebesgue. Thus ∫ T
0
1s≤τnHs dWs −→∫ T
0
1s≤τHs dWs
in L2(Ω,A, P ), hence P -a.s. along a subsequence, say (τnk). Consequently∫ τ
0
Hs dWs = limk→∞
∫ τnk
0
Hs dWs = limk→∞
∫ T
0
1s≤τnkHs dWs =
∫ T
0
1s≤τHs dWs
P -a.s. and (21) is shown.
45
Define
H := (Hs)s∈[0,T ] | H is (Ft)-adapted and
∫ T
0
H2s ds <∞ P -a.s..
In the following proposition we extend the stochastic integral to integrand processes inH. In this case the stochastic integral is not necessarily a martingale.
Proposition 3.24 There exists a unique linear map
J : H −→ C = continuous processes on [0, T ]
such that:
(1) Extension property: If (Ht)t∈[0,T ] is a simple process, then
P(J(H)t = I(H)t ∀t ∈ [0, T ]
)= 1.
(2) Continuity property: If (Hn)n∈N ⊂ H, then
P( ∫ T
0
(Hns )2 ds ≥ ε
)−→n→∞
0 ∀ε > 0⇒ P(
supt∈[0,T ]
|J(Hn)t| ≥ ε)−→n→∞
0 ∀ε > 0.
For H ∈ H, we set as before∫ t
0
Hs dWs := J(H)t, t ∈ [0, T ].
Proof Existence: Let H ∈ H and
Tn := infs ∈ [0, T ] |
∫ s
0
H2u du ≥ n
, inf ∅ :=∞.
Then Tn ∞ P -a.s. as n∞. Since Tn ≤ t⇔∫ t
0H2u du ≥ n, we get
Tn ≤ t =∫ t
0
H2u du ≥ n
.
Thus it is enough to show that( ∫ t
0H2u du
)t∈[0,T ]
is adapted in order to show that Tn is a
stopping time. If H ∈ H this is true, since ∃Hn simple with∫ t
0
(Hnu )2 du︸ ︷︷ ︸
adapted
−→n→∞
∫ t
0
H2u du P -a.s. for t = T hence ∀t ∈ [0, T ].
46
If H ∈ H then this is also true since∫ t
0
(|Hu| ∧ k)2 du︸ ︷︷ ︸adapted, since (|Hu|∧k)u∈[0,T ]
is in H
−→k→∞
∫ t
0
H2u du P -a.s. ∀t ∈ [0, T ].
Furthermore
E[ ∫ T
0
H2s1s≤Tn ds
]= E
[ ∫ T∧Tn
0
H2s ds
]≤ n,
thusHns := Hs1s≤Tn is in H ∀n ≥ 1.
Note that 1s≤Tn is Fs-mb since Tn is a stopping time. Furthermore for any n, k ≥ 1
∫ t
0
Hns dWs =
(21)
∫ T
0
1s≤tHns dWs =
∫ T
0
1s≤t 1s≤Tn
=Hn+ks︷ ︸︸ ︷
1s≤Tn+kHs︸ ︷︷ ︸=Hn
s since Tn≤Tn+k
dWs
=
∫ T
0
1s≤t∧TnHn+ks dWs =
(21)
∫ t∧Tn
0
Hn+ks dWs.
Thus on Tn > T we have P -a.s.
J(Hn)t = J(Hn+k)t ∀t ≤ T. (24)
Since Tn > T =∫ T
0H2u du < n
Ω (up to some P -zero set) we can define almost
surely a process J(H) by localization, i.e. for n ∈ N:
∀t ≤ T J(H)t := J(Hn)t on∫ T
0
H2u du < n
.
Then,
- J(H)t is (almost surely) continuous since J(Hn)t is continuous for any n, and J(H)is well defined by (24)
- the extension property holds by construction, since if H is a simple process(bounded !), then P -a.s. H = Hn = Hn0 for ∀n ≥ n0 for some n0 (since Tn0 = ∞for some n0 ⇒ Tn =∞ ∀n ≥ n0). Thus
J(H)t = J(Hn0)t = I(H)t ∀t ≤ T P -a.s.
47
Next we show the continuity property. For any N ∈ N
P(
supt∈[0,T ]
|J(H)t| ≥ ε)≤ P
(∫ T
0
H2s ds ≥
1
N
)+ P
(1∫ T0 H2
s ds<1N supt∈[0,T ]
|J(H)t| ≥ ε)
(25)
and
T 1N
= infs ∈ [0, T ] |
∫ s
0
H2u du ≥
1
N
≤ T1 and
∫ T
0
H2u du <
1
N
= T 1
N> T.
Hence ∀t ≤ T we have P -a.s.
1T 1N>TJ(H)t = 1T 1
N>T1T1>TJ(H)t = 1T 1
N>T1T1>TJ(H1)t
= 1T 1N>T
∫ t
0
1s≤T1Hs dWs = 1T 1N>T
∫ t∧T 1N
0
1s≤T1Hs dWs
=(21)
1T 1N>T
∫ T
0
1s≤T11s≤t∧T 1NHs dWs
=(21)
1T 1N>T
∫ t
0
1s≤T 1NHs dWs.
Therefore
P(1T 1
N>T sup
t∈[0,T ]
|J(H)t| ≥ ε)
= P(
1T 1N>T sup
t∈[0,T ]
∣∣ ∫ t
0
1s≤T 1NHs dWs
∣∣ ≥ ε)
≤Chebychev′sineq. and (20)
4
ε2E[ ∫ T
0
H2s1s≤T 1
N ds]≤ 4
ε2N.
Thus by (25)
P(
supt∈[0,T ]
|J(H)t| ≥ ε)≤ P
(∫ T
0
H2s ds ≥
1
N
)+
4
ε2Nfor any N ∈ N.
As a result, the continuity property follows. Next we show linearity. First note that theextension property and the continuity property imply (exercise)
J(H) = J(H) ∀H ∈ H.
Thus linearity holds for J on H. Let H,K ∈ H, then with Hns = 1s≤TnHs and Kn
s =
1s≤TnKs, Tn := infs ∈ [0, T ] |
∫ s0K2u du ≥ n
, we have∫ T
0
(Hns −Hs)
2 ds,
∫ T
0
(Kns −Ks)
2 ds −→n→∞
0 P -a.s.
48
and the convergence is also in probability. Then for α, β ∈ R, by the continuity propertyJ(αHn+βKn)→ J(αH+βK), J(Hn)→ J(H) and J(Kn)→ J(K) in probability, henceP -a.s. along a common subsequence, say (nl). It follows P -a.s.
J(αH + βK)t = liml→∞
J(αHnl + βKnl)t = liml→∞
(αJ(Hnl)t + βJ(Knl)t)
= liml→∞
(αJ(Hnl)t + βJ(Knl)t) = αJ(H)t + βJ(K)t.
The uniqueness of J follows similarly (exercise).
Summary: For
(Wt)t≥0 (Ft)-BM,
(Ht)t≥0 (Ft)-adapted process with P(∫ T
0
H2s ds <∞
)= 1,
we can define the stochastic integral(∫ T
0
Hs ds <∞)t∈[0,T ]
,
and we know that the stochastic integral is a martingale, if
E[ ∫ T
0
H2s ds
]<∞.
Moreover, it holds (using adequate stopping)
E[ ∫ T
0
H2s ds
]<∞ ⇔ E
[supt∈[0,T ]
∣∣∣ ∫ t
0
Hs dWs
∣∣∣2] <∞,and in either of these cases one has
E[( ∫ T
0
Hs dWs
)2]= E
[ ∫ T
0
H2s ds
].
3.4.2 Ito calculus
Definition 3.25 Let (Wt)t≥0 be an (Ft)-BM on (Ω,F, P ). An R-valued process (Xt)t∈[0,T ]
is called an Ito process, if
P(Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds ∀t ∈ [0, T ])
= 1,
where
49
- X0 is F0-measurable,
- (Ht)t∈[0,T ], and (Kt)t∈[0,T ] are (Ft)-adapted processes,
-∫ T
0|Ks| ds <∞ P -a.s.
-∫ T
0H2s ds <∞ P -a.s.
The following proposition implies that the decomposition of an Ito process is unique.
Proposition 3.26 If (M)t∈[0,T ] is a continuous martingale, such that
Mt =
∫ t
0
Ks ds, t ∈ [0, T ], and P(∫ T
0
|Ks| ds <∞)
= 1,
thenP (Mt = 0 ∀t ∈ [0, T ]) = 1.
(”A continuous martingale with finite variation is P -a.s. constant”.)
Proof Exercise.
Proposition 3.26 implies with adequate stopping:
- an Ito process decomposition is unique, i.e. if P -a.s for all t ≤ T
Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds = Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds,
thenP (X0 = X0) = 1, H = H and K = K ds⊗ dP -a.e.
- if (Xt)t∈[0,T ] is a martingale, i.e. X0 +∫ t
0Hs dWs +
∫ t0Ks ds is a martingale, then
K = 0 ds⊗ dP -a.e.
Theorem 3.27 (1-dimensional Ito formula for Ito processes) Let (Xt)t∈[0,T ] be anIto process of the form
Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds,
50
and f ∈ C2(R) (f is twice continuously differentiable function on R). Then P -a.s.
f(Xt) = f(X0) +
∫ t
0
f ′(Xs) dXs +1
2
∫ t
0
f ′′(Xs) d〈X,X〉s
where
〈X,X〉t := 〈X〉t :=
∫ t
0
H2s ds, t ∈ [0, T ],
and ∫ t
0
f ′(Xs) dXs :=
∫ t
0
f ′(Xs)Hs dWs +
∫ t
0
f ′(Xs)Ks ds, t ∈ [0, T ].
Likewise, if f ∈ C1,2(R+ × R) (twice continuously differentiable w.r.t. x ∈ R and oncecontinuously differentiable w.r.t. t ∈ R+ = [0,∞)), then
f(t,Xt) = f(0, X0) +
∫ t
0
f ′s(s,Xs) ds+
∫ t
0
f ′x(s,Xs) dXs +1
2
∫ t
0
f ′′x (s,Xs) d〈X,X〉s.
Proof See e.g. Karatzas/Shreve.
3.4.3 Examples: Ito formula in practice
For f(x) = x2, Xt = Wt, Ks ≡ 0, Hs ≡ 1:
W 2t = W 2
0︸︷︷︸=0
+
∫ t
0
2Ws dWs +1
2
∫ t
0
2 d 〈W,W 〉s︸ ︷︷ ︸=∫ s0 1 du=s
,= 2
∫ t
0
Ws dWs + t
and so
W 2t − t = 2
∫ t
0
Ws dWs.
Since E[ ∫ T
0W 2s ds
]=︸︷︷︸
Fubini
∫ T0E[W 2
s ] ds =∫ T
0s ds = T 2
2< ∞, it follows from Proposition
3.22 that( ∫ t
0Ws dWs
)t≥0
is a martingale. Thus we get a representation of the martingale
(W 2t − t)t≥0 from Proposition 3.12(2) as a stochastic integral.
Now, we want to find the solutions (if any) (St)t≥0 of
St = x0 +
∫ t
0
Ss(µ ds+ σ dWs), µ, σ, x0 ∈ R fixed.
51
In symbolic form this is written asdSt = St(µ dt+ σ dWt),
S0 = x0.(26)
Precisely, we are looking for an adapted and continuous process (St)t≥0 such that for P -a.e
ω ∈ Ω, (∫ t
0Ss(ω) ds) exists in the Lebesgue-Stieltjes sense, (
∫ t0Ss dWs) exists as stochastic
integral, and
St = x0 +
∫ t
0
σSs dWs +
∫ t
0
µSs ds P -a.s. ∀t ≥ 0.
In order to obtain a hint on how a solution might look like, let us suppose that wecan apply Ito’s formula with f(x) = log(x) /∈ C2(R) and that log(x0) is defined. Since〈S〉s =
∫ s0σ2S2
u du, this gives
log(St) = log(x0) +
∫ t
0
1
SsdSs −
1
2
∫ t
0
1
S2s
d〈S〉s
= log(x0) +
∫ t
0
σ dWs +
∫ t
0
µ ds− 1
2
∫ t
0
σ2 ds
= log(x0) + σWt +(µ− σ2
2
)t,
thus
St = x0 exp(σWt +
(µ− σ2
2
)t).
Since our procedure was not justified, we proceed backwards and show that this (St) reallyprovides a solution for any x0 ∈ R. We have
St = f(t,Wt) with f(t, x) = x0 exp(σx+
(µ− σ2
2
)t), t ≥ 0,
and since f ∈ C1,2(R+ × R) the time dependent Ito formula applies:
St = f(t,Wt)
= f(0,W0) +
∫ t
0
∂tf(s,Ws)︸ ︷︷ ︸=(µ−σ2
2
)Ss
ds+
∫ t
0
∂xf(s,Ws)︸ ︷︷ ︸=σSs
dWs +1
2
∫ t
0
∂xxf(s,Ws)︸ ︷︷ ︸=σ2Ss
d 〈W 〉s︸ ︷︷ ︸=s
= x0 +
∫ t
0
σSs dWs +
∫ t
0
µSs ds. (27)
52
Moreover, (St) is adapted and continuous. One can also check that (∫ t
0µSs ds) exists path-
wise P -a.s. since (St) is continuous, and that (∫ t
0σSs dWs) exists. Therefore the stochastic
differential equation (26) admits at least one solution.
Remark 3.28 One could also have checked that (St) is a solution by applying the Itoformula to the Ito process Zt = σWt +
(µ− σ2
2
)t with the C2-function f(x) = x0 exp(x).
We will establish uniqueness of a solution to (26). First we need an integration by partsformula.
Proposition 3.29 Let (Xt) and (Yt) be two Ito processes,
Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds
and
Yt = Y0 +
∫ t
0
H ′s dWs +
∫ t
0
K ′s ds.
Then
XtYt = X0Y0 +
∫ t
0
Xs dYs +
∫ t
0
YsdXs + 〈X, Y 〉t,
with the convention
〈X, Y 〉t =
∫ t
0
HsH′s ds, t ≥ 0.
Proof By Ito’s formula
(1):
(Xt + Yt)2 = (X0 + Y0)2 +
∫ t
0
2(Xs + Ys) d(Xs + Ys) +
∫ t
0
(Hs +H ′s)2 ds,
(2):
X2t = X2
0 +
∫ t
0
2Xs dXs +
∫ t
0
H2s ds
(3):
Y 2t = Y 2
0 +
∫ t
0
2Ys dYs +
∫ t
0
(H ′s)2 ds
53
(1)− ((2) + (3)) : 2XtYt = 2X0Y0 + 2
∫ t
0
XsdYs + 2
∫ t
0
Ys dXs + 2
∫ t
0
HsH′s ds
Now, we return to the uniqueness question. Let (Xt) be another solution to (26) withS0 = X0 = x0 6= 0. Define
Zt :=x0
St= exp
(− σWt +
(− µ+
σ2
2
)t)
= exp(σ′Wt +
(µ′ − (σ′)2
2
)t),
where σ′ = −σ and µ′ = −µ+ σ2. We know from (27) applied to (Zt) that
Zt = 1 +
∫ t
0
Zs(σ′ dWs + µ′ ds) = 1 +
∫ t
0
Zs(−σ dWs + (−µ+ σ2)ds).
From 3.29 we getd(XtZt) = Xt dZt + Zt dXt + d 〈X,Z〉t,
where
〈X,Z〉t =
∫ t
0
(σXs)(−σZs) ds = −∫ t
0
σ2XsZs ds.
Thus
d(XtZt) = XtZt(−σdWt + (−µ+ σ2)dt) + ZtXt(σdWt + µdt)− σ2XtZtdt = 0,
andXtZt = X0Z0 = x0 · 1 = x0 ∀t ≥ 0.
Therefore
Xt = x0Z−1t = x0
Stx0
= St P -a.s. ∀t ≥ 0.
⇒ P (Xt = St ∀t ≥ 0) = 1, i.e. X and S are indistinguishable.
We obtain:
Theorem 3.30 Let σ, µ, x0 ∈ R, T > 0, and (Wt)t≥0 be a BM. Then there exists a uniqueIto process (St)t∈[0,T ] that satisfies
St = x0 +
∫ t
0
Ss(σdWs + µds) P -a.s. ∀t ∈ [0, T ].
This process is given by
St = x0 exp(σWt +
(µ− σ2
2
)t), t ∈ [0, T ]
54
Proof We have proven the theorem for x0 6= 0. For x0 = 0 uniqueness can also be shown.
Remark 3.31 (i) The process (St) of 3.30 will model the evolution of the stock pricein the Black-Scholes model.
(ii) If µ = 0 then (St) is martingale (see 3.12).
Remark 3.32 Let U ⊂ R be an open set and (Xt)t∈[0,T ] and Ito process with P (Xt ∈U ∀t ∈ [0, T ]) = 1. If f ∈ C2(U), then the Ito formula holds:
f(Xt) = f(X0) +
∫ t
0
f ′(Xs) dXs +1
2
∫ t
0
f ′′(Xs) d〈X,X〉s P -a.s. ∀t ≥ 0.
This result allows us to apply e.g. Ito’s formula to log(Xt), if (Xt)t∈[0,T ] is a strictly positiveprocess.
3.4.4 Multidimensional Ito formula
Definition 3.33 A standard p-dimensional (Ft)-BM is an Rp-valued (Ft)-adapted process
Wt = (W 1t , . . . ,W
pt ), t ≥ 0,
where all the (W it ), 1 ≤ i ≤ p are standard independent (Ft)-BMs. In particular, it can be
proved that in this case Wt −Ws is independent of Fs, whenever 0 ≤ s < t.
Definition 3.34 An Ito process w.r.t. (Wt,Ft) where (Wt) is a standard p-dimensional(Ft)-BM, is a stochastic process (Xt)t∈[0,T ] of the form
Xt = X0 +
p∑i=1
∫ t
0
H is dW
is +
∫ t
0
Ks ds,
where:
• (Kt) and all the processes (H it), 1 ≤ i ≤ p are (Ft)-adapted,
•∫ T
0|Ks| ds <∞ P -a.s,
•∫ T
0(H i
s)2ds <∞ P -a.s, 1 ≤ i ≤ p.
55
The Ito formula becomes:
Proposition 3.35 Let (X1t ), . . . , (Xn
t ) be n Ito processes (w.r.t. (Wt,Ft)), i.e.
X it = X i
0 +
p∑j=1
∫ t
0
H i,js dW j
s +
∫ t
0
Kis ds, 1 ≤ i ≤ n.
If f ∈ C1,2(R+ × Rn), i.e. f(t, x) is two times continuously differentiable in x ∈ Rn, andcontinuously differentiable in t ∈ R+, then
f(t,X1t , . . . , X
nt ) = f(0, X1
0 , . . . , Xn0 ) +
∫ t
0
∂f
∂s︸︷︷︸=:∂sf
(s,X1s , . . . , X
ns ) ds
+n∑i=1
∫ t
0
∂f
∂xi︸︷︷︸=:∂if
(s,X1s , . . . , X
ns ) dX i
s +1
2
n∑i,j=1
∫ t
0
∂2f
∂xi∂xj︸ ︷︷ ︸=:∂ijf
(s,X1s , . . . , X
ns ) d〈X i, Xj〉s
with:
• dX is =
∑pj=1H
i,js dW j
s +Kis ds,
• d〈X i, Xj〉s =∑p
m=1 Hi,ms Hj,m
s ds.
Remark 3.36 If (Xt)t∈[0,T ] and (Yt)t∈[0,T ] are two Ito processes, we can define the covari-ation of X and Y (denoted by 〈X, Y 〉t) through the following properties:
(1) 〈X, Y 〉t is bilinear and symmetric.
(2) 〈∫ ·
0Ks ds,X.〉t = 0, ∀t ∈ [0, T ].
(3) 〈∫ ·
0Hs dW
is ,∫ ·
0H ′s dW
js 〉t = 0, ∀t ∈ [0, T ] if i 6= j.
(4) 〈∫ ·
0Hs dW
is ,∫ ·
0H ′s dW
is〉t =
∫ t0HsH
′s ds, ∀t ∈ [0, T ], 1 ≤ i ≤ p.
The covariation d〈X i, Xj〉s as defined previously in 3.35 is consistent with this definition.
56
3.5 Existence and uniqueness
3.5.1 Ito’s theorem
Definition 3.37 Fix a filtered probability space (Ω,F∞, (Ft)t>0, P ) that satisfies the “usualconditions”. Let (Bt)t≥0 be an (Ft)-Brownian motion starting from zero. Let
σ : R+ × R→ R, ”dispersion coefficient”
andb : R+ × R→ R, ”drift coefficient”
be measurable functions. Let
ξ : Ω→ R be an F0-measurable r.v. ”initial condition”
A solution to the stochastic differential equation (29) below, is an (Ft)-adapted stochasticprocess X = (Xt)t≥0 that satisfies for all t ≥ 0:∫ t
0
|σ(s,Xs)|2ds+
∫ t
0
|b(s,Xs)|ds <∞, P -a.s. (28)
and
Xt = ξ +
∫ t
0
σ(s,Xs)dBs +
∫ t
0
b(s,Xs)ds P -a.s. (29)
The integral∫ t
0b(s,Xs)ds has to be understood for P -a.e. path as a Lebesgue(-Stieltjes)
integral and∫ t
0σ(s,Xs)dBs is a stochastic integral as defined in subsection 3.4.1. Thus
condition (28) just ensures the existence of the integrals in (29).
Remark 3.38 Formally and for short the integral equation (29) is written asdXt = σ(t,Xt)dBt + b(t,Xt)dt
X0 = ξ(30)
57
Examples 3.39 (i) (Non-uniqueness or more then one solution)
Consider dXt = 3X
2/3t dBt + 3X
1/3t dt
X0 = 0.(31)
For α > 0 let Fα(x) := (x− α)3I[α,∞)(x). Then Fα ∈ C2(R) and
F ′α(x) = 3(x− α)2I[α,∞)(x) = 3Fα(x)2/3,
and1
2Fα′′(x) = 3(x− α)I[α,∞)(x) = 3Fα(x)1/3,
so by Ito’s formula dFα(Bt) = 3Fα(Bt)
2/3dBt + 3Fα(Bt)1/3dt
Fα(B0) = 0.
Thus for any α > 0 Xt := Fα(Bt) is a solution to (31), so (31) has infinitely many solu-tions.
(ii) (Explosion in finite time)
Consider dXt = X2
t dBt +X3t dt
X0 = 1.(32)
By ”Ito’s formula”
d
(1
Xt
)= − 1
X2t
dXt +1
2
2
X3t
d〈X〉t
= −dBt −Xtdt+Xtdt = −dBt.
Thus 1Xt− 1 = Bt, so Xt = 1
1−Bt solves (32) but only up to the first hitting timeT1 := inft > 0 |Bt = 1 of Brownian motion of 1. By Proposition 3.15, we know thatP (T1 <∞) = 1. Thus Xt explodes in finite time.
58
Question: Are there conditions that guarantee existence, so in particular non-explosion,and uniqueness of a solution ?
A first answer will be given in Proposition 3.43 and Theorem 3.45 below.
Definition 3.40 A measurable function g : R+ × R → R is said to satisfy a Lipschitzcondition in x if for any T > 0 there exists a constant KT > 0 such that
|g(s, x)− g(s, y)| ≤ KT |x− y| ∀0 ≤ s ≤ T, x, y ∈ R.
Definition 3.41 Pathwise uniqueness is said to hold for the SDE (30) if whenever(Xt) and (Yt) are two solutions to (30) on (Ω,F∞, (Ft)t>0, P ) with P (X0 = Y0) = 1, then
P (Xt = Yt ∀t ≥ 0) = 1.
Lemma 3.42 [Gronwall] Let g : [0, T ] → R be continuous, h : [0, T ] → R be integrable,and β ≥ 0. If
0 ≤ g(t) ≤ h(t) + β
∫ t
0
g(s)ds ∀t ∈ [0, T ],
then
0 ≤ g(t) ≤ h(t) + β
∫ t
0
h(s)eβ(t−s)ds ∀t ∈ [0, T ],
and if h is an increasing function, then g(t) ≤ h(t)eβt for all t ∈ [0, T ].
Proof By the assumption for all t ∈ [0, T ]
d
dt
(e−βt
∫ t
0
g(s)ds
)=
(g(t)− β
∫ t
0
g(s)ds
)e−βt ≤ h(t)e−βt.
Taking integrals ∫ t
0
g(s)ds ≤∫ t
0
h(s)eβ(t−s)ds ∀t ∈ [0, T ].
Hence again by the assumption
0 ≤ g(t) ≤ h(t) + β
∫ t
0
g(s)ds ≤ h(t) + β
∫ t
0
h(s)eβ(t−s)ds ∀t ∈ [0, T ].
From this the last statement is clear.
59
Proposition 3.43 Suppose that σ, b satisfy both a Lipschitz condition in x. Then pathwiseuniqueness holds for (30).
Proof Let X, Y be two solutions to (30) on the same filtered probability space with sameBrownian motion and P (X0 = Y0) = 1. Let
Tn := inft > 0 | |Xt − Yt| ∨ |Bt| > n.
Then the Tn are stopping times with Tn ∞ P -a.s. By Ito’s formula applied to X − Ywe have P -a.s.
(Xt − Yt)2 = 2
∫ t
0
(Xs − Ys)d(Xs − Ys) + 〈X − Y 〉t
= 2
∫ t
0
(Xs − Ys)(σ(s,Xs)− σ(s, Ys))dBs + 2
∫ t
0
(Xs − Ys)(b(s,Xs)− b(s, Ys))ds
+
∫ t
0
(σ(s,Xs)− σ(s, Ys))2ds =: Zt.
Since P ((Xs − Ys)2 = Zs ∀s ≤ t) = 1 the last also also holds with t replaced by t∧ Tn. Inparticular ∫ t∧Tn
0
(Xs − Ys)(σ(s,Xs)− σ(s, Ys))dBs , t ≥ 0
is a martingale and (Xt∧Tn − Yt∧Tn)2 is bounded for all n. Thus taking expectations andusing the Lipschitz condition we get for all t ≤ T and every n
E[(Xt∧Tn − Yt∧Tn)2] ≤ E[2KT
∫ t∧Tn
0
(Xs − Ys)2ds] +K2TE[
∫ t∧Tn
0
(Xs − Ys)2ds]
= (2KT +K2T )E[
∫ t
0
(Xs∧Tn − Ys∧Tn)2ds].
Fubini= (2KT +K2
T )
∫ t
0
E[(Xs∧Tn − Ys∧Tn)2]ds.
By Gronwall’s lemma 3.42
g(t) := E[(Xt∧Tn − Yt∧Tn)2] = 0 ∀t ≤ T.
Thus P (Xt∧Tn = Yt∧Tn) = 1, and letting n → ∞ we get P (Xt = Yt) = 1 for ant t ≤ T .This holds for any T > 0. Thus pathwise uniqueness now follows from continuity using aSandwich argument.
60
For the existence (in particular non-explosion) of a solution we have to impose somegrowth condition on the coefficients. The proof of existence is similar to the proof forODE’s (ODE=ordinary differential equation). Note also that our typical examples in3.39 were just simple modifications of the well-known examples in the theory of ODE’s(dXtdt
= 3X2/3t , X0 = 0 with the solutions Xα
t = (t − α)3I[α,∞)(t), and dXtdt
= X2t , X0 = 1
with the exploding solution Xt = 11−t).
Definition 3.44 A measurable function g : R+ × R → R is said to satisfy a lineargrowth condition in x if for any T > 0 there exists a constant CT > 0 such that
|g(s, x)| ≤ CT (1 + |x|) ∀0 ≤ s ≤ T, x ∈ R. (33)
Since for all x ≥ 0, 1 + x2 ≤ (1 + x)2 ≤ 2(1 + x2), it follows that (33) is equivalent with
|g(s, x)|2 ≤ C ′T (1 + x2) ∀0 ≤ s ≤ T, x ∈ R.
Theorem 3.45 Let σ, b satisfy both a Lipschitz and a linear growth condition in x. LetE[ξ2] <∞. Then the SDE (30) has a pathwise unique solution.
Proof 1. Step: Assume σ, b bounded in space, i.e. bounded on [0, T ]×R for any T > 0.
We will proceed by Picard iteration. Define X0t ≡ ξ and define inductively for
X i+1t := ξ +
∫ t
0
σ(s,X is)dBs +
∫ t
0
b(s,X is)ds i = 0, 1, 2, ... (34)
Thus for i ≥ 1
X i+1t −X i
t =
∫ t
0
[σ(s,X i
s)− σ(s,X i−1s )
]dBs︸ ︷︷ ︸
=:Ft
+
∫ t
0
[b(s,X i
s)− b(s,X i−1s )
]ds︸ ︷︷ ︸
=:Gt
. (35)
Letgi(t) := E[sup
s≤t|X i+1
s −X is|2].
61
By Doob’s inequality and Fubini and the Lipschitz condition
E[sups≤t|Fs|2] ≤ 4E
[∫ t
0
∣∣σ(s,X is)− σ(s,X i−1
s )∣∣2 ds]
≤ 4K2T
∫ t
0
E[|X i
s −X i−1s |2
]ds
≤ 4K2T
∫ t
0
gi−1(s)]ds. (36)
By Cauchy-Schwarz
E[sups≤t|Gs|2] ≤ tE
[∫ t
0
∣∣b(s,X is)− b(s,X i−1
s )∣∣2 ds]
≤ tK2T
∫ t
0
E[|X i
s −X i−1s |2
]ds
≤ tK2T
∫ t
0
gi−1(s)ds. (37)
By (35), (36), (37) and (x + y)2 = x2 + 2xy + y2 there is a constant C > 0 such that forall i ≥ 1
gi(t) ≤ 2E[sups≤t|Fs|2] + 2E[sup
s≤t|Gs|2]
≤ C(1 + t)
∫ t
0
gi−1(s)ds. (38)
For i = 0 similarly to (36), (37) there is a constant D > 0 such that
g0(t) = E[sups≤t|X1
s − ξ|2]
= E
[sups≤t
∣∣∣∣∫ t
0
σ(s, ξ)dBs +
∫ t
0
b(s, ξ)ds
∣∣∣∣2]≤ D(1 + t). (39)
By (38), (39)
g1(t) ≤ C(1 + t)
∫ t
0
D(1 + s)ds ≤ C(1 + t)D(1 + t)2
2(40)
62
for all t ≤ T and so by (38), (40)
g2(t) ≤ C2(1 + t)2
∫ t
0
D(1 + s)2
2ds ≤ C2(1 + t)2D
(1 + t)3
3!.
Therefore by induction
gi(t) ≤ Ci(1 + t)iD(1 + t)i+1
(i+ 1)!i = 0, 1, 2, ...
for all t ≤ T . It follows
∞∑i=0
gi(t)1/2 < ∞.
Define the norm‖Y ‖ := E[sup
s≤t|Ys|2]1/2.
The for any m > n
‖Xm −Xn‖ ≤m−1∑i=n
‖X i+1 −X i‖ =m−1∑i=n
gi(T )1/2 −→ 0
as m,n → ∞. Thus (Xn)n∈N is a Cauchy sequence for the complete norm ‖ · ‖ and sothere is a process X with ‖Xn −X‖ → 0 as n→∞. In particular there is a subsequencesuch that
sups≤t|Xs −Xnj
s |j→∞−→ 0 P -a.s.
for any t ≤ T and therefore X is continuous. This holds for every T > 0. Finally, lettingni →∞ in (34) we get (30) for σ, b bounded in space.
2. Step: σ, b as in the statement.
Fix T > 0. Let fn ↑, fn ∈ C∞0 (R), fn(x) = 1 for |x| ≤ n fn(x) = 0 on |x| > n + 1.Then σfn, bfn satisfy the assumption of 1. Step. And so there is a corresponding uniquesolution (Xn
t ) of (30) with the coefficients σfn, bfn. Let
Tn := inft > 0 | |Xnt | > n.
63
We claim that Tn ∞, P -a.s. Indeed, let gn(t) := E[sups≤t∧Tn |Xn
s |2]. Then using in
particular the linear growth condition we have for any t ≤ T
gn(t) ≤ 2E[ξ2] + 16E
[∫ t∧Tn
0
|σfn(s,Xns )|2ds
]+ 4tE
[∫ t∧Tn
0
|bfn(s,Xns )|2ds
]≤ 2E[ξ2] + 16E
[∫ t∧Tn
0
C ′T (1 + |Xns |2)ds
]+ 4tE
[∫ t∧Tn
0
C ′T (1 + |Xns |2)ds
]≤ 2E[ξ2] + 16TC ′T (1 + T )︸ ︷︷ ︸
=:CT
+ 16C ′T (1 + T )︸ ︷︷ ︸=:DT
∫ t
0
gn(s)ds.
Thus by Gronwall’s inequality gn(t) ≤ CT eDT ·t for all t ≤ T . But then by Chebychev for
any T > 0
P (Tn ≤ T ) = P (sups≤T|Xn
s | > n)
≤ 1
n2E
[sups≤T|Xn
s |2]≤ 1
n2CT e
DT ·T → 0 as n→∞.
Therefore Tn ∞, P -a.s. Let
Xt := Xnt for t ≤ Tn.
Of course X defined in this way will solve (30), does not explode, and will be the uniquesolution by Proposition 3.43. We only have to check whether X is well-defined. In orderto do so we have to show that
Xnt = Xm
t for t ≤ Tn ∧ Tm =: Tnm (41)
64
holds. But for any t ≤ T similarly to 1. Step one shows that
E
[sup
s≤t∧Tnm|Xn
s −Xms |2]≤ E
[sups≤t|Xn
s∧Tnm −Xms∧Tnm |
2
]≤ 2E
[sups≤t
∣∣∣∣∫ s∧Tnm
0
σfn(u,Xnu )− σfm(u,Xm
u )dBu
∣∣∣∣2]
+2E
[sups≤t
∣∣∣∣∫ s∧Tnm
0
bfn(u,Xnu )− bfm(u,Xm
u )du
∣∣∣∣2]
≤ 8E
[∫ t∧Tnm
0
|σ(u,Xnu )− σ(u,Xm
u )|2du]
+2tE
[∫ t∧Tnm
0
|b(u,Xnu )− b(u,Xm
u )|2du]
≤ C(1 + T )
∫ t
0
E
[sup
s≤t∧Tnm|Xn
s −Xms |2]ds.
Thus (41) follows by Gronwall’s inequality.
3.5.2 The Ornstein-Uhlenbeck process
The OU process is the unique solution ofdXt = σdBt − cXtdt
X0 = x ∈ R
The explicit solution can be obtained as follows : Consider Yt = Xtect. Integration by
parts yields
dYt = ectdXt +Xtd(ect) + d 〈X, ect〉︸ ︷︷ ︸=0
= σectdWt − cectXtdt+ cectXtdt = σectdWt
Thus
Xtect = X0 +
∫ t
0
σecs dBs, and
Xt = e−ctx+ σ
∫ t
0
ec(s−t) dBs
65
Let us compute mean and variance of Xt:
E[Xt] = e−ctx+ σe−ct E[ ∫ t
0
ecs dBs
]︸ ︷︷ ︸
=0
= e−ctx.
E[∫ t
0ecs dBs] = 0, since
∫ t0(ecs)2ds <∞, hence (
∫ t0ecs dBs)t≥0 is a martingale.
Var(Xt) = E[(Xt − E[Xt])2] = E
[(σ
∫ t
0
ec(s−t) dBs
)2]= σ2
∫ t
0
e2c(s−t) ds = σ2 [e2c(s−t)]s=ts=0
2c= σ2 (1− e−2ct)
2c.
3.5.3 Multidimensional stochastic differential equations
On (Ω,F∞, (Ft)t>0, P ) let Wt = (B(1)t , ..., B
(p)t ), t ≥ 0 be an p-dimensional (Ft)-Brownian
motion starting from zero. For 1 ≤ i ≤ n, 1 ≤ j ≤ p let
σij : R+ × Rn → R, bi : R+ × Rn → R
be measurable functions. σ(t, x) = (σij(t, x))1≤i≤n,1≤j≤p is called the dispersion matrix andb(t, x) = (b1(t, x), ..., bn(t, x)) the drift (vector). Let
ξ = (ξ1, ..., ξn) : Ω→ Rn be an F0-measurable random vector (”initial condition”).
A solution to the stochastic differential equation (SDE)dXt = σ(t,Xt)dWt + b(t,Xt)dt
X0 = ξ(42)
also writen as
Xt = ξ +
∫ t
0
σ(s,Xs)dWs +
∫ t
0
b(s,Xs)ds
is an (Ft)-adapted Rn-valued stochastic process Xt = (X(1)t , ..., X
(n)t ) that satisfies for all
t ≥ 0, and i = 1, ..., n:
X(i)t = ξi +
p∑j=1
∫ t
0
σij(s,Xs)dB(j)s +
∫ t
0
bi(s,Xs)ds P -a.s. (43)
66
Of course we implicitely assume that all integrals are well-defined, i.e. P (∫ t
0|σij(s,Xs)|2 +
|bi(s,Xs)|ds <∞) = 1 for all 1 ≤ i ≤ n, 1 ≤ j ≤ p and t ≥ 0.
Consider the (local) martingale part
M it :=
p∑j=1
∫ t
0
σij(s,Xs)dB(j)s
of (43). Then by Remark 3.36
〈M i,Mk〉t =
⟨p∑j=1
∫ ·0
σij(s,Xs)dB(j)s ,
p∑l=1
∫ ·0
σkl(s,Xs)dB(l)s
⟩t
=
∫ t
0
p∑j=1
σijσkj(s,Xs)ds =
∫ t
0
aik(s,Xs)ds,
where a(t, x) = (aij(t, x))1≤i,j≤n = σ(t, x)σT (t, x), is called the diffusion matrix. Let f ∈C1,2(R+ × Rn). The linear operator
Lf(t, x) =n∑
i,j=1
aij(t, x)
2∂i∂jf(t, x) +
n∑i=1
bi(t, x)∂if(t, x) + ∂tf(t, x)
where ∂i = ∂∂xi
, ∂t = ∂∂t
is called the (infinitesimal) generator of the diffusion (t,Xt). Let
|x| := (∑n
i=1 x2i )
1/2 be the euclidean norm of x ∈ Rn. Since for each (t, x) ∈ R+ × Rn
n∑i,j=1
aij(t, x)yiyj = a(t, x)y · y = |σT (t, x)y|2 ≥ 0 ∀y = (y1, ..., yn) ∈ Rn,
the diffusion matrix is nonnegative definite and consequently L is an elliptic differentialoperator.
Define the matrix norm
‖σ(t, x)‖ :=
(n∑i=1
(p∑j=1
σ2ij(t, x)
))1/2
.
Theorem 3.46 Suppose that for any T > 0 there exists a constant KT > 0 such that forall s ≤ T and x, y ∈ Rn:
67
(i) (Lipschitz condition)
‖σ(s, x)− σ(s, y)‖ ≤ KT |x− y| and |b(s, x)− b(s, y)| ≤ KT |x− y|.
(ii) ((Linear growth condition)
‖σ(s, x)‖2 ≤ KT (1 + |x|2) and |b(s, x)|2 ≤ KT (1 + |x|2).
Let E[ξ2] <∞. Then the SDE (42) has a pathwise unique solution.
Proof The proof is nearly exactly the same as the corresponding proofs of 3.43 and 3.45in dimension one. We therefore omit them.
3.5.4 The Markov property of the solution of an SDE
Let σ, b, ξ satisfy the conditions Theorem 3.45. Similarly to the proof of Theorem 3.45 onecan show by Picard iteration that for any s ≥ 0 there exists a unique solution to
Xt = ξ +
∫ t
s
σ(u,Xu) dWu +
∫ t
s
b(u,Xu) du, t ≥ s (44)
Denote this solution by
(Xs,ξt )t≥s ”starting at time s from ξ”
Thus
Xs,ξt = ξ +
∫ t
s
σ(u,Xs,ξu ) dWu +
∫ t
s
b(u,Xs,ξu ) du t ≥ s (45)
In particular (and this is Theorem 3.45) for s = 0, and ξ = x there is a unique solution to
Xt = x+
∫ t
0
σ(u,Xu) dWu +
∫ t
0
b(u,Xu) du, t ≥ 0, x ∈ R. (46)
Correspondingly, this solution is denoted by
(X0,xt )t≥0 ”starting at time 0 from x”
Taking the difference of the solution at time t to the solution at time s in (46) we get
X0,xt = X0,x
s︸︷︷︸∈L2(Ω,Fs,P )!
+
∫ t
s
σ(u,X0,xu ) dWu +
∫ t
s
b(u,X0,xu ) du, t ≥ s (47)
68
Thus by uniqueness of (44) with ξ = X0,xs we get pathwise (i.e. P -a.s.)
Xs,X0,xs
t = X0,xt ∀t ≥ s, x ∈ R (”flow property”).
Now the Markov property can be stated as follows:
Theorem 3.47 Let (Xt)t≥0 = (X0,xs )t≥0 be the unique solution of (46) with x ∈ R, on
the filtered probability space (Ω,F, (Ft)t≥0, P ). Then for any bounded Borel measurablefunction f , we have
E[f(Xt) |Fs] = φ(Xs) P -a.s. ∀t ≥ s,
where φ(x) = E[f(Xs,xt )].
Proof It can be shown (difficult) that for fixed t ≥ s
Xs,xt = Φ(x, (Ws+u −Ws)u≥0), (48)
i.e. the solution is a measurable function Φ of its initial condition and the Brownianincrements (this is natural, if we think of the Picard iteration procedure). Thus by theflow property for t ≥ s
X0,xt = Xs,X0,x
st = Φ(X0,x
s︸︷︷︸Fs-mb
, ( Ws+u −Ws︸ ︷︷ ︸independent of Fs
)u≥0)
Now by [1, Proposition 7.6] for P -a.e. ω ∈ Ω
E[f Φ(X0,xs , (Ws+u −Ws)u≥0) |Fs](ω) = E[f Φ(X0,x
s (ω), (Ws+u −Ws)u≥0)]
=(48)
E[f(Xs,X0,x
s (ω)t )] = φ(X0,x
s )(ω).
3.47 generalizes to:
Theorem 3.48 Let (Xt)t≥0 = (X0,xs )t≥0 and f be as in Theorem 3.47 and let r(s, x) be
non-negative and measurable. Then for t ≥ s
E[e−∫ ts r(u,Xu) duf(Xt) | Fs] = φ(Xs) P -a.s.,
where φ(x) = E[e−∫ ts r(u,X
s,xu ) duf(Xs,x
t )].
69
Remark 3.49 Suppose that σ and b do not depend on time. From (45) with t = s+h, h ≥0 and ξ = x ∈ R we get
Xs,xs+h = x+
∫ s+h
s
σ(Xs,xu ) dWu +
∫ s+h
s
b(Xs,xu ) du
=u=s+v
x+
∫ h
0
σ(Xs,xs+v)d(Ws+v −Ws︸ ︷︷ ︸
=:Bv also BM!
) +
∫ h
0
b(Xs,xs+v) dv. (49)
Since pathwise uniqueness holds for (44), uniqueness in law holds for (44). Thus sincealso (X0,x
h )t≥0 solves (49) but with different BM we get
(X0,xh )h≥0 and (Xs,x
s+h) have same law (”(Xh)h≥0 is homogeneous”).
Therefore, for any measurable function f we have
E[f(X0,xt )] = E[f(Xs,x
t+s)].
This can be again extended, namely for r(x) measurable non-negative, we get
E[e−
∫ s+ts r(Xs,x
u ) duf(Xs,xs+t)]
= E[e−
∫ t0 r(X
0,xu ) duf(X0,x
t )]
In this case 3.48 yields
E[e−
∫ ts r(Xu) duf(Xt) |Fs
]= φ(Xs), t > s,
whereφ(x) = E
[e−
∫ t−s0 r(X0,x
u ) duf(X0,xt−s)].
70
4 The Black-Scholes model
4.1 Description of the model
4.1.1 The behavior of prices
Consider continuous-time model with:
- one riskless asset: (S0t ) (e.g a bond)
- one risky asset: (St) (e.g. a stock price)
We assume that (S0t ) solves the ODE
dS0t = rS0
t dt
S00 = 1,
(50)
where r > 0 is the (instantaneous) interest rate. Then
S0t = S0
0ert = ert, t ≥ 0.
We further suppose, that the behavior of the stock price is determined by the SDE
dSt = σS0 dBt + µSt dt, S0 > 0. (51)
and (Bt)t≥0 is a standard BM, σ and µ are constants. σ is the so-called ”volatility of(St)”. The model will be considered for t ∈ [0, T ], T is the maturity of an option on (St).We know that (51) has the (pathwise) unique solution
St = S0 exp(σBt +
(µ− σ2
2
)t), t ≥ 0.
Because of the explicit representation of (St), such a solution is called a closed-formsolution.
Remarks on St:
(0) (St) is called geometric (or also exponential) Brownian motion.
(1) log(StS0
)is normally distributed for any t ≥ 0, i.e. St
S0is lognormal.
(2) (St)t≥0 is continuous.
71
(3) S0 is the stock price at time 0, so usually considered as strictly positive constant.
(4) the relative increments are independent, i.e. if s ≤ t, then
StSs
= exp(σ(Bt −Bs) +
(µ− σ2
2
)(t− s)
),
or equivalently the relative increment St−SsSs
is independent of Fs, where (Ft)t≥0 isthe natural filtration of (Bt)t≥0 (see Definition 3.11+ Remark 3.10)
(5) the relative increments are stationary, i.e. if s ≤ t, then
St − SsSs
andSt−s − S0
S0
have same law.
Hypotheses (2), (4) and (5) are the hypotheses of Black and Scholes on the behavior ofthe stock price.
4.1.2 Self-financing strategies
Let (Ft)t≥0 be the natural filtration of (Bt)t≥0. A strategy is an R2-valued (Ft)-adaptedprocess
φ = (φt)t∈[0,T ] = (H0t , Ht)t∈[0,T ],
where
H0t = number of units (quantities) of riskless asset held at time t,
Ht = number of units (quantities) of risky asset held at time t.
The value of the portfolio at time t is
Vt(φ) = H0t S
0t +HtSt.
In discrete time (see Remark 2.2) we characterized self-financing strategies through theequality
Vn+1(φ)− Vn(φ) = φn+1(Sn+1 − Sn).
In continuous time the last is expressed through the SDE
dVt(φ) = H0t dS
0t +Ht dSt.
To give a meaning to this SDE, we need of course conditions on the strategy φ = (H0, H).These are stated in the following definition.
72
Definition 4.1 A self-financing strategy is a pair of (Ft)-adapted processes φ =((H0
t )t∈[0,T ], (Ht)t∈[0,T ]
)satisfying:
(i) ∫ T
0
(|H0s |+H2
s ) ds <∞ P -a.s.
(ii) ∀t ∈ [0, T ], we have
H0t S
0t +HtSt︸ ︷︷ ︸
=Vt(φ)
= H00S
00 +H0S0︸ ︷︷ ︸=V0(φ)
+
∫ t
0
H0s dS
0s +
∫ t
0
Hs dSs P -a.s.
By (i) both integrals in (ii) are well defined, since:∫ T
0
H0s dS
0s =
∫ T
0
H0s re
rs ds and
∫ T
0
|H0s re
rs| ds ≤ rerT∫ T
0
|H0s | ds︸ ︷︷ ︸
<∞
and ∫ T
0
Hs dSs = σ
∫ T
0
HsSs dWs + µ
∫ T
0
HsSs ds,
where the first integral exists since∫ T
0
H2s (ω)S2
s (ω)︸ ︷︷ ︸pathwise boundedsince continuous
ds ≤ C(ω)︸ ︷︷ ︸<∞
∫ T
0
H2s (ω) ds <∞
for P -a.e. ω ∈ Ω. Similarly, applying the Cauchy-Schwarz inequality the secondintegral also exists P -a.s.
DefineSt := e−rtSt, t ≥ 0 ”discounted price of risky asset”.
Proposition 4.2 (cf. Proposition 2.3) Let φ = (H0t , Ht)t∈[0,T ] with values in R2 be a
pair of (Ft)-adapted processes. Suppose∫ T
0(|H0
s | + |Hs|2) ds < ∞ P -a.s., and let Vt(φ) =
H0t S
0t +HtSt, and Vt(φ) := e−rtVt(φ), t ∈ [0, T ]. Then φ is a self-financing strategy, if and
only if
Vt(φ) = V0(φ) +
∫ t
0
Hs dSs P -a.s. ∀t ∈ [0, T ]. (52)
73
Proof We have to show that under the given assumptions, we have: (52) ⇔ 4.1(ii).Suppose φ is self-financing, i.e 4.1(ii) holds. Using integration by parts
d Vt(φ)︸ ︷︷ ︸=e−rtVt(φ)
= −re−rtVt(φ) dt+ e−rt dVt(φ) + d〈e−r(·), V·(φ)〉t︸ ︷︷ ︸=0
Thus
dVt(φ) =︸︷︷︸4.1(ii)
−re−rt(H0t e
rt +HtSt) dt+ e−rt(H0t d(ert) +Ht dSt)
= Ht(−re−rtSt dt+ e−rt dSt)
= Ht(St d(e−rt) + e−rt dSt) = Ht dSt
and (52) holds. The converse holds similarly, applying the integration by parts formula toVt(φ) = ertVt(φ).
Remark 4.3 In discrete time we imposed predictability on the strategies. In continuoustime the predictability condition is included in some sense in the definition of stochasticintegral. In the study of complete discrete markets we considered equivalent martingalemeasures. By this, we were able to find replicating strategies. The main tools for the samein continuous time are the Girsanov transformation and Ito’s representation theorem thatwe introduce in the next section.
4.2 Change of probability. Representation of martingales
4.2.1 Equivalent probabilities
Let (Ω,A, P ) be a probability space, and Q be another probability measure on (Ω,A).Then define
QA P (in words: Q is absolutely continuous w.r.t P (on A))
:⇔ A ∈ A and P (A) = 0 ⇒ Q(A) = 0
and
Q ≈A P (in words: Q and P are equivalent (on A))
:⇔ QA P and P A Q.
74
Theorem 4.4 (Radon-Nikodym special case) Let P,Q be probability measures on(Ω,A). Then
Q P ⇔ ∃ r.v. Z : Ω→ R+ with Q(A) =
∫A
Z dP ∀A ∈ A.
Z is called the (Radon-Nikodym) density of Q w.r.t. P and one writes Z = dQdP
. In partic-ular, if Q P , then: Q ≈ P ⇔ P (Z > 0) = 1.
Proof ”⇐” and last statement obvious. ”⇒” see Radon-Nikodym theorem (probabilisticpart of lecture).
4.2.2 The Girsanov theorem
Let (Ω,F, (Ft), P ) be a filtered probability space and (Bt)t∈[0,T ] be a standard (Ft)-BM.Then we have:
Theorem 4.5 (Girsanov transformation) Let (θt)t∈[0,T ] be (Ft)-adapted with∫ T
0θ2s ds <
∞ P -a.s. Suppose
Lt := exp(−∫ t
0
θs dBs −1
2
∫ t
0
θ2s ds), t ∈ [0, T ],
is an (Ft)-martingale w.r.t. P . Let dP (LT ) := LT dP . Then P (LT ) is a probability measurethat is equivalent to P and
Wt := Bt +
∫ t
0
θs ds, t ∈ [0, T ],
is a standard (Ft)-BM under P (LT ).
Remark 4.6 The most well-known sufficient condition on (Lt)t∈[0,T ] to be a martin-gale is Novikov’s condition, i.e.
E[
exp(1
2
∫ T
0
θ2 dt)]
<∞.
75
4.2.3 Representation of Brownian martingales
Let (Bt)t∈[0,T ] be a standard BM on (Ω,F, P ) with natural filtration (Ft)t∈[0,T ]. The fol-lowing is known as Ito’s representation theorem:
Theorem 4.7 Let (Mt)t∈[0,T ] be a square-integrable martingale w.r.t. (Ft) (= natural fil-
tration of (Bt)). Then there exists an (Ft)-adapted process (Ht)t∈[0,T ] with E[ ∫ T
0H2s ds
]<
∞ and
Mt = M0 +
∫ t
0
Hs dBs P -a.s. ∀t ∈ [0, T ]. (53)
In particular: (Mt) is continuous.
Proof See for instance Karatzas/Shreve.
If U is an FT -measurable r.v. and E[U2] <∞, i.e. U is square-integrable, then
U = E[U ] +
∫ T
0
Hs dBs, P -a.s.
for some (Ft)-adapted process with E[ ∫ T
0H2s ds
]<∞. For the proof it suffices to consider
the square-integrable ”Brownian martingale”
Mt := E[U |Ft], t ∈ [0, T ],
and to apply Theorem 4.7.
4.3 Pricing hedging options in the Black-Scholes model
4.3.1 A probability measure under which (St) is a martingale
Consider the model introduced in 4.1. We have
dSt = d(e−rtSt) = e−rt dSt − re−rtSt dt= e−rt(σSt dBt + µSt dt)− re−rtSt dt= St(σ dBt + (µ− r) dt)
= σSt(dBt +µ− rσ
dt)
If we set Wt = Bt + µ−rσt, then
dSt = σSt dWt (54)
76
and from 4.5 applied with θt ≡ µ−rσ
((θt)t∈[0,T ] satisfies Novikov’s condition 4.6) we knowthat
Wt = Bt +
∫ t
0
µ− rσ
dt, t ∈ [0, T ]
is a BM under
dP (LT ) = exp(− µ− r
σBT −
(µ− r)2
2σ2T)dP.
By Theorem 3.30, the unique solution to (54) is given
St = S0 exp(σWt −
σ2
2t).
Since (Wt) is a Brownian motion under P ∗ := P (LT ), we know that (St) is a (squareintegrable) P ∗-martingale.
4.3.2 Pricing
A contingent claim (or option) is defined to be a non-negative FT -measurable r.v. h,e.g.
h = f(ST ) with f(x) =
(x−K)+ ”European call”,
(K − x)+ ”European put”.
Definition 4.8 A strategy φ = (H0t , Ht)t∈[0,T ] is called admissible, if it is self-financing,
and if the discounted value Vt(φ) = H0t + HtSt of the corresponding portfolio is non-
negative for all t, and such that
supt∈[0,T ]
Vt(φ) ∈ L2(Ω,FT ,P∗).
An option (or contingent claim) h is said to be replicable, if there exists an admissiblestrategy φ with
VT (φ) = h. (”φ replicating strategy”)
In particular, if h is replicable then h ∈ L2(P ∗). For a European option (put or call), wealways have h ∈ L2(P ∗), since
((ST −K)+)2 ≤ S2T ∈ L1(P ∗), and ((K − ST )+)2 ≤ K2 ∈ L1(P ∗).
Theorem 4.9 In the Black-Scholes model any contingent claim h, with h ∈ L2(Ω,FT , P∗)
is replicable and the value at time t of any replicating portfolio is
Vt(φ) = Vt = E∗[e−r(T−t)h | Ft]. (55)
77
Remark: In this case one also says that E∗[e−r(T−t)h | Ft] is the value of the option h attime t ∈ [0, T ]. Note that it is independent of φ.Proof Assume first ∃ admissible strategy φ = (H0, H) that replicates the option h, i.e.
VT (φ) = h.
We haveVt := Vt(φ) = H0
t S0t +HtSt,
and for the discounted value Vt = e−rtVt we have
Vt = H0t +HtSt.
Since φ is in particular self-financing, we get
Vt =4.2V0 +
∫ t
0
Hu dSu =(54)
V0 +
∫ t
0
HuσSu dWu.
Since φ is admissible, we have supt≤T Vt ∈ L2(P ∗), hence supt≤T
∣∣∣ ∫ t0 HuσSu dWu
∣∣∣ ∈L2(P ∗). Therefore, (
∫ t0HuσSu dWu)t∈[0,T ], and hence (Vt)t∈[0,T ] is a square-integrable mar-
tingale w.r.t. P ∗ (cf. Assignments). It follows
Vt = E∗[VT | Ft],
thusVt = E∗[e−r(T−t)h | Ft].
We have hence proved: If φ is a replicating strategy, then (55) holds.To complete the proof, we have to show that any option h ∈ L2(Ω,FT , P
∗) is replicable,i.e. ∃ admissible strategy φ = (H0, H) such that VT (φ) = h.Since h ∈ L2(Ω,FT , P
∗) it follows that
Mt := E∗[e−rTh | Ft], t ∈ [0, T ],
is a square integrable (P ∗, (Ft))-martingale. We have
(Ft) natural filtration of (Bt) w.r.t. P =⇒P ∗≈P
(Ft) natural filtration of (Wt) w.r.t. P ∗.
Thus by Ito’s representation theorem 4.7 ∃(Kt)t∈[0,T ] (Ft)-adapted, with E∗[ ∫ T
0K2s ds
]<
∞ and
Mt = M0 +
∫ t
0
Ks dWs P ∗-a.s t ∈ [0, T ].
78
Define
φ = (H0t , Ht) :=
(Mt −
Kt
σ,Kt
σSt
).
Then clearly φ is (Ft)-adapted,∫ T
0(|H0
s |+ |Hs|2) ds <∞ P -a.s. and
Vt(φ) = H0t +HtSt = Mt = M0 +
∫ t
0
Ks dWs. (?)
Since V0(φ) = M0 and ∫ t
0
Hs dSs =(54)
∫ t
0
HsσSs dWs =
∫ t
0
Ks dWs,
we get
Vt(φ) = V0(φ) +
∫ t
0
Hs dSs =⇒4.2
φ is self-financing.
FurthermoreVt = Vt(φ) =
(?)ertMt = E∗[e−r(T−t)h | Ft].
This implies Vt ≥ 0, supt≤T Vt(φ) ∈ L2(P ∗) (Doob’s inequality applied to (Vt(φ))t∈[0,T ]),and VT = h, hence φ is admissible and replicating.
Remark 4.10 Suppose h in Theorem 4.9 can be written as h = f(ST ). Then
ST = S0 exp(σBT +
(µ− σ2
2
)T)
= S0 exp(σBt +
(µ− σ2
2
)t)
︸ ︷︷ ︸=St
exp(σ(BT −Bt︸ ︷︷ ︸
=WT−Wt−µ−rσ (T−t)
) +(µ− σ2
2
)(T − t)
)
= St exp(σ(WT −Wt) +
(r − σ2
2
)(T − t)
)Now
Vt =4.9
E∗[e−r(T−t)f(ST ) | Ft]
= E∗[e−r(T−t)f
(St︸︷︷︸
Ft-mb
exp(σ(WT −Wt︸ ︷︷ ︸
indep. of Ft
) +(r − σ2
2
)(T − t)
))︸ ︷︷ ︸
=φt(St,WT−Wt)
| Ft]
= F (t, St),
79
by Proposition 7.6 of the Probabilistic Background, where
F (t, x) = E∗[e−r(T−t)f
(x exp
(σ(WT −Wt) +
(r − σ2
2
)(T − t)
))]. (56)
Under P ∗ we have WT −Wt ∼ N(0, T − t). Thus for t < T
F (t, x) = e−r(T−t)1√
2π(T − t)
∫Rf(xe(r−σ
2
2)(T−t)eσy)e−
y2
2(T−t) dy
=z= y√
T−t
e−r(T−t)1√2π
∫Rf(xe(r−σ
2
2)(T−t)eσz
√T−t)e−
z2
2 dz. (∗)
Explicit calulcation of F in case of calls and puts: In case of a European call, we havef(x) = (x−K)+, thus
F (t, x) =(∗)
1√2π
∫ ∞−∞
(xeσz
√T−t−σ
2
2)(T−t) −Ke−r(T−t)
)+e−
z2
2 dz
= E∗[(xeσ
√θZ−σ
2θ2 −Ke−rθ
)+],
where Z ∼ N(0, 1) and θ := T − t. Set
d1 :=log(xK
)+(r + σ2
2
)θ
σ√θ
, d2 := d1 − σ√θ.
Then
F (t, x) = E∗[(xeσ
√θZ−σ
2θ2 −Ke−rθ
)1Z≥−d2
]=
∫ ∞−d2
(xeσ
√θz−σ
2θ2 −Ke−rθ
)e− z22√2π
dz
=
∫ d2
−∞
(xe−σ
√θz−σ
2θ2 −Ke−rθ
)e− z22√2π
dz
=
∫ d2
−∞xe−
(z−σ√θ)2
2dz√2π−Ke−rθN(d2),
with
N(d) :=1√2π
∫ d
∞e−
x2
2 dx.
Finally, with the change of the variable z = y + σ√θ, one gets
F (t, x) = xN(d1)−Ke−rθN(d2) (57)
and similarly for a European put, i.e. f(x) = (K − x)+, we obtain
F (t, x) = Ke−rθN(−d2)− xN(−d1).
80
4.3.3 Hedging calls and puts
In the proof of Theorem 4.9, we used Ito’s representation theorem to show the existenceof a replicating strategy and in Remark 4.10, we derived a pricing formula for Europeanoptions, or more generally for options of the form h = f(ST ). After having sold the optionfor the price F (0, S0), the writer of the option needs an explicit replicating strategy,so that he can generate explicitly the replicating portfolio. We will construct an explicitreplicating strategy in case h is given by h = f(ST ): At any time t a replicating portfoliomust satisfy
Vt = e−rtF (t, St), (where F is as in (56)).
One can show (not elementary): f ∈ Cb(R) =⇒ F ∈ C∞([0, T ) × R). This also holds forf(x) = (x − K)+ and f(x) = (K − x)+. We hence fix f such that F ∈ C∞([0, T ) × R)and define
F (t, x) := e−rtF (t, xert).
Then F (t, St) = e−rtF (t, St) = Vt, and for t < T by Ito’s formula
F (t, St) = F (0, S0) +
∫ t
0
∂xF (u, Su) dSu +1
2
∫ t
0
∂xxF (u, Su) d〈S〉u +
∫ t
0
∂tF (u, Su) du,
where d〈S〉u := d〈S, S〉u =(54)
σ2S2u du. Therefore
Vt = F (t, St) = F (0, S0) +
∫ t
0
σ∂xF (u, Su)Su dWu +
∫ t
0
Ku du
with∫ T
0|Ku| du < ∞ P ∗-a.s. Since Vt is a martingale under P ∗, we have
∫ t0Ku du = 0
P ∗-a.s. by Assignment no. 9, Exercise 2. It follows
Vt = F (t, St) = F (0, S0) +
∫ t
0
∂xF (u, Su) dSu = V0 +
∫ t
0
∂xF (u, Su) dSu.
Thus by (52), the natural candidate for H is
Ht = ∂xF (t, St) = ∂xF (t, St) (continuous in t).
Since F (t, St) = Vt = H0t +HtSt we then have to put
H0t = F (t, St)− ∂xF (t, St)St (continuous in t).
Then φ = (H0, H) is self-financing and Vt(φ) = Vt = F (t, St) as desired.
81
Remark 4.11 We have just shown (without Ito’s representation theorem), that in caseh = f(ST ), we can find an explicit replicating strategy.
Remark 4.12 We have, with the notations of Remark 4.10
∂xF (t, x) = N(d1) for a European call
and∂xF (t, x) = −N(−d1) for a European put.
∂xF (t, x) is called ”delta of the option”. More generally, if Vt = Ψ(t, St), then we havethe following terminologies :
∂xΨ(t, St) is called ”delta of the portfolio” (measures the sensitivity of the portfolioVt w.r.t. stock price fluctuations)
∂xxΨ(t, St) is called ”gamma of the portfolio”,
∂tΨ(t, St) is called ”theta of the portfolio”,
∂σΨ(t, St) is called ”vega of the portfolio”.
4.4 Implied volatility and volatility models
One of the main features of the Black-Scholes model :
”hedging and pricing formulas only depend on one parameter σ.”
The drift µ disappears by Girsanov, and r,K, S0 are given. The volatility σ is hence alsosaid to be non-observable. In practice, two methods are used to evaluate σ :
1. The historical method : In the present model σ2T is the variance of log(ST/S0
)The r.v.’s
log(ST/S0
), log
(S2T/ST
), . . . , log
(SNT/S(N−1)T
)are independent and identically distributed. Therefore σ can be estimated by statisticalmethods using stock prices of the past (for example by calculating empirical variances(?)).
2. The implied volatility method : Some options are quoted on organized markets.The price of options (calls and puts) is an increasing function of σ, so we can invert the
82
Black-Scholes pricing formula and associate an ”implied volatililty” to each quoted option.More precisely, let Cobs(S0, K, T ) be the observed option price of a benchmark (reference)option on the market. Then find σ that solves
Cobs(S0, K, T ) = F (0, S0) = Fσ(0, S0, K, T ).
Imperfection of the Black-Scholes model : Important differences between historicaland implied volatility are observed.
The implied volatility appears to depend on the strike price and time to maturity. In fact,if one plots a graph of implied volatility in dependence of strike prices one sees a ”volatilitysmile” (graph is convex). The graphs becomes ”more and more” convex the shorter thetime to maturity is. This contradicts the Black-Scholes model or in other terms : TheBlack-Scholes model does not describe the market correctly !? Nonetheless, the model isregarded as a standard reference. One possibility to construct a more consistent model isto consider a time dependent volatility (σt)t∈[0,T ].
(?) Empirical variances : given stock prices S0, ST , S2T , . . . , SNT , let
δi := log(S(i+1)T
SiT
), i = 0, . . . , N − 1
Empirical mean : δ :=1
N
N−1∑i=0
δi
Empirical variance : σhist :=1√T
( 1
N − 1
N−1∑i=0
(δi − δ)2) 1
2(”unbiased estimator”)
5 Option pricing and partial differential equations
5.1 European option pricing and diffusions
European option = ”vanilla option” = European call or put
In the black-Scholes model
price of European option at time t ∈ [0, T ] = Vt = E[e−r(T−t)f(ST )|Ft],
83
where E[ · ] is the risk-neutral expectation, (i.e. St = e−rtSt is a martingale), f(x) =(x−K)+ (call) or (K − x)+ (put), and
St = S0e(r−σ
2
2)t+σWt
solvesdSt = σStdWt + rStdt.
We will see below that the pricing of an European option is a special case of the followingproblem:
Suppose a diffusion (Xu) solves for u > t, x ∈ R++
Xu = x+
∫ u
t
σ(s,Xs)dWs +
∫ u
t
b(x,Xs)ds (58)
with
σ, b satisfying a Lipschitz and linear growth condition in x (cf. Section 3.5.4)
(the conditions ensure the existence of a pathwise unique solution also denoted (X t,xu )u>t
starting from x at time t (cf. Section 3.5.4)) and let more generally
r(t, x) = bounded, continuous (modelling the riskless interest rate).
How does one compute
Vt := E[e−∫ Tt r(s,X0,x
s )dsf(X0,xT )|Ft] ?
Answer: As in the Black-Scholes model
Vt = F (t,X0,xt ),
where (cf. Theorem 3.48)
F (t, x) = E[e−∫ Tt r(s,Xt,x
s )dsf(X t,xT )].
Under some regularity assumptions on σ and b, F (t, x) is the unique solution u(t, x) ofthe following partial differential equation (hereafter PDE) with terminal condition f , i.e.
∂tu+ Au− ru = 0 in [0, T )× Ru(T, ·) = f in R,
(59)
where
Au(t, x) =σ2(t, x)
2∂xxu(t, x) + b(t, x)∂xu(t, x).
This will be shown below.
84
5.1.1 The infinitesimal generator of a diffusion
For simplicity we let here σ, b only depend on space. Let (Xt) solve
dXt = σ(Xt)dWt + b(Xt)dt (60)
with the conditions specified in Theorem 3.45.
Proposition 5.1 Let f ∈ C2(R), f ′ be bounded and Af = σ2
2f ′′ + bf ′. Then
Mt := f(Xt)−∫ t
0
Af(Xs)ds, t ∈ [0, T ]
is an (Ft)-martingale.
Proof By Ito’s formula,
f(Xt) = f(X0) +
∫ t
0
f ′(Xs)dXs +1
2
∫ t
0
f ′′(Xs)d〈X〉s
= f(X0) +
∫ t
0
f ′(Xs)σ(Xs)dWs +
∫ t
0
f ′(Xs)b(Xs) +1
2f ′′(Xs)σ
2(Xs)︸ ︷︷ ︸=Af(Xs)
ds.
Since σ has sublinear growth and f ′ is bounded, say |f ′| 6 K we get
E[
∫ t
0
|f ′(Xs)|2|σ(Xs)|2ds] 6 K2 E[
∫ t
0
C ′T (1 + |Xs|2)ds] 6 K2C ′TT (1 + E[sups6T|Xs|2]︸ ︷︷ ︸
<∞ (exercise)
) <∞
Thus (∫ t
0f ′(Xs)σ(Xs)dWs)t∈[0,T ] is a martingale and so is (Mt)t∈[0,T ].
Remark 5.2 Let (Xxt ) be the unique solution of (60) with Xx
0 = x (starting from x att = 0). By the proof of Proposition 5.1, we know
E[f(Xxt )] = f(x) + E
[∫ t
0
Af(Xxs )ds
]. (61)
Suppose additionally f ′′ bounded, say also |f ′′| 6 K. Then
E[sups6T|Af(Xx
s )|] 6 E[sups6T
( K|b(Xxs )|︸ ︷︷ ︸
62(K2+|b(Xxs )|2)
+K
2|σ2(Xx
s )|)
6 E[2K2 + max(2,K
2) sups6T
2C ′T (1 + |Xxs |2)] <∞.
85
Hence ∣∣∣∣1t∫ t
0
Af(Xxs )ds
∣∣∣∣ 6 sups6T|Af(Xx
s )| ∈ L1.
Since s 7→ Af(Xxs ) is continuous a.s., we have
1
t
∫ t
0
Af(Xxs )ds →
t→0Af(Xx
0 ) = Af(x) a.s.
Thus by Lebesgue and (61)
limt→0
E[f(Xxt )]− f(x)
t= lim
t→0E[
1
t
∫ t
0
Af(Xxs )ds
]= Af(x).
Therefore A is called the infinitesimal generator of the diffusion (Xt). Similarly (alsounder some regularity assumptions) one can show that
Lu(t, x) := ∂tu(t, x) +σ2
2(t, x)∂xxu(t, x) + b(t, x)∂xu(t, x)
is the infinitesimal generator of the diffusion Yt = (t,Xt) where Xt solves
dXt = σ(t,Xt)dWt + b(t,Xt)dt.
New we assume again that σ, b are time-dependent and satisfy the usual conditions thatguarantee the existence of a pathwise unique solution to (58). Let
Au(t, x) :=σ2
2(t, x)∂xxu(t, x) + b(t, x)∂xu(t, x).
Proposition 5.3 Let u ∈ C1,2([0,∞)×R) with bounded first derivative in x. If Xt(= Xxt
starting from x at time 0) is the unique solution of (58), then
Mt := u(t,Xt)−∫ t
0
(∂tu+ A)(s,Xs)ds, t ∈ [0, T ]
is a martingale.
Proof By the time dependent Ito formula, we have
u(t,Xt) = u(0, x) +
∫ t
0
σ(s,Xs)∂xu(s,Xs)dWs +
∫ t
0
(∂t + A)u(s,Xs)ds.
86
By our assumptions, similarly to the proof of Proposition 5.1∫ t
0
σ(s,Xs)∂xu(s,Xs)dWs, t ∈ [0, T ]
is a martingale. Hence so is (Mt)t∈[0,T ].
Proposition 5.4 Let u and (Xt) be as in Proposition 5.3. Let r(t, x) be bounded andcontinuous, i.e. r ∈ Cb([0,∞)× R). Then
Mt := e−∫ t0 r(s,Xs)dsu(t,Xt)−
∫ t
0
e−∫ s0 r(u,Xu)du(∂t + A− r)u(s,Xs)ds, t ∈ [0, T ]
is a martingale.
Proof Apply Ito’s product rule Proposition 3.29 to
e−∫ t0 r(s,Xs)ds︸ ︷︷ ︸=:Yt
u(t,Xt)︸ ︷︷ ︸=:Zt
.
Then apply Ito’s formula to u(t,Xt) and conclude similarly to Propositions 5.1, 5.3.
Next we consider the multidimensional case with
σ(t, x) = (σij(t, x))16i6n,16j6p and b(t, x) = (b1(t, x), ..., bn(t, x))
fulfilling Lipschitz and linear growth conditions (see Theorem 3.46). For each t ∈ [0, T ],and x = (x1, ..., xn) ∈ Rn, there exists a pathwise unique solution Xu = (X1
u, ..., Xnu ),
u > t to the system
X iu = xi +
p∑j=1
∫ u
t
σij(s,Xs)dWjs +
∫ u
t
bi(s,Xs)ds, 1 6 i 6 n. (62)
Denote this solution by (X t,xu )u>t. If t = 0, we also write (Xx
u)u>0, and if t = 0 and nospecific initial condition is assumed, we write (Xu)u>0. We always assume E[|X0|2] < ∞(see Theorem 3.46).For f ∈ C1,2(R+ × Rn), let
Af(t, x) :=1
2
n∑i,j=1
aij(t, x)∂ijf(t, x) +n∑j=1
bj(t, x)∂jf(t, x),
87
where ∂ij = ∂2
∂xi∂xj, ∂j = ∂
∂xj, and
aij(t, x) =
p∑k=1
σik(t, x)σjk(t, x),
i.e. a(t, x) = σ(t, x) σT (t, x) where σT (t, x) is the transposed matrix of σ(t, x).
Proposition 5.5 Let (Xt)t∈[0,T ] be a solution to (62), i.e. with E[|X0|2] < ∞. Let u ∈C1,2(R+×Rn) with bounded partial derivatives ∂iu, 1 6 i 6 n and r ∈ Cb(R+×Rn). Then
Mt := e−∫ t0 r(s,Xs)dsu(t,Xt)−
∫ t
0
e−∫ s0 r(u,Xu)du(∂tu+ Au− ru)(s,Xs)ds, t ∈ [0, T ]
is a martingale.
Proof Similar to Propositions 5.1, 5.3, 5.4. In this case we apply the multidimensionaltime dependent Ito formula Proposition 3.35.
Remark 5.6 The infinitesimal generator (∂t + A) of (t,Xt) is also called the Dynkinoperator of the diffusion (t,Xt).
5.1.2 Conditional expectations and PDE’s
Let (X t,xu )u∈[t,T ], and X0,x
t = Xxt , t ∈ [0, T ], be the unique solutions of (62) as described
after Proposition 5.4. Let f : Rn → R, and r ∈ Cb(R+ × R). From the Markov propertyTheorem 3.48 (which also holds in the multidimensional case), it follows that
Vt := E[e−∫ Tt r(s,Xx
s )dsf(XxT )|Ft] = F (t,Xx
t ), t ∈ [0, T ],
whereF (t, x) := E[e−
∫ Tt r(s,Xt,x
s )dsf(X t,xT )].
Actually, the function F that has a probabilistic(stochastic) representation solves a PDEwith terminal condition f .
Theorem 5.7 Let u ∈ C1,2([0, T ) × Rn), and ∂iu, 1 6 i 6 n, be bounded. Suppose usolves
∂tu+ Au− ru = 0 in [0, T )× Rn,
u(T, ·) = f on Rn.(63)
Thenu(t, x) = E[e−
∫ Tt r(s,Xt,x
s )dsf(X t,xT )] (= F (t, x)).
88
Proof We only show the statement for t = 0. By Proposition 5.5
Mt = e−∫ t0 r(s,X
0,xs )dsu(t,X0,x
t )
is a martingale. Therefore
u(0, x) = E[M0] = E[MT ] = E[e−∫ T0 r(s,X0,x
s )ds u(T,X0,xT )︸ ︷︷ ︸
=f(X0,xT )
].
The proof for t > 0 is similar. Indeed, in this case we apply Ito’s formula to (62) witht > 0.
Remark 5.8 Theorem 5.7 suggests the following method for option pricing: In order tocompute
F (t, x) = E[e−∫ Tt r(s,Xt,x
s )dsf(X t,xT )].
We have just to find a classical solution u (i.e. u ∈ C1,2) to the Cauchy problem (63) withterminal condition f . We therefore rely on existence/uniqueness and regularity resultsfor (63). Sufficient conditions for classical solutions generally require A to be (globally)strictly elliptic, i.e. ∃C > 0 with
n∑i,j=1
aij(t, x)ξiξj > C|ξ|2, ∀ξ = (ξ1, ..., ξn) ∈ Rn. (64)
However, it is possible to relax the conditions by using e.g. the concept of viscosity solutions(but we will not do this here).
5.1.3 Application to the Black-Scholes model
With respect to the risk-neutral probability P ∗ the asset price satisfies
dSt = σStdWt + rStdt,
where (Wt)t∈[0,T ] is a standard Brownian motion. By Remark 4.10, the call price is giventhrough
F (t, x) = xN(d1)−Ke−r(T−t)N(d1 − σ√T − t)
89
where
d1 :=log( x
K) + (r + σ2
2)(T − t)
σ√T − t
, and N(d) :=1√2π
∫ d
−∞e−
x2
2 dx, d ∈ R.
We checked (in the exercises of FiMa1) that F solves∂tu+ Absu− ru = 0 in [0, T )× (0,∞)
u(T, ·) = (· −K)+ in (0,∞).
where
Absf(x) =σ2
2x2f ′′(x) + rxf ′(x), f ∈ C2([0,∞))
is the infinitesimal generator of (St)t>0.Note that Abs is not (globally) stritly elliptic (cf. (64)). However, since the pathwise unique
solution St = S0e(r−σ
2
2)t+σWt is strictly positive, we can consider Xt = log(St). Then
Xt = log(St) = log(S0) + (r − σ2
2)t+ σWt = x0 +
∫ t
0
σdWs +
∫ t
0
(r − σ2
2)ds,
thus
dXt = σdWt + (r − σ2
2)dt
and (Xt)t∈[0,T ] has the infinitesimal generator
Alog(bs)f =σ2
2f ′′ + (r − σ2
2)f ′
which is (globally) strictly elliptic (since σ > 0) and has coefficients that satisfy Lipschitzand linear growth conditions. In order to obtain the fair price F (t, x) for an Europeanoption in the Black-Scholes model, we first have to find a classical solution v to
∂tv + Alog(bs)v − rv = 0 in [0, T )× R,v(T, ·) = f exp(·) in R,
(65)
where f(x) = (x−K)+.
Since by Theorem 5.7v(t, x) = E[e−r(T−t)f exp(X t,x
T )]
90
andXt,log(x)T = log(St,xT ),
we then obtain
v(t, log(x)) = E[e−r(T−t)f exp(Xt,log(x)T )] = E[e−r(T−t)f(St,xT )] = F (t, x)
for f(x) = (x−K)+.
5.1.4 PDEs on a bounded open set and computation of expectations
In this section, we let σ, b, r only depend on x.
Af(x) :=σ2(x)
2f ′′(x) + b(x)f ′(x)
andAf(x) := Af(x)− r(x)f(x) discount operator.
Goal: Solve ∂tu+ Au = 0 on [0, T )× R,u(T, ·) = f on R.
(66)
on open interval O = (a, b) instead of R. In this case one needs additional boundary condi-tions (otherwise the problem not well-posed). The most common boundary conditions areDirichlet and Neumann boundary conditions. (66) on O = (a, b) with Dirichlet boundaryconditions becomes
∂tu+ Au = 0 on [0, T )× O,
u(t, a) = u(t, b) = 0 ∀t ∈ (0, T ),
u(T, ·) = f on O.
(67)
Theorem 5.9 Let u ∈ C([0, T ]× [a, b])∩C1,2((0, T )×O), and ∂xu := ∂∂xu be bounded on
(0, T )× O. If u solves (67), then
u(t, x) = E[1∀s∈[t,T ],Xt,xs ∈Oe
−∫ Tt r(Xt,x
s )dsf(X t,xT )]
for all (t, x) ∈ [0, T ]× O.
91
Proof As in the proof of Theorem 5.7, we only show the statement for t = 0. We shallaccept: there exists an extension u ∈ C1,2([0, T ]×R) of u as in the assertion of the theorem.By Proposition 5.4,
Mt := e−∫ t0 r(X
0,xs )dsu(t,X0,x
t )−∫ t
0
e−∫ s0 r(X
0,xv )dv(∂tu+ Au)(s,X0,x
s )ds︸ ︷︷ ︸=0 by assumption
is a martingale. Define
τx := infs ∈ [0, T ]|X0,xs /∈ O ∧ T, (inf ∅ := +∞).
Obviously, τx is bounded, and a stopping time, since
τx = τxa ∧ τxb ∧ T
where τx` := infs ∈ [0, T ]|X0,xs = ` (stopping time by the proof of Proposition 3.15).
By the optional sampling theorem E[M0] = E[Mτx ] and since (∂tu+ Au)(s,X0,xs ) = 0 for
s ∈ [0, τx) (even [0, τx] by contunuity), we obtain using the Dirichlet boundary conditions
u(0, x) = E[M0] = E[Mτx ]
= E[e−∫ τx0 r(X0,x
s )dsu(τx, X0,xτx )]
= E[1∀s ∈ [0, T ], X0,xs ∈ O︸ ︷︷ ︸
⇒τx=T
e−∫ T0 r(X0,x
s )ds u(T,X0,xT )︸ ︷︷ ︸
=f(X0,xT )
]
+ E[1∃s ∈ [0, T ], X0,xs /∈ O︸ ︷︷ ︸
⇒X0,xτx∈a,b
e−∫ τx0 r(X0,x
s )ds u(τx, X0,xτx )︸ ︷︷ ︸
=0
].
Remark 5.10 An option defined by the FT -measurable random variable
1∀s∈[t,T ],Xt,xs ∈Oe
−∫ Tt r(Xt,x
s )dsf(X t,xT )
is called extinguishable (as soon as the asset price exists the open set O it becomes zero,i.e. worthless).
92
5.2 Solving parabolic equations numerically
How to solve (66) numerically ? We shall illustrate the finite difference method. Thismethod is useful for general diffusions (for Black-Scholes with European options not nec-essary because we know that there exists a closed-form solution). First we need to localize(66).
5.2.1 Localization
Localize on O` := (−`, `) (` is a constant that has later to be chosen in accordance withthe algorithm). Typically, the boundary conditions are
Dirichlet bounday conditions, i.e. u(t, `) = u(t,−`) = 0 ∀t,or
Neumann boundary conditions, i.e. ∂xu(t, `) = ∂xu(t,−`) = 0 ∀t.
Consider (67) with O = O`, i.e.∂tu+ Au = 0 on (0, T )× O`,
u(t, `) = u(t,−`) = 0 ∀t ∈ (0, T ),
u(T, ·) = f on O`.
(68)
Estimate the error that is made by restricting R to O`.
For later application we shall work in a Black-Scholes environment. Thus Xt := log(St),St =asset price solves
dXt = σdWt + (r − σ2
2)dt.
We want to compute the price difference for options with payoff f(ST ) = f(eXT ) on Rand O`. Define
f(x) := f(eX).
Assume that the solutions u of (66) and u` of (68) with f replaced by f and Af =σ2
2f ′′ + (r − σ2
2)f ′ in (66) and (68) are smooth enough, so that by Theorems 5.7 and 5.9
u(t, x) = E[e−r(T−t)f(X t,xT )],
andu`(t, x) = E[1∀s∈[t,T ],Xt,x
s ∈(−`,`)e−r(T−t)f(X t,x
T )],
93
with
X t,xs = x+ σ(Ws −Wt) + (r − σ2
2)(s− t), s ∈ [t, T ].
Assume: |f | 6M (⇒ |f | 6M) and r > 0.
Claim: We have
|u(t, x)− u`(t, x)| 6M · P(∃s ∈ [t, T ], X t,xs /∈ (−`, `))
and for large `, such that `− |r′T | > 0, where r′ := r − σ2
2, it holds
P(∃s ∈ [t, T ], X t,xs /∈ (−`, `)) ≤ e−(
`−|r′T |−xσ
)2 12T + e−(
`−|r′T |+xσ
)2 12T
In particularlim`→∞
u`(t, x) = u(t, x), ∀(t, x) (locally uniformly in x).
Proof Let . Then
∃s ∈ [t, T ], X t,xs /∈ (−`, `) ⊂ sup
s∈[t,T ]
|x+ σ(Ws −Wt) + r′(s− t)| > `
⊂ sups∈[t,T ]
|x+ σ(Ws −Wt)| > `− |r′T |.
Therefore
|u(t, x)− u`(t, x)| ≤ E[1∃s∈[t,T ],Xt,xs /∈(−`,`) e
−r(T−t)︸ ︷︷ ︸61
|f(X t,xT )|︸ ︷︷ ︸
6M
]
6M · P(∃s ∈ [t, T ], X t,xs /∈ (−`, `))
6M · P( sups∈[t,T ]
|x+ σ(Ws −Xt︸ ︷︷ ︸∼Ws−t (same law)
)| > `− |r′T |)
= M · P( sups∈[0,T−t]
|x+ σWs| > `− |r′T |)
6M · P( sups∈[0,T ]
|x+ σWs| > `− |r′T |).
By Proposition 3.15,
E[e−λTa ] = e−√
2λ|a|,∀λ > 0,
whereTa = infs > 0,Ws = a, a ∈ R.
94
Thus ∀a > 0, λ > 0
P( sups∈[0,T ]
Ws > a) = P(Ta 6 T ) 6 E[1Ta6T eλ(T−Ta)︸ ︷︷ ︸>1
] 6 eλT E[eλTa ] = eλT e−√
2λa.
Minimizing with respect to λ yields
P( sups∈[0,T ]
Ws > a) 6 e−a2
2T ,
henceP( sup
s∈[0,T ]
(x+ σWs) > a) 6 e−(a−xσ
)2 12T .
Since (−Wt)t>0 is also BM,
P( infs∈[0,T ]
(x+ σWs) 6 −a) = P( sups∈[0,T ]
(−x+ σ(−Ws)) > a) 6 e−(a+xσ
)2 12T .
Hence
P( sups∈[0,T ]
|x+ σWs| > a) 6 P( sups∈[0,T ]
(x+ σWs) > a) + P( infs∈[0,T ]
(x+ σWs) 6 −a)
6 e−(a−xσ
)2 12T + e−(a+x
σ)2 1
2T .
Finally, for ` so large that `− |r′T | > 0, it holds
|u(t, x)− u`(t, x)| 6M(e−(`−|r′T |−x
σ)2 1
2T + e−(`−|r′T |+x
σ)2 1
2T ),
hencelim`→∞
u`(t, x) = u(t, x), ∀(t, x) (locally uniformly in x).
Remark 5.11
- The reflection principle P(sups∈[0,T ] Ws > a) = 2P(WT > a) leads to a slightly betterestimate than above.
- Localization method can be used for American options where numerical approxima-tion is an efficient tool.
95
5.2.2 The finite difference method
Let A, and its discount operator A, be as at the beginning of Subsection 5.1.4., i.e. timeindependent. Consider the localized PDE with Dirichlet boundary conditions
∂tu+ Au = 0 on [0, T )× O`,
u(t,−`) = u(t, `) = 0 ∀t ∈ [0, T ],
u(T, ·) = f on O`.
(69)
The finite difference method is a discretization of (69) in time and space.First we discretize the space (for any fixed time t):
xi := −`+ i · 2`
N + 1, i = 0, ..., N + 1,
and define
h :=2`
N + 1= ”space step”.
Discretize the terminal condition as
f ih := f(xi), i = 1, ...N, and fh = (f ih)i=1,...N .
Letuh(t) = uh = (uih)i=1,...N ∈ RN
be a discrete approximation of u(t, xi), i.e.
uih(t) = uih ≈ u(t, xi).
Replace ∂xu(t, xi) with
∂huih(t) = ∂hu
ih :=
ui+1h − ui−1
h
2h”symmetric derivative”,
and ∂xxu(t, xi) with
∂2hu
ih(t) = ∂2
huih :=
ui+1h −uihh− uih−u
i−1h
h
h=ui+1h − 2uih + ui−1
h
h2,
where u0h(t) = u0
h ≈ u(t,−`) and uN+1h (t) = uN+1
h ≈ u(t, `).
The Dirichlet boundary conditions read as
u0h = uN+1
h = 0
96
For Neumann boundary conditions one would impose u0h = u1
h and uNh = uN+1h , i.e.
∂xu(t, `) = 0 = ∂xu(t,−`).
The space-discretized operator Ah is defined by
(Ahuh(t))i = (Ahuh)i :=σ2(xi)
2∂2hu
ih + b(xi)∂hu
ih − ruih, i = 1, ..., N.
Examples 5.12 In the Black-Scholes model (with logarithmic change of variables)
Alog(bs)u(x) =σ2
2∂xxu(x) + (r − σ2
2)∂xu(x)− ru(x)
is discretized as
(Ahuh)i =σ2
2
(ui+1h − 2uih + ui−1
h
h2
)+
(r − σ2
2
)ui+1h − ui−1
h
2h− ruih
=[ σ2
2h2−(r − σ2
2
)1
2h
]︸ ︷︷ ︸
=:α
ui−1h −
[σ2
h2+ r]
︸ ︷︷ ︸=:β
uih +[ σ2
2h2+
(r − σ2
2
)1
2h
]︸ ︷︷ ︸
=:γ
ui+1h ,
where i = 1, ..., N , and the Dirichlet boundary conditions u0h = uN+1
h = 0 lead to
Ah = ((Ah)i,j)i,j=1,...,N =
β γ 0 . . . 0α β γ . . . 0
0 α. . . . . . 0
.... . . . . . β γ
0 . . . 0 α β
.
With Neumann boundary conditions u0h = u1
h, uNh = uN+1h , one would get
Ah = ((Ah)i,j)i,j=1,...,N =
β + α γ 0 . . . 0α β γ . . . 0
0 α. . . . . . 0
.... . . . . . β γ
0 . . . 0 α β + γ
.
The space discretization transforms (69) into an ODE∂tuh(t) + Ahuh(t) = 0 t ∈ [0, T ]
uh(T ) = fh.(70)
97
Next we discretize time in (70) using θ-schemes, θ ∈ [0, 1]. Let
k =T
M, M ∈ N, M large, k = ”time-step”.
Approximate uh(nk) by unh,k, i.e. unh,k ≈ uh(nk), where uh,k := (unh,k)n=0,...,M ∈ R(M+1)N
solves the backward recursive equationsuMh,k = fh = (f(xi))16i6Nun+1h,k −u
nh,k
k+ θAhu
nh,k + (1− θ)Ahun+1
h,k = 0, n = M − 1, ..., 0(71)
which are solved backwards starting from n = M − 1 and going down to n = 0. (unh,k)iapproximates u(nk, xi), i.e.
unh,k ≈ (u(nk, xi))16i6N .
Remark 5.13
θ = 0 : ”scheme is explicit” unh,k directly computed from un+1h,k .
θ > 0 : ”scheme more complex and time consuming” but good convergence properties.
At each step n solve the system Runh,k = G, where
R = I − θkAh (R tridiagonal matrix)
G = (I + (1− θ)kAh)un+1h,k .
θ = 12
: ”Crank and Nicholson scheme” good if e.g. b = 0 and σ =constant.
θ = 1 : ”scheme completely implicit”
Notations
uh,k = solution of (71) (uh,k = (unh,k)n ∈ R(M+1)N)
u = solution of (69) (function)
Define the function ukh on [0, T )× O` by
ukh(t, x) = (unh,k)i for t ∈ [(n− 1)k, nk), x ∈[xi −
h
2, xi +
h
2
),
δφ(x) :=1
h
[φ
(x+
h
2
)− φ
(x− h
2
)]”approximate centered derivative of φ”,
(·, ·)L2(O`) = inner product and ‖ · ‖L2(O`) = norm in L2(O`).
98
Theorem 5.14 Let σ, b be Lipschitz continuous, and r be continuous r > 0. Let A beelliptic, i.e. ∃ε > 0 such that ∀g ∈ C2
0(O`) (”C2 and compact support in O`”)
(−Ag, g)L2(O`) > ε(‖g‖L2(O`) + ‖g′‖L2(O`)).
Then:
• If θ ∈ [12, 1] : As h, k → 0,
ukh → u and δukh → ∂xu in L2([0, T ]× O`).
• If θ ∈ [0, 12) : As h, k → 0 such that h
k2→ 0
ukh → u and δukh → ∂xu in L2([0, T ]× O`).
Proof Without, see e.g. reference given in textbook.
Remark 5.15
• θ ∈ [0, 12) : ”scheme is conditionally convergent” (because with k and h, k
h2must also
converge to 0). In this case the numerical implementation is complicated, exceptwhen θ = 0.
• θ ∈ [12, 1] : ”scheme is unconditionally convergent” (because it converges as soon as
h and k converge to 0).
How to solve (71) in practice:
At each step n solve RX = G, whereX = unh,k = solution
G = (I + (1− θ)kAh)un+1h,k , (un+1
h,k given)
R = I − kθAh, (tridiagonal matrix, given)
One possibility: Use the ”Gauss method”
(Solves the system with a number of operations proportional to N)
99
Illustration: Let
b1 c1 0 . . . 0a2 b2 c2 . . . 0
0. . . . . . . . . 0
... 0 aN−1 bN−1 cN−1
0 . . . 0 aN bN
︸ ︷︷ ︸
=R
x1
x2......xN
︸ ︷︷ ︸
=X
=
g1
g2......gN
︸ ︷︷ ︸
=G
.
First transform R into a lower triangular matrix R′ using the Gauss method from bottomto top, i.e. set
b′N = bN
g′N = gN and for i = N − 1, ..., 1 decreasing, set
b′i = bi −ciai+1
b′i+1
, g′i = gi −cig′i+1
b′i+1
, ”upward elimination”.
Then one obtains the equivalent system R′X = G′, where
R′ =
b′1 0 . . . . . . 0a2 b′2 0 . . . 0
0. . . . . . . . . 0
.... . . . . . b′N−1 0
0 . . . 0 aN b′N
and G′ =
g′1g′2......g′N
Gauss elimination works like this:∣∣∣∣∣∣∣∣∣∣∣
b1 c1 0 . . . 0a2 b2 c2 . . . 0
0. . . . . . . . . 0
... 0 aN−1 bN−1 cN−1
0 . . . 0 aN bN
∣∣∣∣∣∣∣∣∣∣∣
g1
g2...
gN−1
gN
← L1
← L2...
← LN−1
← LN
etc...
(LN−2 − constN−2LN−1) =: LN−2
(LN−1 − constN−1LN) =: LN−1
LN
where constj have to be determined for j = N − 1, ..., 1.
100
Finally compute X starting from the top of R′
x1 =g′1b′1
and for i = 2, ...N increasing
xi =g′i − aixi−1
b′i”downward”.
Remark 5.16 Algorithm only works if R invertible (otherwise division through 0 willoccur at some point). Sufficient condition for invertibility is
|ai|+ |ci| 6 |bi|, ∀i ”dominant diagonal”.
In Black-Scholes model satisfied if: |r − σ2
2| 6 σ2
h, hence OK for small h.
5.3 Solving the Black-Scholes equation with the Fourier trans-form method
In this subsection we consider again the Black-Scholes equation from the beginning ofSubsection 5.1.3:
∂tF + σ2
2x2∂xxF + rx∂xF − rF = 0 in [0, T )× (0,∞)
F (T, ·) = (· −K)+ in (0,∞).(72)
Our goal is to solve (72) explicitly, using change of variables and Fourier transform meth-ods. For this, we introduce new variables τ, y, v through
x = Key ⇔ y = log( xK
)(x ∈ (0,∞)⇒ y ∈ R),
t = T − 2τ
σ2⇔ τ =
σ2
2(T − t)
(t ∈ [0, τ)⇒ τ ∈
(0,σ2
2T
]),
and
F (t, x) = Kv(τ, y) = Kv
(σ2
2(T − t), log
( xK
)).
Then
∂xF = K
vτ · ∂x(σ2
2(T − t)
)︸ ︷︷ ︸
=0
+vy · ∂x log( xK
)︸ ︷︷ ︸
= 1x
= vyK
x= e−yvy
101
where vτ = ∂τv, vy = ∂yv. Similarly (with vyy = ∂yyv)
∂tF = −Kσ2
2vτ ,
∂xxF =e−2y
K(vyy − vy).
Introducing the new variables in (72), we obtain
−Kσ2
2vτ +
σ2
2K2e2y e
−2y
K(vyy − vy) + rKeye−yvy − rKv
= −Kσ2
2vτ +K
σ2
2vyy +
(rK −Kσ2
2
)vy − rKv = 0.
Multiplying with − 1rK
, we get
σ2
2rvτ −
σ2
2rvyy +
(σ2
2r− 1
)vy + v = 0.
Multiplying with k := 2rσ2 gives
vτ − vyy + (1− k)vy + kv = 0.
The terminal condition in (72) becomes an initial condition, since
t = T ⇒ τ = 0,
and
v(0, y) =1
KF (T, x) =
1
K(x−K)+ = (ey − 1)+.
We obtain vτ − vyy + (1− k)vy + kv = 0 on (0, σ
2
2T ]× R
v(0, y) = (ey − 1)+ y ∈ R.(73)
Next, we want to transform (73) into a heat equation:Let u(τ, y) be defined through
v(τ, y) = eαy+βτu(τ, y)
Then, we get
vy = (αu+ uy)eαy+βτ
vyy = (α2u+ αuy + αuy + uyy)eαy+βτ
vτ = (βu+ uτ )eαy+βτ .
102
Introducing this in (73), we get
βu+ uτ − (α2u+ 2αuy + uyy) + (1− k)(αu+ uy) + ku
= uτ + (β − α2 + (1− k)α + k)u+ ((1− k)− 2α)uy − uyy = 0.
Solving
1− k − 2α = 0⇔ α =1− k
2and
0 = β − α2 + (1− k)α + k
= β − (1− k)2
4+
(1− k)2
2+ k
=(1− k)2
4+ k + β ⇒ β = −(1 + k)2
4,
we get with this choice of α and β that
uτ = uyy,
with initial condition
u(0, y) = e−αyv(0, y)
= ek−12y(ey − 1)+
= · · · =(ek+12y − e
k−12y)+
(exercise).
Altogether, we hence obtain uτ = uyy on(
0, σ2
2T]× R,
v(0, y) =(ek+12y − e k−1
2y)+
y ∈ R,(74)
where k = 2rσ2 .
We will solve (74) using the Fourier transform method. For sufficiently regular
f :
(0,σ2
2T
]× R→ R,
define
f(τ, w) :=
∫Rf(τ, y)e−iwydy, w ∈ R.
103
Suppose a solution to (74) exists and is sufficiently regular. Then
du
dτ= uτ = uyy = (iw)2u = −w2u.
Solving this ODE by separation of variables, we obtain a solution of the form
u(τ, w) = Ce−w2τ ,
where C is some constant which is given by
u(0, w) = C = Fourier transform of initial condition.
Then the Fourier transformed solution to (74) is
u(τ, w) = u(0, w)e−w2τ . (75)
Now, we will transform (75) back.
Using F−1(f · g) = F−1(f) ∗F−1(g) and F−1(e−w2τ )(y) = 1
2√πτe−
y2
4τ for the inverse Fouriertransform
F−1(f(w))(y) =1
2π
∫Rf(w)eiwydw, y ∈ R,
we obtain
u(τ, y) = F−1(u(τ, w))
= F−1(u(0, w)e−w2τ )
= F−1(u(0, w)) ∗ F−1(e−w2τ )
= u(0, y) ∗ 1
2√πτe−
y2
4τ
=1
2√πτ
∫R
(ek+12z − e
k−12z)+
e−(y−z)2
4τ dz
=1√2π
∫R
(ek+12 (y+
√2τx) − e
k−12 (y+
√2τx))+
e−x2
2 dx (z = y +√
2τx)
=ek+12ye
(k+1)2τ4
√2π
∫ ∞− y√
2τ
e−(x− k+1
2
√2τ)
2
2 dx− ek−12ye
(k−1)2τ4
∫ ∞− y√
2τ
e−(x− k−1
2
√2τ)
2
2 dx.
Make the substitution
w = x− k + 1
2
√2τ in the first integral
104
and
w′ = x− k − 1
2
√2τ in the second integral
and obtain
u(τ, y) = ek+12y+
(k+1)2τ4 φ
(y√2τ
+k + 1
2
√2τ
)−e
k−12y+
(k−1)2τ4 φ
(y√2τ
+k − 1
2
√2τ
)(76)
where
φ(d) :=1√2π
∫ d
−∞e−
x2
2 dx, d ∈ R.
Now, we reverse the change of variables. First
v(τ, y) = eαy+βτu(τ, y)
= e1−k2y− (1+k)2τ
4 u(τ, y)
= eyφ
(y√2τ
+k + 1
2
√2τ
)− e−kτφ
(y√2τ
+k − 1
2
√2τ
)and
y√2τ
+k + 1
2
√2τ =
log( xK
)
σ√T − t
+2rσ2 + 1
2σ√T − t︸ ︷︷ ︸
=(r+σ22 )(T−t)
σ√T−t︸ ︷︷ ︸
=:w
.
Furthery√2τ
+k − 1
2
√2τ = w − σ
√T − t,
kτ =2r
σ2
σ2
2(T − t) = r(T − t),
ey =x
K,
and soF (t, x) = Kv(τ, y) = xφ(w)−Ke−r(T−t)φ
(w − σ
√T − t
).
The value of a put can now be found using the put-call parity (exercise).
105
6 Interest rate models
The most basic interest rate contract is described as follows:
• borrow K dollar at time t (t = loan emission date)
• payback F (t, T ) dollar at time T (T =maturity)
With constant instantaneous interest rate r > 0, we know that
F (t, T ) = K · limn→∞
(1 + r
T − tn
)n= Ke(T−t)·r.
Here
r = interest rate for one time unit (e.g. a year)
n periods, n→∞
after one period: K +K · r T−tn
= K(1 + r T−t
n
)etc...
after n-periods: K(1 + r T−t
n
)nNow suppose that r is variable, i.e. r is given as a deterministic (= given, non-random)positive function and
(R(t, T ))t∈[0,T ] is some average of r over the time interval [t, T ].
If we borrow one dollar at time t, then the payback in dollar at time T should be
F (t, T ) = e(T−t)R(t,T ).
In order to have no arbitrage, we must have (exercise)
F (t, s) = F (t, u)F (u, s), ∀t < u < s. (77)
From (77) and F (t, t) = 1, if F is regular enough, it follows that ∃ (r(t))t∈[0,T ] with
F (t, T ) = e∫ Tt r(s)ds, ∀t ∈ [0, T ].
Thus
R(t, T ) =1
T − t
∫ T
t
r(s)ds,
106
where r(s) = ”instantaneous interest rate at time s”.In general the future is not deterministic, i.e. it is unknown. Thus R(t, T ) will not bedeterministic and so r(s) will also be unknown. We will therefore later regard r(s) as astochastic process which will be a model for the future. In practice one has then to askthe question whether this model is realistic or not.First we continue with deterministic assumptions. We can see our basic interest contractalso the from the perspective of the writer: lend one dollar at time t and receive morethan one dollar at time T > t. So one dollar today is more than one dollar tomorrow.This is expressed by zero-coupon bonds:A zero-coupon bond is a security paying one dollar at maturity time T . Its value at timet is denoted by P (t, T ). Hence P (T, T ) = 1. In our deterministic model (where the futureis certain)
P (t, T ) = e−∫ Tt r(s)ds, t ∈ [0, T ], (78)
is a zero-coupon bond with maturity time T (”lend P (t, T ) dollars a time t and receiveone dollar at time T”).
6.0.1 Yield curve for an uncertain future
As already said for an uncertain future, we regard the instantaneous interest rate r(s) asa random (stochastic) process. For this let
(Ω,F,P, (Ft)t∈[0,T ])
be a filtered probability space. The time horizon is denoted by T , because we want toconsider different maturities 0 6 T 6 T . Assume (Ft)t∈[0,T ] is the natural filtration of astandard BM (Wt)t∈[0,T ], and F = FT . As in the models before introduce a riskless assetwhose price at time t is
S0t = e
∫ t0 r(s)ds,
where (r(t))t∈[0,T ] is an adapted process satisfying∫ T
0|r(s)|ds < ∞ a.s. The terminology
”riskless asset” is a little unprecise since r is random, but we will see later that such an as-set is less risky then the risky asset. The risky assets are zero-coupon bonds (P (t, T ))t∈[0,T ]
with maturity T 6 T . We assume that (P (t, T )) is (Ft)-adapted.
We will make the following (no-arbitrage) hypothesis:
(H) ∃ probability measure P∗ ≈ P on (Ω,F), such that for any T ≤ T
P (t, T )︸ ︷︷ ︸”discounted zero
-coupon bond price”
:= e−∫ t0 r(s)ds · P (t, T )︸ ︷︷ ︸
”price of the zero-coupon bond at time t”
, t ∈ [0, T ],
107
is an (P∗, (Ft))-martingale.
(H)⇒ P (t, T ) = E∗[P (T, T )|Ft] = E∗[e−∫ T0 r(s)ds|Ft] = e−
∫ t0 r(s)ds︸ ︷︷ ︸∈Ft
E∗[e−∫ Tt r(s)ds|Ft],
hence (compare this with (78))
P (t, T ) = E∗[e−∫ Tt r(s)ds|Ft]. (79)
Note that the prices P (t, T ) only depend on the behavior of (r(s))s∈[0,T ] under P∗. Let
LT =dP∗
dPbe the density of P∗ w.r.t. P on (Ω,F).
Then LT ∈ L1(P) and ∀X > 0, X random variable: E∗[X] = E[XLT ] and so
X ∈ Ft ⇒ E∗[X] = E[XLT ] = E [X E[LT |Ft]] .
Thus Lt := E[LT |Ft] =dP∗|FtdP|Ft
, t ∈ [0, T ].
Proposition 6.1 There exists an adapted process (q(t))t∈[0,T ] such that ∀t ∈ [0, T ]
Lt = exp
(∫ t
0
q(s)dWs −1
2
∫ t
0
q(s)2ds
)P-a.s. (80)
Proof The process (Lt)t∈[0,T ] is an L1(Ω,F,P)-martingale w.r.t. the natural filtration(Ft) of the BM (Wt). The Martingale Representation Theorem 4.7 also holds in a similarform for not necessarily square integrable martingales, i.e. if the martingale in Theorem4.7 is only in L1(Ω,F,P), then the adapted process (Ht) only satisfies
P
(∫ T
0
H2sds <∞
)= 1.
Thus applying this more general result to (Lt) we obtain the existence of an (Ft)-adapted
process (Ht)t∈[0,T ] such that∫ T
0H2sds <∞ P-a.s. and such that
Lt = L0 +
∫ t
0
HsdWs P-a.s.
108
Any F0-measurable random variable is P-a.s. constant. Thus E(L0) = L0 P-a.s. Since Lt,t ∈ [0, T ], is an equivalent probability density on Ft, we have P(Lt > 0) = 1 ∀t ∈ [0, T ]and moreover 1 = E[LT ] = E[L0] = L0. From this it is easy to see that
P(Lt > 0 ∀t ∈ [0, T ]) = 1 (exercise).
Thus by Remark 3.32, we may apply Ito’s formula to log(Lt), hence
log(Lt) = log(L0)︸ ︷︷ ︸=0 P-a.s.
+
∫ t
0
1
LsHsdWs −
1
2
∫ t
0
1
L2s
H2sds P-a.s.
With q(s) := HsLs
, s ∈ [0, T ], we obtain (80).
Corollary 6.2 The price of a zero-coupon bond with maturity T > t can be expressed as
P (t, T ) = E[
exp(−∫ T
t
r(s)ds)
exp(∫ T
t
q(s)dWs −1
2
∫ T
t
q(s)2ds)|Ft]
(81)
Proof For all A ∈ Ft and B ∈ FT , T 6 T ,
E∗[1A1
LtE[1BLT |Ft]︸ ︷︷ ︸∈Ft
] =dP∗|Ft=LtdP|Ft
E[1A E[1BLT |Ft]] =A∈Ft
E[1A1B︸ ︷︷ ︸∈FT
LT ] = E∗[1A 1B︸︷︷︸∈L1(P∗)
],
hence1
LtE[1BLT |Ft] = E∗[1B|Ft] P∗-a.s.
The last extends to any positive or integrable FT -measurable random variable Z, hence
E∗[Z|Ft] =1
LtE[ZLT |Ft] P∗-a.s. (82)
It follows
P (t, T ) =(79)
E∗[e−∫ Tt r(s)ds︸ ︷︷ ︸
∈FT , >0
|Ft] =(82)
1
LtE[e−
∫ Tt r(s)dsLT |Ft]
=Prop 6.1
E[e−∫ Tt r(s)dse
∫ Tt q(s)dWs− 1
2
∫ Tt q(s)2ds|Ft].
109
Proposition 6.3 For each maturity T 6 T there is an (Ft)-adapted process (σTt )t∈[0,T ]
such thatdP (t, T )
P (t, T )= σTt dWt +
(r(t)− σTt q(t)
)dt on [0, T ]. (83)
Proof (P (t, T ))t∈[0,T ] is a martingale under P∗. Using (82) with T 6 T and Z = P (T, T )(obviously, (82) also holds for Z ∈ L1(Ω,FT ,P)), we obtain
P (t, T ) = E∗[P (T, T )|Ft] =1
LtE[P (T, T )LT |Ft], P∗-a.s.
hence since P ≈ P∗
E[P (T, T )LT |Ft] = P (t, T )Lt, P-a.s. ∀t ∈ [0, T ].
This together with P (t, T )Lt ∈ L1(Ω,Ft,P) implies that (P (t, T )Lt)t∈[0,T ] is a martingalew.r.t. P. By Corollary 6.2 and the proof of Proposition 6.1, we know that
P(P (t, T )Lt > 0 ∀t ∈ [0, T ]) = 1.
Then using the same arguments as in the proof of Proposition 6.1, we see that there existsan (Ft)-adapted process (θTt )t∈[0,T ] with
∫ T0
(θTt )2dt <∞ P-a.s. and
P (t, T )Lt = P (0, T )e∫ t0 θ
Ts dWs− 1
2
∫ t0 (θTs )2ds. (84)
Hence
P (t, T ) = e∫ t0 r(s)dsP (t, T ) =
(84)
1
LtP (0, T )︸ ︷︷ ︸=P (0,T )
e∫ t0 r(s)ds+
∫ t0 θ
Ts dWs− 1
2
∫ t0 (θTs )2ds
=Proposition 6.1
P (0, T ) exp
∫ t
0
r(s)ds+
∫ t
0
(θTs − q(s))dWs −1
2
∫ t
0
((θTs )2 − q(s)2
)ds︸ ︷︷ ︸
=:Xt
.
110
Applying Ito’s formula to f(Xt) = P (t, T ), i.e. f(x) = P (0, T )ex, we obtain
P (t, T ) = P (0, T ) +
∫ t
0
P (s, T )dXs +1
2
∫ t
0
P (s, T )d〈X〉s
= P (0, T ) +
∫ t
0
P (s, T )
(r(s)− 1
2[(θTs )2 − q(s)2] +
1
2
(θTs − q(s)
)2)ds
+
∫ t
0
P (s, T )(θTs − q(s)
)dWs
= P (0, T ) +
∫ t
0
P (s, T )(θTs − q(s)
)︸ ︷︷ ︸=:σTs
dWs +
∫ t
0
P (s, T )(r(s) + q(s)(q(s)− θTs )
)ds.
Remark 6.4
• the riskless asset S0t satisfies dS0
t = r(t)S0t dt
• the risky asset P (t, T ) satisfies (83)
The term dWt in (83) makes P (t, T ) ”riskier”.
r(t)− σTt q(t) ≈ ”local return rate of the bond” or ”local yield”,
because
E[dP (t, T )
P (t, T )
]≈ E
[∫ t
0
[r(s)− σTs q(s)]ds]
= ”average return rate” or ”average yield”.
Terminologies:
• −σTt q(t) ≈ ”difference between local return rate and riskless rate” = ”risk pre-mium”
• −q(t) ≈ ”risk premium per unit of volatility” = ”market price of risk”
By Proposition 6.1 and the Girsanov theorem Wt := Wt −∫ t
0q(s)ds is a BM under P∗,
hence under P∗dP (t, T )
P (t, T )= σTt dWt + r(t)dt. (85)
Therefore under P∗ the local return rate of the (zero-coupon) bond is equal to the risklessinterest rate. For this reason P∗ is called risk-neutral probability measure. Note that
P (t, T ) = P (0, T )e∫ t0 r(s)dse
∫ t0 σ
Ts dWs− 1
2
∫ t0 (σTs )2ds (86)
solves (85).
111
6.0.2 Bond options
As underlying risky asset (like (St) before), consider
P (t, T ) = price of a zero-coupon bond at time t with maturity T .
Consider further a European option with maturity θ, θ ∈ [0, T ], T 6 T on (P (t, T ))t∈[0,T ].E.g.
(p(θ, T )−K)+ for a call with strike price K.
The ”riskless” asset is still considered as
S0t = e
∫ t0 r(s)ds, t ∈ [0, T ], (r(s) = ”riskless” rate).
We want to hedge the European option with a strategy
(H0t , Ht)t∈[0,θ] (adapted process in R× R),
(H0t , Ht) = ”portfolio at time t”,
where
• H0t is the amount of S0
t held at time t
• Ht is the amount of P (t, T ) held at time t.
Thus the value of the portfolio at time t satisfies
Vt = H0t S
0t +HtP (t, T ) = H0
t e∫ t0 r(s)ds +HtP (t, T ).
The self-financing condition is expressed (cf. Section 4.1.2) through
dVt = H0t dS
0t +HtdP (t, T ),
with integrability conditions:
cf. Definition 4.1
∫ θ
0|H0
t r(t)|dt <∞ a.s.
∫ θ0|Htσ
Tt |2dt <∞ a.s. ← cf. Proposition 6.3
112
Definition 6.5 A strategy φ = ((H0t , Ht))t∈[0,θ] is called admissible, if it is self-financing
and if the discounted value of its portfolio
Vt = Vt(φ)︸ ︷︷ ︸=e−
∫ t0 r(s)dsVt(φ)
= H0t +HtP (t, T )
satisfies Vt > 0 a.s. ∀t ∈ [0, θ], and supt∈[0,θ] Vt ∈ L2(Ω,P∗).
By the following proposition it is possible to hedge all European options (under reasonabletechnical assumptions) if θ < T .
Proposition 6.6 Assume supt∈[0,T ] |r(t)| <∞ a.s. and σTt 6= 0 a.s. ∀t ∈ [0, θ]. Let θ < T ,
and h ∈ Fθ such that he−∫ θ0 r(s)ds ∈ L2(Ω,Fθ,P∗). Then there exists an admissible strategy
φ such that
Vt = Vt(φ) = E∗[e−∫ θt r(s)dsh|Ft] a.s. ∀t 6 θ. (87)
Moreover for any admissible strategy φ with Vθ = h its value at time t 6 θ is given by(87).
Remark: Obviously Vθ = h, thus φ replicates h. For European options, we have h =f(P (θ, T )) with f(x) = (x−K)+ or f(x) = (K − x)+.
Proof (of Proposition 6.6) The method of proof is the same as the one in Proposition 4.2and Theorem 4.9. Let φ = (H0
t , Ht)t be any admissible strategy. Then
dVt(φ) = dVt = d(e−
∫ t0 r(s)dsVt
)= −r(t)e−
∫ t0 r(s)dsVtdt+ e−
∫ t0 r(s)dsdVt
= −r(t)e−∫ t0 r(s)ds
(H0t S
0t +HtP (t, T )
)dt+ e−
∫ t0 r(s)ds
H0t dS
0t︸︷︷︸
=r(t)S0t dt
+ HtdP (t, T )
= Ht
(− r(t)e−
∫ t0 r(s)dsP (t, T )dt+ e−
∫ t0 r(s)ds dP (t, T )︸ ︷︷ ︸=
(85)P (t,T )σTt dWt+P (t,T )r(t)dt
under P∗
)
= HtP (t, T )σTt dWt.
Since φ admissible we have supt∈[0,T ] Vt ∈ L2(Ω,P∗), thus Vt is a square-integrable mar-tingale and so
Vt = E∗[Vθ|Ft] ∀t 6 θ.
113
If we impose Vθ = h, we obtain Vt = e−∫ t0 r(s)dsVt = E∗[e−
∫ θ0 r(s)dsh|Ft] and so
Vt = E∗[e−∫ θt r(s)dsh|Ft]
as claimed.It remains to prove the existence of an admissible strategy satisfying (87).
”Sketch”: Since he−∫ θ0 r(s)ds ∈ L2(Ω,Fθ,P∗) we have that
Mt := E∗[he−∫ θ0 r(s)ds|Ft], t 6 θ
is a square integrable (Ft)-martingale w.r.t. P∗. We want to represent Mt through astochastic integral with Wt. This would directly follow from the martingale represen-tation theorem (cf. Theorem 4.7), if (Ft) is the natural filtration of (Wt). However, this isnot the case in general. Thus we make a detour:As in the proof of Proposition 6.3 (LtMt) is an (Ft)-martingale w.r.t. P. Then applying thegeneral (L1-)martingale representation theorem (cf. proof of Proposition 6.1) ∃Hs ∈ Fs
with∫ θ
0H2sds <∞ a.s. and
LθMθ︸ ︷︷ ︸=:Mθ
= L0M0 +
∫ θ
0
HsdWs.
Applying Ito’s product rule one can see (after some calculations)
L−1θ Mθ︸ ︷︷ ︸=Mt
= L−10 M0︸ ︷︷ ︸=M0
+
∫ θ
0
MsdL−1s +
∫ θ
0
L−1s dMs + 〈L−1, M〉θ
= M0 +
∫ θ
0
L−1s q(s)(Hs −Ms)︸ ︷︷ ︸
=:Js and∫ t0 J
2s ds<∞ a.s.
dWs a.s.
Now set
Ht :=Jt
P (t, T )σTt, and H0
t := Mt −JtσTt, t 6 θ.
Then φ = ((H0t , Ht))t∈[0,T ] defines an admissible strategy satisfying (87). The condition
supt∈[0,T ] |r(t)| <∞ implies that∫ θ
0|r(s)H0
s |ds <∞.
In view of Proposition 6.6, we define the fair price of the option h at time t by
E∗[e−∫ θt r(s)dsh|Ft], t ∈ [0, θ].
114
Remark 6.7 It can be shown that P∗ is the unique probability measure, equivalent to P,under which (P (t, T ))t∈[0,T ] is a martingale, iff σTt 6= 0 dtdP-a.e. (slightly weaker than thecondition in Proposition 6.6).
6.0.3 Forward measures and change of numeraire
Definition 6.8 For a fixed maturity date θ ∈ [0, T ], the θ-forward measure Pθ is definedby
Pθ(A) = E∗[
1Ae−
∫ θ0 r(s)ds
P (0, θ)
], A ∈ F(= FT ).
Pθ is a probability measure since P (0, θ) = E∗[e−∫ θ0 r(s)ds]. Moreover, since the density of
Pθ w.r.t. P∗dPθ
dP∗=e−
∫ θ0 r(s)ds
P (0, θ)
is strictly positive, we havePθ ≈ P∗ ≈ P.
Then
E∗[dPθ
dP∗|Ft]
=E∗[e−
∫ θ0 r(s)ds|Ft]
P (0, θ)=e−
∫ t0 r(s)ds E∗[e−
∫ θt r(s)ds|Ft]
P (0, θ)
=(79)
P (t, θ)
P (0, θ)= ”density of Pθ w.r.t. P∗ on Ft” (88)
To compute the price of an option with maturity θ, we can use the θ-forward measure:
Proposition 6.9 Let θ ∈ [0, T ] be a maturity date.
(1) If h ∈ FT , h > 0 a.s., then we have for t ∈ [0, θ]
E∗[e−∫ θt r(s)dsh|Ft] = P (t, θ)Eθ[h|Ft].
(2) If (Xt)t∈[0,θ] is an adapted stochastic process, then(Xt
S0t
)t∈[0,θ]
is a P∗-martingale ⇐⇒(
Xt
P (t, θ)
)t∈[0,θ]
is a Pθ-martingale.
115
Proof (1) From Bayes rule for conditional expectations (82), we get
Eθ[h|Ft] =1
LθtE∗[hLθθ|Ft] where Lθt =
P (t, θ)
P (0, θ).
Hence
P (t, θ)Eθ[h|Ft] =P (0, θ)
e−∫ t0 r(s)ds
E∗[he−
∫ θ0 r(s)dsP (θ, θ)
P (0, θ)|Ft
]= E∗[he−
∫ θt r(s)ds|Ft].
(2) We have (cf. proof of Proposition 6.3 or the exercises):
(Mt)t∈[0,θ] is a Pθ-martingale⇐⇒ (MtLθt )t∈[0,θ] is a P∗-martingale.
Thus (Xt
S0t
)is a P∗-martingale ⇐⇒ Xt
S0t
(Lθt)−1
=XtP (0, θ)
e∫ t0 r(s)dsP (t, θ)
=Xt
P (t, θ)P (0, θ) is a Pθ-martingale
⇐⇒(
Xt
P (t, θ)
)is a Pθ-martingale.
(since P(0, θ) = E∗[e−∫ θ0 r(s)ds] ∈ (0, 1) is an F0-measurable ”constant”)
Remark 6.10Proposition 6.9(2) = ”change of numeraire theorem”
numeraire = ”an asset that is used as price unit”
E.g. if St is the price of the risky asset at time t, then
St =StS0t
= ”discounted price”
= ”price of the risky asset, where the riskless asset S0t is used as numeraire”.
For t 6 θ:
StP (t, θ)
= ”price of the risky asset with the θ-bond as numeraire”
= ”θ-forward price of St”,
116
deliversSθ = one unit of St at time t = θ.
In our case θ ∈ [0, T ], T 6 T , and
S0t = e
∫ t0 r(s)ds, St = P (t, T ).
Thus by Proposition 6.9(2)
P (t, T ) =P (t, T )
e∫ t0 r(s)ds
is a P∗-martingale︸ ︷︷ ︸holds by (H)
⇐⇒ P (t, T )
P (t, θ)is a Pθ-martingale (t 6 θ).
Define
P θ(t, T ) :=P (t, T )
P (t, θ)= θ-forward price of the T -bond (P (t, T ))t∈[0,T ].
Proposition 6.11 Given maturity dates θ, T , with θ ∈ [0, T ], T 6 T , then Pθ(t, T )satisfies
dP θ(t, T )
P θ(t, T )= (σTt − σθt )dW θ
t , t ∈ [0, θ], (89)
where (σTt )t∈[0,θ] is as in Proposition 6.3 and
W θt = Wt −
∫ t
0
σθsds, t ∈ [0, θ],
with Wt as in Remark 6.4, is a standard BM under (the θ-forward measure) P θ.
Proof First show that W θ is a BM: We have
exp
(∫ t
0
σθsdWs −1
2
∫ t
0
(σθs)2ds
)=
(86)
P (t, θ)
P (0, θ)= Lθt , t ∈ [0, θ], (90)
is the local density of Pθ w.r.t. P∗ on Ft, see (88). By (H), (Lθt )t∈[0,θ] is a P∗-martingale,and by (88)
dPθ
dP∗= Lθθ.
Hence by Girsanov’s theorem
W θt := Wt −
∫ t
0
σθsds, t ∈ [0, θ],
117
is a standard BM under P θ = LθθdP∗. From (86) and (90)
P θ(t, T ) =P (t, T )e−
∫ t0 r(s)ds
P (t, θ)=P (0, T )
P (0, θ)exp
(∫ t
0
(σTs − σθs)dWs −1
2
∫ t
0
[(σTs )2 − (σθs)2]ds
)
= P θ(0, T ) exp
∫ t
0
(σTs − σθs)dW θs −
1
2
∫ t
0
(σTs − σθs)2ds︸ ︷︷ ︸=:Xt
(91)
As in the proof of Proposition 6.3 by Ito’s formula applied to f(x) = P θ(0, T ) exp(x), weget
P θ(t, T ) = P θ(0, T ) +
∫ t
0
P θ(s, T )dXs +1
2
∫ t
0
P θ(s, T )d〈X〉s
= P θ(0, T ) +
∫ t
0
P θ(s, T )(σTs − σθs)dW θs
and the result follows.
Remark 6.12 (θ-forward price of a European call) By Proposition 6.9, the value ofa European call at time t on a T -bond (P (t, T ))t∈[0,T ] with strike price K and maturity θcan be written as
E∗[e−∫ θt r(s)ds(P (θ, T )−K)+|Ft] = P (t, θ)Eθ[(P (θ, T )−K)+|Ft]︸ ︷︷ ︸
=:Cθt
.
Moreover for any t ∈ [0, θ], we have
P (θ, T ) =P (θ, T )
P (θ, θ)= P θ(θ, T )
=(91)
P θ(0, T )︸ ︷︷ ︸∈F0
exp
∫ t
0
(σTs − σθs)dW θs −
1
2
∫ t
0
(σTs − σθs)2ds︸ ︷︷ ︸∈Ft
︸ ︷︷ ︸
=P θ(t,T )∈Ft
·
118
· exp
∫ θ
t
(σTs − σθs)dW θs −
1
2
∫ θ
t
(σTs − σθs)2ds︸ ︷︷ ︸=:Z(t,θ)
.
ThereforeP (θ, T ) = P θ(t, T )︸ ︷︷ ︸
∈Ft
exp(Z(t, θ)).
Assume that (σTs ) and (σθs) are determisistic. Then Z(t, θ) is independent of Ft and
Z(t, θ) ∼ N
−1
2
∫ θ
t
(σTs − σθs)2ds,
∫ θ
t
(σTs − σθs)2ds︸ ︷︷ ︸=:Σ2(t,θ)
.
Thus by [1, Proposition 7.6], we have
Cθt (w) = Eθ
[(P θ(t, T )(w) exp(Z(t, θ))−K
)+]
and soCθt = B(t, P θ(t, T )), (92)
with
B(t, x) = Eθ[(x exp(Z(t, θ))−K)+] =
Rem. 4.10xN (d1(t, x))−KN (d2(t, x)) ,
where
d1(t, x) =log(xK
)+ Σ2(t,θ)
2
Σ(t, θ), d2(t, x) = d1(t, x)− Σ(t, θ),
and
N(y) =1√2π
∫ y
−∞e−
x2
2 dx.
(92) is known as Black’s formula (BS-formula with no interest rate). Now the value ofthe European call at time t is
Ct = P (t, θ)Cθt = P (t, θ)B
(t, P θ(t, T )
)= P (t, T )HT
t + P (t, θ)Hθt ,
whereHTt = N(d1(t, P θ(t, T )), Hθ
t = −KN(d2(t, P θ(t, T )).
Therefore, the option can be hedged by holding HTt T -bonds and Hθ
t θ-bonds at time t.
119
6.1 Some classical models
From (79) and (81) we know that (under (H))
P (t, T ) = E∗[e−∫ Tt r(s)ds|Ft]
andP (t, T ) = E
[e−
∫ Tt r(s)dse
∫ Tt q(s)dWs− 1
2
∫ Tt (q(s))2ds|Ft
],
where
Lt =dP∗|FtdP|Ft
= e∫ t0 q(s)dWs− 1
2
∫ t0 q(s)ds and (q(s))s∈[0,T ] is some adapted process.
Therefore, in order to calculate the price of a θ-bond ot T -bond, we need to know
either the law of r(t) under P∗ or the law of (r(t), q(t)) under P.
The first models describe the dynamics of r(t) under P by a SDE and determine the formthat q(t) should have in order to get a similar SDE under P∗.
6.1.1 The Vasicek model
In the model r(t) satisfies
dr(t) = σdWt + a(b− r(t))dt, (93)
where σ, a, b > 0 and r(0) ≥ 0 are constants. We assume further that q(t) = −λ, withsome λ ∈ R. Since Wt = Wt + λt is a BM under P∗, we get
dr(t) = σdWt + a
b− σλ
a︸ ︷︷ ︸=:b∗
−r(t)
dt (94)
under P∗. We have
d (r(t)− b) = dr(t) =(93)
σdWt − a (r(t)− b) dt,
thus Xt := r(t)− b is an Ornstein-Uhlenbeck process. It follows (see Section 3.5.2) that
Xt = X0e−at + σe−at
∫ t
0
easdWs,
120
and so
r(t) = r(0)e−at + b(1− e−at) + σe−at∫ t
0
easdWs. (95)
Thus r(t) is normally distributed with
E[r(t)] = r(0)e−at + b(1− e−at),
var(r(t)) = E[(r(t)− E[r(t)])2
]= E
[(σe−at
∫ t
0
easdWs)2
]= σ2e−2at
∫ t
0
e2asds = σ2e−2at 1
2a(e2at − 1) =
σ2
2a(1− e−at).
From (95) it follows that r(t) can be negative with positive probability (not fully satis-factory). As t→∞, r(t) converges in law to N(b, σ
2
2a).
For the calculation of zero-coupon bonds, we use (94) under P∗. By (79)
P (t, T ) = E∗[e−
∫ Tt r(s)ds|Ft
]= e−b
∗(T−t) E∗[e−
∫ Tt X∗s ds|Ft
](96)
with X∗t = r(t)− b∗. Since X∗t solves (94) it is an OU-process w.r.t. P∗. Thus its drift anddispersion coefficients do not depend on time and we get (see Remark 3.49)
(X∗0,xh )h>0 ∼same law
(X∗s,xs+h)h>0 (97)
andE∗[e−
∫ Tt X∗udu|Ft
]=
Markov propertyφ(X∗t ), a.s.
with
φ(x) =Theorem 3.47
E∗[e−
∫ Tt X∗t,xu du
]=
(97)E∗[e−
∫ Tt X∗0,xu−t du
]= E∗
[e−
∫ T−t0 X∗0,xu du
].
Thus
E∗[e−
∫ Tt X∗udu|Ft
]= φ(X∗t ) = E∗
[e−
∫ T−t0 X
∗0,r(t)−b∗u du
]=: F (T − t, r(t)− b∗) (98)
Note thatF (θ, x) = E∗
[e−
∫ θ0 X
xs ds],
where Xxs (= X∗0,xs ) is the unique solution of
dXt = σdWt − aXtdt
121
with initial condition Xx0 = x. We calculate F (θ, x) explicitly:
(Xxt ) Gaussian and continuous
(see chapter 3)⇒
∫ θ0Xxs ds = limn→∞
∑ti∈τn X
xti
(ti+1 − ti)︸ ︷︷ ︸Gaussian
is normally distributed.
⇒ E∗[e−
∫ θ0 X
xs ds]
︸ ︷︷ ︸L(
∫ θ0 X
xs ds)(1)
”Laplace transform”
=calculate
exp
(−E∗
[∫ θ
0
Xxs ds
]+
1
2var
(∫ θ
0
Xxs ds
)).
Since E∗[sups6θ |Xxs |2] <∞ it follows Xx
· ∈ L1([0, θ]×Ω, dtdP∗), thus by Fubini and since
Xxt = xe−at + σe−at
∫ t
0
easdWs,
we get
E∗[∫ θ
0
Xxs ds
]=
∫ θ
0
E∗ [Xxs ]︸ ︷︷ ︸
=xe−as
ds = x
(−1
a(e−aθ − 1)
)= x
1− e−aθ
a.
For the computation of the variance we use the formula
cov
(∫ θ
0
Xxs ds,
∫ θ
0
Xxs ds
)=(a)
∫ θ
0
∫ θ
0
cov (Xxt , X
xu) dtdu.
Thus
var
(∫ θ
0
Xxs ds
)= cov
(∫ θ
0
Xxs ds,
∫ θ
0
Xxs ds
)=
∫ θ
0
∫ θ
0
E∗[
(Xxt − E∗[Xx
t ]) · (Xxu − E∗[Xx
u ])]dtdu
=
∫ θ
0
∫ θ
0
σ2e−a(t+u) E∗[∫ t
0
easdWs
∫ u
0
easdWs
]︸ ︷︷ ︸
=(b)
∫ t∧u0 e2asds
dtdu
=
∫ θ
0
∫ θ
0
σ2e−a(t+u) e2a(t∧u) − 1
2adtdu
=calculate
σ2θ
a2− σ2
a3
(1− e−aθ
)− σ2
2a3
(1− e−aθ
)2
122
(a) and (b) above are left as an exercise. It follows
F (θ, x) = exp
(−x(
1− e−aθ
a
)+
1
2
(σ2θ
a2− σ2
a3
(1− e−aθ
)− σ2
2a3
(1− e−aθ
)2))
.
From (96) and (98) we get
P (t, T ) = e−b∗(T−t)F (T − t, r(t)− b∗)
= exp(−(T − t) · R (T − t, r(t))︸ ︷︷ ︸=”average interest rate on [t,T ]”
)
where
R(θ, r) = R∞ −1
aθ
[(R∞ − r)(1− e−aθ)−
σ2
4a2(1− e−aθ)2
]with
R∞ = limθ→∞
R(θ, r) = b∗ − σ2
2a2.
R∞ can be interpreted as ”long-term rate” (exercise). Note that it does not depend on theshort rate r(t), which can be seen as a drawback.
Remark 6.13 In practice parameters must estimated. E.g. r(t) is observed as short rate(e.g. over night). From r(t) the parameters b, a, σ are then estimated by statistical meth-ods. Then λ is obtained from the knowledge of r(t), b, a, σ.
6.1.2 The CIR-model (Cox-Ingersoll-Ross, 1985)
In the CIR-model the instantaneous rate r(t) is given as the unique (see below) solutionof
dr(t) = σ√|r(t)|dWt + (a− br(t))dt (99)
with constants σ > 0, a ≥ 0, b ∈ R, and q(t) = −α√|r(t)|, α ∈ R. Note that x 7→ σ
√|x|
is not Lipschitz (only Holder). However, by standard results on pathwise uniquess indimension one, (99) admits a unique solution (see. e.g. [4, Chapter 9,§3.]). In particular,if r(0) = x > 0 a.s., then r(t) > 0 a.s. ∀t > 0 (without proof). Hence if r(0) = x > 0(which we always assume) then we may replace
√|r(t)| by
√r(t) in (99).
Denote by (Xxt ), x > 0 the unique solution of (99) with Xx
0 = x. Define the stopping time
τx0 := inft > 0|Xxt = 0,
with the convention inf ∅ := +∞. Then:
123
Proposition 6.14 It holds:
(1) a > σ2
2⇒ P(τx0 = +∞) = 1, ∀x > 0.
(2) a ∈ [0, σ2
2), b > 0 ⇒ P(τx0 <∞) = 1, ∀x > 0.
(3) a ∈ [0, σ2
2), b < 0 ⇒ P(τx0 <∞) ∈ (0, 1), ∀x > 0.
Proof For x,M > 0 define the stopping time
τxM := inft > 0|Xxt = M.
Let
g(x) :=
∫ x
1
e2by
σ2 y−2aσ2 dy, x ∈ (0,∞).
Then g is harmonic for the CIR process, i.e. we have g ∈ C2(0,∞), and
Lg :=σ2
2xg′′ + (a− bx)g′ = 0 on (0,∞).
Let ε ∈ (0,M), x ∈ (ε,M), and define the stopping time
τxε,M := τxε ∧ τxM .
Let f ∈ C∞0 (R), supp(f) ⊂ (0,∞), f ≡ 1 on [ε,M ]. Then gf ∈ C∞0 (R), hence by Ito’sformula
gf(Xxt ) = g(x) +
∫ t
0
(gf)′(Xxs )σ√Xxs dWs +
∫ t
0
L(gf)(Xxs )ds
=⇒ gf(Xxt∧τxε,M
)︸ ︷︷ ︸=g(Xx
t∧τxε,M
)
= g(x) +
∫ t∧τxε,M
0
(gf)′(Xxs )σ√Xxs dWs︸ ︷︷ ︸
=∫ t∧τxε,M0 g′(Xx
s )σ√Xxs dWs
+
∫ t∧τxε,M
0
L(gf)(Xxs )ds︸ ︷︷ ︸
=0
, (100)
since Xxs ∈ [ε,M ] for s 6 τxε,M . It follows
E[(g(Xxt∧τxε,M︸ ︷︷ ︸∈[ε,M ]
)− g(x))2] = E
[(∫ t∧τxε,M
0
g′(Xxs )σ√Xxs dWs
)2]
= E[∫ t∧τxε,M
0
σ2g′(Xxs )2Xx
s ds
]> inf
y∈[ε,M ]σ2g′(y)2y︸ ︷︷ ︸>0
E[t ∧ τxε,M
],
124
thusE[t ∧ τxε,M ] 6 constant sup
y∈[ε,M ]
(g(y)− g(x))2
︸ ︷︷ ︸independent of t
<∞, ∀t > 0
and soE[τxε,M ] =
B. Levisupt>0
E[t ∧ τxε,M ] 6 constant,
henceP(τxε,M <∞
)= 1.
Taking expectations in (100) we get
E[g(Xxt∧τxε,M
)]= g(x). (101)
Since Xxt∧τxε,M
∈ [ε,M ], we have |g(Xxt∧τxε,M
)| 6 |g(M)|+ |g(ε)| a.s. Since τxε,M is a.s. finite
we obtain limt→∞ g(Xxt∧τxε,M
) = g(Xxτxε,M
) a.s. Thus by Lebesgue
E[g(Xxτxε,M
)]= lim
t→∞E[g(Xxt∧τxε,M
)]= g(x),
and sog(x) = E[1τxε <τxMg( Xx
τxε︸︷︷︸=ε a.s.
) + 1τxε >τxMg( XτxM︸︷︷︸=M a.s.
)]
= g(ε)P (τxε < τxM) + g(M)P (τxε > τxM) . (102)
If 2aσ2 ∈ [1,∞), then limx→0 g(x) = −∞. Since P (τxε < τx0 ) = 1, we get P (τx0 < τxM) 6
P (τxε < τxM). Hence for ε < 1
g(x) 6 g(ε)︸︷︷︸<0
P (τx0 < τxM) + g(M)P (τxε > τxM)︸ ︷︷ ︸6constant (uniformly in ε)
.
By letting ε 0, we see that P (τx0 < τxM) 6= 0 is impossible, because g(x) is finite.Therefore
P (τx0 < τxM) = 0 for all x,M > 0, if2a
σ2∈ [1,∞).
Since P (Xxt ∈ R ∀t ≥ 0) = 1 ∀x > 0, the last implies P(τx0 < ∞) = 0 ∀x > 0. Thus (1)
holds.If 2a
σ2 ∈ [0, 1), then ∃g(0) := limx→0 g(x) < 0. Let hn ∈ (0, ε), hn ε as n → ∞. Sinceτxε < τxM ⊂ τxε < τxM for ε 6 ε, it follows as n→∞
τxε−hn < τxM ⋂n≥1
τxε−hn < τxM ⊃ τx0 < τxM,
125
and
ω ∈ τx0 > τxM =⇒ ω ∈ τxε−hn > τxM for some n ≥ 1 =⇒ ω ∈⋃n≥1
τxε−hn > τxM,
hence also ⋂n≥1
τxε−hn < τxM ⊂ τx0 < τxM.
Consequently
limn→∞
P(τxε−hn < τxM
)= P
(⋂n≥1
τxε−hn < τxM
)= P (τx0 < τxM) ,
and so by (102)
g(x) = limn→∞
[g(ε− hn)P
(τxε−hn < τxM
)+ g(M)
(1− P
(τxε−hn < τxM
))]= g(0)P (τx0 < τxM) + g(M) (1− P (τx0 < τxM)) .
It follows
P(τx0 < τxM) =g(x)− g(M)
g(0)− g(M).
If ω ∈ τx0 < ∞, then ∃M > 0, with ω ∈ τx0 < τM, i.e. ω ∈⋃M>0τx0 < τxM, and if
ω ∈ τx0 < τxM for some M > 0, then ω ∈ τx0 <∞. Since
τx0 < τxM ⋃M>0
τx0 < τxM,
we get
P(τx0 <∞) = P
(⋃M>0
τx0 < τxM
)= lim
M→∞P(τx0 < τxM)
=g(x)− g(∞)
g(0)− g(∞)=
1 if b > 0 since g(M)∞∈ (0, 1) if b < 0 since g(M) constant.
Thus (2) and (3) hold.
The joint law of
(Xxt ,∫ t
0Xxs ds)
is characterized by:
126
Proposition 6.15 For λ, µ > 0, we have
E[e−λXxt e−µ
∫ t0 X
xs ds] = exp(−aφλ,µ(t)) exp(−xψλ,µ(t)),
where
φλ,µ(t) = − 2
σ2log
(2γe
t(γ+b)2
σ2λ(eγt − 1) + (γ − b) + eγt(γ + b)
)and
ψλ,µ(t) =λ [γ + b+ eγt(γ − b)] + 2µ(eγt − 1)
σ2λ(eγt − 1) + (γ − b) + eγt(γ + b), with γ =
√b2 + 2σ2µ.
Proof (Sketch) Suppose that F ∈ C1,2(R+ × R) has bounded derivatives and solves−∂tF + σ2
2x∂xxF + (a− bx)∂xF = µxF,
F (0, x) = e−λx.(103)
As in the proof of Theorem 5.7, by Ito’s formula it follows for any T > 0, that
Mt := e−µ∫ t0 X
xs dsF (T − t,Xx
t ), t ∈ [0, T ],
is a martingale. Hence ∀T > 0
E[e−µ∫ T0 Xx
s ds e−λXxT︸ ︷︷ ︸
=F (0,XxT )
] = E[MT ] = E[M0] = E[F (T,Xx0 )] = F (T, x).
By additivity properties of (Xxt ) w.r.t. a and x (more precisely of the squared Bessel
process w.r.t. initial condition and dimension, see [4]), we know that
F (t, x) = E[e−µ∫ t0 X
xs dse−λX
xt ] = exp(−aφ(t)− xψ(t))
for some functions φ, ψ depending only on time. Thus (103) above becomesF (t, x)
(aφ′(t) + xψ′(t) + σ2
2xψ2(t)− (a− bx)ψ(t)− µx
)= 0,
F (0, x) = exp(−aφ(0)− xψ(0)) = e−λx.
It follows φ(0) = 0, ψ(0) = λ, and
φ′(t) = ψ(t), −ψ′(t) =σ2
2ψ2(t) + bψ(t)− µ.
127
By solving the two differential equations given the initial conditions φ(0) = 0 and ψ(0) = λwe obtain the expressions for φλ,µ, ψλ,µ.
Applying Proposition 6.15 with µ = 0 we obtain for b > 0
φλ,0(t) = − 2
σ2log
(2betb
σ2λ(ebt − 1) + ebt2b
)= − 2
σ2log
(b
σ2
2λ(1− e−bt) + b
)
ψλ,0(t) =λ(2b)
σ2λ(ebt − 1) + ebt2b=
λbe−bt
σ2
2λ(1− e−bt) + b
.
Thus the Laplace transform of Xxt is given by
E[e−λXxt ] = exp(−aφλ,0(t)) exp(−xψλ,0(t))
=
(b
σ2
2λ(1− e−bt) + b
) 2aσ2
exp
(−x λbe−bt
σ2
2λ(1− e−bt) + b
)
=
(1
2λσ2
4b(1− e−bt)︸ ︷︷ ︸
=:L
+1
) 2aσ2
exp
(−λ
=L︷ ︸︸ ︷σ2
4be−bt(ebt − 1) ·
=:ζ︷ ︸︸ ︷4xb
σ2(ebt − 1)
2λL+ 1
)
=1
(2λL+ 1)2aσ2
exp
(− λLζ
2λL+ 1
).
Then the Laplace transform ofXxt
Lis given by
E[e−λLXxt ] =
1
(2λ+ 1)2aσ2
exp
(− λζ
2λ+ 1
)= g 4a
σ2,ζ(λ),
where
gδ,ζ(λ) :=1
(2λ+ 1)δ2
exp
(− λζ
2λ+ 1
).
It is known that gδ,ζ is the Laplace transform of the non-central chi-square distributionwith δ degrees of freedom and parameter ζ. The density of this distribution w.r.t. theLebesgue measure on R+ is
fδ,ζ(x) =e−
ζ2
2ζδ4− 1
2
e−x2
2 xδ4− 1
2 I δ2−1(√xζ), x > 0
128
where Iν is the first-order modified Bessel function with index ν, i.e.
Iν(x) =(x
2
)ν∑n>0
(x2
)2n
n!Γ(ν + n+ 1).
Iν solvesx2y′′(x) + xy′(x)− x2y(x) = ν2y(x), x > 0.
By the assumption q(t) = −α√r(t) and since Wt := Wt −
∫ t0q(s)ds is a BM w.r.t. P∗ we
get
dr(t) = σ√r(t)d
(Wt − α
∫ t
0
√r(s)ds
)+ (a− br(t))dt
= σ√r(t)dWt + (a− (b+ σα)︸ ︷︷ ︸
=:b∗
r(t))dt under P∗. (104)
Now the price of a zero-coupon bond with maturity T at time 0 is
P (0, T ) = E∗[e−
∫ T0 r(s)ds
]= e−aφ
∗(T )−r(0)ψ∗(T ), (105)
where according to Proposition 6.15 and pathwise uniqueness
φ∗(t) = φ∗0,1(t) = − 2
σ2log
(2γ∗e
t(γ∗+b∗)2
γ∗ − b∗ + eγ∗t(γ∗ + b∗)
)
and
ψ∗(t) = ψ∗0,1(t) =2(eγ
∗t − 1)
γ∗ − b∗ + eγ∗t(γ∗ + b∗)
with γ∗ =√
(b∗)2 + 2σ2.
Let X0,xt solution of (104) with X0,x
0 = x. Then the price at time t is given by
P (t, T ) = E∗[e−
∫ Tt r(u)du|Ft
]= E∗
[e−
∫ Tt X
0,r(0)u du|Ft
]cf. (96)(98)
= E∗[e−
∫ T−t0 X
0,r(t)u du
]=
Prop. 6.15exp(−aφ∗(T − t)− r(t)ψ∗(T − t)).
129
Now we want to price a European call with excercise price K on a T -bond. The price attime t = 0 is given by
C0 = P (0, θ)Eθ[(P (θ, T )−K)+]
where Pθ is the θ-forward measure (see Propositions 6.6 and 6.9). We have
P (θ, T ) > K = e−aφ∗(T−θ)−r(θ)ψ∗(T−θ) > K= −aφ∗(T − θ)− r(θ)ψ∗(T − θ) > logK
=
r(θ) < −aφ
∗(T − θ) + log(K)
ψ∗(T − θ)︸ ︷︷ ︸=:r∗
.
Hence
C0 = P (0, θ)Eθ[P (θ, T )1r(θ)<r∗]−KP (0, θ)Pθ(r(θ) < r∗)
= P (0, θ)Eθ[P (θ, T )] · Pθ,T (r(θ) < r∗)−KP (0, θ)Pθ(r(θ) < r∗),
where
dPθ,T =P (θ, T )
Eθ[P (θ, T )]dPθ.
Now
P (t, T )
S0t
= P (t, T ), t ∈ [0, θ] is a P∗-martingale⇐⇒6.9(2)
(P (t, T )
P (t, θ)
)t∈[0,θ]
is a Pθ-martingale.
Therefore
Eθ[P (θ, T )] =P (0, T )
P (0, θ).
It followsC0 = P (0, T )Pθ,T (r(θ) < r∗)−KP (0, θ)Pθ(r(θ) < r∗).
Using Proposition 6.15 and our previopus results on the Laplace transform of Xxt , one can
see that with
Lθ :=σ2
2
er∗θ − 1
γ∗(eγ∗θ + 1) + b∗(eγ∗θ − 1),
and
Lθ,T =σ2
2
er∗θ − 1
γ∗(eγ∗θ + 1) + (σ2ψ∗(T − θ) + b∗)(eγ∗θ − 1)
the law ofr(θ)
Lθunder Pθ
(resp.
r(θ)
Lθ,Tunder Pθ,T
)130
is a non-central chi-squared distribution with 4aσ2 degrees of freedom and parameter equal
to
ζθ =8r(0)(γ∗)2eγ
∗θ
σ2(eγ∗θ − 1)(γ∗(eγ∗θ + 1) + b∗)(eγ∗θ − 1)(resp. ζθ,T =
8r(0)(γ∗)2eγ∗θ
σ2(eγ∗θ − 1)(γ∗(eγ∗θ + 1) + (σ2ψ∗(T − θ) + b∗)(eγ∗θ − 1))
).
Thus introducing the distribution function
Fδ,ζ(x) :=
∫ x
0
fδ,ζ(y)dy
we have
C0 = P (0, T )F 4aσ2,ζθ,T
(r∗
Lθ,T
)−KP (0, θ)F 4a
σ2,ζθ
(r∗
Lθ
).
Remark 6.16 In the CIR-model the price of a zero-coupon bond P (0, T ) is the exponen-tial of an affine function of the interest rate r(0) (see (105)). The same holds for theVasicek-model (see formula before Remark 6.13). Such models are said to have an affineterm structure, i.e.
P (t, T ) = exp(−A(t, T )−B(t, T )r(t))
for some functions A(t, T ) and B(t, T ). Affine term structure models have diffusion anddrift coefficients of the following form
σ2(t, x) = a(t) + α(t)x, b(t, x) = b(t) + β(t)x.
6.1.3 The Heath-Jarrow-Morton methodology
Main drawback of Vasicek and CIR-model:
P (t, T ) is a function of the instantaneous interest rate r(t).
This can make it difficult to calibrate the model to the initially observed term structure.In the Heath-Jarrow-Morton model (1992) the forward instantaneous interest rate f(t, s),s > t is characterized by
P (t, T ) = exp
(−∫ T
t
f(t, s)ds
), t ∈ [0, T ], ∀T 6 T . (106)
f(t, s) = ” instantaneous interest rate at time s predicted by the market at time t 6 s”
We then make the following assumptions:
131
• (f(t, T ))t∈[0,T ] is adapted for any T ≤ T ,
• f(t, t) = r(t), t ∈ [0, T ],
• f(t, s), t 6 s is continuous in t and s.
The next step in the modelling is to assume that for each T ≤ T and given integrableinitial forward curve s 7→ f(0, s), s 6 T the process (f(t, T ))t∈[0,T ] solves
f(t, T ) = f(0, T ) +
∫ t
0
β(v, T )dWv +
∫ t
0
α(v, T )dv (107)
where β(t, T ), α(t, T ) are continuous in t and T , and adapted in t, i.e. α(t, T ), β(t, T ) are(Ft)-adapted for t ∈ [0, T ] and any T ≤ T .In order to have a reasonable model we have to make sure that condition (H) holds. Thisimplies additional conditions on α, β, that we are going to determine. Let
Xt := −∫ T
t
f(t, s)ds, t ∈ [0, T ], hence P (t, T ) = eXt .
Then
Xt =
∫ T
t
(−f(s, s) + f(s, s)− f(0, s) − f(t, s)− f(0, s)) ds
=(107)−∫ T
t
f(s, s)ds+
∫ T
t
(∫ s
t
β(v, s)dWv
)ds+
∫ T
t
(∫ s
t
α(v, s)dv
)ds︸ ︷︷ ︸
=∫ Tt
∫ Tt 1s6T 1v6sα(v,s)dsdv
=Fubini and
Stochastic Fubini
−∫ T
t
f(s, s)ds+
∫ T
t
(∫ T
v
β(v, s)ds
)dWv +
∫ T
t
(∫ T
v
α(v, s)ds
)dv
= X0 +
∫ t
0
f(s, s)ds−∫ T
0
f(s, s)− f(0, s)ds
+
∫ T
t
(∫ T
v
β(v, s)ds
)dWv +
∫ T
t
(∫ T
v
α(v, s)ds
)dv
= X0 +
∫ t
0
f(s, s)ds−∫ t
0
(∫ T
v
β(v, s)ds
)dWv −
∫ t
0
(∫ T
v
α(v, s)ds
)dv
= X0 +
∫ t
0
(−∫ T
v
β(v, s)ds
)dWv +
∫ t
0
(f(v, v)−
∫ T
v
α(v, s)ds
)dv, (108)
132
thus
dXt = −(∫ T
t
β(t, s)ds
)dWt +
(f(t, t)−
∫ T
t
α(t, s)ds
)dt.
By Ito’s formula
dP (t, T ) = P (t, T )dXt +1
2P (t, T )d〈X〉t
and so
dP (t, T )
P (t, T )= −
(∫ T
t
β(t, s)ds
)dWt +
(f(t, t)−
∫ T
t
α(t, s)ds+1
2
(∫ T
t
β(t, s)ds
)2)dt.
(109)If (H) holds then by Proposition 6.3 and since f(t, t) = r(t)
dP (t, T )
P (t, T )= σTt dWt + (f(t, t)− σTt q(t))dt. (110)
Comparing (109) with (110) it follows that
σTt = −∫ T
t
β(t, s)ds, and σTt q(t) =
∫ T
t
α(t, s)ds− 1
2
(∫ T
t
β(t, s)ds
)2
.
The last gives ∫ T
t
α(t, s)ds =1
2
(∫ T
t
β(t, s)ds
)2
− q(t)∫ T
t
β(t, s)ds.
Differentiating w.r.t. T leads to
α(t, T ) = β(t, T )
(∫ T
t
β(t, s)ds− q(t)).
Since Wt = Wt −∫ t
0q(s)ds is a BM under P∗ we can rewrite (107) as
df(t, T ) = β(t, T )dWt + β(t, T )
(∫ T
t
β(t, s)ds
)dt
under P∗. Thus the dynamics of the instantaneous forward rates f(t, T ) under P∗ doesnot depend on the drift α under P in (107). The model can be specified with the naturalassumption
β(t, T ) = σ(f(t, T ))
133
with σ ∈ C(R) (one can also consider time dependent σ(t, x) but we don’t do this here).In this case (f(t, T ))t∈[0,T ] satisfies
df(t, T ) = σ(f(t, T ))dWt + σ(f(t, T ))
(∫ T
t
σ(f(t, s))ds
)dt. (111)
The following theorem gives sufficient conditions under which (111) posseses a uniquesolution:
Theorem 6.17 (Heath, Jarrow, Morton (1992)) If σ is Lipschitz continuous and boundedand φ ∈ C([0, T ],R+) then there exists a unique continuous process with two indices(f(t, T ))06t6T6T such that (f(t, T ))t∈[0,T ] is adapted and solves (111) with f(0, T ) = φ(T ),∀T 6 T .
Proof Without, see original article of Heath, Jarrow, Morton (1992).
Under the conditions of Proposition 6.17, for any continuous process (q(t)) one can build amodel of the form (107). Indeed, let (f(t, T ))t∈[0,T ] be the unique solution of (111). Define
α(t, T ) := σ(f(t, T ))
(∫ T
t
σ(f(t, s))ds− q(t)).
Then (f(t, T )) satisfies (107) with β(v, T ) = σ(f(v, T )).Another interesting aspect of this model is that the law of the forward rates (f(t, T ))under P∗ only depends on σ (cf. (111)). This implies that the option prices will also onlydepend on σ (cf. (106), (111)). This is similar to the Black-Scholes model.If σ is a constant, then the volatilities of the zero-coupon bonds (σTt )t∈[0,T ] are deterministicso that the price of a call can be calculated similarly to Remark 6.12 (Exercise).
Final remark: The boundedness of σ is essential. For instance if σ(x) = x (a very niceLipschitz function), then it can be shown that (111) has no solution.
6.1.4 The forward LIBOR (London Inter Bank Offered Rate) model
In the HJM model the forward rates f(t, s), s > t are not directly observable (nor easyto estimate) but typical interest rate derivatives involve forward rates over a finite timeinterval. Thus one may want to model other rate such as forward LIBORs directly.A (δ-period) forward LIBOR for the expiry T is a simply compounded interest rate over
134
the time interval [T, T + δ], where δ is called the tenor. Its value L(t, T ) at time t ∈ [0, T ]is given by
simple compounding︷ ︸︸ ︷1 + δ L(t, T )︸ ︷︷ ︸
interest rate
=P (t, T )
P (t, T + δ)= exp
(∫ T+δ
T
f(t, s)ds
)where δ is a time period (e.g. 3 months (δ = 1
4), 1 day (δ = 1
365)). In other words it is
given by
L(t, T ) =1
δ
(P (t, T )
P (t, T + δ)− 1
) (−→δ0
f(t, T )
).
A typical interest rate option is a caplet:
a caplet pays δ(L(T, T )− K︸︷︷︸fixed rate
)+ at time T + δ.
An agent who borrows 1 · $ at time T has to payback (1 + δL(T, T )) · $ at time T + δ. Ifthe agent holds a caplet with fixed rate K, the net payment in $ at time T + δ is
1 + δL(T, T )− δ(L(T, T )−K)+ = 1 + δmin(L(T, T ), K)
⇒ the interest rate reduces to min(L(T, T ), K).
In case of several payment dates Ti = T + iδ, i = 1, ..., n, the interest rate can be ”locked”at the level K by holding a cap (i.e. a collection of caplets). The holder of a cap receivesδ(L(Ti, Ti) − K)+ at times Ti, i = 1, ..., n. Thus the interest rate is at all times 6 K.Denote
Ct = price at time t of a caplet that pays δ(L(T, T )−K)+ at time T + δ.
From Propositions 6.6 and 6.9 we know
Ct = P (t, T + δ)ET+δ[δ(L(T, T )−K)+|Ft], t ∈ [0, T + δ].
Using, thatP (t, T + δ)
P (0, T + δ)dP∗|Ft = dPT+δ|Ft ,
i.e. P (t,T+δ)P (0,T+δ)
is the local density on Ft of PT+δ w.r.t. P∗ one directly shows that(P (t, T )
P (t, T + δ)
)t∈[0,T ]
135
is a PT+δ-martingale. Therefore (L(t, T ))t∈[0,T ] is also a PT+δ-martingale and it is reason-able to assume that the dynamics of L(t, T ) is determined by
dL(t, T )
L(t, T )= γ(t, T )dW T+δ
t , t ∈ [0, T ], (112)
where γ(t, T ) is deterministic and bounded, and W T+δ is a standard PT+δ-BM. If (112)holds, then similarly to Remark 6.12 one can develop a closed-form solution for Ct. Thisis known a Black’s caplet formula. One can further construct a consistent model in whicheach L(t, Ti), i = 1, ..., n satisfies (112) and derive cap prices as sums of caplet prices.
Final Remark: This chapter was a rather short introduction to term-structure models. Inparticular multifactor models were not treated. A good reference on the subject is
Brigo/Mercurio: Interest models-Theory and practice, Springerand
Filipovic: Term-structure models. A graduate course, Springer
7 Asset models with jumps
Why discontinuous models? In continuous models large sudden movements of prices aredifficult to realize and need very large volatilities. But large movement of prices can occur,after e.g.
- the release of unexpected economic data,
- major political changes,
- a natural disaster in a major economy.
Such rare events can be modeled by discontinuous stochastic processes.
7.1 Poisson process
Definition 7.1 Let Ti : Ω → R+, i ∈ N, be a sequence of independent identically expo-nentially distributed random variables with parameter λ > 0, i.e.
P(Ti 6 x) =
∫ x
0
λe−λtdt, i ∈ N.
136
Let τn :=∑n
i=1 Ti, τ0 := 0. Then
Nt :=∑n>1
1τn6t, t ≥ 0,
is called a Poisson process with intensity λ.
Remark 7.2 Nt represents the number of elements of the sequence (τn)n∈N which are lessor equal to t. For n ∈ N ∪ 0, we have τn 6 t < τn+1 = Nt = n, hence
τn = inft > 0|Nt = n
andNt =
∑n>1
n1τn6t<τn+1.
Proposition 7.3 Let (Nt)t>0 be a Poisson process with intensity λ. Then for any t > 0the random variable Nt follows a Poisson law with parameter λ · t, i.e.
P(Nt = n) = e−λt(λt)n
n!, n ∈ N ∪ 0.
In particular, we have
E[Nt] = λt, and var(Nt) := E[N2t ]− E[Nt]
2 = λt.
Moreover for s > 0E[sNt ] = exp(λt(s− 1)).
Proof We first show that τn, n ≥ 1, has a Gamma distribution with parameters (n, λ),i.e.
P(τn 6 x) =
∫ x
0
λe−λt(λt)n−1
(n− 1)!dt.
Indeed, the Laplace transform of T1 is given by
E[e−αT1 ] =
∫ ∞0
e−αtλe−λtdt =
∫ ∞0
(− λ
α + λe−(α+λ)t
)′dt =
λ
α + λ.
By independence, the Laplace transform of τn satisfies
E[e−ατn ] = E
[n∏i=1
e−αTi
]=
n∏i=1
E[e−αTi
].
137
Since (Ti)i∈N is identically distributed, we get
E[e−αTi ] = E[e−αT1 ], ∀i ∈ N.
Consequently
E[e−ατn ] = E[e−αT1 ]n =
(λ
α + λ
)n= L(α).
L(α) is known to be the Laplace transform of the Gamma distribution with parameters(n, λ). Thus by uniqueness of the Laplace transform the first claim follows. Now for n > 1
P(Nt = n) = P(τn 6 t)− P(τn+1 6 t)
=
∫ t
0
λe−λx(λx)n−1
(n− 1)!dx−
∫ t
0
λe−λx(λx)n
n!dx =
(λt)n
n!e−λt
and for n = 0
P(Nt = 0) = P(τ0 6 t)− P(τ1 6 t) = 1−∫ t
0
λe−λxdx = e−λt.
Proposition 7.4 Let (Nt)t>0 be a Poisson process with intensity λ, and Ft := σ(Ns; s 6t). Then (Nt)t>0 is a process with independent stationary increments, i.e. for any t > 0we have
- independence: if s > 0, Nt+s −Nt is independent of Ft
- stationarity: the law of Nt+s −Nt is identical to the law of Ns −N0 = Ns.
Proof (without).
Remark 7.5 The jump times τn are stopping times. Indeed τn 6 t = Nt > n ∈ Ft.A random variable T with exponential law satisfies
P(T > t+ s|T > t) =P(T > t+ s ∩ T > t)
P(T > t)= P(T > s).
Therefore exponentially distributed random variables are said to be ”memory less”. (Theindependence of the increments in Proposition 7.4 is a consequence of this property ofexponential laws).
138
Remark 7.6 (without proof) The law of a Poisson process with intensity λ is character-ized by either of the two following properties:
- (Nt)t>0 is a right continuous homogeneous Markov process with left limits, such that
P(Nt = n) = e−λt(λt)n
n!, t > 0, n ∈ N.
- (Nt)t>0 is a right continuous, non-decreasing process with independent and stationaryincrements, N0 = 0 and jumps of amplitude one.
7.2 Dynamics of the risky asset
The price of the riskless asset at time t is modeled by S0t = ert, r > 0.
The risky asset is modeled as follows: on a probability space (Ω,F,P) consider
- (Wt)t>0 a standard BM,
- (Nt)t>0 a Poisson process with intensity λ,
- (Uj)j>1 a sequence of i.i.d. random variables Uj : Ω→ (−1,∞).
Assume that σ((Wt)t>0), σ((Nt)t>0), and σ((Uj)j>1) are independent. Let
Ft := σ(Ws, Ns, , s 6 t, Uj1j6Nt, j > 1), t > 0.
It can be shown:
- (Wt)t>0 is a standard BM w.r.t. (Ft)t>0,
- (Nt)t>0 is (Ft)t>0-adapted and for all t > s: Nt −Ns is independent of Fs.
The dynamics of Xt = ”price of the risky asset at time t” is determined by
- (Xt)t>0 is an (Ft)-adapted process,
- On the time intervals [τj, τj+1): dXt = Xt(σdWt + µdt), where τj := inft > 0|Nt =j, j ≥ 1, τ0 := 0, and σ, µ are constants with σ > 0,
- at t = τj, j ≥ 1, the jump of Xτj is given by
∆Xτj := Xτj −Xτ−j= Xτ−j
Uj,
where Xτ−j:= limsτj Xs, i.e. at τj the jump is proportional to Uj, so that Xτj =
Xτ−j(1 + Uj).
139
We have
Xt = X0eσWt+(µ−σ
2
2)t, for t ∈ [0, τ1),
thus the left-hand limit at τ1 is given by
Xτ−1= X0e
σWτ1+(µ−σ2
2)τ1 ,
and
Xτ1 = Xτ−1(1 + U1) = X0(1 + U1)eσWτ1+(µ−σ
2
2)τ1 (113)
Then for t ∈ [τ1, τ2)
Xt = Xτ1eσ(Wt−Wτ1 )+(µ−σ
2
2)(t−τ1)
=(113)
X0(1 + U1)eσWt+(µ−σ2
2)t.
From this we get
Xτ−2= X0(1 + U1)eσWτ2+(µ−σ
2
2)τ2 ,
and
Xτ2 = Xτ−2(1 + U2) = X0(1 + U1)(1 + U2)eσWτ2+(µ−σ
2
2)τ2 . (114)
Then for t ∈ [τ2, τ3)
Xt = Xτ2eσ(Wt−Wτ2 )+(µ−σ
2
2)(t−τ2)
=(114)
X0(1 + U1)(1 + U2)eσWt+(µ−σ2
2)t,
and so for t ∈ [τk, τk+1) where k is arbitrary, we get
Xt = X0
(k∏j=1
(1 + Uj)
)eσWt+(µ−σ
2
2)t.
Since Nt = k ⇔ t ∈ [τk, τk+1), we get
Xt = X0
(Nt∏j=1
(1 + Uj)
)eσWt+(µ−σ
2
2)t, t > 0,
with the usual convention (that the empty product)∏0
j=1 := 1. By our previous assump-tions (Xt)t>0 is right-continuous, adapted, and has only finitely many jumps on finite
140
intervals.It can then be shown that for all t > 0
Xt = X0 +
∫ t
0
Xs(σdWs + µds) +Nt∑j=1
Xτ−jUj, P-a.s. (115)
This model is called jump-diffusion model. In differential form, (115) is written as
dXt
Xt−= σdWt + µdt+ dZt
where Zt =∑Nt
j=1 Uj. The process (Zt) is called compound Poisson process.
7.3 Martingales in a jump diffusion model
Proposition 7.7 Suppose E[|U1|] < ∞. The discounted price process (Xt := e−rtXt)t>0
is a martingale, if and only ifµ = r − λE[U1].
In order to calculate E[Xt|Fs], we need the following lemma.
Lemma 7.8 Let for s > 0
Gs := σ ((Wt+s −Ws)t>0, (Nt+s −Ns)t>0, (UNs+j)j>1) .
Then Gs and Fs, s > 0 are independent.
Proof LetWs := σ((Wt+s −Ws)t>0), Ns := σ((Nt+s −Ns)t>0),
Ws := σ(Wu|u ∈ [0, s]), Ns := σ(Nu|u ∈ [0, s]).
In order to show independence of Gs and Fs, we need to show
P(A ∩B) = P(A)P(B) ∀A ∈ Gs, ∀B ∈ Fs. (116)
By what is known about Dynkin systems it is enough to show (116) for all A ∈ A0,B ∈ B0, where A0 (resp. B0) is a class of events that is stable under finite intersectionssuch that σ(A0) = Gs (resp. σ(B0) = Fs). Take
A = AW︸︷︷︸∈WS
∩ AN︸︷︷︸∈Ns
∩(UNs+1, ..., UNs+k) ∈ G︸︷︷︸∈B(Rk)
, k > 1, (A ∈ A0)
141
andB = BW︸︷︷︸
∈Ws
∩ BN︸︷︷︸∈Ns
∩(U1, ..., Ud) ∈ F︸︷︷︸∈B(Rd)
∩ d 6 Ns, d > 1. (B ∈ B0)
We have
A ∩B = AW ∩BW ∩ AN ∩BN ∩ (UNs+1, ..., UNs+k) ∈ G ∩ (U1, ..., Ud) ∈ F ∩ d 6 Ns
=∞⋃p=d
AW ∩BW ∩ AN ∩BN ∩ Ns = p︸ ︷︷ ︸=:BpN
∩(Up+1, ..., Up+k) ∈ G ∩ (U1, ..., Ud) ∈ F︸ ︷︷ ︸=:Cp
.
Thus by independence of (Wt), (Nt) and (Uj), we obtain
P(A ∩B) =∞∑p=d
P(AW ∩BW )P(AN ∩BpN)P(Cp)
and by independence of Ws and Ws, and independence of Ns and Ns, and independenceof (Uj)j>1 we get
P(AW ∩BW ) = P(AW )P(BW ), and P(AN ∩BpN) = P(AN)P(Bp
N),
andP(Cp) =
p>dP((Up+1, ..., Up+k) ∈ G)︸ ︷︷ ︸
=P((U1,...,Uk)∈G)since identically distributed
P((U1, ..., Ud) ∈ F ).
Therefore by independence of (Nt) and (Wt)
P(A∩B) = P(AW )P(BW )P(AN)P((U1, ..., Uk) ∈ G)P((U1, ..., Ud) ∈ F )∞∑p=d
P(BpN)︸ ︷︷ ︸
P(BN∩d6Ns)
(117)
= P(AW )P(AN)P((U1, ..., Uk) ∈ G)P((U1, ..., Ud) ∈ F )P(BN ∩BW ∩ d 6 Ns).Now we choose AW , AN , BW , BN = Ω, F = Rd, and
G = R× · · · × R× H︸︷︷︸j-th coordinate
× R× · · · × R
with H ∈ B(R). Then (117) becomes
P(UNs+j ∈ H ∩ d 6 Ns) = P(Uj ∈ H)P(d 6 Ns) for any j > 1.
142
The last impliesP(UNs+j ∈ H) = P(Uj ∈ H) for any j > 1.
Thus for any j > 1, UNs+j and Uj have same law. Using the independence of (Wt), (Nt)and (Uj) we can now see that (117) implies P(A ∩B) = P(A)P(B) as desired.
Proof of Proposition 7.7: We have
Xt = X0
(Ns∏j=1
(1 + Uj)
)eσWs+
(µ−r−σ
2
2
)s
︸ ︷︷ ︸=Xs∈Fs
(Nt∏
j=Ns+1
(1 + Uj)
)eσ(Wt−Ws)+
(µ−r−σ
2
2
)(t−s)
and so
E[Xt|Fs] = Xs E
[eσ(Wt−Ws)+
(µ−r−σ
2
2
)(t−s)
(Nt−Ns∏j=1
(1 + UNs+j)
)|Fs
]
=7.8Xs E
[eσ(Wt−Ws)+
(µ−r−σ
2
2
)(t−s)
(Nt−Ns∏j=1
(1 + UNs+j)
)]
=UNs+j∼Uj
Xs E
[eσ(Wt−Ws)+
(µ−r−σ
2
2
)(t−s)
(Nt−Ns∏j=1
(1 + Uj)
)]
= Xse(µ−r)(t−s) E
[Nt∏
j=Ns+1
(1 + Uj)
]=
exerciseXse
(µ−r)(t−s)eλ(t−s)E[U1] = Xs, iff µ− r = λE[U1].
Let ν denote the common law of the random variables Uj, i.e.
ν(A) := P(Uj ∈ A), A ∈ B ((−1,∞)) .
In order to deal with jumps in hedging strategies we need 2 lemmas.
Lemma 7.9 Let φ : Rd × R → R be measurable, and such that φ(·, z) : Rd → R iscontinuous for any z ∈ R. Let (Yt)t>0 be a left-continuous (Ft)-adapted process in Rd. If
E[∫ t
0
∫ ∞−1
φ2(Ys, z)ν(dz)ds
]<∞, ∀t ≥ 0,
143
then
Mt :=Nt∑j=1
φ(Yτj , Uj)− λ∫ t
0
∫ ∞−1
φ(Ys, z)ν(dz)ds, t ≥ 0,
(0∑j=1
:= 1
)is a square-integrable martingale and
M2t − λ
∫ t
0
∫ ∞−1
φ2(Ys, z)ν(dz)ds, t ≥ 0,
is a martingale.
Proof Assume that φ is bounded and let
C := sup(y,z)∈Rd×R
|φ(y, z)|.
Then ∣∣∣∣∣Nt∑j=1
φ(Yτj , Uj)
∣∣∣∣∣ 6 C ·Nt ∈ L2,
hence (Mt)t>0 is square-integrable.Fix s, t with s < t and a partition ρ := s = s0 < s1 < ... < sm = t of [s, t] and let
Z :=Nt∑
j=Ns+1
φ(Yτj , Uj),
Zρ :=m−1∑i=0
Nsi+1∑j=Nsi+1
φ(Ysi , Uj).
If ρ := max16i6m |si−si−1| → 0, then Zρ → Z a.s, since s 7→ φ(Ys, z) is a.s. left-continuous.Since |Zρ| 6 C(Nt −Ns) ∈ L2, we then get Zρ → Z in L2 by Lebesgue. We have
E[Zρ|Fs] = E
[m−1∑i=0
E
[ Nsi+1∑j=Nsi+1
φ(Ysi , Uj)︸ ︷︷ ︸=∑Nsi+1−Nsij=1 φ(Ysi ,UNsi+j
)=:Zi+1
|Fsi
]|Fs
]. (118)
Note that Ysi ∈ Fsi and that UNsi+j, j > 1, is independent of Fsi by Lemma 7.8. It hencefollows by Proposition 7.6 of [1] that
E[Zi+1|Fsi ] = φi(Ysi)
144
with
φi(y) = E
Nsi+1−Nsi∑j=1
φ(y, UNsi+j)
=(exercise)
λ(si+1 − si)∫φ(y, z)ν(dz).
Thus (118) can be rewritten as
E[Zρ|Fs] = E
[m−1∑i=0
φi(Ysi)|Fs
]= E
[m−1∑i=0
λ(si+1 − si)∫φ(Ysi , z)ν(dz)|Fs
].
Since
E[Z|Fs] = lim|ρ|→0
E[Zρ|Fs] =
[λ
∫ t
s
∫φ(Yu, z)ν(dz)du|Fs
],
we get
E[Mt|Fs] = E
[Ns∑j=1
φ(Yτj , Uj) +Nt∑
j=Ns+1
φ(Yτj , Uj)− λ∫ t
0
∫φ(Yu, z)ν(dz)du|Fs
]
=Ns∑j=1
φ(Yτj , Uj) + lim|ρ|→0
E[Zρ|Fs]− λE[∫ t
s
∫φ(Yu, z)ν(dz)du|Fs
]︸ ︷︷ ︸
=0
− λ∫ s
0
∫φ(Yu, z)ν(dz)du
= Ms.
Hence (Mt)t>0 is a martingale. Define
Zρ :=m−1∑i=0
φi(Ysi) =m−1∑i=0
λ(si+1 − si)∫φ(Ysi , z)ν(dz).
Note that Zρ =∑m−1
i=0 E[Zi+1|Fsi ] and Zρ =∑m−1
i=0 Zi+1, hence (since the mixed termsdisappear)
E[(Zρ − Zρ)2|Fs] = E
(m−1∑i=0
(Zi+1 − E[Zi+1|Fsi ])
)2
|Fs
= E
[m−1∑i=0
(Zi+1 − E[Zi+1|Fsi ])2︸ ︷︷ ︸E[(Zi+1−E[Zi+1|Fsi ])2|Fsi ]=V (Ysi )
|Fs
]
145
with (exercise)
V (y) = var
Nsi+1−Nsi∑j=1
φ(y, UNsi+j)
= λ(si+1 − si)∫φ2(y, z)ν(dz).
Since
Zρ − Zρ −→|ρ|→0
Z − λ∫ t
s
∫φ(Yu, z)ν(dz)du = Mt −Ms in L2,
we get
E[(Mt −Ms)2|Fs] = lim
|ρ|→0E
[m−1∑i=0
λ(si+1 − si)∫φ2(Ysi , z)ν(dz)|Fs
]
= E[λ
∫ t
s
∫φ2(Yu, z)ν(dz)du|Fs
].
MoreoverE[(Mt −Ms)
2|Fs] = E[M2t −M2
s |Fs]and therefore
M2t − λ
∫ t
0
∫φ2(Yu, z)ν(dz)du, t > 0,
is a martingale. If φ is not bounded, then approximate φ with
φn := (−n) ∨ (φ ∧ n), n > 1.
The assertions of the lemma hold for M(n)t defined through φn, n > 1. Taking limits and
noting that ∫ t
0
∫(φn(Yu, z)− φ(Yu, z))
2 ν(dz)du −→n→∞
0,
(M(n)t )n≥1 is L2-Cauchy and M
(n)t →Mt a.s, the assertion then follows for general φ.
Lemma 7.10 Let (Mt)t>0 be as in Lemma 7.9 (under the same hypotheses). Let (At)t>0
be an adapted process such that
E[∫ t
0
A2sds
]<∞, ∀t > 0.
Define (Lt :=∫ t
0AsdWs)t>0. Then (LtMt)t>0 is a martingale.
146
Proof Using the same approximation as at the end of the proof of Lemma 7.9, wemay assume that φ is bounded by C := sup(y,z) |φ(y, z)| < ∞. Fix again s < t and letρ = s = s0 < ... < sm = t be a partition of [s, t]. Then
E[LtMt − LsMs|Fs] = E
[m−1∑i=0
E[Lsi+1Msi+1
− LsiMsi |Fsi ]︸ ︷︷ ︸=E[(Lsi+1−Lsi )(Msi+1−Msi )|Fsi ]
|Fs
]
= E[m−1∑i=0
(Lsi+1− Lsi)(Msi+1
−Msi)︸ ︷︷ ︸=:Λρ
|Fs].
We have
|Λρ| 6 sup06i6m−1
|Lsi+1− Lsi |
m−1∑i=0
|Msi+1−Msi |
and
m−1∑i=0
|Msi+1−Msi | 6
Nt∑j=Ns+1
|φ(Yτj , Uj)|︸ ︷︷ ︸≤C
+λ
∫ t
s
∫|φ(Yu, z)|︸ ︷︷ ︸
6C
ν(dz)du
6 C(Nt −Ns) + λC(t− s).
By continuity of (Lt)t>0 it then follows
|Λρ| −→|ρ|→0
0 a.s.
Moreover|Λρ| 6 2C sup
u∈[s,t]
|Lu|(Nt −Ns + λ(t− s)).
It then follows from Doob’s inequality that
supu∈[s,t]
|Lu| ∈ L2.
Since Nt −Ns ∈ L2, we get by Lebesgue
Λρ −→|ρ|→0
0 in L1.
ConsequentlyE[LtMt − LsMs|Fs] = 0.
147
7.4 Pricing options in a jump-diffusion model
Consider the jump-diffusion model (for the risky asset)
Xt = X0
(Nt∏j=1
(1 + Uj)
)eσWt+
(µ−σ
2
2
)t, t ∈ [0, T ] (119)
of Section 7.2. Define
dP∗ := eθWT− θ2
2TdP where θ =
r − µ− λE[U1]
σ. (120)
From Girsanov’s theorem we know that W ∗t := Wt−θt, t ∈ [0, T ] is a standard BM under
P∗. Moreover (W ∗t ), (Nt), (Uj) are independent under P∗.
Illustration:
P∗( A︸︷︷︸dep. on(W∗t )
∩ B︸︷︷︸dep. on(Nt)
∩ C︸︷︷︸dep. on(Uj)
) = E[1B1C 1AeθWT− θ
2
2T︸ ︷︷ ︸
dep. on (Wt)
]
= E[1B]E[1C ]P∗(A) (by indep. of (Wt), (Nt), (Uj) under P)
= E[1B]E[eθWT− θ
2
2T]
︸ ︷︷ ︸=1 mart. propertyof Girsanov density
E[1C ]E[eθWT− θ
2
2T]
︸ ︷︷ ︸=1
P∗(A)
=(indep.)
E[1BeθWT− θ
2
2T ]E[1Ce
θWT− θ2
2T ]P∗(A)
= P∗(B)P∗(C)P∗(A).
In particular, we can see from the illustration that the laws of (Nt), (Uj) are the sameunder P and P∗ by choosing B = Nt ∈ B, and C = Uj ∈ C, e.g.
P(Uj ∈ C) = E[1C ]E[eθWT− θ2
2T ]︸ ︷︷ ︸
=1
=indep.
E[1CeθWT− θ
2
2T ] = P∗(Uj ∈ C).
Now, we have from (119)
Xt = X0
(Nt∏j=1
(1 + Uj)
)eσW
∗t +(µ∗−σ
2)t,
withµ∗ = µ+ σθ =
(120)r − λE[U1] = r − λE∗[U1].
148
Therefore by Proposition 7.7, the discounted prices Xt = e−rtXt, t ∈ [0, T ] form a martin-gale under P∗. It turns out (exercise) that there are infinitely many equivalent martingalemeasures. Up to this end, we assume that
µ = r − λE[U1] = r − λ∫ ∞−1
zν(dz). (121)
In other words we are choosing P as risk-neutral probability measure. For technical reasons,we will assume that
E∗[(U1)2] = E[(U1)2] <∞.
We have
E[X2t ] = X2
0 E
(exp
(σWt +
(µ− σ2
2
)t
) Nt∏j=1
(1 + Uj)
)2 (122)
Thus from independence
E[X2t ] = X2
0 E[exp(2σWt)]︸ ︷︷ ︸=e2σ2t
e(2µ−σ2)t E
[Nt∏j=1
(1 + Uj)2
]
=exercise
X20e
σ2te2µt exp(λt (E[(1 + U1)2]− 1)︸ ︷︷ ︸
=E[U21 ]+2E[U1] =
(121)E[U2
1 ]+2 r−µλ
)= X2
0e(σ2+2r)t exp(λtE[U2
1 ]).
Thus under the technical assumption U1 ∈ L2(P), we obtain that the discounted priceprocess is a square-integrable martingale.
7.4.1 Admissible strategies
A trading strategy is an adapted process φ = ((H0t , Ht))t∈[0,T ] with values in R2 such that
H0t = amount of riskless asset S0
t = ert held at time t,
andHt = amount of risky asset Xt held at time t,
are left-continuous (because Xt is right-continuous, Stieltjes-integration).The value at time t of the strategy φ is then Vt = H0
t ert + HtXt and the strategy is said
to be self-financing ifdVt = HtdXt +H0
t rertdt.
149
By (115) between the jump times, i.e. if t ∈ [τj, τj+1)
dVt = HtXt(σdWt + µdt) +H0t re
rtdt
and at jump time τj, Vt jumps by an amount of
∆Vτj = Hτj∆Xτj = HτjUjXτ−j.
Thus by (115), the self-financing condition can be written as
Vt = V0 +
∫ t
0
HsXs(σdWs + µds) +
∫ t
0
H0s re
rsds+Nt∑j=1
HτjUjXτ−j. (123)
For (123) to make sense it suffices to assume∫ T
0
(H2s + |H0
s |)ds <∞ a.s.
However in order to ensure the square integrability of Vt one needs a stronger conditionon (Ht)t∈[0,T ], namely:
Definition 7.11 An admissible strategy is an adapted, left-continuous, R2-valued processφ = ((H0
t , Ht))t∈[0,T ] that satisfies (123) a.s. for all t ∈ [0, T ],∫ T
0|H0
s |ds < ∞ a.s. and
E[∫ T
0H2sX
2sds] <∞.
Note: In contrast to the Black-scholes model, we do not assume that the value of theprotfolio w.r.t. an admissible strategy is at all times non-negative.
Proposition 7.12 (cf. Proposition 4.2) Let (Ht)t∈[0,T ] be an adapted left-continuous pro-cess such that
E[∫ T
0
H2sX
2sds
]<∞,
and let V0 ∈ R. There exists a unique process (H0t )t∈[0,T ], such that the pair ((H0
t , Ht))t∈[0,T ]
defines an admissible strategy with initial value V0. The discounted value Vt = e−rtVt isgiven by
Vt = V0 +
∫ t
0
HsXsσdWs +Nt∑j=1
HτjUjXτ−j− λE[U1]
∫ t
0
HsXsds.
150
Proof If (H0t , Ht) is an admissible strategy then by (123)
Vt = Yt + Zt
with Yt = V0 +∫ t
0HsXs(σdWs + µds) +
∫ t0H0s re
rsds, and Zt =∑Nt
j=1 HτjUjXτ−j. By Ito’s
product rule, we get
e−rtVt = e−rtYt + e−rtZt
= V0 +
∫ t
0
e−rsdYs +
∫ t
0
(−re−rs)Ysds+ e−rtZt. (124)
Further,
e−rtZt =Nt∑j=1
e−rtHτjUjXτ−j
=Nt∑j=1
(e−rτj +
∫ t
τj
(−re−rs)ds
)HτjUjXτ−j
=Nt∑j=1
e−rτjHτjUjXτ−j+
Nt∑j=1
∫ t
0
1τj 6 s︸ ︷︷ ︸=Ns>j
(−re−rs)dsHτjUjXτ−j
=Nt∑j=1
e−rτjHτjUjXτ−j+
∫ t
0
(Ns∑j=1
HτjUjXτ−j
)(−re−rs)ds
=Nt∑j=1
e−rτjHτjUjXτ−j+
∫ t
0
(−re−rs)Zsds.
151
Writing this in (124) with explicit Yt we get
Vt = V0 +
∫ t
0
HsXs(σdWs + µds) +
∫ t
0
H0s rds+
∫ t
0
(−re−rs)( Vs︸︷︷︸=H0
s ers+HsXs
−Zs )ds
+Nt∑j=1
e−rτjHτjUjXτ−j+
∫ t
0
(−re−rs)Zsds
= V0 +
∫ t
0
HsXs(σdWs + µds) +
∫ t
0
H0s rds−
∫ t
0
r(H0s +HsXs)ds
+Nt∑j=1
e−rτjHτjUjXτ−j
= V0 +
∫ t
0
HsXs(σdWs + (µ− r)ds) +Nt∑j=1
e−rτjHτjUjXτ−j,
and so by (121)
Vt = V0 +
∫ t
0
HsXsσdWs +Nt∑j=1
HτjUjXτ−j− λE[U1]
∫ t
0
HsXsds. (125)
Thus if (H0t , Ht) is admissible and V0 and Ht are given then Vt must have the form (125).
It follows
H0t = Vt −HtXt
=(125)−HtXt + V0 +
∫ t
0
HsXsσdWs +Nt∑j=1
HτjUjXτ−j− λE[U1]
∫ t
0
HsXsds.
From this formula one can see that (H0t ) is adapted, has left-hand limit at any t and
H0t = H0
t− (it holds P(Nt = Nt−) = 1 and P(Xt = Xt−) = 1 for any t > 0). Indeed, ift is not a jump time τj, H
0t = H0
t− is clear, and if t = τj then (since Nt = n = t ∈[τn, τn+1), Nτj = j, j ≥ 1, and so Nτ−j
= j − 1)
H0τj−H0
τ−j= −Hτj∆Xτj +HτjUjXτ−j
= 0.
One can also see∫ T
0|H0
s |2ds <∞ a.s. (cf. Remark 7.13 below). Moreover writing H0t e
rt +
HtXt = ert(H0t + HtXt) and integrating by parts as before one can see that (H0
t , Ht) isadmissible with initial value V0.
152
Remark 7.13 The condition E[∫ T
0H2s X
2sds]<∞ implies both that the stochastic inte-
gral(∫ t
0HsXsσdWs
)t∈[0,T ]
and
Mt : =Nt∑j=1
HτjUjXτ−j− λE[U1]
∫ t
0
HsXsds, t ∈ [0, T ],
are square-integrable martingales. Indeed, since (Xt) has only finitely many jumps in [0, T ],setting φ ((y1, y2), z) = y1 · y2 · z, we have
Mt =Nt∑j=1
φ(
(Hτj , Xτ−j), Uj
)− λ
∫ t
0
∫RHsXs−zν(dz)ds
=Nt∑j=1
φ(
(Hτj , Xτ−j), Uj
)− λ
∫ t
0
∫φ(
(Hs, Xs−)︸ ︷︷ ︸
=:Ys left-cont. adapted
, z)ν(dz)ds
with
E[∫ T
0
∫φ2(Ys, z)ν(dz)ds
]= E
[∫ T
0
H2s X
2s−ds
] ∫z2ν(dz)
= E[∫ T
0
H2s X
2sds
]E[U2
1 ] <∞.
Thus (Mt)t∈[0,T ] is a square-integrable martingale by Lemma 7.9. Consequently, (Vt)t∈[0,T ]
is also a square integrable martingale.
7.4.2 Pricing
Consider an European option with maturity T defined by a square integrable FT -measurablerandom variable h, i.e. h ∈ L2(Ω,FT , P ).Consider the following scenario:
- a writer sells the option for the price V0 at time 0
- he follows an admissible strategy up to time T
Note that the strategy is completely determined by (Ht)t∈[0,T ]. Now, at time T : ifh− VT 6 0 writer loses money
h− VT > 0 writer earns money
153
Introduce
RT0 : = E
[(e−rT (h− VT ))2
]which measures the risk at maturity for both, the writer and the owner of the option. Usingthe formula E[Z2] = E[Z]2 + var(Z) and that by the martingale property E[VT ] = V0, weget
RT0 = E
[e−rTh− V0
]2+ E
[(e−rT (h− VT )− (E[e−rTh]− V0)
)2]
=(E[e−rTh]− V0
)2
+ E[(e−rTh− E[e−rTh]− ( VT − V0︸ ︷︷ ︸
only dep. on (Ht)by Prop. 7.12
))2]
(126)
Now in order to minimize RT0 , one has to ask for the premium
V0 = E[e−rTh]
which represents the initial value of any strategy that minimizes the risk at maturity.Thus we define in a natural way E[e−rTh] to be the price of the option h at time zero. Bya similar argument the price at time t ∈ (0, T ) in order to minimize pointwise (for P -a.e.ω ∈ Ω)
RTt := E
[(e−r(T−t)(h− VT )
)2
|Ft]
isVt := E
[e−r(T−t)h|Ft
]. (127)
Indeed, in this case, we may use the formula
E[X2|Ft] = E[(X − E[X|Ft])2|Ft] + E[X|Ft]2,
holding for X ∈ L2(Ω,FT , P ). We hence define the price of the option at time t ∈ (0, T )by E
[e−r(T−t)h|Ft
].
154
7.4.3 Prices of calls and puts
We want to derive an explicit formula for the price of an European call or put with strikeprice K, i.e. we assume h = f(XT ) with f(x) = (x−K)+ or f(x) = (K − x)+. We have
Vt =(127)
E[e−r(T−t)h|Ft
]= E
[e−r(T−t)f
(X0
(NT∏j=1
(1 + Uj)
)eσWT−(µ−σ
2
2)T
)|Ft
]
= E
[e−r(T−t)f
(Xt
(NT∏
j=Nt+1
(1 + Uj)
)eσ(WT−Wt)−(µ−σ
2
2)(T−t)
)|Ft
]
= E
[e−r(T−t)f
(Xt
(NT−Nt∏j=1
(1 + UNt+j)
)eσ(WT−Wt)+(µ−σ
2
2)(T−t)
︸ ︷︷ ︸independent of Ft by Lemma 7.12
)|Ft
].
Therefore by Proposition 7.6 of [1], we have
E[e−r(T−t)f(XT )|Ft] = F (t,Xt)
(=
(127)Vt
)(128)
where
F (t, x) = E
[e−r(T−t)f
(x
NT−Nt∏j=1
(1 + UNt+j)eσ(WT−Wt)+(µ−σ
2
2)(T−t)
)]
= E
[e−r(T−t)f
(x
NT−t∏j=1
(1 + Uj)eσWT−t+(r−λE[U1]−σ
2
2)(T−t)
)]
since NT − Nt ∼ NT−t, WT −Wt ∼ WT−t and UNt+j ∼ Uj see proof of Lemma 7.8 andµ = r − λE[U1] by (121). Let
F0(t, x) := E[e−r(T−t)f(xeσWT−t+(r−σ
2
2)(T−t))
],
which determines the price of the option in the Black-Scholes model. Then
F (t, x) = E
[F0
(t, x
(NT−t∏j=1
(1 + Uj)
)e−λE[U1](T−t)
)]. (129)
155
Since NT−t has Poisson distribution with parameter λ(T − t) and is independent of Uj,we obtain
F (t, x) =∞∑n=0
E
[1NT−t=nF0
(t, x
n∏j=1
(1 + Uj)e−λE[U1](T−t)
)]
=∞∑n=0
E[1NT−t=n
]E
[F0
(t, x
n∏j=1
(1 + Uj)e−λE[U1](T−t)
)]
=∞∑n=0
e−λ(T−t)λn(T − t)n
n!E
[F0
(t, x
n∏j=1
(1 + Uj)e−λE[U1](T−t)
)].
In general each term of this series is computed numerically by simulating the laws of theUj’s. But for some laws the series can be calculated explicitly.
7.4.4 Hedging calls and puts
Consider the hedging problem for h = f(XT ) with f(x) = (x−K)+ or (K − x)+.We have seen in Proposition 7.12 and Remark 7.13: any strategy that minimizes the riskRT
0 at maturity has initial value V0 = E[e−rTh] = F (0, X0). For any such strategy
RT0 = E[(e−rTh− VT )2].
Now, we want to describe the process (Ht)t∈[0,T ], where φ = ((H0t , Ht)) minimizes the risk
RT0 at maturity.
Proposition 7.14 Let ((H0t , Ht))t∈[0,T ] be an admissible strategy with value Vt at time
t. Assume V0 = E[e−rTf(XT )] = F (0, X0) (where F is as in Section 7.4.3). Then thequadratic risk at maturity RT
0 satisfies
RT0 = E[(e−rT (f(XT )− VT ))2]
= E[∫ T
0
(∂xF (s,Xs)−Hs)2X2
sσ2ds
]+ E
[∫ T
0
λ
∫ ∞−1
e−2rs(F (s, (1 + z)Xs)− F (s,Xs)−HszXs
)2
ν(dz)ds
].
Proof By Proposition 7.12 for t ∈ [0, T ] we have
Vt = F (0, X0) +
∫ t
0
HsXsσdWs +Nt∑j=1
HτjUjXτ−j− λE[U1]
∫ t
0
HsXsds. (130)
156
By (128), we have h := e−rTf(XT ) = e−rTF (T,XT ). Define
F (t, x) := e−rtF (t, xert).
ThenF (t, Xt) = e−rtF (t,Xt) =
(128)E[e−rTf(XT )︸ ︷︷ ︸
=h
|Ft]
is the discounted price of the option at time t. From (129) one deduces (exercise) thatF ∈ C2([0, T ) × R+). Hence using the time-dependent Ito’s formula between the jumptimes at t ∈ [τj, τj+1), j ≥ 1, and adding the jumps at times t = τj, we get
F (t, Xt) = F (0, X0) +
∫ t
0
∂xF (s, Xs)dXs +1
2
∫ t
0
∂xxF (s, Xs)d〈X〉s
+
∫ t
0
∂tF (s, Xs)ds+Nt∑j=1
(F (τj, Xτj)− F (τj, Xτ−j
)).
Since Xt = X0 +∫ t
0Xs(σdWs − λE[U1]ds) by (115), (121) and Ito’s product rule, we get
F (t, Xt) = F (0, X0) +
∫ t
0
∂xF (s, Xs)Xs(σdWs − λE[U1]ds) +1
2
∫ t
0
∂xxF (s, Xs)X2sσ
2ds
+
∫ t
0
∂tF (s, Xs)ds+Nt∑j=1
(F (τj, Xτj)− F (τj, Xτ−j
)). (131)
Let θ := T − t. Since f is Lipschitz, we have
|F (t, x)− F (t, y)| 6 E
[e−rθ
∣∣∣f(xeσWθ+(r−λE[U1]−σ2
2)θ)
Nθ∏j=1
(1 + Uj)
− f(yeσWθ+(r−λE[U1]−σ2
2)θ)
Nθ∏j=1
(1 + Uj)∣∣∣]
6 |x− y|E
[eσWθ−σ
2
2θe−λE[U1]θ
Nθ∏j=1
(1 + Uj)
]
=independence
|x− y|E[eσWθ−σ2
2θ]︸ ︷︷ ︸
=1
E
[e−λE[U1]θ
Nθ∏j=1
(1 + Uj)
]︸ ︷︷ ︸
=1 (by Ex. 4 Ass. no. 8)
.
157
It follows that
E[∫ T
0
∫ ∣∣∣F (s, (1 + z)Xs)− F (s, Xs)∣∣∣2ν(dz)ds
]6 E
[∫ T
0
X2sds
∫z2ν(dz)
]<∞.
Hence by Lemma 7.9 the process
Mt :=Nt∑j=1
(F (τj, Xτj)− F (τj, Xτ−j
))− λ
∫ t
0
∫ (F (s, (1 + z)Xs)− F (s, Xs)
)ν(dz)ds
is a square-integrable martingale. We know that (F (t, Xt))t∈[0,T ] is a martingale, hence
(F (t, Xt)−Mt)t∈[0,T ] is also a martingale and from (131) it is an Ito process.
By Exercise 2(iv) of Assignment no. 9 of Financial Mathematics 1, the Ito process F (t, Xt)−Mt can be uniquely written as stochastic integral and so by (131) and uniqueness
F (t, Xt)−Mt = F (0, X0) +
∫ t
0
∂xF (s, Xs)XsσdWs. (132)
By (130) and (132)
Vt = F (t, Xt)−Mt −∫ t
0
∂xF (s, Xs)XsσdWs +
∫ t
0
HsXsσdWs
+Nt∑j=1
HτjUjXτ−j− λE[U1]
∫ t
0
HsXsds
and so
F (t, Xt)− Vt =
∫ t
0
(∂xF (s, Xs)−Hs
)σXsdWs︸ ︷︷ ︸
=:Mt
+Nt∑j=1
(F (τj, Xτj)− F (τj, Xτ−j
)−HτjUjXτ−j
)− λ
∫ t
0
∫ (F (s, (1 + z)Xs)− F (s, Xs)−HszXs
)ν(dz)ds︸ ︷︷ ︸
=:Mt
By Lemma 7.10 (MtMt)t∈[0,T ] is a martingale and so E[MTMT ] = E[M0M0] = 0. It follows
RT0 = E
[(F (T, XT )− VT
)2]
= E[M2T ] + E[M2
T ]
= E[∫ T
0
(∂xF (s, Xs)−Hs)2σ2X2
sds
]+ E[M2
T ].
158
Applying Lemma 7.9
E[M2T ] = E
[λ
∫ T
0
∫ (F (s, (1 + z)Xs)− F (s, Xs)−HszXs
)2
ν(dz)ds
].
Thus RT0 has the desired form.
It follows from Proposition 7.14 that the minimal risk is obtained when Hs satisfies(
∂xF (s, Xs)−Hs
)X2sσ
2 + λ
∫ (F (s, (1 + z)Xs)− F (s, Xs)−HszXs
)zXsν(dz) = 0
P-a.s. Indeed, it suffices to minimize the integrand w.r.t. ds. This yields
Hs = ∆(s,Xs−)
where
∆(s, x) =1
σ2 + λ∫z2ν(dz)
(σ2∂xF (s, x) + λ
∫F (s, (1 + z)x)− F (s, x)
xzν(dz)
)By this (Hs) also satisfies
E[∫ T
0
H2s X
2sds
]<∞
and is therefore an admissible strategy that minimizes the risk at maturity. If there is nojump (i.e. λ = 0), we recover the hedging formula of the Black-Scholes model which isperfect, i.e. RT
0 = 0. If λ > 0 the minimal risk in general satisfies RT0 > 0.
Remark 7.15 In practice the parameters of the model as well as the law of the Ui’sneed to be determined. There are two approaches: (1) statistical approach from historicaldata (2) implied approach from prices of options quoted on an organized market. For theapproach (2) jump models give a better fit to market prices than the Black-Scholes model.
159
8 Credit risk models
credit risk ≈ ”risk of default of a counterparty”
8.1 Structural models
In structural models (also called firm value models) a default event is modeled by relatingit to the value of the firm.
Merton’s model (1974): (Pioneering model)
Vt = value of the firm at time t follows a geometric BM, i.e.
dVt = Vt(σWt + (r − k) dt), V0 > 0, (133)
where (Wt) is a standard BM under the risk neutral probability measure P , with constants
r = instantaneous interest rate
k = expenditure rate
0 < σ = volatility.
Definition 8.1 A zero-coupon bond with face value L is a security paying L (e.g. dollar)at maturity time T . If the instantaneous interest rate (r(s))s∈[0,T ] is deterministic, its priceat time t ∈ [0, T ] is
P (t, T ) = L · e−∫ Tt r(s)ds,
(“lend (pay) L · e−∫ Tt r(s)ds at time t and receive L at time T”). If (r(s))s∈[0,T ] is random,
then we have to take conditional expectations, see (136) below.
The debt Dt of the company (=firm) at time t is supposed to be a zero-coupon bond withmaturity T and face value L > 0, i.e. at time T the firm has to pay back the amount ofL.
The company defaults, if: VT ≤ L = DT at time T .
In case of default: the debtors (= the zero-coupon bond holders) take control of thecompany. In general the bond holders payoff at time T is
VT ∧ L =
VT if company defaults
L if company does not default and hence pays back the debt L at time T.
160
The value at time t ∈ [0, T ] of this defaultable coupon which represents the debt valueof the firm is given by
Dt = E[e−r(T−t)VT ∧ L | Ft].
Here (Ft)t∈[0,T ] is the natural filtration of (Vt)t∈[0,T ] and hence also of (Wt)t∈[0,T ]. SinceVT ∧ L = L− (L− VT )+ we have
Dt = Le−r(T−t) − E[e−r(T−t)(L− VT )+ | Ft].
For the computation of Dt it is hence enough to compute the value of a put on the firmvalue Vt with strike price L. Let
ET := VT − VT ∧ L = (VT − L)+ = equity value of the firm at time T.
The price of ET at time t is hence given by E[e−r(T−t)(VT −L)+ | Ft], i.e. the ”shareholdersof the firm are holder of a call option on Vt with strike price L”.
In Merton’s model: default may only occur at the terminal time T .
In first passage models: default occurs when the firm value hits a time-dependentbarrier (deterministic curve) H(t) for the first time. Then
”default time” = τ := inft ≥ 0 | Vt ≤ H(t)
E.g. Black and Cox model (1976):
H(t) =
Le−γ(T−t) for t ∈ [0, T ) with L < L and γ ≥ 0 const.
L for t = T.
8.2 Intensity-based models
8.2.1 The hazard rate of a random time
Proposition 8.2 Let τ be a r.v. with P (τ > 0) = 1 and P (τ > t) > 0 ∀t > 0. Let the(cumulative) distribution function F (t) := P (τ ≤ t) of τ be continuously differentiable.Then ∃λ : [0,∞) −→ [0,∞), λ continuous with
P (τ > t) = e−∫ t0 λ(s) ds, t > 0.
λ is called the hazard rate of τ .
161
Proof We have P (τ > t) = 1 − F (t) > 0 for all t ≥ 0, and 1 − F (t) decreases. Byassumption λ must satisfy
λ(t) =(− ln
(1− F (t)
))′=
F′(t)
1− F (t), ∀t ≥ 0. (?)
Defining λ by (?) we obtain that λ is continuous, positive and
P (τ > t) =F (0)=0
eln(1−F (t))−ln(1−F (0)) = e−∫ t0 λ(s) ds.
Note: P (τ =∞) = limt→∞ P (τ > t) = limt→∞ e−∫ t0 λ(s) ds. Hence∫ ∞
0
λ(s) ds = +∞⇒ P (τ <∞) = 1
and for δ > 0, since P (τ = t) = 0,
P (τ < t+ δ | τ > t) =P (τ ∈ (t, t+ δ])
P (τ > t)=P (τ > t)− P (τ > t+ δ)
P (τ > t)= 1− e−
∫ t+δt λ(s) ds
Hence
λ(t) = limδ→0−e−∫ t+δt λ(s) ds − 1
δ= lim
δ→0
P (τ < t+ δ | τ > t)
δ.
Thus if τ =”default time”, then hazard rate λ(t) =”rate of occurence of default justafter time t, given the default has not occured before time t”
Remark 8.3 τ has exponential distribution with parameter α > 0 ⇔ λ ≡ α > 0. On theother hand, if ξ is exponentially distributed with parameter 1, i.e. P (ξ ≤ t) = 1− e−t, andλ : [0,∞) −→ [0,∞) is continuous, then
τ := inft ≥ 0 |
∫ t
0
λ(s) ds ≥ ξ
(134)
satisfies
P (τ ≤ t) = P
(∫ t
0
λ(s) ds ≥ ξ
)= 1− e−
∫ t0 λ(s) ds,
and so the hazard rate of τ is λ. (134) provides a way of constructing a random time withgiven hazard rate.
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8.2.2 Intensity and defaultable zero-coupon bonds
General framework of intensity models :
(Ω,A, P ) Probability space
(Ft)t≥0 filtration on (Ω,A, P ), called information relative to the default freemarket
S0t = e
∫ t0 r(s) ds riskless asset with rate r(s)
S0t , r(t) are (Ft)-adapted
τ default time (in contrast to the structural approach not necessarily a stoppingtime w.r.t. (Ft)t≥0, but w.r.t. (Gt), see below).
τ is an exogenous (= ”from outside”) r.v. and default may occur as a surprise. At timet, the investors know whether default has occured or not. Thus the total information attime t is given by
Gt = Ft ∨ σ(τ ≤ s, s ≤ t).
Here C ∨B = smallest σ-algebra containing C and B.
Note: τ is a stopping time w.r.t. (Gt)t≥0.
A pricing measure P ∗ is a probability measure equivalent to P and under which thediscounted prices of the risky assets are (Gt)t≥0-martingales. Risky assets are assumed tobe (Gt)t≥0-adapted, since they may be affected by default.
A defaultable zero-coupon bond with maturity T is given (compare this with (79))by
D(t, T ) = E∗[1τ>Te
−∫ Tt r(s) ds | Gt
], t ∈ [0, T ].
Thus the payoff at maturity T is
D(T, T ) = 1τ>T =
1 if default has not occured,
0 if default has occured.
The following Proposition relates the computation of conditional expectations given Gt toconditional expectations given Ft.
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Proposition 8.4 Let X ≥ 0 be a r.v. It holds P ∗-a.s.
1τ>t E∗[X | Gt] = 1τ>tE∗[X1τ>t | Ft]E∗[1τ>t | Ft]
.
Proof Since E∗[1A | Ft] > 0 P ∗-a.s. on A, i.e P ∗(E∗[1A | Ft] > 0 ∩A) = P ∗(A) for anyA ∈ A (cf. Exercises), the r.v.
Y := 1τ>tE∗[X1τ>t | Ft]E∗[1τ>t | Ft]
∈ Gt
is well-defined as a product of Gt-mb r.v’s. We have
1τ>t︸ ︷︷ ︸∈Gt
E∗[X | Gt] = E∗[X1τ>t | Gt].
In order to prove Y = E∗[X1τ>t | Gt] it is enough to show that
E∗[X1τ>t1A] = E∗[Y 1A], ∀A ∈ C, (?)
where C ⊂ Gt is⋂
-stable and σ(C) = Gt (cf. Assignment no. 8, Dynkin systems !). Sucha class is given by
C = A =τ ≤ s ∩B, B ∈ Ft, s ∈ [0, t] ∪ +∞
.
For s ∈ [0, t], we have E∗[X1τ>t1τ≤s∩B] = 0 = E∗[Y 1τ≤s∩B]. For s = ∞, we haveA = B, and since B ∈ Ft
E∗[X1τ>t1B] = E∗[E∗[X1τ>t | Ft]1B
], (??)
hence
E∗[Y 1B] = E∗[1τ>t
E∗[X1τ>t | Ft]E∗[1τ>t | Ft]
1B︸ ︷︷ ︸∈Ft
]
= E∗[E∗[1τ>t | Ft]
E∗[X1τ>t | Ft]E∗[1τ>t | Ft]
1B
]=
(??)E∗[X1τ>t1B]
and so (?) holds.
Suppose the conditional hazard rate of τ given the default free filtration (Ft) is known,i.e. ∃ (λ(t))t≥0 (Ft)-adapted, with
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(a) P ∗(τ > t | Ft) = E∗[1τ>t | Ft] = e−∫ t0 λ(s) ds, ∀t ≥ 0.
Then, we have:
Proposition 8.5 The value at time t ∈ [0, T ] before default of a defaultable zero-couponbond with maturity T is
D(t, T ) := 1τ>tD(t, T ) = 1τ>t E∗[e−
∫ Tt (r(s)+λ(s)) ds | Ft
], (135)
where (λ(t))t∈[0,T ] is the conditional hazard rate given (Ft).
Proof Using Proposition 8.4 with X = 1τ>Te−∫ Tt r(s) ds, we obtain
1τ>tD(t, T )︸ ︷︷ ︸”value of defaultablebond before default”
= 1τ>t E∗[X | Gt]
= 1τ>tE∗[1τ>Te−
∫ Tt r(s) ds1τ>t | Ft]
E∗[1τ>t | Ft]
=(a)
1τ>tE∗[1τ>Te−
∫ Tt r(s) ds | Ft]
e−∫ t0 λ(s) ds
= 1τ>tE∗[E∗[1τ>T
FT−mb︷ ︸︸ ︷e−
∫ Tt r(s) ds | FT ] | Ft
]e−
∫ t0 λ(s) ds
=(a)
1τ>tE∗[e−
∫ T0 λ(s) dse−
∫ Tt r(s) ds | Ft]
e−∫ t0 λ(s) ds
= 1τ>t E∗[e−∫ Tt (λ(s)+r(s)) ds | Ft].
(λ(t))t∈[0,T ] is also called the intensity of the default. A default free zero-coupon bondhas value
P (t, T ) = E∗[e−
∫ Tt r(s) ds | Ft
]. (136)
The intensity appears thus as additional term to the interest rate in (135). Therefore λ(t)is also called the credit spread at time t. If the intensity λ(t) is deterministic then
D(t, T ) = 1τ>te−∫ Tt λ(s) dsP (t, T ).
Remark 8.6 The intensity is deterministic, iff τ > t is independent of Ft for all t, e.g.if τ is independent of Ft for all t. Note also that given a non-negative (Ft)-adapted process(λ(t)) one can construct a random time with intensity (λ(t)) by (134) with ξ independentof (Ft).
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8.2.3 Credit risk swaps
A credit default swap (CDS) is a credit derivative that offers protection against default.
Investor Afixed payments at times (Ti)i=1,...,n , if τ>Ti (”premium leg”)
−−−−−−−−−−−−−−−−−−−−−− > B(wants to be A Bprotected against Ndefault) < −−−−−−−−−−−−−−−−−−−−−−
payment if τ≤Tn (”protection leg”)K
Premium leg: cashflow from A to B at times Ti, i = 1, . . . n, T0 := 0.
sN(Ti − Ti−1)1τ>Ti,
s = spread of the swap (has to be determined, see below),N = nominal of the swap (usually the face value of the bond).
Protection leg: cashflow from B to A at time τ .
N(1−R)1τ≤Tn, R = recovery rate.
Idea: A holds a bond with nominal N (= face value) issued by a company that maydefault. In case of default A recovers N ·R instead of N , thus together with the protectionleg he recovers N ·R +N(1−R) = N . A typical recovery rate is R = 40%.
Remark 8.7 In the premium leg there is usually an additional payment that we omit, theso-called accrual premium
n∑i=1
s(τ − Ti−1)1τ∈[Ti−1,Ti)
if default occurs in between dates Ti−1 and Ti.
Evaluation of premium leg at time 0: expectation of the discounted cashflows underthe pricing measure
P ∗A→B = E∗[
n∑i=1
sN(Ti − Ti−1)1τ>Tie−∫ Ti0 r(s) ds
].
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Evaluation of protection leg at time 0: same
P ∗B→A = E∗[N(1−R)1τ≤Tne
−∫ τ0 r(s) ds
].
The fair value for the spread s is such that P ∗A→B = P ∗B→A, i.e.
s =E∗[(1−R)1τ≤Tne
−∫ τ0 r(s) ds
]E∗[∑n
i=1(Ti − Ti−1)1τ>Tie−∫ Ti0 r(s) ds
] . (137)
It holds
E∗[1τ>Tie
−∫ Ti0 r(s) ds
]= E∗
[E∗[1τ>Ti e
−∫ Ti0 r(s) ds︸ ︷︷ ︸∈FTi
| FTi]]
= E∗[e−
∫ Ti0 λ(s) dse−
∫ Ti0 r(s) ds
].
Also, one can show (exercise)
E∗[1τ≤Tne
−∫ τ0 r(s) ds
]= E∗
[∫ Tn
0
λ(u)e−∫ u0 (λ(t)+r(t)) dt du
],
and so
s =(1−R)
∫ Tn0
E∗[λ(u)e−
∫ u0 (λ(t)+r(t)) dt
]du∑n
i=1(Ti − Ti−1)E∗[e−
∫ Ti0 (λ(s)+r(s)) ds
] .
If the intensity is deterministic, then (in terms of default free zero-coupon bond prices)
E∗[e−
∫ Ti0 (λ(s)+r(s)) ds
]= e−
∫ Ti0 λ(s) dsP (0, Ti) = P ∗(τ > Ti)P (0, Ti)
andE∗[λ(u)e−
∫ u0 (λ(t)+r(t)) dt
]= λ(u)e−
∫ u0 λ(t) dtP (0, u).
Hence ∫ Tn
0
E∗[λ(u)e−
∫ u0 (λ(t)+r(t)) dt
]du =
∫ Tn
0
λ(u)e−∫ u0 λ(s) dsP (0, u)du
and (137) rewrites as
s =(1−R)
∫ Tn0λ(u)e−
∫ u0 λ(s) dsP (0, u)du∑n
i=1(Ti − Ti−1)P ∗(τ > Ti)P (0, Ti). (138)
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Remark 8.8 In practice, the spreads of CDS with various maturities are used to deriveimplied default probabilities. For this let
sj = ”spread of a swap with payment dates T1, . . . , Tj”
and use the following discretization of (138)
sj =(1−R)
∑ji=1
(P ∗(τ > Ti−1)− P ∗(τ > Ti)
)· P (0, Ti)∑j
i=1(Ti − Ti−1)P ∗(τ > Ti)P (0, Ti). (139)
Indeed for i = 1, . . . , n, it holds
P ∗(τ > Ti−1)− P ∗(τ > Ti) =
∫ Ti
Ti−1
λ(u)e−∫ u0 λ(s) ds du.
From the values sj, j = 1, . . . , n, one can then compute P ∗(τ > Ti), i = 1, . . . , n, anddefine a piecewise constant hazard rate. In fact (139) can also be seen as spread assumingthat in case of default, the protection payment is made not at τ , but at the payment dateright after τ .
References
[1] Lecture Financial Mathematics 1, Probabilistic Background, 2015.
[2] Karatzas, I.; Shreve, S. E. Brownian motion and stochastic calculus. Graduate Textsin Mathematics, 113. Springer-Verlag, New York, (1991).
[3] Lamberton, D.; Lapeyre, B.: Introduction to stochastic calculus applied to finance.Second edition. Chapman & Hall/CRC Financial Mathematics Series, Boca Raton,FL, 2008.
[4] D. Revuz, M. Yor, Continuous martingales and Brownian motion, Springer Verlag,(2005).
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