option pricing and hedging in jump diffusion models
TRANSCRIPT
U.U.D.M. Project Report 2010:7
Examensarbete i matematik, 30 hpHandledare och examinator: Johan Tysk
Maj 2010
Department of MathematicsUppsala University
Option pricing and hedging in jump diffusion models
Yu Zhou
Master thesis of Mathematics
Specialization in Financial Mathematics
Option Pricing and hedging in
Jump-diffusion Models
1
Yu Zhou
Advising professor: Johan Tysk
Department of Mathematics, Uppsala University
May, 2010
2
Abstract
The aim of this article is to solve European Option pricing and hedging in a
jump-diffusion framework. To better describe the reality, some major events, for example,
may lead to dramatic change in stock price; we impose a jump model to classic Black
Scholes model.
Under the assumption that the underlying asset is driven by a simple
jump-diffusion process, along the fact that risk free rate and volatility are
deterministic functions of time, by changing the probability measure to risk-neutral
measure Q, we obtain the pricing formula of European options. The hedging strategy
is defined in order to minimize the risk of hedging under a risk neutral measure.
Key Words: Black-Scholes model Jump-diffusion model option pricing
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4
Contents
1. Introduction
1.1 Jump-Diffusion Model………………………………………………………… 3
1.2 Incomplete Market ..........................................................................…...3
1.3 Mathematical Tools……………………………………………………………..4
1.4 About this article……………………………………………………………….4
2. Option Pricing and Hedging in the Black-Scholes Model
2.1 Introduction to Option Pricing…………………………………...…………….5
2.2 Option Pricing in Black-Scholes Model………………………………………….6
2.3 Pricing………………………………………………………………………..7
2.4 Hedging…………………………………………………………...………….8
3. Option Pricing and Hedging in the Jump-Diffusion Model
3.1Dynamics of the Underlying Asset……………………………………………… 9
5
3.2 Conditions for S�t to be a Martingale………………………………………….10
3.3 European Options Pricing…………………………………………………….12
3.3.1 Admissible Portfolios…………………………………………………...12
3.3.2 European Options Pricing………………………………………………14
3.4 Hedging European Options…………………………………………………...16
4. Reference……………………………………………………………………….20
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Chapter 1 Introduction
1.1 Jump-Diffusion models
In the Black-Scholes Model, the stock price is driven by Brownian motion and is
based on a continuous function of time. However, in reality this is not often the case,
since certain important events can lead to dramatic change in the stock price.
To model such phenomenon, many studies introduced discontinuous stochastic
7
processes by adding jump-diffusion to the classic model. To price options, we must
notice that a jump diffusion model leads to an incomplete market. Therefore, the
classic hedging methods are not applicable here.
There are rather extensive studies in the jump-diffusion model. As early as 1976,
Robert Merton noted that when major events happen, stock price change
discontinuously, or, jump. He also noted that the stock price is driven by Brownian
motion (1976) as well as Poisson process. Having taken that into consideration,
Merton created the Jump-diffusion model. [4]
1.2 Incomplete market
In an arbitrage-free market, there is an equivalent probability measure such that
the underlying asset is a martingale under risk neutral measure, which implies that
the asset is a semi-martingale under the objective measure P. Mathematically, a
complete market means that any contingent claim can be replicated as a stochastic
integral of a sequence of semi-martingales. The integrant in such replications
provides a sequence of hedging strategies which are self-financing or, super hedging.
Whilst in Jump-diffusion models, more random variables are added to the
market, making it Incomplete. In this case, a general claim is not necessarily a
stochastic integral of the underlying asset. Economically speaking, this claim has an
8
intrinsic risk that we only hope to reduce to this minimal component. Thus the
problem is to find and characterize these strategies which minimize the risk.
1.3 Mathematical tools
The main mathematical tools to apply to this field are (1) Stochastic Differential
Equations, (2) Monte Carlo (simulation) and (3) Martingale methods. As the first
two tools are rather often used in previous papers in our department, this article
would try to derive the same results with Martingale methods.
In the more general martingale approach, one specifies a stochastic process for
the underlying asset. Then one can choose an equivalent probability measure turning
the discounted underlying into a (possibly local) martingale and computes the
derivative's value as the conditional expectation of its discounted payoff under this
risk-neutral measure. If the model has a Markovian structure, then this value turns
out to be some function u, say, of the state variables. In the PDE approach, one can
describe the state variables by a stochastic differential equation (SDE) and then
derives for the function u based on the underlying martingale valuation a PDE
involving the coefficients of the given SDE. [5]
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1.4 About this article
Our study will be divided into three parts.
(1) The first chapter is the introduction, in which we present the background of
the jump-diffusion models and hedging in incomplete market.
(2) In the second chapter we begin with the basic theory of option pricing,
Black-Scholes framework and hedging, where r is constant.
(3) The third chapter we assume that the jump follows a Poisson distribution,
and that the interest rate is time deterministic. We firstly find the sufficient and
necessary conditions for the underlying asset to be a martingale. Under such
conditions, by changing the measure, we can obtain the pricing formula under
risk-neutral probability measure Q, and then be able to derive the hedging strategy
by minimizing the risk.
Overall, this article presents an alternative way to solve option pricing and
hedging problem, in the Jump-diffusions model and incomplete market.
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Chapter 2 Option Pricing and Hedging in
the Black-Scholes Model
2.1 Introduction to Option Pricing
2.1.1 Options
An option is a contract between a buyer and a seller that gives the buyer the
right, (but not the obligation), to buy or to sell a particular asset on or before the
options expiration time, at an agreed price (the strike price). In return for granting
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the option, the seller collects a payment (the premium) from the buyer.
A call option gives the buyer the right to buy the underlying asset and a put
option gives the buyer of the option the right to sell the underlying asset. If the buyer
chooses to exercise this right, the seller is obliged to sell or buy the asset at the agreed
price.
The buyer of a call option wants the price of the underlying instrument to rise in
the future; the seller either expects that it will not, or is willing to give up some of the
upside (profit) from a price rise in return for the premium and retaining the
opportunity to make a gain up to the strike price.
In case of a put option, buyer acquires a short position by purchasing the right
to sell the underlying asset to the seller of the option for specified price during a
specified period of time. If the option buyer exercises their right, the seller is
obligated to buy the underlying instrument from them at the agreed upon strike
price, regardless of the current market price. [Wiki]
2.1.2 Option Pricing
In exchange for having an option, the buyer pays the seller or option writer a fee,
known as the option premium. The option premium is the maximum loss in an option
exchange, of the party taking long position of the underlying asset, which depend on
12
the price change of the underlying asset.
Since the underlying asset is usually a risky asset (e.g. a stock), the price of
underlying asset is usually a stochastic process. But when the underlying asset price
is settled down, the options written on that is also settled down. Mathematically, for
option price Ft, there exists a binomial function F(t, S) such that the price of the
option Ft = F(t, S). At the examining date, Ft is deterministic.
FT = �(ST − K)+ (call)(K − ST)+ (put)
�
Option pricing problem is about to derive the above Ft, hence it is an inverse
problem.
2.2 Black Scholes Model option pricing and hedging
2.2.1 The Model
Black Scholes is a model based on continuous time. There are two different types
of assets on the market:
The risk free asset, which has price St0 at time t, and satisfies the equation:
dSt0 = rSt
0dt
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where r ≥ 0 is the risk free rate; thereforeSt0 = S0
0ert , from now on we assume
S00 = 1 for convenience. Then,
St0 = ert , ∀t ≥ 0.
And the risky asset, which has price St at time t, and satisfies the equation:
dSt = μStdt + σStdBt
where μ and σ > 0 are constant, and that Bt is a standard Brownian motion.
Now define the present value of St as S�t, i.e.,
S�t = e−rt St,
By Ito formula,
dS�t = −re−rt Stdt + e−rt dSt = S�t�(μ− r)dt + σdBt� = S�tσdWt
where Wt = Bt + θtσ, θt = (μ−t)t
σ.
We define LT = exp �−∫ θsds − 12∫ θs
2dsT0
T0 �, dQ = LTdP
Then Q and P are equivalent, then we know from Girsanov Theorem that Wt is
a Standard Brownian Motion under Q.
By Ito formula, S�t = S�0exp(σWt −σ2
2t) , thus S�t is a martingale under Q.
Therefore Q is so called the risk neutral measure or equivalent martingale measure.
2.2 Self financing Portfolio
14
Consider a portfolio ϕ = (ϕt)0≤t≤T = (ht0, ht), where ht
0 and ht represent the
number of shares of risk-free asset and risky asset, respectively. Thus this portfolio is,
at time t, worth
Vt(ϕ) = ht0S1
0 + htSt
Furthermore, its discounted value is
V�t(ϕ) = e−rt Vt(ϕ) = ht0 + htS�t.
where S�t = e−rt St is the underlying asset price under measure Q that we are going to
discuss later.
Definition 2.2.1 An ℱtS adapted portfolio ϕ = (ϕt)0≤t≤T = (ht
0, ht) is called
self-financing if the value process Vt satisfies the conditions
dVt(ϕ) = ht0dS1
0 + htdSt
� |ht0|dt
T
0+ � ht
2dtT
0< ∞ (2.1)
where the latter condition makes the portfolio admissible, which is known as the
integrable condition.
Definition 2.2.2 Portfolio ϕ = (ϕt)0≤t≤T = (ht0, ht) is called attainable if it is
self-financing, as well as its discounted value V�t(ϕ) is non-negative and
square-integrable.
2.3 European options pricing
15
To price a European option, we define its contingent claim h = f(ST), where
f(S) = (S − K)+ for a call option and f(S) = (K − S)+ for a put option.
A portfolio is called can be replicated if for its contingent claim h there exists a
strategy ϕ such that the value of the portfolio strategy is equal to h, i.e. Vt(ϕ) = h.
This portfolio strategy is thus called the replicate strategy.
Now we denote F(t, S) be the price of the option at time t. so,
F(t, S) = E∗�e−r(T−t)h�ℱtS� = E∗�e−r(T−t)f(ST)�ℱt
S�
For European call:
C(t, S) = SN(d1) − Ke−r(T−t)N(d2)
where d2 =ln S
K +�r+σ2
2 �(T−t)
σ√T−t, d2 = d1 − σ√T − t, and N(x) is standard normal
distribution
For European put:
P(t, S) = ke−r(T−t) − S + C(t, S)
where it has the same notations as in a call option.
Results of price expressions in this section refer to source [1].
2.4 Hedging
With the previous definition for replication, suppose our claim h admits an Ito
16
representation for the form
h = h0 + � htdSt
T
0 (2.2)
where h satisfies equality (2.1).
Clearly, strategy Vt(ϕ) = h0 + ∫ htdStT
0 is admissible. Moreover, it is
self-financing. Thus, Ito representation (2.2) leads to a strategy which produces h
from initial value of h0 with no risk involved.
Now, we introduce the standard arbitrage-free assumptions, namely the
risk-neutral martingale measure, or the equivalent martingale measure. We assume a
martingale measure Q such that
∂Q∂P
∈ ℒ2(Ω,ℱ, P)
and St is a martingale under Q.
And ht is obtained in a Randon-Nikodym derivative [6]
ht =∂F∂St
(t, St) ≜ Δ(t, St)
where Δ = Fs(t, s(t)) is called its Δ hedging.
Note that in the Black Scholes Model ht is the partial derivative of the value of
the option at time t with respect to the underlying asset price at time t.
So far we have summarized the mathematical construction of hedging in a
complete financial market, where every claim is attainable. In such case, hedging
allows total elimination of risk in handling an option.
17
18
Chapter 3 Option Pricing and Hedging in
the Jump-Diffusion Model
To introduce “jump”, we have to consider a striking feature that distinguishes it
from the standard Black-Scholes model: this makes the market incomplete, and there
is no perfect hedging of options in this case. It is no longer possible to price options
using a replicating portfolio, and the set of probability measures under which the
discounted stock price is a martingale is infinite.
In this chapter, we will discuss the computation of European option prices and
examine hedging strategies that minimize the quadratic risk under the pricing
measure that we choose.
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3.1 Dynamics of the Underlying Asset
Let us consider a financial market without any transaction fee, in which there
are two types of assets, the risk free asset and the risky asset; and the time interval
[0, T], with the maturity time T. We assume the risk-free asset value is Pt; short rate
is a time-deterministic function rt; risky asset value is St.
Risk-free asset value Pt follows the stochastic derivative function
dPt = rtdt, P0 = 1
The risky asset consists two parts, a continuous part modeled by a geometric
Brownian motion, and a jump part, with the jump size modeled by a jump function
Ut and the jump time modeled by a Poisson process, i.e.,
dSt = μt Stdt + σtStdWt + UtStdNt (3.1)
where μt and σt are time deterministic functions; Wt is a standard Brownian
motion; Nt is a Poisson process with parameter λ, which is independent of Wt; Ut is
the jump function; and τj is the j-th stay time. So, Ut is the relative jumping
strength of S, i.e., Uj = �Sτj − Sτj−� /Sτj
−. Here the risky asset price is obviously left
continuous, hence τj− is used to distinguish that.
Note that {Uj} is a sequence of independent identical distributions. We define
the distribution of {Uj} follows a function υ.
We take integration of (3.1) and get the dynamics of the asset,
St = S0 + � Su [μudu + σudWu]t
0+ � Su Uu dNu
t
0
The last term can be further transformed as,
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� SuUudNu
t
0= �� Su Uud(u− τj)
t
0
Nt
j=1
= � Sτj−Uj
Nt
j=1
= �[Sτj − Sτj−]
Nt
j=1
Hence,
St = S0 +� Su [μu du + σu dWu]t
0+ �[Sτj − Sτj
−]Nt
j=1
(3.2)
Using the fact that Ut is the relative jumping size, there’s another version for
this equation
St = S0 ��(1 + Uj) exp �� �μu −12σu
2�du + � σudWu
t
0
t
0�
Nt
j=1
� (3.3)
3.2 Conditions for S�t to be a Martingale
In this section we will provide a formal proof to the Sufficient and Necessary
Conditions for S�t to be a martingale under Q, which enables us to change measure.
Lemma 3.2.1 For all s > 0, denote by 𝒢𝒢s the σ-algebras σ�UNs +1, UNs +2, UNs +3, … �
are independent to ℱs .
Lemma 3.2.2 Let φ(y, z) be a measurable function from Rd × R to R, such that for
any real number z the function y ↦ φ(y, z) is continuous on Rd , and let(Yt)t≤0 be a
left-continuous process, taking values in Rd , adapted to the filtration (ℱt)t≥0.
Assume that, for all t > 0,
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E∗ �� dst
0�υ (dz)φ2(Ys, z)� < ∞
Then the process Mt defined by
Mt = �φ(Yτj, Uj)
Nt
j=1
− λ� dst
0�υ(dz)φ(Ys, z)
is a square-integrable martingale and
Mt2 − λ� ds
t
0�υ(dz)φ(Ys, z)
is a martingale. Note that by convention∑ = 10j=1 .
Lemma 3.2.3 We keep the hypotheses and notations of Lemma 3.2.3. Let (At)t≥0 be
an adapted process such that E∗ �∫ As2ds
t0 � < ∞ for all t > 0. Let Lt = ∫ AsdsdWs
t0 .
Then the product (LtMt)t≥0 is a martingale.
Proofs for Lemma 3.2.1, 3.2.2, 3.2.3 refer to source [2].
Consider that Ft is a deterministic function that satisfies the condition
− 1
2∫ Ft
2t0 dt < ∞. Now let dW�t = dWt + Ftdt , we can find a non-negative random
variable Lt in the following expression,
Lt = exp[−∫ FudWu −1
2∫ Fu
2dut0
t0 ] .
Given this expression, Lt is naturally the solution of the SDE,
dLt = FtLtdWt
By Girsanov Theorem we know that that W�t is a standard Brownian motion
under risk neutral measure Q. [2]
Thus, for ∀0 < s < t ≤ T we have the risk-neutral E∗�S� t�ℱs� as
E∗�S� t�ℱs� = E∗ �e−∫ rtdut0 S0 ���1 + Uj� exp �� �μu −
12σu
2�du + � σudWu
t
0
t
0�
Nt
j=1
� |ℱs�
22
= E∗ �S0��1 + Uj� exp �� �μu − ru −12σu
2�du + � σu(dW�u − Fudu)t
0
t
0�
Nt
j=1
|ℱs�
= E∗ �S0��1 + Uj� exp ��� (μu − ru − σuFu)t
0du−�
12σu
2dut
0+ � σudW�u
t
0��
Nt
j=1
|ℱs�
= S� sE∗ � � �1 + Uj� exp �� �μu − ru −
12σuFu�du
t
s�
Nt
j=Ns+1
�
= S� seλEU1(t−s) exp �� �μu − ru −
12σuFu�du
t
s�
= S� s exp �� �μu − ru −12σuFu + λEU1�du
t
s�
Hence, S� t is a martingale under Q if and only if
� �μu − ru −12σuFu + λEU1�du
t
s= 0
or,
∫ �μu − ru −1
2σuFu�du
ts = −λEU1(t− s)
or,
1
t−s∫ �μu − ru −
1
2σuFu�du
ts = −λEU1.
Consider the last equation, ∀0 < s𝑡𝑡≤ t ≤ T, let t → s, we have that
μt − rt − σtFt = −λEU1
where,
Ft =μt − rt + λEU1
σt (3.4)
Clearly, S� t is a martingale under Q if and only if (3.4) is satisfied, sufficiently
and necessarily, given the integrable condition of course,
� Ft2dt
T
0< ∞
Under such conditions, we have
23
S� t = S0��1 + Uj�Nt
j=1
exp�−� �λEUj +12σu
2�du + � σu dWu
t
0
t
0�
We are going to use these results in the following sections.
3.3 European Option Pricing
3.3.1 Admissible Portfolios
In this section we will define a trading strategy, as in the Black-Scholes model,
by a portfolio ϕ = (ht0, ht), 0 ≤ t ≤ T , that has amount ht
0 and ht in two assets at
time t, respectively, taking values in R2, representing the amounts of assets held over
time. But, to take the jump into account, we will constrain the processes ht0 and ht
to be left-continuous. The value at time t of the strategy ϕ is given in the following
definition.
Definition 3.3.1 The ℱtS adapted portfolio ϕ = (ht
0, ht), 0 ≤ t ≤ T is called
self-financing if,
dVt = ht0rte∫ rs dst
0 dt + htdSt
Taking into account dVt = ht0rte∫ rs dss
0 dt + htSt(μtdt + σtdWt) between the
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jump times and, at a jump time τj, Vt jumps by an amount ΔVτj = hτj Sτj−Uj.
Precisely, the self-financing condition is
Vt = V0 + � hs0dBs
t
0+ � hsSs(μsds + σsdWs)
t
0+ �hτj Sτj
−Uj
Nt
j=1
or,
Vt = V0 + � hs0rse∫ ru dus
0 dst
0+� hsSs�(μs − Fsσs)ds + σsdW�s�
t
0
+ �hτj Sτj−Uj
Nt
j=1
(3.5)
Definition 3.3.2 A portfolio ϕ = (ht0, ht), 0 ≤ t ≤ T is called admissible if it satisfies
the equality (3.5) and satisfies the conditions,
� |hs0|ds
T
0< ∞ 𝑎𝑎. 𝑠𝑠. ,𝐸𝐸 �� hs
2Ss2ds
T
0� < ∞
Actually, in order to make sure the value of a hedging strategy square-integrable,
we have to impose the previous conditions, which are stronger than the class of
self-financing strategies. [2]
Proposition 3.3.3 In an admissible portfolio ϕ = (ht0, ht), 0 ≤ t ≤ T with initial value
V0, for any ht, 0 ≤ t ≤ T, there exists a unique ht0 that keeps the portfolio admissible.
The discounted value for this portfolio is given by
V�t == V0 + � hsS�sσsdW� s
t
0+ �hτj Sτj
−Uj
Nt
j=1
− λ� dst
0hsS�s �υ(dz)z.
25
Proof of Proposition 3.3.3 The value of the portfolio ϕ at time t can be written as
the decomposition Vt = Yt + Zt, in which
Yt = V0 + � hs0rse∫ ru dus
0 dst
0+ � hsSs�(μs − Fsσs)ds + σsdW� s�
t
0
Zt = �hτj Sτj−Uj
Nt
j=1
Then, differentiating the discounted product Y�t, we have
dY�t = de−∫ ru dut0 Yt = −rte−∫ ru dut
0 Ytdt + e−∫ ru dut0 dYt
Moreover, it can be written as
Y�t = V0 + � −rse−∫ ru dus0 Ysds
t
0+� e−∫ ru dus
0 dYs
t
0
= V0 + � −rse−∫ ru dus0 Ysds
t
0
+ � e−∫ ru dus0 �hsrse∫ ru
t0 du ds + hsSs�(μs − Fsσs)ds + σsdW�s��
t
0
= V0 + � −rse−∫ ru dus0 Ysds
t
0+ � e−∫ ru dus
0 hsSs�(μs − Fsσs)ds + σsdW�s�t
0 (3.6)
Continue by doing the same transform to Z� t, we have
Z� t = e−∫ ru dut0 Zt = � e−∫ ru dut
0 hτj Sτj−Uj
Nt
j=1
= ��e−∫ ru duτj
0 + e−∫ ru dus
τj 0 �hτj Sτj−Uj
Nt
j=1
= � e−∫ ru duτj
0 hτj Sτj−Uj
Nt
j=1
+ ��� −e−∫ ru dus
τj 0 rsdst
τj
�hτj Sτj−Uj
Nt
j=1
where in the last term, the integration would only take place on the points of �s ≥ τj�,
thus we have,
Z� t = � e−∫ ru duτj
0 hτj Sτj−Uj
Nt
j=1
+ � �−e−∫ ru dus0 �hτj Sτj
−Uj
Nt
j=1
�t
0ds
= � e−∫ ru duτj
0 hτj Sτj−Uj
Nt
j=1
+ � −e−∫ ru dus0 rsZsds
t
0 (3.7)
26
We get V�t = Y�t + Z� t by combining (3.6) and (3.7). After rearrangement, we
obtain,
V�t = V0 + � −rs(Y�s + Z�s)dst
0+ � hs
0rsdst
0+ � hsS�s�(μs − Fsσs)ds + σsdW� s�
t
0
+ �hτj S�τj−Uj
Nt
j=1
To simplify the upper formula, we substitute V�t by ht0 + htS�t in the right hand
side of the equality and the terms with h0 cancels out, which yields
V�t = V0 + � hsS�s[(μs − Fsσs)ds + σsdW� s]t
0+ �hτj S�τj
−Uj
Nt
j=1
Finally we take into account the equality (3.4) and get the following one that
we need to prove. We will continue to use this in the next sections.
V�t = V0 −� λEU1hsS�sdst
0+ � hsS�sσsdW�s
t
0+ �hτj S�τj
−Uj
Nt
j=1
= V(0) + � hsS�sσsdW�s
t
0+ �hτj S�τj
−Uj
Nt
j=1
− λ� dst
0hsS�s �υ(dz)z (3.8)∎
Remark 3.3.4 By Proposition 3.3.3 it is clear that the processes ht and ht0, together
with the initial value V0, we can uniquely determine one given the other two.
Remark 3.3.5 Note that there is no ht0 term in the right hand side of equality (3.8),
this indicates that the admissible portfolio ϕ = (ht0, ht) with initial value V0 is
completely determined by the process (ht)0≤t≤T representing the amount of the risky
asset.
27
3.3.2 European Options Pricing
Let us consider a European option with maturity T; its contingent claim be h; it
is ℱT-measurable and square-integrable. The seller of this option receives V0 at time
0 and holds an admissible portfolio between times 0 and T. If Vt represents the
value of this portfolio at time t, the payoff for the seller at maturity is given by
VT − h, which is also the hedging mismatch. If this quantity is positive, the seller of
the option earns money, otherwise loses some. Its present value is then V�T −
e−∫ ru dut0 h
In order to better replicate the contingent claim of European option, we need to
minimize the payoff VT − h, namely the risk. One way to measure the risk is to define
the expected square, i.e.,
R0T = E∗(V�T − e−∫ rdut
0 h)2
Applying the identity E(X2) = E2(X) + E(X − E(X))2 to the variable X =
e−∫ rdut0 (VT − h) we obotain,
R0T = E2 �V�T − e−∫ rdut
0 h� + E∗ �V�T − e−∫ rdut0 h− E(V�T − e−∫ rdut
0 h)�2
Since, the condition 𝐸𝐸 �∫ hs2Ss
2dsT0 � < ∞ in Definition 3.3.2 implies that the
discounted value V�t is a square-integrable martingale, we have that E∗�V�T� = V0.
Moreover, we have, E∗ �V�T − e−∫ rdut0 h� = V0 − E∗ �e−∫ rdut
0 h�
By applying this identity to the R0T representation we obtain,
28
R0T = �V0 − E∗ �e−∫ rdut
0 h��2
+ E∗ �e−∫ rdut0 h− E∗ �e−∫ rdut
0 h�+ V�T − V0�2
According to Remark 3.3.5, the quantity V�T − V0 depends only on the process
ht. Then it is obvious that when minimizing R0T, the seller will expect a premiumV0 =
E �e−∫ rduT0 h�. This appears to be the initial value of any strategy designed to
minimize the risk at maturity. For the same argument, when selling the option at
time t > 0, one wants to minimize the quantity RtT = E∗(V�T − e−∫ rduT
t h|ℱt)2 will
expect for a premium V(t) = E(e−∫ rdut0 h|ℱt).
Hence, we will use this quantity to define the price of the option, i.e.,
V0 = E �e−∫ rduT0 h� is the price of the European option at time 0 and Vt =
E(e−∫ rdut0 h|ℱt) is that at time t.
Now, we will give an explicit expression for the price of the European call or put.
By the definition of European options, the contingent claim h can be specified as
follows,
h = f(ST), call: f(x) = (x − k)+ , put: f(x) = (k − x)+
V(t) = E �e−∫ rduTt f(ST)�ℱt�
= E∗ �e−∫ rduTt f�St � �1 + UNt +j�e∫ �μ−1
2σ2−σFu du �+∫ σdW�(u)T
tT
t
NT−Nt
j=1
� �ℱt�
= F(t, St)
With Lemma 3.2.1 (UNs )s≥t and ℱt are independent, we can rewrite the
equality as,
F(t, x) = E∗ �e−∫ rduTt f�x � �1 + UNt +j�e∫ �μ−1
2σ2−σF(u)du �+∫ σdW�(u)T
tT
t
NT−Nt
j=1
��
29
Then we plug in the option price for the Black-Scholes model,
F0(t, x) = E∗ �e−∫ rduTt f �xe∫ �μ−1
2σ2−σF(u)du�+∫ σdW�T
tT
t ��
getting,
F(t, x) = E∗ �F0 �t, xe−λEUj (T−t) � �1 + Uj�N(T−t)
j=1
��
= � E∗ �F0 �t, xe−λEUj (T−t) ��1 + Uj�n
j=1
��∞
n=0
e−λ(T−t)λn(T− t)n
n! (3.9)
For different distributions of jumping size, we can compute the corresponding
EUj numerically and then can derive the expression for F(t, x) explicitly.
3.4 Hedging European options
We continue to use the notations that we defined in the last section.
h = f(ST), V0 = F(0, S0), R0t = E∗(e−∫ r(u)duT
0 f(ST − V�T))2
In section (3.3.1) we have already reached a conclusion that given two out of the
three variables ht, ht0, and V0, we can determine the third one explicitly. In section
(3.3.2) we have seen that the initial value of any admissible strategy aiming at
minimizing the risk R0T at maturity is gien by V0 = E �e−∫ rduT
0 h�.
Now we will determine the process (ht)0≤t≤T representing the amount of the
risky asset over the period [0, T], so as to minimize R0T. Specifically, we consider an
30
admissible portfolio ϕ = (ht0, ht), 0 ≤ t ≤ T, with the value Vt at time t and that
the initial value satisfies V0 = E �e−∫ rduT0 h�. Then we try to give the interpretation
for quadratic risk at maturity R0T.
Firstly, take into account the equality (3.8):
Vt = F(0, S0) + � hsS�sσsdW�s)t
0+ �hτj S�τj
−Uj
Nt
j=1
− λ� dst
0hsS�s �υ(dz)z
We also have that h� = e−∫ r(u)duT0 f(ST) = e−∫ rduT
0 F(T, ST). Let us define a function
F(t, x) = e−∫ rduT0 F(t, xe∫ rduT
0 ), so that
F��t, S�t� = e−∫ rduT0 F(t, St) = e−∫ rdut
0 E∗(e−∫ rduT0 h|ℱt)
Hence, F��t, S�t� is a martingale under Q.
Let Yt = S�t− and φ(Yt, z) = F��t, Yt(1 + z)� − F�(t, Yt)
In our case,
φ�Yτj , Uj� = F� �τ, S�τj−�1 + Uj�� − F� �τ, S�τj
−�
= F� �τj , S�τj� − F� �τj , S�τj−�
We denote
Mt = ∑ φ�Yτj , Uj�Ntj=1 − λ∫ ds∫ ν(dz)t
0 φ(Ys, z)
= ��F� �τj , S�τj� − F� �τj , S�τj−��
Nt
j=1
− λ� ds�ν(dz)t
0�F��s, S�s−(1 + z)� − F��s, S�s−��
= �[F� �τj , S�τj� − F� �τj , S�τj−�]
Nt
j=1
− λ� ds�ν(dz)t
0�F� �s, S�s(1 + z)� − F��s, S�s�� (3.10)
in which, according to Lemma (3.2.2), we know that the process Mt is a
31
square-integrable martingale.
Now let us consider equality (3.9), F��t, S�t� is C2 [0, T) × R+ on we can apply
Ito formula in each fragment between the jump times.
Apply Ito formula to F��t, S�t�,
F��t, S�t� = F(0, S0) + �∂F�∂S �
s, S�s�σs2Ss
2dst
0+ �
∂F�∂x �
s, S�s�[(μs − rs − σsFs)ds + σsdW� s]t
0
+12�
∂2F�∂x2 �s, S�s�σs
2Ss2ds
t
0+ �F� �τj , S�τj� − F� �τj , S�τj
−�Nt
j=1
Together with the equality (3.10) , we get,
F��t, S�t� − Mt = F(0, S0) + �∂F�∂x �
s, S�s�σs2Ss
2dW�s
t
0
+ � �∂F�∂x �
s, S�s��−λEUj�S�s +∂F�∂x �
s, S�s�+12∂2F�∂x2 �s, S�s�σs
2Ss2
t
0
+ λ� ds�ν(dz)t
0�F� �s, S�s(1 + z)� − F��s, S�s���ds
We know that F��t, S�t� and Mt are martingales under Q; therefore the process
F��t, S�t� − Mt is also a martingale, which implies that the third term of the right hand
side in the above equality should be zero.
Therefore,
F��t, S�t� − Mt = F(0, S0) + �∂F�∂x �
s, S�s�σs2Ss
2dW�s
t
0
Then,
32
h� − V�T = F��T, S�T� − V�T = F��T, S�T� − MT + MT − V�T
= F(0, S0) + �∂F�∂x�s, S�s�σsS�sdW� s
T
0
+ �−V0 + λ� dshsS�s �ν(dz)z−� hsS�sσsdW� s
T
0
T
0−�hτj S�τj
−Uj
Nt
j=1
�
+ ��F� �τj , S�τj� − F� �τj , S�τj−��
Nt
j=1
− λ� ds�ν(dz)T
0�F� �s, S�s(1 + z)� − F��s, S�s��
= � �∂F�∂x�s, S�s� − hs� S�sσsdW�s
T
0+ ��F� �τj , S�τj� − F� �τj , S�τj
−� − hτj S�τj−Uj�
Nt
j=1
− λ� ds�ν(dz)T
0�F� �s, S�s(1 + z)� − F��s, S�s� − zhsS�s� ≜ MT
(1) + MT(2)
where,
MT(1) = � �
∂F�∂x �
s, S�s� − hs� S�sσsdW�s
T
0
MT(2) = ��F� �τj , S�τj� − F� �τj , S�τj
−� − hτj S�τj−Uj�
Nt
j=1
− λ� ds�ν(dz)T
0�F� �s, S�s(1 + z)� − F��s, S�s� − zhsS�s�
According to Lemma 3.2.3 Mt(1) ∙ Mt
(2) is a martingale under the risk neutral
measure Q, and it follows,
E∗ �Mt(1) ∙ Mt
(2)� = M0(1) ∙ M0
(2) = 0
Whence,
R0T = E∗(h� − V�T)2 = E∗ �MT
(1)�2
+ E∗ �MT(2)�
2
where,
E∗ �MT(1)�
2= E∗(� [
∂F�∂x �
s, S�s� − hs]S�sσsdsT
0)
E∗ �MT(2)�
2= E∗ {λ� ds�ν(dz)
T
0�F� �s, S�s(1 + z)� − F��s, S�s� − zhsS�s�
2}
33
The risk at the maturity is then given by,
R0T = E∗ �� �
∂F�∂x �
s, S�s� − hs� S�sσsdsT
0�
+ E∗ �λ� ds�ν(dz)T
0�F� �s, S�s(1 + z)� − F��s, S�s� − zhsS�s�
2�
Using the identities (which are obvious),
∂F�∂x �
s, S�s� =∂F∂x
(s, Ss), F� �s, S�s(1 + z)� = e−∫ ru dus0 F�s, Ss(1 + z)�,
F��s, S�s� = e−∫ ru dus0 F(s, Ss)
= E∗ �� e−∫ ru dus0
T
0��∂F∂x
(s, Ss) − hs�2
Ss2σs
2
+ λ� ds�ν(dz)T
0�F�s, Ss(1 + z)� − F(s, Ss) − zhsSs�
2�ds�
It follows that the minimal risk R0T is obtained by finding a hs that gives
minimal value for the function in brackets [∗] above.
Take the derivative of [∙] with respect to hs and let it equal to zero,
∂∂hs
[∗] = −2Ss2σs
2 �∂F∂x
(s, Ss) − hs� − 2λ� z Ss�F�s, Ss(1 + z)� − F(s, Ss) − zhsSs�dν(z)
≜ 0
Solving this equation for hs yields, since (ht)0≤t≤T is left-continuous,
hs = Δ(s, Ss−)
where,
Δ(s, x) =1
σs2 + λ ∫ ν (dz)z2 [σs
2 ∂F∂x
(s, x) + λ�ν (dz)zF�s, x(1 + z)� − F(s, x)
x]
By using this hs, we obtain a process which determines an admissible portfolio
minimizing the risk at maturity. Note that when there is no jump, i.e. λ = 0, this
recovers Δ hedging of the Black-Scholes Model.
Results in this section have been checked with source [2].
34
Remark 3.4.1 In the Black-Scholes framework, the hedging is perfect, i.e. R0T = 0.
However, in the jump-diffusion case, where the market is incomplete, the minimized
risk is generally positive.
35
Reference
[1] Tomas Björk Arbitrage theory in Continuous Time, second edition Oxford
University Press 2004
[2] Damien Lamberton & Bernard Lapeyre Introduction to Stochastic Calculus
Applied to Finance, second edition Chapman & Hall 2007
[3] S. G. Kou A Jump Diffusion Model for Option Pricing with Three Properties:
Leptokurtic Feature, Volatility Smile, and Analytical Tractability Columbia
University 1999
36
[4] R. C. Merton Option Pricing When Underlying Stock Returns Are Discontinuous
Journal of Finance Economics 1976
[5] David Heath Martingale versus PDEs in Finance: An Equivalence Result with
Examples Univ. of Tech., Sydney
[6] Hans Föllmer, Martin Schweizer Hedging of Contingent Claims under Incomplete
Information 1990