Download - Pg p 12 Handbook
Computational Finance (PGP)
Area: Finance Instructor: Michael Carter Term: 6 Objective: This hands-on course aims to develop tools and techniques to implement and analyze the core models of modern finance, as applied in asset pricing and risk management. Financial models will be implemented in Excel, supplemented where appropriate by Visual Basic for Applications (VBA). (It is important to appreciate that it is primarily a course in computation, not a course in programming. Excel and VBA are the vehicles, not the objective.) The course will both enhance understanding of the theory and provide relevant tools for practitioners. The aim is not to produce programmers, but to enable managers to become informed users of the numbers produced for them by others. Outline:
Fixed income assets 1. Basic bond pricing 2. Interest rate swaps 3. Estimating the term structure of interest rates
Derivatives: lattice methods 4. Binomial and trinomial trees 5. Implied trees 6. American options
Derivatives: simulation 7. Random number generation 8. Variance reduction 9. American options 10. Low-discrepancy sequences 11. Non-Gaussian processes
Derivatives: Exotic options 12. Asian options 13. Barrier options 14. Basket and spread options 15. Variance swaps
Interest rate derivatives 16. Black’s model for bond options, caps and swaptions 17. The Black-Derman-Toy model
Pedagogy: Course meetings will combine lecture and practical work. Students must bring their laptop computer to each session. Sessions required: 25 Assessment: Assignments (2) 50% Final exam 40% Homework 10% Prerequisites: Core courses only. Restriction on class size: 30 Relationship with other courses: There is potential overlap with (a) Futures, Options and Risk Management and (b) Fixed Income Securities. However, the computational focus of this course differentiates it from other courses. Bibliography: There is no specific text for the course. Course materials are provided online. Additional student expenses: Nil.
PV Present value of an annuity
NPV Net present value of periodic cash flows
FV Future value of an annuity
RATE Rate of return of an annuity
IRR Internal rate of return of periodic c
PRICE Price of a coupon bond
PRICEDISC Price of a discount bond
TBILLPRICE Price of T-bill (special case of PRICEDISC)
YIELD Yield of coupon bond
YIELDDISC Yield of discount bond
TBILLYIELD Yield of T-bill
ACCRINT Accrued interest
COUPNUM Number of coupons remaining
COUPNCD Next coupon date
COUPPCD Previous coupon date
COUPDAYS Number of days in current coupon
COUPDAYBS Number of days between previous settlement
COUPDAYSNC Number of days between settlemencoupon
DURATION Duration of a coupon bond MDURATION Modified duration
EFFECT Effective annual interest rate TBILLEQ Bond equivalent yield of a T-bill
Useful financial functions in Excel
Formula auditing toolbar
The Formula Auditing Toolbar enables you to trace graphically the relationships between cells. It also allows you to monitor cell contents by placing them in a Watch Window.
To display the formula auditing toolbar
View > Toolbars > Formula auditing
To trace a cell's precedents
1. Select a cell containing a formula 2. Click on the Trace Precedents button 3. Click on the Trace Precedents button again to display the previous level of precedents. 4. Remove tracer arrows one level at a time by clicking Remove Precedent Arrows
To trace a cell's dependents
1. Select a cell containing a formula 2. Click on the Trace Dependents button 3. Click on the Trace Dependents button again to display the previous level of dependents. 4. Remove tracer arrows one level at a time by clicking Remove Dependent Arrows
To select the cell at the other end of an arrow
Double click the arrow
To remove all tracer arrows
Click the Remove All Arrows button.
To display all the relationships in a worksheet
1. In an empty cell, type = 2. Then click the Select All button and evaluate the cell with Ctrl-Enter 3. Click the Trace Precedents button twice.
To display a formula in a cell
Select the cell and press F2
To display all formulae
Click Ctrl-~
To add a cell to the watch window
1. Open the Watch Window by clicking on the Watch Window button in the Formula Auditing Toolbar. 2. Select the cells you want to monitor. 3. Click on the Add Watch button in the Watch Window.
Basic bond pricingIn principal, pricing a risk-free bond is deceptively simple - the price or value of a bond is the present value of the futurecash flows, discounted at the prevailing rate of interest, which is known as the yield.
P = ât=1
T
C1
1 + r
t
+ R1
1 + r
T
where P is the price, C is the coupon, R is the redemption value (principal) and T is the term. Alternatively, the yield of abond is the internal rate of return at which the discounted value is equal to market price. Bonds are known as fixed income
assets, because the timing and magnitude of the future cash flows are fixed. Their value however varies inversely with theyield. Bonds of similar risk and term will attract similar yields.
In practice, bond pricing is more complicated because
æ coupons are paid more frequently than annually, typically every six months.
æ a price is required between coupon periods necessitating discounting for fractional periods.
æ interest rates (yields) may be expected to change during the term of the bond.
The first complication is dealt with by treating the coupon period (e.g. 6 months) as the discounting period. If there are mcoupons per year,
P = ât=1
m T C
m
1
1 +r
m
t
+ R1
1 +r
m
m T
Treatment of fractional periods is a matter of market convention. In particular, various markets employ different day
count conventions for calculating the fraction of the coupon period which as elapsed on a given day. Similar conventionsare employed for pricing zero coupon bonds. However, zero coupon bonds issued with a maturity less than one year(notes) are priced with yet another convention. Computation of bond prices and yields requires being familiar with theprevailing conventions.
Changing interest rates (the yield curve) can be accommodated by discounting each cash flow at the appropriate spot rate.Credit risk can be incorporated in a simple way by discounting at a higher rate than the yield on risk-free bonds. Thisdifference, known as the spread, depends upon the credit rating of the issuer. More sophisticated measures employ creditrisk models to allow for the possibility of default and ratings changes during a given horizon. Sophisticated measures willalso account directly for the options embedded in many bonds, as for example in a callable bond.
Day count conventions
Coupons Day count Basis SettlementCorporate bondsGermany +2India Actual/ 365 3Japan 2UK 2 Actual/Actual 1 +7US 2 30/360 0 +2
Government bondsGermany 1 Actual/365 3 +2India 2 30/360 0 +1 Basis Day countJapan 2 Actual/365 3 0 US (NASD) 30/360UK 2 Actual/Actual 1 +1 1 Actual/actualUS 2 Actual/Actual 1 +1 2 Actual/360US ‐ municipal 30/360 0 3 Actual/365
4 European 30/360Government billsGermany Actual/360 2 +2India Actual/365 3 +2US Actual/Actual 1 +1
Money market (LIBOR)Germany Actual/360 2India Actual/365 3Japan Actual/360 2UK Actual/365 3US Actual/360 2
Eurolibor 30/360 0
Basis is an Excel parameterencoding the date count.
From RBI FAQs
Bond market: The day count convention followed is 30/360, which means that irrespective of the actual number of days in a month, the number of days in a month is taken as 30 and the number of days in a year is taken as 360.
Money market: The day count convention followed is actual/365, which means that the actual number of days in a month is taken for number of days(numerator) whereas the number of days in a year is taken as 365 days. Hence, in the case of Treasury bills, which are essentially money market instruments, money market convention is followed.
Day counts are market conventions, which are subject to change and exceptions.The information collected here has been assembled from a variety of sources. It is tentative and provided for educational purposes only. The information needs to be verified before being used for commercial purposes
Duration and convexity
Duration and SensitivityAssuming annual coupons, the price of a coupon bond is the discounted value of cash flows
P = ât=1
T
C1
1 + r
t
+ R1
1 + r
T
where P is the full or dirty price, C is the annual coupon, R is the redemption value and T is the term. This
can be rewritten as
P = ât=1
T
C H1 + rL-t+ R H1 + rL-T
Differentiating with respect to the yield gives
¶ P
¶r= â
t=1
T
-t C H1 + rL-t-1- T R H1 + rL-T-1
which can be written as
(1)¶ P
¶r= -
1
1 + rât=1
T
C1
1 + r
t
t + R1
1 + r
T
T
The (Macauley) duration of the bond is
Dur = ât=1
T C I 1
1+rMt
Pt +
R I 1
1+rMT
PT
so that
(2)P � Dur = ât=1
T
C1
1 + r
t
t + R1
1 + r
T
T
which is precisely the term inside the brackets in equation (1). Substituting equation (2) into equation (1)
gives
¶ P
¶r= -
1
1 + rDur � P
With m coupons per year, this becomes (see below)
¶ P
¶r= -
1
1 +r
m
Dur � P
To simplify, we call the product on the left modified duration. That is, defining
MDur =
1
1 +r
m
Dur
we have
dP
dr= -MDur � P
For small changes in interest rate, we have
DP
Dr» -MDur � P
or
DP
P» -MDur � Dr
A one percentage point increase in yield will lead to (approx.) MDur fall in price.
Practitioners often express duration (that is, interest-rate sensitivity) in terms of the dollar value of a basis
point (DV01) or more generally price value of a basis point (PV01). This is defined as
PV01 = MDur � P � 0.01 � 0.01
Note that, strictly speaking, it is the invoice or dirty price that should be used for P in this calculation.
à Multiannual coupons
If there are m coupons per year, the price of a bond is
P = ât=1
m T C
m
1
1 +r
m
t
+ R1
1 +r
m
m T
= ât=1
m T C
m1 +
r
m
-t
+ R 1 +
r
m
-m T
Differentiating with respect to the yield
¶ P
¶r= - â
t=1
m T
tC
m1 +
r
m
-t-1 1
m- m T R 1 +
r
m
-m T-1 1
m
= -
1
I1 +r
mM
ât=1
m T t
m
C
m
1
1 +r
m
t
+ T R1
1 +r
m
m T
2 Duration.nb
= -
1
I1 +r
mM
ât=1
m T t
m
C
m
1
1+r
m
t
P+ T
R1
1+r
m
m T
PP
= -
1
I1 +r
mM
Dur � P
= mDur � P
A closed formula for durationInverting the previous equation, the duration of a bond is
(3)Dur = -
I1 +r
mM
P
¶ P
¶r
where
P = ât=1
m T C
m
1
1 +r
m
t
+ R1
1 +r
m
m T
By summing the geometric series, the price of the bond can be written in closed form as
P =
C
r1 -
1
I1 +r
mMm T
+
1
I1 +r
mMm T
R
Differentiating this expression and substituting in (2), we obtain a closed formula for the duration of a
bond
(4)Dur =
1 +r
m
r-
T I C
R- r M + I1 +
r
mM
J C
RJI1 +
r
mMm T
- 1N + r N
When the bond is at par, C � R = r , and this simplifies to
Dur =
1 +r
m
r1 -
1
I1 +r
mMm T
The limit of duration for long term bondsAs T goes to infinity, the second term in equation (3) goes to zero. Therefore, the duration of a long-term
bond converges to
Duration.nb 3
limT ® ¥
Dur =
1 +r
m
r
For example, with a yield of 5%, the duration of a biannual converges to 1+
5 %
2
5 %= 20.5
ConvexityDuration is related to the first derivative of bond price with respect to yield. Convexity is a measure of the
second derivative, normalised by bond price.
C =
â2P
âr2
P
Though it is possible to derive a formula for convexity, by differentiating the above formula for â P � â r,
we would need to incorporate the complications date count conventions for mid-coupon bonds. Alterna-
tively, we can estimate convexity accurately by numerical differentiation
C =
PHr + â rL - 2 PHrL + PHr - â rL
P â r2
where âr is a small change in interest rate (e.g. 0.0001 for 1 basis point).
Alternatively, we can compute convexity from the first derivative of duration (this is useful if we have a
formula for duration, as in Excel). From above
P C =
â2 P
â r2=
â I âP
ârM
â r
But
â P
â r= - P D
where D is modified duration. Substituting and using the product rule
P C =
=
â H-P DL
â r
= -Dâ P
â r- P
â D
â r
= P D2- P
â D
â r
so that
C = D2-
â D
â r
4 Duration.nb
â D � â r can itself be calculated by numerical differentiation.
Duration.nb 5
Numerical differentiationThe derivative of a function f HxL is
f ' HxL = lim„xØ0
f Hx + „ xL - f HxL„ x
An obvious method to approximate the derivative is to compute
∑ f
∑ xº
f Hx + „ xL - f HxL„ x
for small „ x. This is known as the forward difference. A better alternative (though more costly to compute) is
∑ f
∑ xº
f Hx + „ xL - f Hx - „ xL2 „ x
which is known as the central difference.
Using central differences, the second derivative can be estimated by
∑2 f
∑ x2º
f ' Ix + 12„ xM - f ' Ix - 1
2„ xM
„ x=
f Hx+„xL- f Hx L„x
-f HxL- f Hx-„xL
„x
„ x=
f Hx+„xL- f Hx L„x
-f HxL- f Hx-„xL
„x
„ x
=f Hx + „ xL - 2 f Hx L + f Hx - „ xL
H„ xL2
Numerical Recipes (Press et al., 2007: 229) discuss the numerical issues in computing numerical derivatives. Inparticular, choice of „x can be crucial.
Interest rates swapsThe present value of the floating side of a swap (assuming a notional principal of one) is
PVfloating = âi=1
n
∆i DTi ri-1,i
where ∆i is the discount factor at the end of period i, DTi is the elapsed time adjusted for day count, ri-1,i is
forward rate fixed at the end of period i-1 and payable at the end of period i, and n is the number of floatingrate payments. The forward rate is
ri-1,i =
∆i-1
∆i- 1
DTi
Substituting
PVfloating = âi=1
n
∆i DTi
∆i-1
∆i- 1
DTi
= âi=1
n
∆i
∆i-1
∆i
- 1
= âi=1
n
H∆i-1 - ∆iL = H∆0 - ∆1L + H∆1 - ∆2L + º + H∆n-1 - ∆nL
= 1 - ∆n
since ∆0 = 1. That is
(1)PVfloating = 1 - ∆n
Let N denote the number of fixed payments, ∆ j the discount factor applicable to the jth fixed payment, and DT j
the time over which the jth fixed payment is accrued. The present value of the fixed side (assuming a notionalprincipal of one) and annual fixed payments
(2)PVfixed = âj=1
N
∆ j DT j sN = sN QN
where
QN = âj=1
N
∆ j DT j = QN-1 + ∆N DTN
Note that SN DT j is the “dollar” amount of the fixed payment at time T j.
Consequently, the net present value of a swap is
(3)NPV = HsN QN - H1 - ∆N L L ´ Principal
At fair value (NPV = 0)
1 - ∆N = sN QN
= sN HQN-1 + ∆N DTN L
= sN QN-1 + sN ∆N DTN
Assuming previous discount factors ∆1 º∆N-1 are determined, so is QN-1, and we can solve for ∆N
(4)∆N =
1 - sN QN-1
1 + sN DTN
where
QN-1 = âj=1
N-1
∆ j DT j = QN-2 + ∆N-1 DTN-1
This provides a general bootstrapping procedure from inferring discount factors from swap rates.
Generalising, consider a forward swap starting in period t + 1 and ending in period T . The present value of thefloating side is
PVfloating = âi=t+1
T
∆i DTi
∆i-1
∆i- 1
DTi
= âi=t+1
T
∆i
∆i-1
∆i
- 1
= âi=t+1
T
H∆i-1 - ∆iL = H∆0 - ∆1L + H∆1 - ∆2L + º + H∆n-1 - ∆nL
= ∆t - ∆T
The present value of the fixed side (assuming a notional principal of one) is
PVfixed = âj=t+1
T
∆ j DT j sN = sN âj=1
T
∆ j DT j - âj=1
t
∆ j DT j = sN HQT - QtL
where the sum is taken over all fixed payments between times t + 1 and T . Equating fixed and floating sides,the forward swap rate (the fixed rate of a swap starting in t+1 and ending in period T) is given by
(5)sN =
∆t - ∆T
QT - Qt
2 InterestRateSwaps.nb
BootstrappingWith annual compounding, the price of a unit par bond with n years remaining is given by
c P1 + c P2 + … + c Pi-1 + H1 + cL Pt = 1
where c is the coupon (yield) and Pi is the discount factor (price of a t-year zero-coupon bond). This can be solved succes-
sively to give the prices of zero-coupon bonds to match a given yield curve.
Pt =
1 - c Úi=1t-1 Pi
1 + c
For semi-annual coupons, the analogous equations are
c
2P 1
2
+
c
2P1 + … +
c
2P
t-1
2
+ 1 +
c
2Pt = 1
and
Pt =
1 -c
2Ú
i=1
2
t-1
2 Pi
1 +c
2
Estimating the term structure
The basic bond pricing equation is
(1)P = âi=1
n C �m
H1 + sti �mLm ti+
R
H1 + stn �mLm tn
where
P = price Hfull or dirty priceLC = annual coupon
R = redemption payment HprincipalLm = frequency of coupons
n = number of remaining coupons
This can be written in terms of the discount factors
P =
C
mâi=1
n
∆ti + ∆tn R
where
∆ti =
1
1 + sti �m
m ti
The spot rates or discount factors also determine the forward rates. Let rti denote the forward (short) rate
1 +
s
m
m ti
= 1 +
s
m
m ti-1
1 +
rti
m
so that
1 +
rti
m=
I1 +s
mM
m ti
I1 +s
mM
m ti-1=
∆ti-1
∆ti
rti = m∆ti-1 - ∆ti
∆ti
= mD∆ti
∆ti
If there is an active market in zero-coupon bonds, these can be used to give immediate market estimates of
the discount rate at various terms. However, such instruments are traded only in the U.K. and U.S.
treasury markets. Moreover, even in these markets, they are usually disregarded because of restricted
maturities, limited liquidity and tax complications.
In principle, discount factors H∆ti L can be inferred from the prices of coupon bonds inverting (1). In turn,
these can be used to infer the spot rate (zti ) and forward rate Hrti L curves. The inversion process is known
as bootstrapping.
In practice, estimation of the spot rate curve is complicated by two basic problems:
æ Bonds of the same maturity may be selling at different yields, due to market imperfections,
limited liquidity, tax etc.
æ There may be no data on bonds of other maturities.
These problems are tackled (with varying degrees of success) by statistical estimation and interpolation.
The basic approach is to assume a specific functional form for the forward rate or discount function, and
then adjust the parameters until the best fit is obtained. Simple polynomial functions such as
(2)f HtL = Α0 + Α1 t + Α2 t2+ Α3 t3
have been found not to be very suitable, since they imply that rates go to plus or minus infinity as t ® ¥.
Two basic generalizations are found - exponential functions and polynomial or exponential splines.
à Parsimonious functional forms
The most straightforward generalization of (2) is to substitute an exponential for each power of t, fitting a
model of the form
f HtL = Α0 + Α1 ã-k1 t
+ Α2 ã-k2 t
+ Α3 ã-k3 t
+ …
This is the exponential yield model adopted by J.P Morgan.
The most popular model of this form is due to Nelson and Siegel (1987). They observe that the second
order exponential model is the general solution to a second-order differential equation (assuming real
unequal roots)
f HtL = Β0 + Β1 ã-
t
Τ1 + Β2 ã-
t
Τ2
where Τ1, Τ2 are the rates of decay. Finding that this is overparameterized, they adopt the general solution
for the case of equal roots
(3)f HtL = Β0 + Β1 ã-
t
Τ + Β2
t
Τ
ã-
t
Τ
The short rate is Β0 + Β1, while the long rate is lim t® ¥ f HtL = Β0. Β1 can be interpreted as the weight
attached to the short term component, and Β2 as the weight of the medium term. Τ determines the rate of
decay.
The spot rate, the average of the forward rates, can be obtained by integrating this equation, giving
(4)sHtL = à0
t 1
tf HtL â t = Β0 + H Β1 + Β2L
Τ
tJ1 - ã
-t
Τ N - Β2 ã-
t
Τ
Given values for the parameters Β0, Β1, Β3 and Τ, bonds can be valued using the continuous analogue of
(1)
(5)P = âi=1
n
ã-sHtL t
C
m+ ã
-sHtnL tn R
This is the model adopted by the National Stock Exchange of India for estimating its published spot rate
series.
Svennson (1994) extended this specification by adding an additional term for greater flexibility,
specifically
2 EstimatingTermStructure.nb
Svennson (1994) extended this specification by adding an additional term for greater flexibility,
specifically
f HtL = Β0 + Β1 ã-
t
Τ1 + Β2
t
Τ1
ã-
t
Τ1 + Β3
t
Τ2
ã-
t
Τ2
The corresponding spot rate curve is
sHtL = Β0 + H Β1 + Β2LΤ1
tK1 - ã
-t
Τ1 O - Β2 ã-
t
Τ1 + Β3
Τ2
tK1 - ã
-t
Τ2 O - Β3 ã-
t
Τ2
This is the model used by the Deutsche Bundesbank for estimating its published spot rate series.
� Example: National Stock Exchange of India
Estimating the Nelson-Siegel model for bonds traded on 26 June 2004 yields the following parameter
estimates
Β0 = 0.0727, Β1 = -0.0231, Β2 = -0.0210, Τ = 2.8601
5 10 15 20
2
3
4
5
6
7
8
Spot
Forward
� Example: Deutsche Bundesbank
For the 15 September 2004, the Deutsche Bundesbank estimated the following parameters for the Svenn-
son model:
Β0 = 5.4596, Β1 = -3.53042, Β2 = -0.37788, Β3 = -0.98812, Τ1 = 2.70411, Τ2 = 2.53479
These parameters imply the following spot rates.
EstimatingTermStructure.nb 3
1 2.303112 2.643363 2.946344 3.212465 3.444256 3.645147 3.818868 3.969059 4.0990410 4.21179
The spot and forward curves are illustrated in the following graph.
5 10 15 20
2
3
4
5
6
à Spline
A cubic spline comprises a sequence of cubic functions between chosen points called knots, the coeffi-
cients of the cubic functions chosen so that their first and second derivatives match at each of the knots.
This makes the resulting spline smooth.
Author Instrument Estimation Knots
McCullough Discount LS n
Fisher Forward NLLS n �3
Waggoner Forward NLLS n �3
à Evaluation
A recent comprehensive review by Ioannides (2003) found that the parsimonious functional forms out-
performed corresponding spline methods, with the Svennson specification preferred over that of Nelson
and Siegel. However, we note that the Bank of England recently drew the opposite conclusion, switching
from Svensson's method to a spline method (Anderson and Sleath, 1999).
4 EstimatingTermStructure.nb
'Implementation of Nelson-Siegel method for estimating forward rate curve ' Michael Carter, 2004 Function Getformula(ThisCell) Getformula = ThisCell.Formula End Function ' Discount function Function df(t As Double, b0 As Double, b1 As Double, b2 As Double, tau As Double) As Double df = Exp(-t * (b0 + (b1 + b2) * (1 - Exp(-t / tau)) * (tau / t) - b2 * Exp(-t / tau))) End Function 'Bond price function Function Pr(t As Double, C As Double, n As Integer, b0 As Double, b1 As Double, b2 As Double, tau As Double) As Double Dim i As Integer Dim P As Double P = 0 For i = 1 To n P = P + df(t + (i - 1) / 2, b0, b1, b2, tau) * (100 * C / 2) Next i Pr = P + df(t + (n - 1) / 2, b0, b1, b2, tau) * 100 End Function
BIS Papers No 25 xi
Table 1
The term structure of interest rates - estimation details
Central bank Estimation method
Minimised error
Shortest maturity in estimation
Adjustments for tax
distortions
Relevant maturity spectrum
Belgium Svensson or Nelson-Siegel
Weighted prices Treasury certificates: > few days
Bonds: > one year
No Couple of days to 16 years
Canada Merrill Lynch Exponential Spline
Weighted prices Bills: 1 to 12 months
Bonds: > 12 months
Effectively by excluding bonds
3 months to 30 years
Finland Nelson-Siegel Weighted prices ≥ 1 day No 1 to 12 years
France Svensson or Nelson-Siegel
Weighted prices Treasury bills: all Treasury
Notes: : ≥ 1 month
Bonds: : ≥ 1 year
No Up to 10 years
Germany Svensson Yields > 3 months No 1 to 10 years
Italy Nelson-Siegel Weighted prices Money market rates: O/N and Libor rates from 1 to 12 months
Bonds: > 1 year
No Up to 30 years
Up to 10 years (before February 2002)
Japan Smoothing splines
Prices ≥ 1 day Effectively by price adjustments for bills
1 to 10 years
Norway Svensson Yields Money market rates: > 30 days
Bonds: > 2 years
No Up to 10 years
Spain Svensson
Nelson-Siegel (before 1995)
Weighted prices
Prices
≥ 1 day
≥ 1 day
Yes
No
Up to 10 years
Up to 10 years
Sweden Smoothing splines and Svensson
Yields ≥ 1 day No Up to 10 years
Switzerland Svensson Yields Money market rates: ≥ 1 day
Bonds: ≥ 1 year
No 1 to 30 years
xii BIS Papers No 25
Table 1 cont
The term structure of interest rates - estimation details
Central bank Estimation method
Minimised error
Shortest maturity in estimation
Adjustments for tax
distortions
Relevant maturity spectrum
United Kingdom1
VRP (government nominal)
VRP (government real/implied inflation)
VRP (bank liability curve)
Yields
Yields
Yields
1 week (GC repo yield)
1.4 years
1 week
No
No
No
Up to around 30 years
Up to around 30 years
Up to around 30 years
United States Smoothing splines (two curves)
Bills: weighted prices
Bonds: prices
–
≥ 30 days
No
No
Up to 1 year
1 to 10 years
1 The United Kingdom used the Svensson method between January 1982 and April 1998.
3. Zero-coupon yield curves available from the BIS
Table 2 provides an overview of the term structure information available from the BIS Data Bank. Most central banks estimate term structures at a daily frequency. With the exception of the United Kingdom, central banks which use Nelson and Siegel-related models report estimated parameters to the BIS Data Bank. Moreover, Germany and Switzerland provide both estimated parameters and spot rates from the estimated term structures. Canada, the United States and Japan, which use the smoothing splines approach, provide a selection of spot rates. With the exception of France, Italy and Spain, the central banks report their data in percentage notation. Specific information on the retrieval of term structure of interest rates data from the BIS Data Bank can be obtained from BIS Data Bank Services.
U.S. Treasury - Treasury Yield Curve Methodology
Treasury Yield Curve Methodology
This description was revised and updated on February 9, 2006.
The Treasury’s yield curve is derived using a quasi-cubic hermite spline function. Our inputs are the COB bid yields for the on-the-run securities. Because the on-the-run securities typically trade close to par, those securities are designated as the knot points in the quasi-cubic hermite spline algorithm and the resulting yield curve is considered a par curve. However, Treasury reserves the option to input additional bid yields if there is no on-the-run security available for a given maturity range that we deem necessary for deriving a good fit for the quasi-cubic hermite spline curve. In particular, we are currently using inputs that are not on-the-run securities. These are two composite rates in the 20-year range reflecting market yields available in that time tranche. Previously, a rolled-down 10-year note with a remaining maturity nearest to 7 years was also used as an additional input. That input was discontinued on May 26, 2005.
More specifically, the current inputs are the most recently auctioned 4-, 13- and 26-week bills, plus the most recently auctioned 2-, 3-, 5-, and 10-year notes and the most recently auctioned 30-year bond, plus the off-the-runs in the 20-year maturity range. The quotes for these securities are obtained at or near the 3:30 PM close each trading day. The long-term composite inputs are the arithmetic averages of the bid yields on bonds with 18 - 22 years remaining to maturity; and those with 20 years and over remaining to maturity, each inputted at their average maturity. The inputs for the three bills are their bond equivalent yields.
To reduce volatility in the 1-year CMT rate, and due to the fact that there is no on-the-run issue between 6-months and 2-years, Treasury uses an additional input to insure that the 1-year rate is consistent with on-the-run yields on either side of it’s maturity range. Thus, Treasury interpolates between the secondary bond equivalent yield on the most recently auctioned 26-week bill and the secondary market yield on the most recently auctioned 2-year note and inputs the resulting yield as an additional knot point for the derivation of the daily Treasury Yield Curve. The result of this step is that the 1-year CMT is generally the same as the interpolated rate. Treasury has used this interpolated methodology since August 6, 2004.
Treasury does not provide the computer formulation of our quasi-cubic hermite spline yield curve derivation program. However, we have found that most researchers have been able to reasonably match our results using alternative cubic spline formulas.
Treasury reviews its yield curve derivation methodology on a regular basis and reserves the right to modify, adjust or improve the methodology at its option. If Treasury determines that the methodology needs to be changed or updated, Treasury will revise the above description to reflect such changes.
Yield curve rates are normally available at Treasury’s interest rate web sites as early as 5:00 PM and usually no later than 6:00 PM each trading day.
Office of Debt Management Department of the Treasury
Daily Treasury Yield Curve Rates
Daily Treasury Long-Term Rates
Daily Treasury Real Yield Curve Rates
Daily Treasury Real Long-Term Rates
Weekly Aa Corporate Bond Index
file:///C|/Documents%20and%20Settings/MC/Desktop/U_...ry%20-%20Treasury%20Yield%20Curve%20Methodology.htm (1 of 2)23/11/06 9:35:09 AM
The binomial modelMichael CarterA derivative is an asset the value of which depends upon another underlying asset. Consider the simplestpossible scenario, in which the underlying has two possible future states "up" and "down". The value of thederivative in these two states is Vu and Vd respectively.
Underlying
S
u S
d S
Derivative
V
Vu
Vd
The current value of the derivative is enforced by the possibility of arbitrage between the derivative and theunderlying asset. Consider a portfolio comprising x shares and short one option.
Portfolio
x S - V
x u S - Vu
x d S - Vd
=
By choosing x appropriately, we can make the portfolio risk-free. That is, choosing x so that
x u S - Vu = x d S - Vd
we have
x S =Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
ü Exercise
Suppose S = 100, u = 1.05, d = 0.95, Vu = 5 and Vd = 0. Calculate the risk-free hedge. Show that it isrisk-free by comparing the value of the portfolio in the two states.
ü
ü
Substituting for x S, the value of the portfolio at time T in either state is
u x S - Vu = u J Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
N - Vu
=u Vu - u Vd - u Vu + d VuÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
u - d
=d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
u - d
The value of the portfolio at time 0 is
x S - V = d Hu x S - VuL = d ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
y{zzwhere d is the discount factor. Let R = 1 ê d. Solving for V
V = x S - d ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
y{zz=
Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
-1ÅÅÅÅÅÅR
ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
y{zz=
1ÅÅÅÅÅÅR
ikjj R Vu - R Vd - d Vu + u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
y{zz=
1ÅÅÅÅÅÅR
ikjj R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
Vu +u - RÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
Vdy{zz
Letting
p =R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
and 1 - p = 1 -R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
=u - d - R + dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
u - d=
u - RÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
we obtain the fundamental option valuation equation
V =1ÅÅÅÅÅÅR
Hp Vu + H1 - pL VdLThe value of the option at time 0 is the discounted expected value of the payoff, where the expectation is takenwith respect to the synthetic or risk-neutral probabilities (defined above) and discounted at the risk-free rate.
2 BinomialModel.nb
This value is enforced by arbitrage. To see this, suppose that option is selling at a premium above its true value.
V >1ÅÅÅÅÅÅR
Hp Vu + H1 - pL VdLAn arbitrageur can sell n options and buy n x shares, borrowing the net cost n Hx S - V L. At time T , the portfoliois worth nHx u S - VuL in the "up" state and (equally) nHx d S - VdL in the "down" state. Repaying the loan plusinterest of R n Hx S - V L, the arbitrageur makes a risk-free profit of
profit = payoff - loan= n Hx u S - VuL - R n Hx S - V L
= n ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
y{zz - R n J Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
- V N= n R JV -
1ÅÅÅÅÅÅR
Hp Vu + H1 - pL VdLNConversely, if the option is selling at a discount, a risk-free profit can be made by reversing this transaction,buying options and selling shares.
ü Exercise
Suppose S = 100, u = 1.05, d = 0.95, Vu = 5, Vd = 0 and R = 1.01. Calculate the true value of the option.Suppose that the option is priced at 3.10. Find a profitable arbitrage.
ü
ü
ü Remarks
æ R is the risk-free total return for the period T . It is given either by R = 1 + r T or R = ‰r T where r is therisk-free (spot) rate for the period T . It is common to use continuous compounding in option evaluation,although discrete compounding is convenient (and appropriate) for the binomial model.
æ The risk-neutral probabilities p and 1 - p are those probabilities at which the expected growth rate of theunderlying asset is equal to the risk-free rate, that is
p u S + H1 - pL d S = R S
Solving for p,
p Hu - d L S + d S = R Sp Hu - d L S = HR - dL S
p =R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
In the language of probability, p makes the discounted asset price a martingale.
BinomialModel.nb 3
ü Exercise
What condition is required to ensure the existence of this equivalent martingale measure (probability)?
æ The current asset price S will depend upon the real probabilities q. The expected rate of return
m =q u S + H1 - qL d S - SÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
S= q u + H1 - qL d - 1
must be sufficient to induce investors to hold the asset.
æ The hedge ratio x is equal to delta of the option, the sensitivity of the option price to changes in the price of theunderlying
x =Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅHu - dL S =
D VÅÅÅÅÅÅÅÅÅÅÅÅÅD S
æ For a vanilla call option at maturity with a strike price of K
Vu = max Hu S - K, 0L and Vd = max Hd S - K, 0LFor a vanilla put option at maturity with a strike price of K
Vu = max HK - u S, 0L and Vd = max HK - d S, 0LFor a vanilla European option prior to maturity, Vu and Vd are the discounted expected values of the option inthe "up" and "down" states respectively.
For a vanilla American option prior to maturity, Vu and Vd are the maximum of the intrinsic values and dis-counted expected values of the option in the "up" and "down" states respectively.
4 BinomialModel.nb
The Black-Scholes formula for stock indices, currencies and futuresMichael CarterThe standard Black-Scholes formula is
c = S0 NHd1L - K ‰-r T NHd2Lwhere
d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=lnHS0 ê KL + Hr + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 = d1 - s è!!!!T
This can be rewritten as
c = S0 NHd1L - K ‰-r T NHd2L= ‰-r T HS0 „r T NHd1L - K NHd2LL= ‰-r T HF0 N Hd1L - K N Hd2LL
where F0 = S0 ‰r T is the expected forward price of S determined at time 0 under the risk-neutral distribution. Astraightforward proof is given in the appendix.
à Continuous dividend
If the underlying assets pays a continuous dividend yield at the rate q, its forward price is
F0 = S0 ‰Hr-qL T
and therefore the call option value is
c = ‰-r T HS0 ‰-Hr-qL T NHd1L - K NHd2LL = S0 ‰- q T NHd1L - K ‰-r T NHd2Lwith
d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=lnHS0 ê KL + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 = d1 - s è!!!!T
à Foreign currency
The forward price of a foreign currency is given by
F0 = S0 ‰Hr-r f L T
which is known as covered interest parity. Therefore, the value of a foreign currency option is
c = ‰-r T HS0 ‰Hr- f f L T NHd1L - K NHd2LL = S0 ‰- r f T NHd1L - K ‰-r T NHd2Lwith
d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=lnHS0 ê KL + Hr - r f + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 = d1 - s è!!!!T
In effect, the foreign currency is a dividend yield q = r f .
à Future
The value of a call option on a future is given directly by
c = ‰-r T HF0 NHd1L - K NHd2LLwith
d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 = d1 - s è!!!!T
à Generalized Black-Scholes formula
All these cases can be subsumed in a generalized Black-Scholes formula
c = S0 ‰Hb-rL T NHd1L - K ‰-r T NHd2Lwhere
d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=lnHS0 ê KL + Hb + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 = d1 - s è!!!!T
where b is the cost-of-carry of holding the underlying security, with
b = r non-dividend paying stock
b = r - q stock with dividend yield q
b = r f currency option
b = 0 futures options
Put-call parity gives
p + S0 ‰Hb-rL T = c + K ‰- r T
so that
2 GeneralizedBlackScholes.nb
p = H S0 ‰Hb-rL T NHd1L - K ‰-r T NHd2LL + K ‰- r T - S0 ‰Hb-rL T
= K ‰-r T KH1 - NHd2LL - S0 ‰Hb-rL T H1 - NHd1L L= K ‰-r T NH-d1L - S0 ‰Hb-rL T NH-d1L
Traditionally, the Black-Scholes model is implemented in dividend yield form
c = S0 ‰- q T NHd1L - K ‰-r T NHd2L
p = K ‰-r T NH-d2L - S0 ‰- q T NH-d1L
d1 =lnHS0 ê KL + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 = d1 - s è!!!!T
with the specific cases being obtained with the following substitutions
q = 0 non-dividend paying stock
q = q stock with dividend yield q
q = r f currency option
q = r futures options
Note that even if the dividend yield is not constant, the formulae still hold with q equal to the average annualizeddividend yield during the life of the option.
à Appendix
THEOREM. If S is lognormally distributed and the standard deviation of ln S is s then
PrHS > KL = NHd2Land
EHS » S > KL = EHSL NHd1Lwhere
d1 =lnHEHSL ê KL + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s
d2 =lnHEHSL ê KL - s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s
Consequently
(1)E@maxHS - K, 0LD = EHSL NHd1L - K NHd2LProof:
PrHS > KL = ProbHln S > ln KL = NHd2L
GeneralizedBlackScholes.nb 3
For the second part, see Hull (2003: 262-263).
Recognising that (under Black-Scholes assumptions) EHST L = F0 = S0 ‰r T and s = s è!!!!T , the Black-Scholes
formula for a call option
c = ‰-r T HF0 NHd1L - K NHd2LL = ‰-r T HS0 ‰r T NHd1L - K NHd2LL = S0 NHd1L - K ‰-r T NHd2Lis immediate.
4 GeneralizedBlackScholes.nb
Implementation of Black-Scholes option pricing Michael Carter, 2004
Option Explicit ' ************************************************************ ' Option values Function BSCall(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double Dim d2 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) d2 = d1 - sigma * Sqr(T) BSCall = S * Exp(-q * T) * Application.NormSDist(d1) - K * Exp(-r * T) * Application.NormSDist(d2) End Function Function BSPut(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double Dim d2 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) d2 = d1 - sigma * Sqr(T) BSPut = K * Exp(-r * T) * Application.NormSDist(-d2) - S * Exp(-q * T) * Application.NormSDist(-d1) End Function ' ************************************************************ ' The Greeks Function BSCallDelta(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) BSCallDelta = Exp(-q * T) * Application.NormSDist(d1) End Function Function BSPutDelta(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) BSPutDelta = Exp(-q * T) * (Application.NormSDist(d1) - 1) End Function
Function BSCallGamma(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) Debug.Print d1 BSCallGamma = Exp(-q * T) * Application.NormDist(d1, 0, 1, False) / (S * sigma * Sqr(T)) End Function Function BSPutGamma(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double BSPutGamma = BSCallGamma(S, K, r, q, sigma, T) End Function
Dealing with dividendsMichael Carter
European optionsThe Black-Scholes formula is readily adapted to continuous dividends yields (see The Black-Scholes formula forstock indices, currencies and futures).
The price of a dividend paying stock typically falls when the stock goes ex-dividend. A common approach todealing with discrete dividends is to subtract the present value of the dividends from the current stock pricebefore applying the Black-Scholes formula (Hull 2003: 253). For example, if dividends d1, d2, …, dn areanticipated at times t1, t2, …, tn, the present value of the dividends is
D = ‚i=1
n
‰r ti di
and the option is valued as
cHS - D, K, r, s, tL or pHS - D, K, r, s, tLwhere c and p are the Black-Scholes formulae for call and put options respectively.
This is problematic, not the least because historical volatility measures refer to the stock price including divi-dends (Fischling 2002).
Bos and Vandemark (2002) propose a simple modification that closely matches numerical results. Instead ofsubtracting the full present value of future dividends from the current stock price, they propose apportioningeach dividend between the current price and the strike price in proportion to the relative time. Specifically, ifdividends d1, d2, …, dn are anticipated at times t1, t2, …, tn, they compute "near" and "far" components
Dn = ‚i=1
n T - tiÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅT
‰r ti di and D f = ‚i=1
n tiÅÅÅÅÅÅÅT
‰r ti di
The option is valued as
cHS - Dn, K + D f , r, s, tL or pHS - Dn, K + D f , r, s, tLwhere c and p are the Black-Scholes formulae for call and put options respectively.
American optionsDealing with dividends for American options is more complicated, since dividends are closely interwined withthe incentives for early exercise. This is discussed in the complementary note American options.
The binomial modelIn a risk-neutral world, the total return from the stock must be r. If dividends provide a continuous yield of q, theexpected growth rate in the stock price must be r - q. The risk-neutral process for the stock price therefore is
„ S = Hr - qL S „ t + s S „ z
The can be approximated in the simple binomial model by adjusting the risk-neutral probabilities, so that
p u S0 + H1 - pL d S0 = S0 ‰Hr-qL Dt
or
p =‰Hr-qL Dt - d
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d
With this amendment, the binomial model can be used to value European and American options on indices,currencies and futures.
Discrete proportional dividends are also straightforward to incorporate into the binomial model. Whenever thestock pays a proportional dividend, the stock price tree must be adjusted downwards when the stock goesex-dividend (Hull 2003: 402).
Discrete cash dividends are more difficult, since the adjusted tree becomes non-recombining for nodes afterdividend date. This leads to an impractical increase in the number of nodes. We can finesse this problem in ananalogous way to the treatment of cash dividends with the Black-Scholes formula.
Assume that the stock price S has two components - a risky component S* with volatility s* and the dividendstream ‰- r t D. Develop a binomial tree to represent the stochastic part S* with
S0* = S0 - ‰- r t D, p =
‰r Dt - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
u - d, u = ‰ s* Dt, d = ‰ -s* Dt
Then add back the present value of the dividends to each node (prior to the ex-dividend date) to obtain a bino-mial tree representation of S, which can then be used to value contingent claims in the usual way.
This procedure could be enhanced by apportioning the dividends between current price and strike price accord-ing to the procedure of Bos and Vandemark discussed above.
2 Dividends.nb
Hedging strategiesMichael Carter
à Preliminaries
IntroductionConsider a derivative (or portfolio of derivatives) on a single underlying asset. Its value depends upon thecurrent asset price S and its volatility s, the risk-free interest rate r, and the time to maturity t. That is,V = f HS, r, s, tL. (It also depends upon constants like the strike price K.) Taking a Taylor series expansion, thechange in value over a small time period can be approximated by
(1)dV º
∑ fÅÅÅÅÅÅÅÅÅÅÅ∑S
dS +∑ fÅÅÅÅÅÅÅÅÅÅÅ∑r
dr +∑ fÅÅÅÅÅÅÅÅÅÅÅ∑s
ds +∑ fÅÅÅÅÅÅÅÅÅÅÅ∑ t
dt +1ÅÅÅÅÅ2
∑2 fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 dS2
+ other second order terms+ higher order terms
The partial derivatives in this expansion are known collectively as "the Greeks". They measure the sensitivity ofa portfolio to changes in the underlying parameters. Specifically
D =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑S
Delta measures the sensitivity of the portfolio value to changes in the price of the underlying
r =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑r
Rho measures the sensitivity of the portfolio value to changes in the interest rate
v =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑s
Vega measures the sensitivity
of the portfolio value to changes in the volatility of the underlying
Q =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑ t
Theta measures the sensitivity of the portfolio value to the passage of time
G =∑2 fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 =
∑ DÅÅÅÅÅÅÅÅÅÅÅ∑S
Gamma measures the sensitivity of delta to changes in the price of the underlying,
or the curvature of the S - V curve.
Substituting in (1), the change in value of the portfolio can be approximated by
(2)dV º DdS + rdr + v ds + Qdt +1ÅÅÅÅÅ2
GdS2
Because differentiation is a linear operator, the hedge parameters of a portfolio are equal to a weighted averageof the hedge parameters of its components. In particular, the hedge parameters of a short position are the nega-tive of the hedge parameters of a long position. Consequently, (2) applies equally to a portfolio as to an individ-ual asset. The sensitivity of a portfolio to the risk factors (S, r, s) can be altered by changing the compositionof the portfolio. It can be reduced by adding assets with offsetting parameters.
The Greeks are not independent. Any derivative (or portfolio of derivatives) V = f HS, r, s, tL must satisfy theBlack-Scholes differential equation
∑ fÅÅÅÅÅÅÅÅÅÅÅ∑ t
+ r S∑ fÅÅÅÅÅÅÅÅÅÅÅ∑S
+1ÅÅÅÅÅ2
s2 S2 ∑2 fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 = r V
Substituting
∑ PÅÅÅÅÅÅÅÅÅÅÅ∑ t
= Q∑ PÅÅÅÅÅÅÅÅÅÅÅ∑S
= D∑2 PÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 = G
it follows that the Greeks must satisfy the following relationship
(3)Q + r S D +1ÅÅÅÅÅ2
s2 S2 G = r V
Computing the GreeksThe Greeks of vanilla European options have straightforward formulae, which can be derived from the Black-Sc-holes formula. The generalized Black-Scholes formulae for European options are
c = S ‰-q T NHd1L - K ‰-r T NHd2Lp = K ‰-r T NH-d2L - S ‰-q T NH-d1L
where
d1 =lnHS ê KL + Hr - q + s2 TL ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 =lnHS ê KL + Hr - q - s2 TL ê2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
= d1 - s
The partial derivatives ("the Greeks") are
Call PutDelta ‰-q T NHd1L ‰-q T HNHd1L - 1LGamma
‰-q T N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS s
è!!!!T
‰-q T N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS s
è!!!!T
Rho K T ‰-r T NHd2L - K T ‰-r T NH-d2LVega ‰-q T S è!!!!T N ' Hd1L ‰-q T S è!!!!T N ' Hd1LTheta -
‰-q T S s N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 è!!!!T
-‰-q T S s N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
2 è!!!!T
+ q ‰-q T S NHd1L - q ‰-q T S NH-d1L-r K ‰-r T NHd2L - r K ‰-r T NH-d2L
2 HedgingStrategies.nb
As an example of the derivation, for a call option
G =∑DÅÅÅÅÅÅÅÅÅÅÅ∑S
= ‰-q T N ' Hd1L ∑d1ÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S
= ‰-q T N ' Hd1L 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS s
è!!!!T
Calculating vega from the Black-Scholes formula is an approximation, since the formula is derived under theassumption that volatility is constant. Fortunately, it can be shown that it is a good approximation to the vegacalculated from a stochastic volatility model (Hull 2003: 318).
Some exotic options (e.g. barrier options) have analogous formulae. However, for most exotic options andvanilla options, the Greeks must be estimated by numerical techniques. Since these are the type of options forwhich institutions require such information, this motivates are interest in the accurate computation of optionvalues and sensitivities.
In principle, the Greeks can be estimated by numerical differentiation. For example,
D =cHS1L - cHS0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
S1 - S0and G =
DHS1L - DHS0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS1 - S0
However, this is not always the most appropriate method, as the small size of the denominator in the limitmagnifies errors in the numerator.
HedgingIn the previous section, we showed the sensitivity of the value of a portfolio of derivatives of a single underlyingto its risk factors can be approximated by
(4)dV º DdS + rdr + v ds + Qdt +1ÅÅÅÅÅ2
GdS2
Hedging is the process of modifying the portfolio to reduce or eliminate the stochastic elements on the right-hand side. Delta-hedging eliminates the first-term on the right-hand side by making the portfolio delta neutral (D= 0). This can be done by taking an offsetting position in the underlying asset, as represented by the tangent tothe portfolio at the current asset price.
HedgingStrategies.nb 3
90 100 110 120
2.55
7.510
12.515
17.520
Delta-gamma hedging also eliminates the last term in (4) by making the portfolio gamma neutral (G = 0). Sincethe underlying is gamma neutral, delta-gamma hedging requires the addition of other derivatives to the portfolio.Curvature (Gamma) increases as an option approaches maturity, especially for at-the-money options.
90 100 110 120
2.5
5
7.5
10
12.5
15
17.5
20Increasing curvature approaching maturity - 1, 3 ,6 months
Time
4 HedgingStrategies.nb
1 2 3 4 5 6Months to expiry
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Call Gamma over time
Out of the money
At the money
In the money
90 100 110 120
2.55
7.510
12.515
17.520
Hedge option
Recall the fundamental relationship (3)
Q + r S D +1ÅÅÅÅÅ2
s2 S2 G = r V
For a delta-gamma-neutral portfolio, this reduces to
Q = r V
The portfolio earns the risk-free rate.
The closer that hedging option matches the target option, the more robust will be the hedge provided (i.e. thewider the range of parameter variation that will be neutralised). The hedge may be improved by combining twoor more options. For example, combining two options, one with a shorter and one with a longer time to maturity
HedgingStrategies.nb 5
would a more accurate match to the gamma of the target option. There is a tradeoff between the robustness ofthe hedge (the frequency of hedge adjustments) and the number of options that must be purchased and managed.The actual performance of a hedge may not reach its theoretical potential (for example, because of model errorsand transaction costs). Consequently, adding too many options to the hedge may give results that are better onpaper than in reality.
A hedge comprising at least two derivatives, in addition to the underlying, can be used to eliminate three termsin equation (3). A hedge comprising three derivatives, in addition to the underlying, can be used to neutralize allfour stochastic terms in equation (3), eliminating all risk to a first-order approximation.
In principle, a hedge can be found by solving a system of linear equations. Suppose there are m potential hedg-ing instruments. Let x1, x2, …, xm denote the amount of hedging instrument j, and let xS denote the amountinvested in the underlying. Then, we seek a solution to the following system of equations.
xS + x1 D1 + x2 D2 + … + xm Dm = Dx1 G1 + x2 G2 + … + xm Gm = Gx1 v1 + x2 v2 + … + xm vm = vx1 r1 + x2 r2 + … + xm rm = r
Provided that the Greeks of the hedging instruments are linearly independent, there will be a unique solution ifm = 3 and multiple solutions if m > 3. However, the solutions may not be economically sensible.
Since T appears explicitly in the formula for vega, options of different maturities will be most effective inhedging against volatility risk. Although interest rate risk can be hedged by options, it may be cost-effective andcertainly more straightforward to hedge interest rate risk by trading bond future contracts, since they are purerho instruments, with no impact on delta, gamma or vega.
ü Example
6 HedgingStrategies.nb
Rules of thumbConsider a call option that is at-the-money forward, that is
K = F0 = S0 ‰r T
Then the Black-Scholes formula (assuming no dividend yield) simplifies to
c = S0HNHd1L - NHd2LLwhere
d1 =ln HS0 ê F0L + Hr + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=ln HS0 ê S0L - r T + Hr + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=1ÅÅÅÅÅ2
s è!!!!T
d2 = d1 - s è!!!!T = -
1ÅÅÅÅÅ2
s è!!!!T
Therefore the call option value is
c = S0JNJ 1ÅÅÅÅÅ2
s è!!!!T N - NJ-
1ÅÅÅÅÅ2
s è!!!!T NN
Provided that s è!!!!T is small, this can be approximated by
c = S0 ä0.4 s è!!!!T
Since the peak of the standard normal density function is 1 ëè!!!!!!!2 p º 0.4, the area can be approximated by arectangle of height 0.4.
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4The standard normal distribution
This formula can be inverted to obtain a "rough and ready" estimate of the implied velocity from quotedoption prices, using the average of the two nearest-the-money call options.
s = 2.5c
ÅÅÅÅÅÅÅÅS0
1
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!T
Greeks and the binomial method
Numerical differentiation
Delta measures the sensitivity of the option value to changes in the price of the underlying. It is defined as
D =
¶V HSL
¶S= lim
dS ®0
V HS + dSL - V HSL
dS
An obvious method to evaluate D is to compute
D »
V HS + dSL - V HSL
dS
for small dS. This is known as the forward difference. A better alternative (though more costly to compute) is
D »
V HS + dSL - V HS - dSL
2 dS
which is known as the central difference. The other first-order Greeks (rho, theta and vega) can be estimated similarly.
Gamma is the derivative of delta, or the second derivative of vHSL. Using central differences, gamma can be estimated by
G »
DIS +1
2dSM - DIS -
1
2dSM
dS=
V HS+dSL-V HS LdS
-V HSL-V HS -dSL
dS
dS=
V HS+dSL-V HS LdS
-V HSL-V HS -dSL
dS
dS
=
V HS + dSL - 2 V HS L + V HS - dSL
dS2
Numerical differentiation and the binomial tree
Numerical differentiation is not the best method to be applied to the binomial tree. The problem is illustrated in the followingdiagram.
82.5 87.5 90 92.5 95 97.5
2
4
6
8
10
12
Black Scholes
82.5 87.5 90 92.5 95 97.5
2
4
6
8
10
12
Binomial
2 Greeks.nb
Implied trees
Michael Carter
� Preliminaries
�
Implied trees are one practical method to deal with the volatility smile. The objective is to build a tree for the underlying thatreproduces the market prices of traded derivatives. In other words, the resulting tree is calibrated to current market data. Thecalibrated tree can then be used to
æ to compute hedge parameters for traded options
æ to price non-standard derivatives (American options and exotics) on the same underlying.
The starting point is current market prices and quotations for calls and puts of various strikes and maturities. Typically, thesemarket data are converted into a table of implied volatilities. This table is then used to interpolate volatilities and marketprices for arbitrary strikes and maturities.
Binomial tree
� The one-step tree
The value of a one-period call option struck at the initial price S is
(1)VcHSL =
1
Rpu HSu - SL
where pu is the risk-neutral probability of an up-tick, Su = u S is the price of the asset following an up-tick and 1 � R is the
one-period discount factor. Similarly, the value of a one-period put option struck at S is
(2)VpHSL =
1
Rpd HS - SdL
where pd = 1 - pu is the probability of a down-tick, and Sd = d S is the resulting asset price.
In the binomial model, the risk-neutral (arbitrage free) probability pu is that which makes the expected return on the asset
equal to the risk-free rate of return, that is
pu Su + H1 - puL Sd = R S
where R = ãHr-qL Dt is the one-period risk-free rate of capital gains.
Solving for pu
(3)pu =
R - d
u - dor pu =
R S - Sd
Su - Sd
If we add an additional constraint such as u = 1 �d, then we can solve for the binomial tree consistent with market data. For
example, if we have the market price Vc, we obtain from solving equations (1) and (3) the following formula for u, from
which we can derive Su, Sd and pu.
u =
∆ S + Vc
∆ R S - Vc
Alternatively, we can use the market price Vp of a put to obtain from (2) and (3)
u =
∆ R S + Vp
∆ S - Vp
Note that ∆ R = 1 if and only if the dividend yield equals zero.
Given u, we can derive
Su =
SH∆ S + VcL
∆ R S - Vc
, Sd =
S H∆ R S - VcL
S ∆ + Vc
, pu =
R S - Sd
Su - Sd
or
Su =
SI∆ R S + VpM
∆ S - Vp
, Sd =
S I∆ S - VpM
∆ R S + Vp
, pu =
R S - Sd
Su - Sd
We can also compute the state contingent prices (discounted probabilities) at the new nodes
Λu = ∆ pu and Λd = ∆ H1 - puL
These will be required in the next step.
We have shown how we can infer a unique one-step binomial tree consistent with given a single market price. Note that theinferred tree will differ depending upon whether we start with a call or a put price, unless the prices are consistent. Inpractice, consistency is usually imposed by starting with an implied volatility for relevant term and strike, and imputingprices for the call or put using a binomial tree with the requisite volatility. This required implied volatility is obtained byinterpolation from the implied volatilities of quoted options.
Assuming that we are dealing with European options, the implied volatility of quoted options is obtained using the Black-Scholes formula. Then, the recommended practice is to use these data to interpolate implied volatilities for the nodes of thetree. The implied prices for options struck at nodes of the tree are then obtained from a binomial model with the requisitevolatility.
� The two - step tree
To complete the second step, we can arbitrarily set the asset price along the middle branch of the tree equal to the startingprice. Given this base line, we can determine the top and bottom nodes of the tree. Suppose we know the value vu of a one-
period call option struck at S = Su, the upper node in the previous step. Then we are in a position analogous to the first step.
From equation (1) we have
vuHSL =
1
Rpu ISup - SM
with
pu =
R - d
u - d=
R S - d S
u S - d S=
R S - Sd
Sup - Sd
Solving for Sup gives
2 ImpliedTrees.nb
Solving for Sup gives
(4)Sup =
S2- Sd H∆ S + vuL
S - H∆ Sd + vuL
where in this case Sd = S0.
Similarly, setting the Su = S0, we can determine the bottom node of the tree from the value vd of a one-period put option
struck at the bottom node in the previous step, solving
vd =
1
RH1 - puL HS - SdownL and pu =
R S - Sdown
Su - Sdown
to give
(5)Sdown =
S2- SuH∆ S - vdL
S - H∆ Su - vdL
It remains to supply the supposed data - the values of the one-period options struck at the Su and Sd . We now show how this
can be inferred from market data. Let VcHSu, 2) denote the value of a call option struck at Su and expiring after two periods.
This value can be interpolated from market data. After one period, there are two possibilities. Either the price increased to Su
or decreased to Sd . In the former case, the option is then worth vu. In the latter case it is worthless, as it is bound to expire out
of the money. Consequently
VcHSu, 2L = ∆ H pu vu + H1 - pL 0L = ∆ pu vu
from which we can infer
vu =
VcHSu, 2L
∆ pu
Similarly the value vpHSd , 2) of a two-period put option struck at Sd is
VpHSd , 2L = ∆ H pu 0 + H1 - pL vdL = ∆ H1 - puL vd
which implies
vd =
VpHSd , 2L
∆ H1 - puL
Throughout this note, we will adopt the convention that a capital V refers to a market value, while a small v refers to animputed value.
� The n-step tree
In essence, building an n-step tree involves repeating the preceding two steps again and again. To facilitate this, we need todefine some more precise notation.
� Notation
Let Si, j denote the asset price at the jth node of the ith step. That is, Hi, jL denotes the node reached after j upward and i - j
downward steps. Si,0 is the minimum price attained after i steps while Si,i is the maximum price. Similarly, let Λi, j denote the
state contingent price at the Hi, jLth node. Let pi, j denote the probability of an uptick at the Hi, jLth node with 1 - pi, j the
probability of a downtick. Finally, vi, j denotes the value of at node i of a one-period (maturing at time i + 1) at-the-money
option (struck at Si, j).
ImpliedTrees.nb 3
Λ0,0
S0,0 p0,0
v0,0
Λ1,0
S1,0 p1,0
v1,0
Λ1,1
S1,1 p1,1
v1,1
Λ2,0
S2,0 p2,0
v2,0
Λ2,1
S2,1 p2,1
v2,1
Λ2,2
S2,2 p2,2
v2,2
Λ3,0
S3,0 p3,0
v3,0
Λ3,1
S3,1 p3,1
v3,1
Λ3,2
S3,2 p3,2
v3,2
Λ3,3
S3,3 p3,3
v3,3
Λ4,0
S4,0 p4,0
v4,0
Λ4,1
S4,1 p4,1
v4,1
Λ4,2
S4,2 p4,2
v4,2
Λ4,3
S4,3 p4,3
v4,3
Λ4,4
S4,4 p4,4
v4,4
� State contingent prices and probabilities
We have seen in a previous lecture how state prices can be calculated inductively from previous values
Λi,0 = ∆I1 - pi-1,0M Λi-1,0, Λi,i = ∆ pi-1,i-1 Λi-1,i-1
Λi, j = ∆ H I1 - pi-1, jM Λi-1, j + pi-1, j-1 Λi-1, j-1L, 0 < j < i
while the previous probabilities are computed from current asset prices
pi-1, j =
R Si-1, j - Si, j
Si, j+1 - Si, j
The sequence of calculation is Si,_ ® pi-1,_ ® Λi,_. It remains to infer the asset prices from the market data.
� Forward option prices
Let VcISi, j, i + 1M denote the value of a call option struck at Si, j at time t0 and maturing at time Hi + 1L Dt , that is maturing after
i + 1steps in the tree. We can infer this value by interpolation from market data. Note that there is no need to specify thecurrent asset price of the multi-period options - it is always the initial asset price S0,0. Now consider the possibilities for this
option one period prior to maturity. Let vi,kISi, jM denote the value of this option at time i Dt if the asset price reaches Si,k . Then
VcISi, j, i + 1M = âk=0
i
Λi,k vi,kISi,kM
which we can write as
(6)
Out At In
VcISi, j, i + 1M = âk=0
j-1
Λi,k vi,kISi,kM + Λi, j vi, jISi,kM + âk= j+1
i
Λi,k vi,kISi,kM
with
(7)vi,kISi, jM = ∆I pi,k max ISi+1,k+1 - Si, j, 0M + I1 - pi,kM maxISi+1,k - Si, j, 0MM
where ∆ is the one-period discount factor and pi,k is the risk-neutral probability applicable at node Hi, kL.
4 ImpliedTrees.nb
where ∆ is the one-period discount factor and pi,k is the risk-neutral probability applicable at node Hi, kL.
For all states k < j, the option must necessarily expire out of the money. Therefore vi,kISi, jM = 0 for all k < j, and the first term
in equation (6) is zero. For all k > j, the option will expire in the money. Consequently, for k > j we can eliminate the max
operator from (7) to give
(8)vi,kISi, jM = ∆I pi,k ISi+1,k+1 - Si, jM + I1 - pi,kM ISi+1,k - Si, jMM = ∆I pi,k Si+1,k+1 + I1 - pi,kM Si+1,kM - ∆ Si, jM
Furthermore, the risk-neutral probability pi,k satisfies the equation
pi,k Si+1,k+1 + I1 - pi,kM Si+1,k = R Si,k
where R = 1 � ∆ equals the one-period risk neutral return. Substituting in (8) gives
vi,kISi, jM = ∆ IR Si,k - Si, jM for k > j
Substituting this in (6) gives
Out At In
VcISi, j, i + 1M = 0 + Λi, j vi, jISi,kM + âk= j+1
i
∆ Λi,k IR Si,k - Si, jM
where vi, j = vi, jISi, jM is the only unknown. (For the at-the-money option vi, j, we can dispense with the argument Si, j, since
there is no ambiguity.) Solving for vi, j= vi, jISi, jM,
(9)vi, j =
VcISi, j, i + 1M - Úk= j+1i
Λi,k ∆ IR Si,k - ∆ Si, jM
Λi, j
Similarly, the value of an Hi + 1L-period put option struck at Si, j is
In At Out
VpISi, j, i + 1M = âk=0
i
Λi,k vi,kISi, jM = âk=0
j-1
Λi,k vi,kISi, jM + Λi, j vi, jISi, jM + âk= j+1
i
Λi,k vi,kISi, jM
where vi,kISi, jM is the value of the option at node Hi, kL. For k < j , the option will expire in the money and
vi,kISi, jM = ∆I pi,k ISi, j - Si+1,k+1M + I1 - pi,kM ISi, j - Si+1,kMM
= ∆ Si, j - ∆I pi,k Si+1,k+1 + I1 - pi,kM Si+1,kM
= ∆ ISi, j - R Si,kM
Conversely, for k > j, the option will expire out of the money with vi,kISi, jM=0. Consequently
VpISi, j, i + 1M = âk=0
j-1
Λi,k ∆I Si, j - R Si,k M + Λi, j vi, j + 0
and
(10)vi, j =
vpISi, j, i + 1M - Úk=0j-1
Λi,k ∆I Si, j - R Si,kM
Λi, j
To summarize, vi, j denotes the implied value at node i, j of an at-the-money call or put option struck at Si, j and expiring after
one-period. The choice of call or put is arbitrary. It is conventional to use (9) to determine vi, j in the top half of the tree and
use (10) determine vi, j in the bottom half of the tree. The choice of call or put along the horizontal is arbitrary.
Note that, once we have determined the asset prices at stage i in the tree, we can then determine state prices and forwardoption prices at the same stage. The sequence of calculation is Si,_ ® pi-1,_ ® Λi,_ ® vi,_.
ImpliedTrees.nb 5
Note that, once we have determined the asset prices at stage i in the tree, we can then determine state prices and forwardoption prices at the same stage. The sequence of calculation is Si,_ ® pi-1,_ ® Λi,_ ® vi,_.
� Asset prices
It remains to determine the asset prices in the next stage of the tree, which we compute from the forward asset prices in theprevious stage. If the stage i is even, then we arbitrarily set asset price of the central node equal to the starting price. That is
Si,i�2 = S0,0, i = 2, 4, 6, …
This is known as the centering condition. If the stage i is odd, we determine the values of the two central nodes in identicalfashion to the one-stage tree above. Let m = di �2t, the largest integer less than i �2. m points to the central node when i iseven, and to the lower of the two central nodes when i is odd.
Λi-1, j
Si-1, j pi-1, j
vi-1, j
Λi, j
Si, j pi, j
vi, j
Λi, j+1
Si, j+1 pi, j+1
vi, j+1
Now consider a node above the central node. A one-period call option struck at the preceding (down) node has the valueë
vi-1, j = ∆ pi- j, j ISi, j+1 - Si-1, jM where pi- j, j =
R Si-1, j - Si, j
Si, j+1 - Si, j
in which the only unknown is Si, j+1. Starting at the central node, we can successively solve for all the nodes above the center
Si, j+1 =
∆ R Si-1, j2
- ∆ Si-1, j Si, j - Si, j vi-1, j
∆ R Si-1, j - ∆ Si, j - vi-1, j
, j = :m + 1, m + 2, …, i , i even
m + 2, m + 3, …, i , i odd
Similarly, we can successively compute the prices at all nodes below the central node using the values of one-period putoptions struck at the preceding (up) node.
vi-1, j = ∆ I1 - pi- j, jM ISi-1, j - Si, jM where pi- j, j =
R Si-1, j - Si, j
Si, j+1 - Si, j
giving
Si, j =
∆ R Si-1, j2
- ∆ Si-1, j Si, j+1 + Si, j+1 vi-1, j
∆ R Si-1, j - ∆ Si, j+1 + vi-1, j
, j = m - 1, m - 2, …, 0
� The resulting tree
There is a unique binomial tree fitting a given set of market data (once the central trunk has been specified). The resultingtree is distorted, as the asset prices adapt to fit the volatility at different nodes in the tree.
6 ImpliedTrees.nb
An implied binomial tree
The procedure can be adapted to apply to American style options and options with dividends (Chriss 1996). Nothing in theprocedure guarantees that the transition probabilities are nonnegative. If and when "wrong" probabilities arise, the tree mustbe adjusted to eliminate them. The same problem arises with trinomial trees.
Trinomial trees
Trinomial trees give additional degrees of freedom. We can build an asset price tree with regularly spaced nodes, and thenuse market data to determine the transition probabilities. This additional flexibility is especially useful where the market datais noisy, or where it is desired to use the fitted tree to price certain exotic options like barrier options.
Let S0,0 denote the root of the tree, with nodes at step i being labelled Si,-i, Si,-i+1, …, Si,0, …, Si,i. In the context of a given
tree, the value of a call option struck at K and expiring at i DT is (S = S0,0 always)
(11)VcHK, i L = âj=-i
i
Λi, j ISi, j - KM+
= âj= j*
i
Λi, j ISi, j - KM
where j* is such that Si, j*> K and Si, j*
-1 £ K. In particular,
VcISi,i-1, iM = Λi,i ISi,i - Si,i-1M
from which we can deduce
Λi,i =
VcISi,i-1, iM
Si,i - Si,i-1
Similarly
VcISi,i-2, iM = Λi,i ISi,i - Si,i-2M + Λi,i-1 ISi,i-1 - Si,i-2M
so that
Λi,i-1 =
VcISi,i-2, iM - Λi,i ISi,i - Si,i-2M
Si,i-1 - Si,i-2
Proceeding in this fashion, we can iteratively deduce the state contingent prices at each node of the tree from the value ofoptions struck at adjacent nodes. Computationally, it is preferable to use call options to calibrate the top of half of the tree,and then switch to put options for the bottom half of the tree.
ImpliedTrees.nb 7
Proceeding in this fashion, we can iteratively deduce the state contingent prices at each node of the tree from the value ofoptions struck at adjacent nodes. Computationally, it is preferable to use call options to calibrate the top of half of the tree,and then switch to put options for the bottom half of the tree.
Given the state contingent prices, we can then deduce the transition probabilities at each nodes. At the root node, for exam-ple,
Λ1,1 = ∆ p0,0u so that p0,0
u=
Λ1,1
∆
Similarly
p0,0d
=
Λ1,-1
∆
and p0,0m
= 1 - p0,0u
- p0,0d
Furthermore
Λ2,2 = ∆ p1,1u
Λ1,1 so that p1,1u
=
1
∆
Λ2,2
Λ1,1
Next we observe that the risk-neutral transition probabilities must satisfy the equations
p1,1u S2,2 + p1,1
m S2,1 + p1,1d S2,0 � R S1,1
p1,1u
+ p1,1m
+ p1,1d
� 1
where R = ãHr-qL Dt is the risk-neutral drift.
Given p1,1u , we can solve these equations for p1,1
m and p1,1d
p1,1m
=
R S1,1 - S2,0 - p1,1u I S2,2 - S2,0M
S2,1 - S2,0
, p1,1d
= 1 - p1,1u
- p1,1u
Next, we can use p1,1m to deduce the value of p1,0
u from the equation
Λ2,1 = ∆ I p1,1m
Λ1,1 + p1,0u
Λ1,0M
which gives
p1,0u
=
Λ2,1 - ∆ p1,1m
Λ1,1
∆ Λ1,0
Iterating this procedure, we can generate all the transition probabilities in the tree, using the following recursive system ofequations.
(12)
8 ImpliedTrees.nb
(12)
pi,iu
=
Λi+1,i+1
∆ Λi,i
pi,ku
=
Λi+1,k+1 - ∆ pi,k+1m
Λi,k+1
∆ Λi,k
, k < i
pi,km
=
R Si,k - Si+1,k-1 - pi,ku I Si+1,k+1 - Si+1,k-1M
Si+1,k - Si+1,k-1
pi,kd
= 1 - pi,ku
- pi,ku
To avoid the accumulation of numerical errors, it is recommended to compute the top half of the tree using (12) and then
compute the bottom half using analogous recursions working from the bottom up, namely
(13)
pi,-id
=
Λi+1,-Hi+1L
∆ Λi,-i
pi,-kd
=
Λi+1,-Hk+1L - ∆ pi,-Hk+1Lm
Λi,-Hk+1L
∆ Λi,-k
, k < i
pi,-km
=
Si+1,-k+1 - R Si,-k - pi,-kd I Si+1,-k+1 - Si+1-,k-1M
Si+1,-k+1 - Si+1,-k
pi,-ku
= 1 - pi,-ku
- pi,-ku
Specifically, in (12) and (13), let k = i, i - 1, …, 0 for successive i = 1, 2, 3, ….
We must e� sure that the transition probabilities are all nonnegative. There are two ways in which negative probabilites canarise
æ the state space is inappropriate
æ the volatility skew is extreme at particular observations
A robust way to avoid the first problem is to build a tree whose volatility is equal to the maximum implied volatility in thedata.
Once a market-consistent tree has been constructed, any contingent claim can be priced in the standard manner. For Euro-pean-style derivatives, we can use the state-contingent prices at maturity; for American-style derivatives, we use backwardinduction.
ImpliedTrees.nb 9
Binomial model - implementationMichael CarterThe following is the pseudocode for an efficient implementation of the binomial method for American andEuropean options.
BinomialAmericanCall (S0, K, r, q, s, t, nL
/* initialize parameters */Dt = t/n;u = ‰s è!!!!!!Dt ; d = 1/u;d = ‰-r Dt ;p = ‰Hr-qL Dt - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu-d ; p = 1-p;pu = d p; pd = d H1 - pL; /* pu and pd are discounted probabilities */
/* allocate storage */vectors s[-n,n], v[-n,n];
/* initialize the tree */s[0] = S0;
/* calculate potential asset prices */for j = 1 to n
s[j] = u * s[j-1];s[-j] = d*s[-j+1];
end;
/* calculate option exercise values */for j = -n to n by 2
v[j] = max (s[j] - K,0);end;
/* calculate option values by backward induction */for i = n - 1 to 0 by - 1
for j = -i to i by 2v[j] = max Hpu * v[j+1] + pd * v[j-1], s[j] - K);
end;end;
/* return option value */BinomialAmericanCall = v[0]
The corresponding put option can be evaluated by taking the negative of S and K .
BinomialEuropeanCall (S0, K, r, q, s, t, nL
/* initialize parameters */
Dt = t/n;u = ‰s
è!!!!!!!Dt ; d = 1/u;
p = ‰Hr-qL Dt - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu-d ; p = 1- pÅÅÅÅÅÅÅÅÅÅÅp ;
/* calculate maximum asset price and probability*/s = un S0;P = pn;EV = P * maxHs - K, 0L;
/* proceed down final asset prices calculating expected values */for j = n - 1 to 0 by -1
P = j+1ÅÅÅÅÅÅÅÅÅÅn- j P p;s = d2 s;IV = maxHs - K, 0L;if (IV > 0)
EV = EV + P * IV;else
break;endif
end;
/* return option value */BinomialEuropeanCall = ‰-r t EV;
NOTE: We have to be careful that P = pn does not cause an underflow error. As T Ø ¶, p Ø 1ÅÅÅÅ2 andH 1ÅÅÅÅ2 L1050
º 10-316. For example, the Excel VBA data type Double has a minimum value of4.94065645841247 * 10-324, so that n = 1000 is a practical maximum for this algorithm. However, thealgorithm could be modified to exclude very low probability final prices and so increase the number ofbranches.
2 BinomialProgramming.nb
Implementation of binomial model for option pricing Michael Carter, 2004
Option Explicit Option Base 0 Enum OptionType callOption = 1 putOption = -1 End Enum Enum OptionStyle American = 1 European = 2 End Enum Function RecursiveBinomialTree(S As Double, K As Double, u As Double, _ d As Double, p As Double, delta As Double, n As Integer) As Double ' A recursive version of the basic option valuation routine ' For illustrative purposes ' Only viable for a small number of stages (e.g. 15) Dim pu As Double, pd As Double pu = delta * p ' pu and pd are discounted prices pd = delta * (1 - p) If n > 15 Then RecursiveBinomialTree = "Too many steps for recursion" ElseIf n = 0 Then RecursiveBinomialTree = WorksheetFunction.Max(S - K, 0) Else RecursiveBinomialTree = (pu * RecursiveBinomialTree(u * S, K, u, d, p, delta, n - 1) + _ pd * RecursiveBinomialTree(d * S, K, u, d, p, delta, n - 1)) End If End Function Function RecursiveBinomialEurCall(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) RecursiveBinomialEurCall = RecursiveBinomialTree(S0, K, u, d, p, delta, n) End Function
Function BinomialTree(S0 As Double, K As Double, u As Double, d As Double, _ p As Double, delta As Double, n As Integer) As Double ' Efficient implementation of binomial tree for American options ' Allocate storage Dim i As Integer, j As Integer Dim pu As Double, pd As Double Dim S() As Double, V() As Double ReDim S(2 * n) ReDim V(2 * n) ' Initialize parameters pu = delta * p ' pu and pd are discounted prices pd = delta * (1 - p) ' Initialize the tree S(n) = S0 ' Calculate potential asset prices For j = n + 1 To 2 * n S(j) = u * S(j - 1) Next j For j = n - 1 To 0 Step -1 S(j) = d * S(j + 1) Next j ' Calculate option values at maturity For j = 0 To 2 * n Step 2 V(j) = Application.Max(S(j) - K, 0) Next j ' Calculate option values by backward induction For i = n - 1 To 0 Step -1 For j = n - i To n + i Step 2 V(j) = Application.Max(pu * V(j + 1) + pd * V(j - 1), S(j) - K) Next j Next i ' Return option value BinomialTree = V(n) End Function
Function BinomialAmerCall(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) BinomialAmerCall = BinomialTree(S0, K, u, d, p, delta, n) End Function Function BinomialAmerPut(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) BinomialAmerPut = BinomialTree(-S0, -K, u, d, p, delta, n) End Function Function BinomialEurCall(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, fp As Double, _ pi As Double, S As Double, EV As Double, IV As Double, j As Integer dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u p = (Exp((r - q) * dt) - d) / (u - d) pi = (1 - p) / p ' calculate maximum asset price and probability S = u ^ n * S0 fp = p ^ n EV = fp * WorksheetFunction.Max(S - K, 0)
' proceed down final asset prices calculating expected values For j = n - 1 To 0 Step -1 fp = (j + 1) / (n - j) * fp * pi S = d ^ 2 * S IV = WorksheetFunction.Max(S - K, 0) If IV > 0 Then EV = EV + fp * IV Else Exit For End If Next j BinomialEurCall = Exp(-r * T) * EV End Function Function BinomialEurPut(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, fp As Double, _ pi As Double, S As Double, EV As Double, IV As Double, j As Integer dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u p = (Exp((r - q) * dt) - d) / (u - d) pi = p / (1 - p) ' calculate minimum asset price and probability S = d ^ n * S0 fp = (1 - p) ^ n EV = fp * WorksheetFunction.Max(K - S, 0) ' proceed up final asset prices calculating expected values For j = 0 To n - 1 fp = (n - j) / (j + 1) * fp * pi S = u ^ 2 * S IV = WorksheetFunction.Max(K - S, 0) If IV > 0 Then EV = EV + fp * IV Else Exit For End If Next j BinomialEurPut = Exp(-r * T) * EV End Function
Function EBinomialTree(S0 As Double, K As Double, u As Double, d As Double, _ p As Double, delta As Double, n As Integer) As Variant 'Extended binomial tree for calculating Greeks n = n + 2 ' extend tree ' Allocate storage Dim A As Variant Dim i As Integer, j As Integer Dim pu As Double, pd As Double Dim S() As Double, V() As Double ReDim S(2 * n) ReDim V(2 * n) ' Initialize parameters pu = delta * p ' pu and pd are discounted prices pd = delta * (1 - p) ' Initialize the tree S(n) = S0 ' Calculate potential asset prices For j = n + 1 To 2 * n S(j) = u * S(j - 1) Next j For j = n - 1 To 0 Step -1 S(j) = d * S(j + 1) Next j ' Calculate option values at maturity For j = 0 To 2 * n Step 2 V(j) = Application.Max(S(j) - K, 0) Next j ' Calculate option values by backward induction For i = n - 1 To 2 Step -1 For j = n - i To n + i Step 2 V(j) = Application.Max(pu * V(j + 1) + pd * V(j - 1), S(j) - K) Next j Next i ' Return option values EBinomialTree = Array(V(n - 2), V(n), V(n + 2)) End Function
Function BinomialAmerCallDelta(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double Dim V As Variant dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) V = EBinomialTree(S0, K, u, d, p, delta, n) BinomialAmerCallDelta = (V(2) - V(0)) / ((u ^ 2 - d ^ 2) * S0) End Function Function BinomialAmerPutDelta(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double Dim V As Variant dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) V = EBinomialTree(-S0, -K, u, d, p, delta, n) BinomialAmerPutDelta = (V(2) - V(0)) / ((u ^ 2 - d ^ 2) * S0) End Function Function BinomialAmerCallGamma(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double Dim V As Variant dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) V = EBinomialTree(S0, K, u, d, p, delta, n) BinomialAmerCallGamma = ((V(2) - V(1)) / ((u ^ 2 - 1) * S0) - (V(1) - V(0)) / ((1 - d ^ 2) * S0)) / ((u ^ 2 - d ^ 2) * S0 / 2) End Function
Improving the binomial methodMichael Carter
IntroductionAs the number of steps are increased, the binomial method converges to the true value (by the Central
Limit Theorem), but the convergence is slow and awkward. This is illustrated in the following graph for
an American out-of-the-money put option (S = 100, K = 90, r = 5 %, Σ = 30 %, T = 1 �2 ). The
horizontal axis represents the true value as calculated with a 50,000 step tree.
50 100 150 200 250
3.32
3.33
3.34
3.35
3.36
3.37
3.38
Binomial convergence - out-of-the-money put
This pattern repeats indefinitely as the number of steps is increased.
500 1000 1500 2000 2500
3.343
3.344
3.345
3.346
3.347
3.348
3.349
Binomial convergence - out-of-the-money put
The next graph illustrates the same option with K = 110.
50 100 150 200 250 300
13.37
13.38
13.39
13.4
13.41
13.42
Binomial convergence - in-the-money put
Clearly, there is a tradeoff between accuracy and efficiency (speed). Various methods are available for
improving the performance of the binomial model. These can be classified into two groups depending on
whether aimed at
æ improving accuracy
æ improving efficiency
Typically, success on one front implies a sacrifice on the other.
2 ImprovingBinomial.nb
Improving accuracy
à Successive averages
A popular technique in practice is to average the results of successive integers, n and n + 1.
50 100 150 200 250 300
3.33
3.34
3.35
3.36
3.37
Binomial convergence - successive averages
à Parameterization
In class, we used the simple parameterization
u = ãΣ Dt , d = ã
-Σ Dt , p =
ãHr-qL Dt
- d
u - d
where Dt = t/n. Some improvement in accuracy (at negligble computational cost) can be attained by
modifying the parameterization. Two possibilities are:
u = ãΝDt +Σ Dt , d = ã
ΝDt -Σ Dt , p =
ãHr-qL Dt
- d
u - d
and
u = ãDx, d = ã
-Dx, Dx = Σ2
Dt + Ν2
Dt2 , p =
1
21 + Ν
Dt
Dx
where Ν = r - q -1
2Σ
2.
Tian (1999) proposed a flexible binomial tree with
u = ãΛ Σ
2Dt +Σ Dt , d = ã
Λ Σ2
Dt -Σ Dt , p =
ãHr-qL Dt
- d
u - d
where Λ is an arbitrarily chosen tilt parameter. Λ can be chosen so as to make one node coincide with the
strike price, thus smoothing convergence.
Another proposal of Tian sounds promising. Convergence of the binomial tree price to the true price
requires that the first and second moments (mean and variance) of the (discrete) binomial distribution of
the nodes match or converge to those of the (continuous) process of the underlying asset, that is
ImprovingBinomial.nb 3
Another proposal of Tian sounds promising. Convergence of the binomial tree price to the true price
requires that the first and second moments (mean and variance) of the (discrete) binomial distribution of
the nodes match or converge to those of the (continuous) process of the underlying asset, that is
p u + H1 - pL d = ãr Dt
p u2+ H1 - pL d2
= ãIΣ
2+2 rM Dt
where ãIΣ2
+2 rM Dt is the second moment of asset returns when the log returns have variance Σ2Dt. (See
Chance 2007 for a thorough discussion.) Tian (1993) suggests matching in addition the third moment to
allow for the skew in the underlying distribution, requiring
p u3+ H1 - pL d3
= ã3 IΣ
2+ rM Dt
This gives three equations which can be solved for the three unknowns u, d and p. However, the outcome
is disappointing.
50 100 150 200 250 300
3.32
3.33
3.34
3.35
3.36
3.37
3.38
Binomial convergence - out-of-the-money put
Tian
à Magic numbers
The oscillations arises from the relationship between the strike price and the terminal nodes of the tree.
The graphs reveal that there are particular choices of n that minimize the error in their neighbourhood.
These magic numbers depend upon the precise parameters of the option. By tailoring the size of the tree
to the particular option, we might obtain more accurate results with smaller trees. This becomes especially
important in applying the binomial method to barrier options.
à Binomial-Black-Scholes
Convergence can be significantly enhanced using the Black-Scholes formula to evaluate the penultimate
nodes. This is known as the Binomial-Black-Scholes method.
4 ImprovingBinomial.nb
50 100 150 200 250 300
3.335
3.34
3.35
3.355
3.36
3.365
Binomial convergence - out-of-the-money put
Binomial Black-Scholes
à Richardson extrapolation
Richardson extrapolation is a method to improve an approximation that depends on a step size. Applied to
the binomial model, extrapolation attempts to estimate and incorporate the improvement of higher n. For
example, suppose we assume that errors decline inversely with n, so that
Pn1» P +
C
n1
Pn2» P +
C
n2
where P is the (unknown) true value, Pn1 and Pn2
are estimates with step size n1 and n2 respectively, and
C is an unknown constant. Solving for P, we have
P »
n2 Pn2- n1 Pn2
n2 - n1
In particular, when n2 = 2 n1 = n, we have
P »
n Pn -n
2Pn�2
n
2
= 2 Pn - Pn�2
which can be alternatively expressed as
P » 2 Pn - Pn�2 = Pn + HPn - Pn�2L
It is not helpful when applied to the pure binomial model, but is very effective when applied after Black-
Scholes smoothing.
ImprovingBinomial.nb 5
50 100 150 200 250 300
3.33
3.34
3.35
3.36
3.37
Binomial convergence - out-of-the-money put
BBS with Richardson Extrapolation
Smoothing and extrapolation also makes a dramatic improvement to the Tian moment-matching tree.
50 100 150 200 250
3.32
3.33
3.34
3.35
3.36
3.37
3.38
Binomial convergence - out-of-the-money put
Tian
In a recent contribution, Widdicks, Andricopoulos, Newton and Duck (2002) have applied extrapolation
to the peaks of the errors, as illustrated in the following diagram from their paper.
6 ImprovingBinomial.nb
à Control variable
A simple and effective technique is to use the binomial method to estimate the early exercise premium, as
measured by the difference between the estimated prices of identical American and European options.
This estimate is added to the Black-Scholes value, to give the estimated value of the American option.
P = p + HAb - EbL
where p is the Black-Scholes value, Ab is the binomial estimate of the American option, and Eb is the
binomial estimate of a corresponding European option. This is known as the control variable technique.
Rewriting the previous equation
P = Ab + Hp - EbL
we observe that the effectiveness of this approach depends upon the degree to which the binomial error in
the American option matches that of the European option in sign and magnitude. Chung and Shackleton
(2005) explore this issue, provide a methodology for determining the optimal control, and discuss other
potential control variables.
ImprovingBinomial.nb 7
50 100 150 200 250 300
3.335
3.34
3.35
3.355
3.36
3.365
Binomial convergence - out-of-the-money put
Black-Scholes control variable
à Comparison
50 100 150 200 250 300
3.335
3.34
3.35
3.355
3.36
Binomial convergence - out-of-the-money put
Binomial Black-Scholes
BBS with RE
Black-Scholes control variable
8 ImprovingBinomial.nb
50 100 150 200 250
3.32
3.33
3.34
3.35
3.36
3.37
3.38
Binomial convergence - out-of-the-money put
Binomial Black-Scholes
BBS with RE
Tian with smoothing and extrapolation
Improving efficiency
à Truncation
Andricopoulos et al. (2004) proposed curtailing the range of nodes that are computed by backward
induction, reporting a near doubling of speed with negligible loss of accuracy.
ImprovingBinomial.nb 9
à The diagonal algorithm
Curran (1995) proposed an innovative diagonal algorithm for evaluating binomial trees, which signifi-
cantly reduced the number of nodes that needed to be evaluated. He reported a 10 to 15-fold increase in
speed (with identical accuracy) over the corresponding standard tree. Note that this algorithm achieves a
pure increase in efficiency, returning the same result as the standard method. It is equally applicable to
extrapolation and control variable techniques.
10 ImprovingBinomial.nb
The diagonal algorithmMichael CarterCurran (1995) proposed an innovative diagonal algorithm for evaluating binomial trees, which significantlyreduced the number of nodes that needed to be evaluated. He reported a 10 to 15-fold increase in speed (withidentical accuracy) over the corresponding standard tree. Note that this algorithm achieves a pure increase inefficiency, returning the same result as the standard method.It is equally applicable to extrapolation and controlvariable techniques.
The diagonal algorithm depends upon two propositions regarding the evolution of option values in a binary tree.They can be illustrated diagramatically as follows:
Proposition 1If it pays to exercise the option in the next period, it pays to exercise immediately.
exercise á
exercise à
exercise
Proposition 2If it pays to hold the option at some time and asset price, then it pays to hold the option at the same asset price atevery earlier price.
? á à
hold hold à á
?
Proposition 1 applies provided q § r. The intuition is that, on average, the asset price will grow, and thereforethe implicit value will decline. If it is worth exercising in the future, it is worth exercising now. Proposition 2applies irrespective of the dividend yield (provided that u d = 1).
These properties of a binary tree enable two forms of acceleration in the tree.
æ By Proposition 2, once an entire diagonal of no exercise (or hold) nodes has been computed, we can jumpimmediately to the origin, since there are no further exercise nodes. We can evaluate the initial value of theoption by computing the discounted expected value of the implicit values along the no-exercise diagonal in amanner similar to computing expected values of the terminal nodes of a European option.
æ Provided that q § r, we can start evaluation along the diagonal starting immediately below the strike price, sincewe know that all nodes below this diagonal will be exercise nodes (Proposition 1), and therefore their value willbe equal to the implicit value.
Proof of Proposition 1: Let S denote the current asset price. Assume that both subsequent nodes are exercisenodes. Then the expected future value is
FV = Hp HK - u SL + H1 - pL HK - d SLL = K - Hp u + H1 - pL dL S
Recall that the risk-neutral probability p is such that
p u + H1 - pL d = ‰Hr-qLDt
Substituting, the expected future value at the subsequent node is
FV = K - ‰Hr-qLDt S
Provided that q § r, the expected future value is less than the current implicit value, that is
FV = K - ‰Hr-qLDt S § K - S
A fortiori, the discounted future value is less than the current implicit value. That is, ‰-rDtHK - SL is more thanyou can expect by waiting. Consequently, the option should be exercised immediately. Note that this is notnecessarily the case if q > r. In this case, expected capital gains are negative. So the option may become morevaluable.
Proposition 2 applies irrespective of the dividend yield. It depends upon the following lemma.
Lemma. Ceteris paribus, the value of an American option increases with time to maturity (Lyuu, Lemma 8.2.1).
Proof of lemma. Suppose otherwise. Sell the more expensive shorter option and buy the one with the longermaturity for a positive cash flow. Let t denote the time at which the shorter option is exercised or expires, and Pt
the value of the longer option at this time (assuming a put for example).
Case 1: Pt > max HK - St, 0L. Sell the longer option.
Case 2: Pt § max HK - St, 0L. In this case, the short option will be exercised. Offset this by exercising thelonger option.
In either case, we have a positive cash flow at time zero, and a nonnegative cash flow at time t.
Proof of Proposition 2. Let Pu and Pd denote the possible values of the option given an asset price of S, and letPu+2 and Pd
+2 denote the possible values of the option two periods later. By assumption, the holding value at time+2 is greater than the exercise value. That is
‰-rDtHp Pu+2 + H1 - pL Pd
+2 L ¥ K - S
By the lemma, the possible values are at least as great as they will be two periods later.
Pu ¥ Pu+2 and Pd ¥ Pd
+2
Therefore, the current holding is at least as great as the current exercise value.
‰-rDtHp Pu + H1 - pL Pd L ¥ ‰-rDtHp Pu+2 + H1 - pL Pd
+2 L ¥ K - S
2 DiagonalAlgorithm.nb
Timing VBA code executionMichael CarterAccurately timing execution speed on a multitasking computer is surprisingly difficult, since the CPU can beregularly interrupted by other processes. It is normal to record different times on repeated runs. So good practicewould be to average (or minimum) over a number of runs. It is also sensible to close other applications whenundertaking timing comparisons.
VBA contains a function Timer() that gives the number of seconds since midnight. By calling Timer() at thebeginning and end of a lengthy computation, it is possible to estimate the time taken by the function as follows:
Dim StartTime, EndTime, ComputationTime As Single
StartTime = TimerDo lengthy computation
EndTime = TimerComputationTime = EndTime − StartTime
More accurate timing can be achieved using the Windows operating system function GetTickCount(), whichreturns the time in milliseconds since the system was started. It is claimed to have a resolution of 10 millisec-onds (approximately). To use this function, it must first be declared as follows:
Declare Function GetTickCount Lib " Kernel32 " HL As LongDim StartTime, EndTime, ComputationTime As Long
StartTime = GetTickCountDo lengthy computation
EndTime = GetTickCountComputationTime = EndTime − StartTime
More information is available in the Microsoft tutorial note How To Use QueryPerformanceCounter to TimeCode.
Random number generationMichael CarterA pseudorandom number generator is a deterministic algorithm that generates a series of numbers in a
manner that appears to be random. It can be described by a recursive function
xi = f Hxi-1L
or more generally
xi = f Hxi-1, xi-2, … , xi-kL
f may be linear or nonlinear. The starting value x0 (or values) is called the seed. By default, the seed is often
set from the system clock.
The most common pseudorandom number generator is the linear congruential generator
xi = Ha xi-i + cL mod m
Rescaling the sequence by dividing my m gives a sequence in the ui = xi �m in the unit interval (0,1).
By careful choice of a, c and m, a sequence of period m can be obtained. Then, the resulting sequence of real
numbers will appear to be uniformly distributed on H0, 1L. However, not that inappropriate choice of a, c and
m can give very poor results.
Excel VBA's RND function is a linear congruential generator with a = 1 140 671 485, c = 12820163 and
m = 224, with a period of 224= 16, 777, 216. Starting with the default seed of 327680, the first values in
the sequence are: 0.705548, 0.533424, 0.579519, 0.289562, and 0.301948. (It is not possible to replicate this
generator in VBA or a spreadsheet because the internal representation of long integers is not accessible.)
Press, Teukolsky, Vetterling and Flannery (1992) is a valuable source of reliable random number generators
that have been well-tested, including the so-called "minimal standard" generator of Park and Miller which
uses
a = 75= 16 807 c = 0 m = 231
- 1 = 2 147 483 647
The period of this generator is 231- 2 » 2.1 � 109. For general use, they recommend ran1 which combines
the minimal standard generator with an additional shuffle to eliminate low-order serial correlations. Where a
longer period is required, they describe ran2 which combines two linear congruential generators with
different periods. They also describe some "quick and dirty alternatives" for less demanding applications.
An increasingly popular algorithm is the Mersenne twister. It has a period of 219,937- 1, is fast to compute
and is guaranteed to have equidistribution properties in at least 633 dimensions. An implementation for Excel
is freely available from NT Technologies.
Excel 2003 implemented a new RAND function based on the Wichman and Hill (1982) algorithm, which
combines three linear congruential generators. It is claimed to have a period of 1013. This is retained in Excel
2007 (see Description of the RAND function in Excel 2007 and in Excel 2003).
Normal random deviatesPseudo-random values from a distribution other than uniform (e.g. normal) are known as random deviates, to
distinguish them from uniform random numbers. Normal random deviates are almost always generated in a
two-step procedure, first generating normal random numbers and then transforming them in some way to
produce numbers that appear to come from a standard normal distribution. A variety of very clever methods
have been proposed for doing this transformation, of which the most common are the Box-Muller method
and normal inverse transformation.
� Box-Muller transformation
à Draw two random numbers u1 and u2 from U @0, 1D such that w = u12
+ u22
< 1
à Calculate c = -2log HwL
w
à z1 = c u1 and z2 = c u2 are normally distributed (N@0, 1D)
The Box-Muller transformation is very efficient. However, there are circumstances in which it is
inappropriate
à It should not be used with quasi-random number sequences, as it will scramble the low-
discrepancy property of the sequence.
à It should not be used in conjunction with a simple LCG generator (Bratley, Fox, Schrage 1982:
223)
� Inverse normal transformation
Let U be uniformly distributed on (0,1). Let FHxL be the distribution function of a continuous distribution.
Then F is strictly increasing on [0,1] and has an inverse F-1 such that
u = FHxL if and only if x = F-1HuL
Therefore, we can compute a random deviate X with distribution FHxL simply by setting X = F-1HU L.Then
PHX £ xL = PIF-1HU L £ xM = PHU £ FHxLL
But since U is uniform, PHU £ uL = u for 0 < u < 1. Therefore
PHX £ xL = PHY £ FHxLL = FHxL
So the problem of generating random deviates is essentially a numerial problem of inverting the distribution
function.
2 RandomNumberGeneration.nb
The cumulative distribution function of the standard normal distribution is
FHxL =
1
2 Π
à-¥
x
ãt2�2
â t
This has no closed form, and so much be computed numerically. Hull (2003) outlines a polynomial approxi-
mation that gives six place decimal accuracy (p. 248).
Similarly, the inverse CDF much be computed numerically. Moro (1995) proposed a rational polynomial
approximation that gives eight place decimal accuracy. It has become very popular in finance applications.
The NORMSINV function in Excel XP has been reported to still give slightly erroneous results (McCullough
and Wilson, 2002).
Multivariate normal distributionGiven two independent standard normal random variables z1 and z2, the following transformation generates
two correlated normal random variables with mean 0, standard deviation 1 and correlation coefficient Ρ.
x1 = z1
x2 = Ρ z1 + 1 - Ρ2 z2
� Exercise
Do it!
RandomNumberGeneration.nb 3
Furthermore, if we set
x1 = Σ1 z1 + Μ1
x2 = Σ2K Ρ z1 + 1 - Ρ2 z2O + Μ2
then Hx1, x2L has a bivariate normal distribution with mean HΜ1, Μ2L and covariance matrix
S =Σ1
2Ρ Σ1 Σ2
Ρ Σ1 Σ2 Σ22
, as can be confirmed by direct computation.
The transformation comes from the Cholesky decomposition of S =Σ1
2Ρ Σ1 Σ2
Ρ Σ1 Σ2 Σ22
.
Any positive definite matrix S can be factored into a product of triangular matrices S = H HT . In this case,
H =
Σ1 0
Ρ Σ2 1 - Ρ2
Σ2
with H HT= S
If Z is a vector of independent normal random variables, that is Z ~ N(0,I), then
X = H Z + Μ
is N(Μ,S) where S = H HT . Conversely, if X ~ N(Μ,S), then
Z = H-1HX - ΜL
is N(0,I).
4 RandomNumberGeneration.nb
Article ID: 828795 - Last Review: January 16, 2007 - Revision: 5.3
Description of the RAND function in Excel 2007 and in Excel 2003
This article describes the modified algorithm that is used in the random numbergenerator function, RAND, in Microsoft Office Excel 2007 and in Microsoft Office Excel 2003.
The RAND function in earlier versions of Excel used a pseudo-random numbergeneration algorithm whose performance on standard tests of randomness was not sufficient. Although this is likely toaffect only those users who have to make a large number of calls to RAND, such as a million or more, and not to be aconcern for almost every user, the pseudo-random number generation algorithm that is described here was firstimplemented for Excel 2003. It passes the same battery of standard tests.
The battery of tests is named Diehard (see note 1). The algorithm that is implemented in Excel 2003 was developed byB.A. Wichman and I.D. Hill (see note 2 and note 3). This random number generator is also used in the RAT-STATSsoftware package that is provided by the Office of the Inspector General, U.S. Department of Health and HumanServices. It has been shown by Rotz et al (see note 4) to pass the DIEHARD tests and additional tests developed by theNational Institute of Standards and Technology (NIST, formerly National Bureau of Standards).
Notes
The basic idea is that if you take three random numbers on [0,1] and sum them, the fractional part of the sum is itself arandom number on [0,1]. The critical statements in the Fortran code listing from the original Wichman and Hill articleare:
Therefore IX, IY, IZ generate integers between 0 and 30268, 0 and 30306, and 0 and 30322 respectively. These arecombined in the last statement to implement the simple principle that was expressed earlier: if you take three randomnumbers on [0,1] and sum them, the fractional part of the sum is itself a random number on [0,1].
Because RAND produces pseudo-random numbers, if a long sequence of them is produced, eventually the sequence willrepeat itself. Combining random numbers as in the Wichman-Hill procedure guarantees that more than 10^13 numberswill be generated before the repetition begins. Several of the Diehard tests produced unsatisfactory results with earlierversions of RAND because the cycle before numbers started repeating was unacceptably short.
Results in Earlier Versions of Excel
The RAND function in earlier versions of Excel was fine in practice for users who did not require a lengthy sequence ofrandom numbers (such as a million). It failed several standard tests of randomness, making its performance an issuewhen a lengthy sequence of random numbers was needed.
Results in Excel 2003
A simple and effective algorithm has been implemented. The new generator passes all standard tests of randomness.
Conclusions
Power users of RAND who require lengthy sequences of random numbers are better offwith the new generator of Excel 2003. Other users should be undeterred from using RAND in earlier versions of Excel.
SUMMARY
MORE INFORMATION
The tests were developed by Professor George Marsaglia, Department of Statistics, Florida State University andare available at the following Web site: http://i.cs.hku.hk/~diehard (http://i.cs.hku.hk/~diehard)
Wichman, B.A. and I.D. Hill, Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator,Applied Statistics, 31, 188-190, 1982.Wichman, B.A. and I.D. Hill, Building a Random-Number Generator, BYTE, pp. 127-128, March 1987.Rotz, W. and E. Falk, D. Wood, and J. Mulrow, A Comparison of Random Number Generators Used in Business,presented at Joint Statistical Meetings, Atlanta, GA, 2001.
C IX, IY, IZ SHOULD BE SET TO INTEGER VALUES BETWEEN 1 AND 30000 BEFORE FIRST ENTRY IX = MOD(171 * IX, 30269) IY
= MOD(172 * IY, 30307) IZ = MOD(170 * IZ, 30323) RANDOM = AMOD(FLOAT(IX) / 30269.0 + FLOAT(IY) / 30307.0 +
FLOAT(IZ) / 30323.0, 1.0)
Description of the RAND function in Excel 2007 and in Excel 2003 http://support.microsoft.com/kb/828795/en-us
1 of 2 05/03/2009 10:52
Help and Support©2009 Microsoft
For more information about an issue that was documented to occur in RAND, click thefollowing article number to view the article in the Microsoft Knowledge Base: 834520 (http://support.microsoft.com
/kb/834520/ ) The RAND function returns negative numbers in Excel 2003
APPLIES TO
Keywords: kbfuncstat kbfunctions kbinfo KB828795
Get Help Now
Contact a support professional by E-mail, Online, or Phone
REFERENCES
Microsoft Office Excel 2007Microsoft Office Excel 2003
Description of the RAND function in Excel 2007 and in Excel 2003 http://support.microsoft.com/kb/828795/en-us
2 of 2 05/03/2009 10:52
1/24/2012
1
Numerical Recipes
Neveruse a LCG generator (alone)use a generator with a period < 2 ≈ 2 × 10use a built‐in generator in C or C++
Avoidgenerators designed for cryptographic usegenerators with period > 10
Numerical Recipes
Alwaysuse a generator that (correctly) combines two unrelated methods
If criticalconfirm results with a different generator
Normal transformation
Don’tuse Box‐Mueller transformation
Insteaduse Moro transformation or alternative
Excel’s NormSInv now OK
Excel
Set seed? Volatile? QualityRnd() Y N WeakRand() N Y UnknownPRand Y Y &N Good
PRand
‘Every‐day’ RNG from Numerical Recipes
Combines 64‐bit XorShift with LCG
Period 2 ≈ 2 × 10
NormMInv () – Moro inverse normal CDF
Uniform Normal Seed
Volatile URand() NRand() SetSeed
Non‐volatile URandNonV() NRandNonV() SetSeedNonV()
Comparative efficiency
NormSInv NormMInv
Rand 3.4 – 4.2 3.6
URand 4.2 3.6
NRand 3.6
vbaNRand 4.7
Time required for 1 million simulations (minutes)
2.1 GHz Pentium
f f f i * o N r E c A R t o s l M U t A T t o N
THE FULL MoNITETronsforming quosi'rondom numbers into useful Monte Corlosimulotion voriobles con be o bit of o gomble. Boris Moroexploins how fo do it quickly ond occurotely
I I onte Carlo simulat ion is often used forf V I pricing European-slyle derivative secu-ri t ies n' i th complicated pavoffs, especiai lvthose dependent upon severa l Bror t 'n ianmotions.
I t invo lves s imulat ing the path o f theunderlf ing security price a number of timesand calculating the expected option value asan average over all prices obtained at maturi-ry. The path depends on the drift and volatil-ity, n'hich measures the size of normalh' dis-tributed random fl uctuations.
In practice, one first has to calculate pseudo-random numbers in the interval (0,1) - or aunit h1'percube, (0,1)s, for an s-factor model -and then transform them into a set of Gaussiandeviates using the Box-Muller algorithm (seeFlannery, Press and Teukolsky (1992), page289 ) . Un fo r t una te l y , i f t he Mon te Ca r l omethod is based on standard pseudo-randomnumbers, it is very slow because the pricingerror decreases only with 1/{N where N is thenumber of simulation runs.
Recently, Jov, Bo1'le and Tan (1994) haveshown that the efficienry of the method canbe greatly increased if quasi-random rather
than pseudo-random numbers are used (Fox(1986)). These are determinist ic sequencesdefined on a unit hvpercube n'hich have aIon'discrepanc\/, thails, they fill the domainnearl;' uniformll'. Unlike a pseudo-randomsequence of finite length, there can be no clus_ters of points, so u'hen the function is calcu-lated by averaging over such a set, the esti-mate is more accurate.
In a Monte Carlo simulation, such num-bers must be transformed into Gaussian devi-ates so that tl're low discrepancy of the origi-nal sequence is preserved. The Box-Mullermethod and its variants scramble the quasi-random sequence and therefore should not beused for that purpose.
Instead, one has to computex(Y), of the cumulative normalfunction
the inverse,distr ibution
t '1 x
Y(x) = -= te 'd t
^l2n -*
u'hich can be done using Newton,s iteration.in practice, howevet this is too slow to be use_ful. The approximation'derived by Hill and
Relotive errors in the foil of distribution1 . 0
I O-rt0-:
I0- l
l 0 *
l 0 -5
l0 -6
l 0 t
l0-8
l0 -e
l 0 - , 0
l 0 - "
l 0 - , r
l 0 - , ,
I 0 - , 0
2 .6 .104 I . 9 . 1 0 - 8 1 . 0 . 1 0 - r r
( :
Davis (1973) also gives the result correct tomachine precision but is not much faster thanNen'ton's ntethod. Rational approximationsfor x(Y) n'ere deriyed by Odeh and Evans
LT.974) and by Beastey and Springer (1977).The l a t t e r p rog ram (ppND) i J f as t bu tbecomes inaccurate in the tails of the distrib_ution.
We obta ined an approx imat ion to theinverse of the cumulative normal distributionfunction n'hich has a high acoracy for all y inthe inten'al [10-r:, !-Lo-L2]and ii faster thanstandard implementations of the Box-Mullermethod.
We shall use symmetry of the inverse ofthe cumulative normal distribution and con_sider onlv the interval (0.5, 1). For smalldeviations (Y < 0.92) we use the functionderived by Beasley and Springer:
H 2 ^La"y
x(y)=v-s* . _ (1)
1+ I bny2 "
w h e r e Y : Y - 0 . 5 .For Y > 0.92 \^/e approximate x(y) by a
polynomial:
n=8
x (Y )= I . n .nn=0
where: . -
kr in(- ln(1- y)) + k, ; an, bn and cnare suitabh' chosen coefficients and the con_stants k, and lg are such that z = _! whenY = 0.92 andz : lwheny = 1 - !O-r2 .
The rat ional approximation (1) has thelargest absolute error of 3x10-s in the interval[0.5, 0.92]. In the tails of the distribution, ouralgori thm retains this accuracy for up toseven standard deviations. High_order poly_nomial approximations which have 14_digitaccuracy in the interval [10-rs, t-tg-rs; cineas i ly be const ructed but they run muchslower.
The table shoH's the time needed for gen-erating 107 double precision normally dis_tributed numbers on a Sun Sparc 10-30 usingthree programs: GASDEV (Box-Muller algol
11B as implemented by Flanne ry et i ly,PIND (Beasley and Springer) and CNDEVu'hich is based on our algorithm. In the firstset of runs, uniformly distributed pseudo_
RISKVOL 8/NO 2/FEBRUARY
. l a M O N T E C A R L O S I M U L A T I O N
randonl numbers from the inten'al (O,1) n'eregenerated by the function DRAND48. In thesecond set of runs (PPND and CNDEV) nor-mal deviates \^'ere produced from uniformlydrarvn numbers from the interval (0,1), ie,rr'ithout function DRAND48. I
Boris Moro is o researcher for TMG FinonciolProducts in Greenwich, Connecticut. He thonks
Professor Phelin Boyle for comments ondsuggesfions
THE CNDEV.C PROGRAM#include <moth.h>
Are you taking enough care?
RISkcareThe Derivative Development & Consultancy Specialists
Experienced independent advice on hedging strategies, investment opportunities, individual & portfolio pricing and ongoing risk analysis.For users, market makers and software developers of derivative products, Riskcare provides
the mathematics, technology, training and business awareness to suit your requirements.
Complex structures need care.Telephone: 44 (0) l7t 251 4748 or Fax: 251 2663
Complete confidentiality guaranteed.
RISK vor. B/No 2/IEBRUARY r ee5
Simulating asset pricesMichael CarterAs a computation method, simulation has two major advantages
æ it can easily deal with multiple random factors, for example random interest rates or volatility, oroptions on multiple assets
æ it can easily incorporate more alternative asset processes, such as jumps or non-normal stochasticprocesses
The major disadvantage is that it is computationally very intensive.
Our starting point is the observation that periodic returns are approximately normally distributed around atrend
CNX Nifty 1990-2006
50010001500200025003000
Prices
-0.1-0.05
00.050.1
Returns
-4 -2 2 4
0.05
0.1
0.15
0.2
0.25
CNX Nifty distribution v. normal
That is, we assume
lnJ StÅÅÅÅÅÅÅÅÅÅÅÅÅSt-1
N ~ NIn Dt, sè!!!!!
Dt M
with mean and variance are proportional to Dt. This implies that the distribution of asset prices is lognormal.This process can be simulated by computing
ln St = ln St-1 + n Dt + sè!!!!!
Dt zt
This equation distinguishes two components of the periodic return - the steady drift n Dt together with thestochastic variation s è!!!!!
Dt zt. The previous equation can be rewritten as
St = ‰n Dt + sè!!!!!!
Dt z St-1
where zt is a standard normal random variable. Since the exponential function is nonlinear
E@ StD = ‰Hn + s2ê2L t S0 = ‰m t S0
where m = n + s2 ê 2 is the expected return of the asset.
More formally, the fundamental assumption underlying Black-Scholes and most other financial models isthat asset prices follow a particular stochastic process called geometric Brownian motion (GBM)
(1)dS = m S dt + s S dz
or
dSÅÅÅÅÅÅÅÅÅS
= m dt + s dz
This implies that the logarithm of the stock price follow
(2)d ln S = n dt + s dz
where n = m - 1ÅÅÅÅ2 s2. The derivation of (2) from (1) is an application of Ito's lemma (Hull: 2003:232-233,Luenberger 1998:313).There are two methods of simulating stock prices, depending upon whether we take a discrete version of (1)or (2).
A discrete version of (1) is
dS = St - St-1 = m St-1 Dt + s St-1 et è!!!!!
Dt
where et is a standard normal random variable, so that
(3)St = I1 + m Dt + s et è!!!!!
Dt M St-1
This is the approach taken by a Hull (2003: 224).
A discrete version of (2) is
d ln S = lnHStL - lnHSt-1L = n Dt + s et è!!!!!
Dt
where et is a standard normal random variable and n = m - s2 ê 2, so that
lnHStL = lnHSt-1L + n Dt + s et è!!!!!
Dt
or
(4)St = ‰n Dt + s et è!!!!!!
Dt St-1
In practice, the second approach is preferred because it is more accurate (although it is computationallyslower). Whereas (3) is only true in the limit as Dt Ø 0, equation (4) is true for all Dt.
2 SimulatingPrices.nb
Monte Carlo simulation
Statistical foundations
The value of a derivative instrument is discounted expected value of the payoff
V0 = EAã-r T V HSt, 0 £ t £ TLE
where the expectation is taken with respect to the risk-neutral distribution. Monte Carlo simulation can be used to estimatethis expected value by averaging over repeated samples from the assumed distribution of the underlying.
V0 = ã- r T E@V D » ã
- r TV1 + V2 + … Vn
n
where Vi is the value of the of the ith realization under the risk-neutral distribution.
The statistical foundations of Monte Carlo simulation rest on the two fundamental theorems of probability. Assume thatHX1, X2, ..., XnL are independent and identically distributed random variables with mean Μ and variance Σ2.
� First fundamental theorem (Law of large numbers)
The sample average converges to the mean, that is
X =
X1 + X2 + ... + Xn
n� Μ
� Variance of independent random variables
Since the variables are independent, their variance is additive, that is
Var@X D = VarB1
nX1F + VarB
1
nX2F + ... + VarB
1
nXnF
=
1
n2Var@X1D +
1
n2Var@X2D + ... +
1
n2Var@XnD
=
1
nVar@X D =
Σ2
n
so that ΣX = Σ � n .
� Second fundamental theorem (Central limit theorem)
Whatever the distribution of X , the sample mean X converges to a normal distribution. That is, for large samples,
X ~ NJΜ, Σ2
nN.
Variance reduction
The principal methods of variance reduction are:
� Antithetic variables
A sufficient condition for variance reduction with uniform or normal random variables is that the outcome function ismonotonic.
� Control variables
With Α selected optimally, that is Α*= Cov@Y , CVD �Var@CVD, the ratio of the variances of the control variable estimator to
the standard estimator is
VarAY - Α*ICV - E@CVDME
Var@Y D= 1 - ΡY ,CV
2
æ The effectiveness of the control variable method depends solely upon the degree of correlation betweentarget variable and the control variable (the sign is absorbed into Α).
æ In practice, the population moments are unknown and Α* is estimated from the sample. This introducessome bias in finite samples. This bias can be eliminated by estimating Α on a different random sample.Typically, the bias is so small that this is not worthwhile.
� Importance sampling
Other methods include:
� Moment matching
� Stratified sampling
� Latin hypercube sampling
Comments:
æ Moment matching can also be achieved by using the moments as control variables. Boyle, Broadie andGlasserman, BS show that including using moments as control variables is asymptotically better thanmoment matching.
æ Low-discrepancy (quasi-random) sequences automatically achieve first-moment matching, stratified andLatin hypercube sampling.
Simulating the Greeks
The delta of a derivative is
D =
d V
d S0
With simulation, this can be estimated by the average of the finite-differences over a sample n of replications.
2 MonteCarloSimulation.nb
D�
=
1
nã
V�
HS0 + ΕL - V�
HS0L
Ε
If we use different random samples for estimating V�
HS0L and V�
HS0 + ΕL, the variance of D�
becomes very large as Ε becomessmall.
A better estimate of D is generally obtained by using the same random numbers in estimating both V�
HS0L and V�
HS0 + ΕL. The
variance of D� is
Var@D�
D =
VarAV�
HS0LE + VarAV�
HS0 + ΕLE - 2 CovAV�
HS0L, V�
HS0 + ΕLE
Ε2
Provided that V�
HS0L and V�
HS0 + ΕL are positively correlated, the estimate obtained from common random numbers will have alower variance.
� Pathwise derivative
The pathwise derivative decomposes the total derivative into two components.
d V�
d S0
=
d V�
d ST
d ST
d S0
Assuming geometric Brownian motion and the risk-neutral distribution
ST = S0 ãKr-q-
Σ2
2O T + Σ T Z
so that
d ST
d S0
= ãKr-q-
Σ2
2O T + Σ T Z
=
ST
S0
The discounted payoff of a vanilla European call option is ã-r T max HST - K, 0L
d V�
d ST
= ã-r T
d
d S0
max HST - K, 0L =ã
-r T , ST > K
0, ST < K
The function is not differentiable at ST = K, but this is a zero-probability event. Combining the two factors, the pathwiseestimator of delta for a vanilla European call option is
d V�
d S0
=
d V�
d ST
d ST
d S0
= ã-r T
ST
S0
I 8SHTL > K<
where I 8SHTL > K< equals one if SHTL > K and zero otherwise. This estimator can easily be computed by simulation. Aminor modification of this procedure can be used to estimate vega, but it is generally inapplicable to estimating gamma.
MonteCarloSimulation.nb 3
The Cholesky decomposition
Michael Carter
� Preliminaries
Multivariate normal distribution
Given two independent standard normal random variables Z1 and Z2, the following transformation generates two correlatednormal random variables with mean 0, standard deviation 1 and correlation coefficient Ρ.
X1 = Z1, X2 = Ρ Z1 + 1 - Ρ2 Z2
Furthermore, if we set
X1 = Μ1 + Σ1 Z1 , X2 = Μ2 + Σ2K Ρ Z1 + 1 - Ρ2 Z2O
then HX1, X2L has a bivariate normal distribution with mean HΜ1, Μ2L and covariance matrix S =Σ1
2Ρ Σ1 Σ2
Ρ Σ1 Σ2 Σ22
, as
can be confirmed by direct computation.
Generalizing, if Z is a n-vector of independent normal random variables (Z ~ N(0,I)) and an L is an n´n square matrix, then
X = Μ + L Z
is N HΜ, SL where S = L LT . Therefore, in order to produce a random sample from a multivariate normal distribution, we
need to find a matrix L with the property that L LT= S. Then
E@XD = Μ + A E@ZD = Μ
EAHX - ΜL HX - ΜLT E = EAL ZHL ZLT E = EAL Z ZT LT E = L EAZ ZT E LT= L I LT
= L LT
The Cholesky decomposition
Any square, symmetric, positive-definite matrix A can be factored into the product of two triangular matrices
A = L LT
For example, the Cholesky decomposition of the matrix2 1
1 2is
2 0
1
2
3
2
since
2 0
1
2
3
2
2 1
2
0 3
2
=2 1
1 2
Now consider the general case for n = 3
L LT=
l1,1 0 0
l2,1 l2,2 0
l3,1 l3,2 l3,3
l1,1 l2,1 l3,1
0 l2,2 l3,2
0 0 l3,3
=
l1,12 l1,1 l2,1 l1,1 l3,1
l1,1 l2,1 l2,12
+ l2,22 l2,1 l3,1 + l2,2 l3,2
l1,1 l3,1 l2,1 l3,1 + l2,2 l3,2 l3,12
+ l3,22
+ l3,32
=
a1,1 a2,1 a3,1
a2,1 a2,2 a3,2
a3,1 a3,2 a3,3
Equating the resulting matrices element by element, we can solve the system row by row
l1,12
� a1,1
l1,1 l2,1 � a2,1
l1,1 l3,1 � a3,1
l1,1 l2,1 � a2,1
l2,12
+ l2,22
� a2,2
l2,1 l3,1 + l2,2 l3,2 � a3,2
l1,1 l3,1 � a3,1
l2,1 l3,1 + l2,2 l3,2 � a3,2
l3,12
+ l3,22
+ l3,32
� a3,3
to give
l1,1 = a1,1 ,
l2,1 =
a2,1
l1,1
, l2,2 = a2,2 - l2,1
l3,1 =
a3,1
l1,1
, l3,2 =
a3,2 - l2,1 l3,1
l2,2
, l3,3 = a3,3 - l3,12
- l3,22
The general case is
li,i = ai,i - âj=1
i-1
li, j2
l j,i =
1
li,iai, j - â
k=1
i-1
li,k l j,k , j = i + 1, …, n
Implementation
2 Cholesky.nb
Asian options
IntroductionAsian options are popular in currency and commodity markets because
æ they offer a cheaper method of hedging exposure to regular periodic cash flows
æ they are less susceptible to manipulation of the spot market.
There are two classes of Asian options
æ average price options (payoff max IS - K, 0M or max IK - S, 0M)
æ average strike options (payoff max IST - S, 0M or max IS - ST , 0M)
Average strike options are sometimes called floating strike options. Vanmaele et al. (2006) provide transforma-tions between average price and average strike options.
In addition, there are two ways of calculating the average S - arithmetic and geometric.
A =
1
m + 1HS0 + S2 + … + SmL ³ G = HS0 �S1 �… �SnL
1
m+1
Note that by convention, the average includes the price on the first day of averaging. Jensen's inequality statesthat the arithmetic mean is larger than the geometric mean, with equality if and only if the observations areequal.
Unfortunately, most Asian options in practice are based on arithmetic averaging, while precise results arereadily available only for geometric averages. In practice, we use the price of an equivalent geometric averageoption as a lever to price its arithmetic counterpart.
� Upper and lower bounds
Jensen's inequality implies that the value of an arithmetic Asian option is bounded below by its geometriccounterpart.
cGHTL £ cAHTL
We can also develop useful upper bounds. The value of the arithmetic Asian option is
cAHTL = ã-r T EAIST - KM
+E
where expectation is taken with respect to the risk neutral distribution. This implies
cAHTL = ã-r T EAIST - KM
+E
= ã-r T EB
1
m + 1âi=0
m
SHtiL - K+
F
=
1
m + 1ã
-r T EB âi=0
m
HSHtiL - KL+
F
£
1
m + 1ã
-r T EBâi=0
m
HSHtiL - KL+F
=
1
m + 1âi=0
m
ã-r T E@HSHtiL - KL+D
£
1
m + 1âi=0
m
cHtiL
The value of an Asian option is less than the value of a portfolio of intermediate options with the same strikeprice. Furthermore, absent dividends, cHtiL £ cHTL and therefore
cGHTL £ cAHTL £
1
m + 1âi=0
m
cHtiL £ cHTL
� Put-call parity
Put-call parity applies to European average price options, whether arithmetic or geometric. To see this, observethat the difference between the terminal values of a put and call option is
cH0L - pH0L = IS - KM+
- IK - SM+
= S - K
Taking risk-neutral expectations
cHTL - pHTL = ã-r T IE@SD - KM
The expectation E@SD can be calculated exactly for both arithmetic and geometric averages. Consequently, itsuffices to compute call values and derive the values of puts from (1).
Geometric average optionsSuppose that St is lognormal with mean Ν t and variance Σ2 t. Then the geometric average
Gm = HS0 �S1 � … � SmL1
m+1
is lognormal with mean
(1)E@GmD = S0 ãKΝ +
2 m+1
m+1
Σ2
6O
T
2
and variance
(2)V @log GmD =
2 m + 1
m + 1
Σ2
6T
See Appendix. Under the risk neutral distribution Ν = r - q - Σ2 �2. Substituting in (1)
E@GmD = S0 ãKr -q -
m+2
m+1
Σ2
6O
T
2
By the fundamental theorem (Hull 2003: 262), the value of a European call option on Gm is
cG = ã-r t E@maxHGm - K, 0LD = ã
-r tHEHGmL NHd1L - K NHd2LL
where NHL denotes the standard normal distribution function and
d1 =
lnI E@GmDK
M +V 2
2T
V, d2 =
lnJ E@GmDK
-V 2
2TN
V
With continuous averaging, m � ¥ and
E@G¥D = S0 ãKr -q -
Σ2
6O
T
2
with
V @log G¥D =
Σ2
3T
2 Asian.nb
Making the substitution
q*=
1
2r + q +
Σ2
6
the expected value is
E@G¥D = S0 ãKr -q -
Σ2
6O
T
2 = S0 ãHr - q*L T
which is the same as the expected value of ST assuming the dividend yield q*.
Consequently, the value of European geometric Asian option with continuous averaging is given by Black-Scholes formula with the substitutions (Hull 2003, p. 444)
dividend yield =
1
2r + q +
Σ2
6volatility =
Σ
3
This also provides a useful approximation for large n.
Moments of the geometric mean under the risk neutral distribution
� Discrete Continuous
E@GD S0 ãKr -q -
m+2
m+1
Σ2
6O
T
2 S0 ãKr -q -
Σ2
6O
T
2
V @log GD I 2 m + 1
m+1M Σ
2
6T Σ
2
3T
Arithmetic average optionsThe arithmetic average of lognormal random variables is not lognormal, and its precise distribution has provedintractable. There are three practical approaches to accurate valuation of arithmetic Asian options:
æ analytical approximation
æ simulation using the geometric average option as a control variate
æ a modified binomial method
� Analytical approximation
Although the distribution of the arithmetic average is A is intractable, its moments EAAkE can be readily
calculated. This has spurred a variety of approximation methods.
� Moments of A
Observe that
S1 = S0 R1
S2 = S1 R2 = S0 R1 R2
Sm = S0 R1 R2 … Rn
âi=0
m
Si = S0H1 + R1 + R1 R2 + … + R1 R2 … RmL
where the gross returns Ri are independent and identically distributed. Let a = E [Ri] . Then
EBâi=0
m
SiF = S0 E@1 + R1 + R1 R2 + … + R1 R2 … RnD = S0I1 + a + a2+ … + amM = S0
1 - am+1
1 - a
Under the risk neutral distribution, a = E@RiD = ãHr-qL T�m. Therefore
Asian.nb 3
E@AmD =
1
m + 1EBâ
i=0
m
SiF =
S0
m + 1
1 - ãHr-qL THm+1L�m
1 - ãHr-qL T�m
For continuous averaging, we have (Hull 2003: 444)
E@A¥D = limn® ¥
E@AnD = limm® ¥
S0
m + 1
1 - ãHr-qL THm+1L�m
1 - ãHr-qL T�m
=
IãHr-qL T
- 1M
Hr - qL TS0
� An upper bound
We have previous observed that G £ A (Jensen's inequality). This implies that CG £ GA, but also that thedifference in payoffs is
HA - KL+- HG - KL+
=
0, G £ A £ K
A - K, G £ K £ A
A - G, K £ G £ A
In every case we have
HA - KL+- HG - KL+
£ A - G
Taking risk-neutral expectations
CAHTL - CGHTL £ ã-r T HE@AD - E@GDL
and therefore
CGHTL £ CAHTL £ CGHTL + ã-r T HE@AD - E@GDL
Example. For S0 = 50, r = 10 %, q = 0, Σ = 40 %, T = 1 and n = 250, the expected values of the arithmeticand geometric means are 52.59 and 51.86 respectively. The value of geometric average call option is 5.13.Therefore, we can deduce that the value of an arithmetic average option CA is bounded as follows
5.13 £ CA £ 5.13 + ã-0.1 H52.37 - 51.87L = 5.79
� Simple modified geometric
The simplest analytical approximation assumes that arithmetic average is lognormally distributed with mean
E@AnD, but assuming the standard deviation of ln An is Σg= Σ/ 3 . By the fundamental theorem (Hull 2003:
262)
CA = ã-r T E@maxHS - K, 0LD = ã
-r T HE@AnD NHd1L - K NHd2LL
where
d1 =
lnHE@AnD �KL + ΣG2 T �2
ΣG T, d2 = d1 - ΣG T
� Other analytical approximations
Extensions of the previous idea include:
æ Levy: Assume An is lognormally distributed with true mean and variance (Hull 2003: 444)
æ Turnbull and Wakeman: Approximate the actual distribution of An using Edgeworth expansions.
There is evidence that Levy approximation is not generally more accurate (and may be less accurate) than thesimple modified geometic method, and that the Turnbull and Wakeman method is no more accurate unlesshigher (third and fourth) moments are used (James 2003: 215-216).
� Bounds
Earlier, we developed upper and lower bounds for an arithmetic average option
4 Asian.nb
CGHTL £ CAHTL £ CGHTL + ã-r T HE@AD - E@GDL
In a recent contribution, Nielsen and Sandmann (2003) develop tighter upper and lower bounds that are easilycalculated. In a numerical analysis of 32 options, they find that the Levy approximation satisfies the bounds inonly eight cases while the Turnbull and Wakeman method satisfies the bounds in only seven cases. Theyconclude "more information is gained from the easily calculated bounds than from the pricing approximations"in their sample. They also show how the bounds can be used to calculate hedge parameters.
Asian.nb 5
Barrier optionsMichael Carter
à Preliminaries
IntroductionA barrier option is similar to a vanilla call or put, but the final payoff depends on whether or not the underly-ing price achieves a particular level (the barrier) during the life of the option. The vanilla option may eitherknock-in or knock-out when the barrier is crossed. Sometimes, a rebate is paid if the vanilla option lapsesbecause the barrier condition is not met. Barrier options are one of the most commonly traded exoticoptions, since they are less expensive than vanilla calls and puts.
European barrier optionsFollowing Björk (1998), we consider the class of derivatives on a single underlying asset where the payoffFHST L depends on ST (and not St, t < T). Throughout, we consider continuous monitoring. Since a portfoliocomprising a knock-in and a knock-out option with the same barrier and strike will always pay the same as avanilla option (e.g. HS - KL+), we can use the in-out parity condition
Knock - in option + Knock - out option = Vanilla option
to deduce the price of one type from the other. A knock-out option can be either down-and-out or up-and-out, depending upon the relationship between the current asset price S and the barrier H .
85 90 95 100 105
2
4
6
8
10
12
14European barrier call options
Barrier
Strike
Vanilla
Down-out
Down-in
Theorem 1 (Down-and-out) Suppose that V HS, F L denotes the value of a European derivative contractF HST L on S maturing at time T. The value of a down-and-out version of the derivative is
VdoHS0, FL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS
, FHy{zzz
where H < S0 is the barrier, n = r - q - s 2 ê2 and
FH HSL = 9 FHSL,0,
for S > Hfor S § H
= FHSL . IHS > LL
FH is F truncated below H .
Proof: Björk 1998: 185-186
ü Proof
ü
Corollary 1 (Down-and-in) Suppose that V HSL denotes the value of a European derivative contract on Smaturing at time T. The value of a down-and-in version of the same option is
VdiHS0, FL = V HS0, FH L + J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, FHy{zzz
where H < S0 is the barrier, n = r - q - s 2 ê2 and
FH HSL = 9 FHSL,0,
for S < Hfor S ¥ H
= FHSL . IHS < LL
Note that Vdi depends upon both FH and FH .
Proof Risk neutral valuation implies that the value functional V is linear in F. Since F = FH + FH
V HS0, FL = V HS0, FH L + V HS0, FH LBy in-out parity
Vdi HS0, FL = V HS0, FL - Vdo HS0, FL
= V HS0, FH L + V HS0, FH L -i
kjjjjj V HS0 FH L - J H
ÅÅÅÅÅÅÅÅS
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS
, FHy{zzz y
{zzzzz
= V HS0, FH L + J HÅÅÅÅÅÅÅÅS
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS
, FHy{zzz
ü Example: Vanilla European call with H £ K
The payout for a vanilla European call is
FHSL = max HS - K, 0LThe truncated payouts FH and FH depend upon whether H § K or H > K. For H § K
FH HSL = FHSL = max HS - K, 0L and FH HSL = 0
2 Barrier.nb
H K
FH FH
and therefore
V HS0, FH L = V HS0, FL and V HS0, FH L = 0
By Theorem 1 and Corollary 1
VdoHS0, FL = V HS0, FL - J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Fy{zzz = cHS0, KL - J H
ÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
cikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz
VdiHS0, FL = V HS0, 0L + J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Fy{zzz = J H
ÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
cikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz
where cHS0, KL is the Black-Scholes formula for a vanilla European call. Note that we can directly use theBlack-Scholes formula for valuing these options. How would you obtain the hedge parameters?
ü Exercise
Show that this is the same as the formula given in Hull (2003:439) for the value of a vanilla down-and-incall, namely
cdi = S0 ‰- q T J HÅÅÅÅÅÅÅÅS0
N2 l
NHyL - K ‰-r T J HÅÅÅÅÅÅÅÅS0
N2 l-2
NIy - s è!!!!T M
where
l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 , y =lnHH2 ê HS0 KLL + Hr - q + s2 ê 2LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=lnHH2 ê HS0 KLLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
+ l s è!!!!T
ü Answer
cdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
cikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz
= J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
‰-q T N HyL - K ‰-r T N Iy - s è!!!!T My
{zzz
= S0 ‰-q T J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2 +2
N HyL - K ‰-r T J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
N Iy - s è!!!!T M
Barrier.nb 3
where
y =lnI H2
ÅÅÅÅÅÅÅÅÅÅÅS0 K M + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
and n = r - q -s2ÅÅÅÅÅÅÅÅÅÅ2
Define
l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 =r - q - s2 ê 2 + s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 =n
ÅÅÅÅÅÅÅÅÅÅs2 + 1
so that
2 l =2 nÅÅÅÅÅÅÅÅÅÅs2 + 2
Substituting, we derive the formula in Hull.
cdiHS0, K, HL = S0 ‰-q T J HÅÅÅÅÅÅÅÅS0
N2 l
NHyL - K ‰-r T J HÅÅÅÅÅÅÅÅS0
N2 l-2
N Iy - s è!!!!T M
l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2
y =lnI H2
ÅÅÅÅÅÅÅÅÅÅÅS0 K M + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=lnI H2
ÅÅÅÅÅÅÅÅÅÅÅS0 K M + s2 l TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
=lnI H2
ÅÅÅÅÅÅÅÅÅÅÅS0 K MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
+ s lè!!!!T
Alternatively, we can start with the formula in Hull. Factoring out the power, this can be written as
cdi = J HÅÅÅÅÅÅÅÅS0
N2 l-2
ikjjjj S0 ‰- q T J H
ÅÅÅÅÅÅÅÅS0
N2
NHy1L - K ‰-r T NIy1 - s è!!!!T My
{zzzz
= J HÅÅÅÅÅÅÅÅS0
N2 l-2
ikjjjikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
y{zzz ‰- q T NHy1L - K ‰-r T NIy1 - s
è!!!!T My{zzz
= J HÅÅÅÅÅÅÅÅS0
N2 Hl-1L
cikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz
where cI H2ÅÅÅÅÅÅÅÅS0
, KM is the value of a vanilla call with underlying price H2 ê S0 and strike K
l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2
y =lnI H2
ÅÅÅÅÅÅÅÅÅÅÅS0 K MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
+ s lè!!!!T
Furthermore
l - 1 =r - q + s2 ê2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 - 1 =r - q + s2 ê2 + s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 =r - q - s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 =n
ÅÅÅÅÅÅÅÅÅÅs2
where n = r - q - s2 ê 2.Therefore, the value of a down-and-in call with barrier H § K is
cdi = J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
cikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz
with
4 Barrier.nb
y =lnI H2
ÅÅÅÅÅÅÅÅÅÅÅS0 K MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
+ s lè!!!!T =
lnI H2ÅÅÅÅÅÅÅÅÅÅÅS0 K M + s2 l T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T=
lnI H2ÅÅÅÅÅÅÅÅÅÅÅS0 K M + Hr - q + s2 ê 2L T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T
which is the same as corollary 1.
ü Example: Vanilla European call H > K
The payout for a vanilla European call is
FHSL = max HS - K, 0L
HK
FH FH
For H > K
FH HSL = maxHS - H , 0L + HH - KL . IHS > HLThe down-truncated payout is a portfolio of a standard call and HH - KL cash-or-nothing binary options bothwith strike H , and therefore
V HS, FH L = cHS, HL + HH - KL bHS, HLwhere bHS, HL is the value of an option that pays 1 if ST > H and zero otherwise.
Substituting in Theorem 1
(1)
cdo HS0, K, HL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, FHy{zzz
=
c HS0, HL + HH - KL b HS0, HL - J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjjc
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzz + HH - KL b
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzzy{zzz
Similarly
FH HSL = FHSL - FH HSL = max HS - K, 0L - maxHS - H , 0L - HH - KL . IHS > HL
Barrier.nb 5
The up-truncated payoff is a portfolio comprising a bull spread together with a short position in H - Kbinary options.
V HS, FH L = cHS, KL - cHS, HL - HH - KL bHS, HLSubstituting in Corollary 1
(2)cdi HS0, K, HL =
cHS0, KL - cHS0, HL - HH - KL bHS0, HL + J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjjcik
jjj H2ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzz + HH - KL bi
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzzy{zzz
Note that
cdi HS0, K, HL + cdo HS0, K, HL = cHS0, KL
ü Exercise
Show that the value of a single binary option is
bHS, HL = ‰-r T NHd2Lwhere N is the cumulative distribution function of the standard normal distribution and
d2 =lnHS0 ê HL + nTÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
sè!!!!T
, n = r - q - s2 ê 2
Note that this is analogous to the second term of the Black-Scholes formula with H in place of K.
ü
ü Exercise
Show that equations (1) and (2) are equivalent to the formulae given in Hull (2003: 439), namely
cdo = S0 ‰-q T NHx1L - K ‰-r T NHx2L - S0 ‰-q T J HÅÅÅÅÅÅÅÅS0
N2 l
NHy1L + J HÅÅÅÅÅÅÅÅS0
N2 l-2
K ‰-r T NHy2L
cdi = c - cdi
where
x1 =lnHS0 ê HL + Hr - q + s2 ê 2L T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T=
lnHS0 ê HLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
+ l s è!!!!T , x2 = x1 - s
è!!!!T
y1 =lnHH ê S0L + Hr - q + s2 ê2L T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T=
lnHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T+ l s
è!!!!T , y2 = y1 - s è!!!!T
6 Barrier.nb
ü Answer
cdoHS0, K, HL = cHS0, HL + HH - KL bHS0, HL - J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjjci
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzz + HH - KL bi
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzzy{zzz
= S0 ‰-q T NHx1L - H ‰-r T NHx2L + HH - KL ‰-r T NHx2L -
J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
‰-q T NHy1L - H ‰-r T NHy2L + HH - KL ‰-r T NHy2Ly{zzz
= S0 ‰-q T NHx1L - K ‰-r T NHx2L - J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
‰-q T NHy1L - K ‰-r T NHy2Ly{zzz
= S0 ‰-q T NHx1L - K ‰-r T NHx2L - J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2 +2
S0 ‰-q T NHy1L + J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
K ‰-r T NHy2L
where
x1 =lnHS0 ê HL + Hr - q + s2 ê 2L T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T, x2 = x1 - s
è!!!!T , n = r - q -s2ÅÅÅÅÅÅÅÅÅÅ2
y1 =lnHH ê S0L + Hr - q + s2 ê2L T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T, y2 = y1 - s
è!!!!T
Define
l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 =r - q - s2 ê 2 + s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 =n
ÅÅÅÅÅÅÅÅÅÅs2 + 1
Substituting and simplifying
cdoHS0, K, HL = S0 ‰-q T NHx1L - K ‰-r T NHx2L - S0 ‰-q T J HÅÅÅÅÅÅÅÅS0
N2 l
NHy1L + J HÅÅÅÅÅÅÅÅS0
N2 l-2
K ‰-r T NHy2L
where
x1 =lnHS0 ê HL + Hr - q + s2 ê 2L T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T=
lnHS0 ê HLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
+ l s è!!!!T , x2 = x1 - s
è!!!!T
y1 =lnHH ê S0L + Hr - q + s2 ê2L T
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T=
lnHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs
è!!!!T+ l s
è!!!!T , y2 = y1 - s è!!!!T
ü
Analogous results for up options HH > S0L follow.
Theorem 2 (Up-and-out) Suppose that V HS, F L denotes the value of a European derivative contract F HST Lon S maturing at time T. The value of an up-and-out version of the derivative is
VuoHS0, FL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS
, FHy{zzz
where H > S0 is the barrier, n = r - q - s 2 ê2 and
FH HSL = 9 FHSL,0,
for S < Hfor S ¥ H
= FHSL . IHS < HL
Barrier.nb 7
Corollary 2 (Up-and-in) Suppose that V HSL denotes the value of a European derivative contract on Smaturing at time T. The value of an up-and-in version of the same option is
VuiHS0, FL = V HS0, FH L + J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, FHy{zzz
where H > S0 is the barrier, n = r - q - s 2 ê2,
FH HSL = FHSL . IHS > HL and FH HSL = FHSL . IHS < HL
ü Example Vanilla European call
When H § K, an up-and-in call is equivalent to a vanilla whereas the value of an up-and-out call is alwayszero. Therefore, we need only consider explicitly the case H > K.
Recall that the up-truncated payoff of a vanilla European call when H > K is a portfolio comprising a bullspread together with a short position in H - K binary options.
V HS, FH L = cHS, KL - cHS, HL - HH - KL bHS, HLSubstituting the payoff in Theorem 2
(3)
cuo HS0, K, HL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS
, FHy{zzz
= cHS0, KL - cHS0, HL - HH - KL bHS0, HL -
J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjj ci
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz - ci
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzz - HH - KL bi
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzzy{zzz
The down-truncated payoff of a vanilla European call when H > K is a portfolio comprising a standard calland HH - KL cash-or-nothing binary option both with strike H .
V HS, FH L = cHS, HL + HH - KL bHS, HLSubstituting the payoff in Corollary 2
(4)
cui HS0, K, HL = V HS0, FH L + J HÅÅÅÅÅÅÅÅS
N2 nÅÅÅÅÅÅÅÅs2
Vikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS
, FHy{zzz
= c HS0, HL + HH - KL b HS0, HL +
J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjj c
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz - c
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzz - HH - KL b
ikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Hy{zzzy{zzz
ü Exercise
Show that equations (3) and (4) are equivalent to the formula on p. 440 of Hull (2003).
ü
8 Barrier.nb
ü
ü Discrete monitoring
The above formulae assume continuous monitoring of the asset price. Many real-world contracts assumemonitoring at discrete intervals, e.g. daily at a specific time. Broadie, Glasserman and Kou (1997) suggestan adjustment to the barrier to account for discrete monitoring. Specifically, the barrier H is adjusted asfollows
H ö9 H ‰+ b s è!!!!!!!!!!Têm , H > S0 HupL
H ‰- b s è!!!!!!!!!!Têm , H < S0 HdownL
where b = 0.5826 and m is the monitoring frequency.
ü Summary
The following table summarizes the valuation formulae for European style barrier call options.
Ñ H £ K H > K
cdi I HÅÅÅÅÅÅÅS0M
2 nÅÅÅÅÅÅÅÅs2 cI H2ÅÅÅÅÅÅÅÅS0
, KM cHS0, KL - cHS0, HL - HH - KL bHS0, HL +
I HÅÅÅÅÅÅÅS0M
2 nÅÅÅÅÅÅÅÅs2 IcI H2ÅÅÅÅÅÅÅÅS0
, HM + HH - KL bI H2ÅÅÅÅÅÅÅÅS0
, HMM
cdo cHS0, KL - I HÅÅÅÅÅÅÅS0M
2 nÅÅÅÅÅÅÅÅs2 cI H2ÅÅÅÅÅÅÅÅS0
, KM cHS0, HL + HH - KL bHS0, HL -
I HÅÅÅÅÅÅÅS0M
2 nÅÅÅÅÅÅÅÅs2 IcI H2ÅÅÅÅÅÅÅÅS0
, HM + HH - KL bI H2ÅÅÅÅÅÅÅÅS0
, HMMcui cHS0, KL cHS0, HL + HH - KL bHS0, HL +
I HÅÅÅÅÅÅÅS0M
2 nÅÅÅÅÅÅÅÅs2 I cI H2ÅÅÅÅÅÅÅÅS0
, KM - cI H2ÅÅÅÅÅÅÅÅS0
, HM - HH - KL bI H2ÅÅÅÅÅÅÅÅS0
, HMMcuo 0 cHS0, KL - cHS0, HL - HH - KL bHS0, HL -
I HÅÅÅÅÅÅÅS0M
2 nÅÅÅÅÅÅÅÅs2 I cI H2ÅÅÅÅÅÅÅÅS0
, KM - cI H2ÅÅÅÅÅÅÅÅS0
, HM - HH - KL bI H2ÅÅÅÅÅÅÅÅS0
, HMM
ü Rebate
For knock-out options, it is customary for the rebate to be paid immediately that the barrier is hit. Thiscomplicates the valuation, since the rebate is paid at a random time.
The discounted expected value of the rebate R is
Vrebate = RikjjjjJ
HÅÅÅÅÅÅÅÅS0
Na+b
NHh zL + J HÅÅÅÅÅÅÅÅS0
Na-b
NIh z - 2 h b s è!!!!T M y
{zzzz
where
a = l - 1, b =è!!!!!!!!!!!!!!!!!!!!!!!
n2 + 2 r s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s2 , z =lnHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
sè!!!!T
+ b s è!!!!T , n = r - q - s2 ê 2
and h = 1 if the barrier is approached from above HS0 > HL and h = -1 if the barrier is approached frombelow HS0 < HL.
ü Proof
Barrier.nb 9
ü
Standard American barrier optionsWe showed above that, for H § K, the value of a down-and-in vanilla European option is related to thevalue of the underlying non-barrier option by the formula
cdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
cikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz
Fortunately, the same relationship applies to knock-in American options (Haug 2001, Dai and Kwok 2004).That is, where H § K,
(5)CdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
Cikjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz
where Cdi is the value of a knock-in option entitling the holder to a standard American option of strike K ifand when the asset price S falls below the barrier H , and C is the value of the underlying American option.More generally
(6)CdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0
N2 nÅÅÅÅÅÅÅÅs2
ikjjjCi
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzz - ci
kjjj H2
ÅÅÅÅÅÅÅÅÅÅÅS0
, Ky{zzzy{zzz + cdiHS, K, HL
provided that
(7)H § max ikjjK,
rÅÅÅÅÅq
Ky{zz
When H § K, cdiHS, K, HL matches the second term inside the brackets.
Analogous (though different formulae) can be given when (7) is not satisfied (Dai and Kwok 2004). Thismeans that we can easily value a knock-in American option by applying our best numerical techniques tocorresponding standard options.
Unfortunately (for computation), in-out parity does not hold for American barrier options. Consequently, wecannot use (6) to compute the value of a knock-out option. Indeed,
CdiHS, K, HL + CdoHS, K, HL > CHS, KLTo see this, consider a portfolio comprising a down-and-in plus a down-and-out with the same strike, barrierand maturity. We show that this portfolio dominates a corresponding nonbarrier call. Suppose that theportfolio holder adopts an exercise policy for the down-and-out identical to that for the nonbarrier option(though this is suboptimal for the down-and-out). The exercise payoff of the portfolio is always higher thanthat of the nonbarrier call because the portfolio has an additional option. Both the portfolio and the optionhave the same payoff at maturity because one of the options will be knocked-out. In all scenarios, theportfolio is worth at least as much as the option and possibly more.
10 Barrier.nb
Numerical methodsThe preceding analysis provides exact formulae for standard European barrier options, plus an efficientapproach for dealing with American knock-in options. However, this does not exhaust the variety of barriersoptions traded in over-the-counter markets. In particular, where
æ the barrier is non-constant (and non-exponential)
æ early exercise is allowed (American knock-out options)
æ it is desired to allow for non-lognormality (volatility skew)
it is necessary to resort to numerical methods. Barrier options pose a special problem for numerical methods,which is illustrated by the binomial method.
à Binomial trees
We have previously examined the convergence of the basic binomial method. The following diagramdepicts the errors for a one-year vanilla European call option (S = 95, K = 100, r = 10 %, q = 0,s = 25 %).
50 100 150 200 250 300
-0.04
-0.02
0.02
0.04
Performance deteriorates dramatically when applied to Barrier options. For a down-and-out version of theprevious option with the barrier at 90, we find
Barrier.nb 11
100 200 300 400
0.25
0.5
0.75
1
1.25
1.5
1.75
2Binomialerror in down-and-out call
The reason is that, when the barrier lies between the nodes of the tree, the effective barrier is different to thetrue barrier. Performance can be significantly improved if we ensure that the barrier lies on or just above arow of nodes. This requires a judicious choice of the number of steps n.
Recall that in the basic binomial model
u = ‰s è!!!!!!!!!Tên , d = ‰-s
è!!!!!!!!!Tên
Barrier H will be precisely aligned with a row of nodes in a tree of size n when
S0 ‰k s è!!!!!!!!!Tên = H
lnS0ÅÅÅÅÅÅÅÅH
= k s $%%%%%%%TÅÅÅÅÅÅn
for some k = ≤1, ≤2, …, ≤n.
n = ikjj k s
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅlnHS0 ê HL
y{zz
2 T , k = ≤1, ≤2,
Barrier H will lie on or just above a row of nodes when n is the largest integer which is smaller than theright-hand side. For the option depicted above, these magic numbers are 21, 85, 192, 342 and 534.
At these magic numbers, the binomial method delivers similar accuracy to that obtained for vanilla options.
12 Barrier.nb
n Error21 0.045089785 0.0228345192 0.00895174342 0.00144772534 0.00352852
100 200 300 400
0.05
0.1
0.15
0.2
0.25
Binomialerror in down-and-out call
Derman, Kani, Ergener and Bardhan (1995) propose an enhancd binomial method to allow for nonconstantbarriers.
à Trinomial trees
We have just seen how the accuracy of binomial method can be preserved by careful selection of the treesize n to align the barrier with the nodes in the tree. Since the trinomial tree has an additional degree offreedom, this alignment can achieved for any n through judicious choice of the "stretch" parameter l. Thestandard parameterization of the trinomial tree is
u = ‰l s è!!!!!!
Dt
d =1ÅÅÅÅÅu
n = r - q -1ÅÅÅÅÅ2
s2
pu =1
ÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l2 +
è!!!!!Dt
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l s
n
pd =1
ÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l2 -
è!!!!!Dt
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l s
n
pm = 1 - pu - pd = 1 -1
ÅÅÅÅÅÅÅÅl2
Barrier.nb 13
Consider an up-and-out barrier option. To find the appropriate value of l, start with l = 1. Let m denote themaximum number of consecutive up moves allowed before breaching the barrier. That is
S0 um < H but S0 um+1 ¥ H
Given l = 1, we have
S0 ‰m s è!!!!!!Dt < H ï m <logHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
sè!!!!!
Dt
Define
m* =logHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
sè!!!!!
Dt
and then choose l > 1 such that l m = m* which implies that
S0 ‰l m s è!!!!!!Dt = H
With this choice of l, the tree is precisely aligned with the barrier after m upward steps.
For example, suppose S0 = 95, H = 125, with s = 25 % and T = 1. For a 5-step tree, Dt = 1 ê 5,m* = 2.45464 and m = 2, so that l = m* ê m = 1.22732. For this choice of l, the tree coincides with thebarrier after precisely 2 steps.
The following graph depicts the convergence of a trinomial tree in valuing a 6-month European up-and-outcall option (S = 100, K = 90, H = 125, r = 5 %, q = 2 %, and s = 30 %L. The red path shows the errorof a trinomial tree with l =
è!!!!!!!!!3 ê 2 against n, while the blue line depicts the error a trinomial tree in which lis chosen optimally for each n so as to align a row of nodes with the barrier.
100 200 300 400
-0.2
0.2
0.4
Trinomial error in up-and-out call
The following graph displays the percentage error.
14 Barrier.nb
100 200 300 400
-10
10
20
Trinomial percentage error in up-and-out call
Barrier.nb 15
Spread options� Preliminaries
Examples� Commodity markets
The soybean crush spread traded on the CBOT is
CS =
48
2000SM
+
11
100SO
- SB
where SM is the price is the futures price of soy meal in dollars per ton, SO is the futures price of soy oil indollars per 100 pounds, and St is the futures price of soybean in dollars per bushel. The payoff of a call optionon the soybean crush is
CSC = HCS - KL+
� Energy markets
The payoff of the 1:1:0 gasoline crack spread call is
CSGC = H42 UG - CO - KL+
where UG is the price of unleaded gasoline ($ / gallon) and CO is the price of crude oil ($ / barrel). The payoff
of the 3:2:1 crack spread call is
CSHC = 422
3UG +
1
3HO - CO - K
+
where HO is the price of heating oil. Crack spread options are traded on NYMEX.
A spark spread is a proxy for cost of converting a specific fuel (usually natural gas) into electricity.
SS = E - Heff NG
where E and NG are the futures prices of electricity and natural gas, and Heff is the heat rate.
� Interest rate markets
The TED spread measures the difference between 3-month Tbills and 3-month LIBOR. The MOB spreadmeasures the difference in yield between municipal and treasury bonds.
Exchange optionsA spread option with a zero strike is known as an option to exchange
payoff = H S1 - S2 L+
for which we have an exact formula (known as the Magrabe formula).
c = ã-q1 T S1 NHd1L - ã
-q2 T S2 NHd2L
where
d1 =
LnHS1 �S2L + Iq2 - q1 + Σ2 �2M T
Σ T
d2 = d1 - Σ T
Σ = Σ12
- 2 Ρ Σ1 Σ1 + Σ22
Note that the value is independent of the risk-free rate. Alternatively, in terms of futures prices, we have
c = ã-r T HF1 NHd1L - F2 NHd2L L
where
d1 =
LnHF1 � F2L + Σ2 T �2
Σ T
d2 = d1 - Σ T
Σ = Σ12
- 2 Ρ Σ1 Σ1 + Σ22
Approximations� The Kirk approximation
The payoff of a spread option is
payoff = H S1 - S2 - KL+
The popular Kirk approximation values the spread option as an exchange between F1 and F2 + K, treating
F2 + K as lognormal with volatilty J F2
F2+KN Σ2.
c = ã-r T H F1 NHd1L - HF2 + KL NHd2L L
where
Fi = ãHr-qiL T Si, i = 1, 2
Σ = Σ12
- 2 b Ρ Σ1 Σ1 + b 2Σ2
2 , b =
F2
F2 + K
d1 =
LnJ F1
F2+KN +
1
2Σ
2 T
Σ T
d2 = d1 - Σ T
Clearly, the Kirk approximation reduces to the Margrabe formula when K = 0, and will be most useful whenK << S2.
� Bjerksund and Stensland
The payoff of the spread can be written as
CHTL = H S1 - S2 - KL+= H S1 - S2 - KL × I HS1HTL ³ S2HTL + KL
where IH L is the indicator function, taking the value one when its argument is true, and zero otherwise. Bjerk-sund and Stensland consider the related derivative with the payoff
cHTL = H S1 - S2 - KL+= H S1 - S2 - KL × I S1HTL ³
a HS2HTLLb
EAHS2HTLLbE
where a = F2 + K, b = F2 � HF2 + KL and F2 is the forward price F2 = S2 ãHr-q2L T . This has two implications:
2 SpreadOptions.nb
æ They can compute the exact value of the related derivative, which they propose as a superior alternativeto the Kirk approximation.
æ The related derivative can be used as a control variable in simulating the value of a spread option.
The exact value is given by the following formula.
c = ã-r T H F1 NHd1L - F2 NHd2L - K NHd3L L
where
Fi = ãHr-qiL T Si, i = 1, 2
Σ = Σ12
- 2 b Ρ Σ1 Σ1 + b 2Σ2
2
d1 =
LnJ F1
aN + I 1
2Σ1
2- b Ρ Σ1 Σ2 +
1
2b2
Σ22M T
Σ T
d2 =
1
Σ TLn
F1
a+ -
1
2Σ1
2+ Ρ Σ1 Σ2 +
1
2b2
Σ22
- b Σ22 T
d2 = d1 -
IΣ12- H1 + bL Ρ Σ1 Σ2 + b Σ22M
Σ
T
d3 = d1 -
IΣ12- b Ρ Σ1 Σ2M
Σ
T
d3 =
LnJ F1
aN + I-
1
2Σ1
2+
1
2b2
Σ22M T
Σ T
GreeksFor the BjS
c = ã-r T H F1 NHd1L - F2 NHd2L - K NHd3L L
First note
¶di
¶ F1
=
1
F1 Σ T
¶c
¶ F1
= ã-r T K NHd1L + HF1 ΦHd1L - F2 ΦHd2L - K ΦHd3LL
¶di
¶ F1
O
= ã-r T NHd1L +
F1 Φ Hd1L - F2 Φ1Hd2L - K Φ Hd3L
F1 Σ T
and therefore
¶c
¶S1
=
¶c
¶ F1
¶ F1
¶S1
= ã-q T NHd1L +
F1 Φ Hd1L - F2 Φ1Hd2L - K Φ Hd3L
F1 Σ T
Computing the derivative with respect to F2 is a a little more difficult, since the adjusted volatility dependsupon b, which is a function of F2.
¶b
¶ F2
=
b2 K
F22
SpreadOptions.nb 3
¶d0
¶ F2
=
1
2 T Σ3
Σ2 HΡ Σ1 - b Σ2LK b2
F22LogB
F1
F2 + KF - T Σ
2-
2 Σ2
F2 + K
¶d1
¶ F2
=
1
2 T Σ3
Σ2 HΡ Σ1 - b Σ2LK b2
F222 LogB
F1
F2 + KF - T Σ
2-
2 Σ2
F2 + K
¶d2
¶ F2
=
¶d1
¶ F2
-
b2 K
F22
H1 - bL I1 - Ρ2M Σ12
Σ22
Σ3
T
4 SpreadOptions.nb
Volatility and Variance swaps
IntroductionA volatility swap is a forward contract on realized volatility. Its payoff is
payoff = HΣR - KvolL ´ N
where ΣR is realised volatility (annualized), Kvol is the specified delivery price, and N is the notional amountof the swap in "dollars" per annualized volatility point. A variance swap is a forward contract on realizedvariance, with payoff
payoff = IΣR2
- KvarM ´ N
where N is expressed in "dollars" per annualized volatility point squared.
For example, a 3-month volatility swap on the S&P 500 struck at 20% with a notional $100,000 would pay(25 - 20) × 100000 = $500000 if the realised volatility was 25%.
A volatility or variance swap is a position on realized volatility. It is related to, but distinct from, options onimplied volatility (VIX) which are traded on the CBOE.
� Questions
æ how to hedge volatility.
æ how to replicate (hence price and hedge) a variance swap?
æ how is the VIX computed.
Hedging volatilityIn the Black-Scholes framework, vega of a vanilla call or put is
Vega =
¶V
¶ Σ
= ã-q T S N ' Hd1L
where
d1 =
LnHS �KL + Ir - q - Σ2 T �2M
Σ T
For this purpose, it is convenient to measure variance vega, which (by the chain rule) is a multiple of vega.
VarVega =
¶V
¶ Σ
¶ Σ
¶ Σ2
=
1
2 Σ
´Vega
The following graph shows vega on traded options on the Nifty on 5 January 2010, allowing for the volatilitysmile. We observe that:
æ (Var)vega is sharply peaked around the strike of an option.
æ We also observe that (var)vega increases with the strike.
4000 5000 6000 7000
500
1000
1500
Vega of a one-month options on the Nifty at selected strikes
This suggests that vega can be hedged by a weighted portfolio of traded options. This is illustrated in thefollowing diagram, where the red line is the (var)vega of a portfolio of options weighted inversely to strikesquared.
Out[6]=
4000 5000 6000 7000
500
1000
1500
In fact, a portfolio of options at all strikes, weighted inversely proportional to strike, will give an exposure tovariance which is independent of asset price. This is precisely what is needed to trade pure variance. It is easyto show that the terminal payoff of such a portfolio comprising puts K £ K* and calls K > K* is
(1)
payoff = à0
K* 1
K2HK - ST L+
â K + àK*
¥ 1
K2HK - ST L+
â K
=
ST - K*
K*
- logST
K*
where log is natural log, and K* is a boundary strike.
Valuing a variance swapFor an asset following geometric Brownian motion, it can be shown that expected (realized) variance overperiod @0, TD is
(2)E@VarD =
2
THr - q L T - EBlog
ST
S0
F
Inverting equation (1) shows that a log contract is equivalent to a forward together with a portfolio of vanillacalls and puts.
2 VarSwaps.nb
logST
K*
=
ST - K*
K*
forward
- à0
K* 1
K2HK - ST L+
â K put options
- àsK*
¥ 1
K2HST - KL+
â K call options
Substituting and setting F = E@ST D = S0 ãHr-qL T
(3)
E@VarD =
2
THr - qL T - log
F
S0
- logK*
F-
F - K*
K*
+ à0
K* 1
K2ã
r T PHKL â K + àK*
¥ 1
K2ã
r T CHKL â K
=
2
T- log
F
K*
-
F - K*
K*
- ãr T à
0
K* 1
K2PHKL â K - ã
r T àK*
¥ 1
K2CHKL â K
=
2
Tà
0
K* 1
K2ã
r T PHKL â K + àK*
¥ 1
K2ã
r T CHKL â K -
1
T
F - K*
K*
2
where we have used the fact that
logF
K*
»
F - K*
K*
To value a variance swap, we need to approximate this with traded options.
� First method: numerical integration
Approximate (3) by
(4)E@VarD =
2
Tâ
K £ K*
DK
K2ã
r T PHKL + âK > K*
DK
K2ã
r T CHKL -
1
T
F - K*
K*
2
This is the basis of the method employed to calculate the VIX.
� Second method: replicating the log contract
Rewriting (2)
E@VarD =
2
THr - q L T - EBlog
ST
S0
F
=
2
THr - q L T -
E@ST D - K*
K*
+
E@ST D - K*
K*
- EBlogK*
S0
+ logST
K*
F
=
2
THr - q L T -
S0 ãHr-qL T
- K*
K*
- logK*
S0
+ E@ f HST LD
where
f HST L =
2
TBST - K*
K*
- logST
K*
f HST L is a payoff function which can be approximated by a portfolio traded vanilla calls and puts, as shown inthe workbook VarSwaps.xlsm.
VarSwaps.nb 3
4500 5000 5500 6000 6500
0.05
0.10
0.15
0.20
The payoff function f HST L
To approximate this, consider first the segment around K*.
Out[25]=
5400 5500 5600 5700
0.001
0.002
0.003
0.004
0.005
A segment of the payoff function f HST L
Computing the VIXThe CBOE VIX is an estimate of the 30-day expected volatility of the S&P 500 index, computed as a weightedaverage estimated volatility for near- and next-term options. Specifically
VIX = 100 ´
T2 - T30
T2 - T1
T1 Σ12
+
T30 - T1
T2 - T1
T2 Σ22
where Σ12 and Σ2
2 are the estimated variance of near- and next-term options respectively, T1 and T2 are their
respective times to maturity and T30 = 30 �365.
Each variance computed by evaluating (4) using all quoted options with active bid price. In the evaluation
æ times to maturity are computed in minutes (although expressed on an annual basis).
æ the forward price F is computed by put-call parity applyed to the nearest-to-the-money strike
F = KNTM + ãr T HCHKL - PHKL
where KNTM is the strike at which the absolute difference between call and put prices is mininized.
æ the boundary K* is selected as the strike immediately below the forward price F .
In computing India-VIX, the NSE uses the same methodology, with the following modifications.
æ the risk-free rate is MIBOR 30 or 90 days. (It appears that the rate is not interpolated to match time tomaturity.)
4 VarSwaps.nb
æ the forward price F is set at the appropriate futures closing price.
æ cubic splines are used to compute mid-price where the bid-ask spread is greater than 30%.
VarSwaps.nb 5
Interest rate derivativesMichael CarterInterest rate derivatives are financial assets whose payoff depends in some way on the level of interest rates.Examples include
æ Bond options
æ Bond futures options
æ Callable and putable bonds
æ Mortgages
æ Mortgate-backed securities
æ Interest rate caps and floors
æ Swaps
æ Swapoptions
Interest rate derivatives are more difficult to value than stock or foreign exchange derivatives, because
æ The behavior of interest rates is more complicated. It is necessary to develop of model describingthe behavior of the entire zero-coupon yield curve.
æ The volatilities at different point of the yield curve are different.
æ Interest rates are used for discounting as well as defining the payoff of the derivative.
Modeling the term structure - discrete timeWe can represent the term structure of interest rates in three equivalent ways:
æ discount function (prices of zero coupon bonds)
æ spot rates (yield to maturity of zero coupon bonds)
æ forward rates
Let R1, R2, …, RT denote the spot rates of interest appropriate to 1, 2, …, T years (we assume annualcompounding for simplicity). Then the current prices of zero coupon bonds are
(1)PH0, 1L =1
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + R1
, PH0, 2L =1
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1 + R2L2 , …, PH0, TL =
1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1 + RT LT
Conversely
(2)RT = J 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL N
1ÅÅÅÅÅT
- 1
Forward rates are rates agreed now for a loan in the future. For example, f3,5 denotes the rate agreed now fora 2-year loan beginning in 3 years. Absence of arbitrage requires that
H1 + R2L2 = H1 + R1L H1 + f1,2Land more generally
H1 + RT LT = H1 + RtLt H1 + ft,T LT-t
Substituting from (1), we have
1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 2L =
1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 1L H1 + f1,2L and
1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL =
1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, tL H1 + ft,T L
so that
f1,2 =PH0, 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 2L - 1 and ft,T =
PH0, tLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL - 1
Note also that
H1 + R3L3 = H1 + R2L2 H1 + f2,3L = H1 + R1L H1 + f1,2L H1 + f2,3Lso that
H1 + RT LT = H1 + R1L H1 + f1,2L H1 + f2,3L … H1 + fT-1,T LIn a discrete model, the short rate r is the one period forward rate. That is
r0 = f0,1 =PH0, 0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 1L - 1 =
1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 1L - 1 = R1, r1 = f1,2, r2 = f2,3, …, rT-1 = fT-1,T
and therefore
(3)H1 + RT LT = H1 + r0L H1 + r1L H1 + r2L … H1 + rT-1L
Modeling the term structure - continuous timeAs in the Black-Scholes model, theoretical work is almost always exposited in continuous time. As withdiscrete compounding, we can represent the term structure of interest rates in three equivalent ways:
æ discount function (prices of zero coupon bonds)
æ spot rates (yield to maturity of zero coupon bonds)
æ forward rates
Each of these familiar interest rate concepts has a continuous time counterpart.
2 InterestRateDerivatives.nb
Discrete Continuous
Discount function PHt, TL = J 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + RHt, TL N
T-tPHt, TL = ‰-RHt,TL HT-tL
Spot rate RHt, TL = PHt, TL 1ÅÅÅÅÅÅÅÅÅÅT-t - 1 RHt, TL = -ln PHt, TLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
T - t
Forward rate f Ht, T , T + 1L =PHt, T + 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
PHt, TL - 1 f Ht, TL = -∑
ÅÅÅÅÅÅÅÅÅÅÅ∑T
ln PHt, TL
To derive the expressions for the forward rate, note that one period forward rates f Ht, t, t + 1L must satisfy
(4)
H1 + RHt, TLLT-t = H1 + RHt, t + 1LLäH1 + f Ht, t + 1, t + 2LL ä… äH1 + f Ht, T - 1, TLL =
‰i=1
T-t
H1 + f Ht, t + i - 1, t + iLL
where f Ht, t, t + 1L = RHt, t + 1L. In particular, this implies that
(5)PHt, T + 1L =PHt, TL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + f Ht, T , T + 1L
which yields
f Ht, T , T + 1L =PHt, TL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPHt, T + 1L - 1
For small time periods, (5) requires
PHt, T + DTL = ‰- f Ht,T ,T+1L DT PHt, TLor
ln PHt, T + DTL = - f Ht, T , T + D TL DT ä ln PHt, TL
f Ht, T , T + D TL = -ln PHt, T + D TL - ln PHt, TL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD T
Letting D T Ø 0, the instantaneous forward rate is
(6)f Ht, TL = limD T Ø 0
-ln PHt, T + D TL - ln PHt, TL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD T
= -∑
ÅÅÅÅÅÅÅÅÅÅÅ∑T
ln PHt, TL
From (1)
PHt, TL = ‰i=1
T-t 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + f Ht, t + i - 1, t + iL
With continuous discounting, this becomes
PHt, TL = ‰i=1
T-t
‰- f Ht,t+i-1,t+iLä1 = expi
kjjjjj ‚
i=1
T-t
- f Ht, t + i - 1, t + iLä1y
{zzzzz
In continuous time, this becomes
PHt, TL = ‰ -ŸtT f Ht,tL „t
InterestRateDerivatives.nb 3
This can be derived more formally by integrating (3)
‡t
Tln PHt, tL „ t = ln PHt, TL - ln PHt, tL = -‡
t
Tf Ht, tL „ t
But since PHt, tL = 1
ln PHt, TL = -‡t
Tf Ht, tL „ t ï PHt, TL = ‰ -Ÿt
T f Ht,tL „t
The price of the discount bond is the final cashflow discounted by instantaneous forward rates.
It follows that the spot rate is the continuous average of forward rates
RHt, TL =1
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅT - t
‡t
Tf Ht, tL „ t
Further
f Ht, TL = -∑
ÅÅÅÅÅÅÅÅÅÅÅ∑T
ln PHt, TL =∑
ÅÅÅÅÅÅÅÅÅÅÅ∑T
HT - tL RHt, TL = RHt, TL + HT - tL ∑ÅÅÅÅÅÅÅÅÅÅÅ∑T
RHt, TL
The short rate r is the rate on instantaneous borrowing or lending, i.e. RHt, tL = f Ht, tL . A sum of $1invested at the short rate at time zero and continuously rolled over is called a money market account. Itsvalue at time t is
pt = ‰Ÿ0t rs ds
Interest rate derivatives as portfolios of bond optionsSeveral popular interest rate derivatives can be valued as portfolios of European options on discount bonds.Therefore, it suffices to develop formulae or algorithms for discount bonds. This is especially useful incalibrating interest-rate models to market data.
à Coupon bonds
Assuming that there is only one factor of uncertainty, so that all rates are positively related to r0, we canvalue options on coupon bonds by treating them as a portfolio of options on zero-coupon bonds. Consider acall option maturing at time T on a coupon bond maturing at time tn > T , where n is the number of couponpayments remaining at time T . The price of the bond at time T will be
B = ‚i=1
n
ci PrT HT , tiL
and the payoff of a call option maturing at time T will be
maxikjjjjj‚
i=1
n
ci PrT HT , tiL - K, 0y{zzzzz
4 InterestRateDerivatives.nb
where rT is the short rate at time T . Note that cn includes the principal repayment. Note that B depends uponthe short rate at time T . Let rK denote the short-rate at time T at which the price of the coupon bond is equalto the strike price. That is
‚i=1
n
ci PrK HT , tiL = K
The option will be exercised provided rT < rK and not exercised if rT ¥ rK . Let Ki denote the value at timeT of a zero-coupon bond paying $1 at time ti given rT = rK
Ki = PrK HT , tiLso that
‚i=1
n
ci Ki = ‚i=1
n
ci PrK HT , tiL = K
Therefore the payoff of the coupon bond option can be written as
maxikjjjjj‚
i=1
n
ci PrT HT , tiL - K, 0y{zzzzz = max
ikjjjjj‚
i=1
n
ci PrT HT , tiL - ‚i=1
n
ci Ki, 0y{zzzzz
= maxikjjjjj‚
i=1
n
ciHPrT HT , tiL - KiL, 0y{zzzzz
= ‚i=1
n
ci maxHPrT HT , tiL - Ki, 0L
Consequently, the option can be valued as a portfolio of options on discount bonds.
à Swaptions
A swaption is an option on an interest rate swap, and can therefore be regarded as an option to exchange acoupon bond for a floating rate bond. At the start of the swap, the value of the floating rate bond equals theprincipal amount of the swap. A European swaption can therefore be regarded as an exchange a couponbond for the principal amount of the swap. If the swaption gives the holder the right to pay fixed and receivefloating (a payer swaption), it is a put option on a fixed rate bond with strike price equal to the notionalprincipal. If the swaption gives the holder the right to pay floating and receive fixed (a receiver swaption), itis a call option on the fixed rate bond.
à Interest rate caps and floors
An interest rate cap is a portfolio of interest rate options. Each component option on a forward rate isknown as a caplet.
We show that an interest rate cap can be valued as a portfolio of European put options on discount bonds.Assuming annual tenor, the payoff from each caplet discounted to the beginning of the period is
payoff =P max Hr - cap, 0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1 + r
where is P is the principal and cap is the cap rate. This is equivalent to
InterestRateDerivatives.nb 5
payoff = P max J r - capÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1 + r, 0N
= P max J 1 + r - H1 + capLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1 + r, 0N
= max JP -PH1 + capLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
1 + r, 0N
The second term in the brackets, PH1 + capL ê H1 + rL, is the value at the beginning of the period of a discountbond that pays P(1+cap) at the end of the period. Therefore, the payoff of the t-period caplet is equal to thepayoff of a put option on a discount bond with a face value of PH1 + capL and a maturity of t + 1. The optionhas a strike price of P and maturity of t. Consequently, the cap can be regarded as a portfolio of Europeanput options on zero-coupon bonds.
Analogously, an interest rate floor can be valued as a portfolio of European call options on discount bonds.A collar is a combination of a long position in a cap and a short poisiton in a floor.
There is a put-call parity relationship between the prices of caps and floors with the same strike place RK ,namely
cap price + floor price = value of swap
where the swap is an agreement to receive floating and pay fixed RK , with no exchange of payments on thefirst reset date.
6 InterestRateDerivatives.nb
Interest rate derivatives: Standard market modelsMichael CarterThe three most popular over-the-counter interest rate derivatives are bond options, interest rate caps and floorsand swap options. Many traders use a Black-Scholes type formula known as Black's model for valuing thesederivatives. It is based on the following key result.
THEOREM. If S is lognormally distributed and the standard deviation of ln S is s then
PrHS > KL = NHd2Land
EHS » S > KL = EHSL NHd1Lwhere
d1 =lnHEHSL ê KL + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s, d2 =
lnHEHSL ê KL - s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s= d1 - s
Consequently
(1)E@maxHS - K, 0LD = EHSL NHd1L - K NHd2LProof: Hull (2003: 262-263)
Recognising that (under Black-Scholes assumptions) EHST L = S0 ‰r T and s = s è!!!!T , the Black-Scholes formula
for a call option
c = ‰-r T E@maxHS - K, 0LD = ‰-r T HS0 ‰r T NHd1L - K NHd2LL = S0 NHd1L - K ‰-r T NHd2Lis immediate.
Consider a European call option on a variable S. Let
T = time to maturity of the optionF = forward price of S with maturity TF0 = value of F at time 0K = strike price of optionP H0, TL = price at time 0 of a discount bond paying $1 at time T .
Assume
æ S is lognormally distributed with the standard deviation of ln S equal to s è!!!!T
æ EHST L = F0
Then the expected payoff of the option at time T is
E@maxHST - K, 0LD = F0 NHd1L - K NHd2LThe present value of the option is
(2)c = PH0, TL E@maxHST - K, 0LD = PH0, TL HF0 NHd1L - K NHd2LLwhere
d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 =lnHF0 ê KL - s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
= d1 - s è!!!!T
Similarly the value of a corresponding put option is
(3)p = PH0, TL HK NH-d2L - F0 NH-d1LLThis is usually referred to as Black's model since the formulas are similar to those in the model suggested byFisher Black for commodity futures. Note that it is not necessary to make any assumption about the stochasticprocess followed by V of F, only that VT is lognormal at time T . Although the volatility parameter s is usuallyreferred to as the volatility of F or the forward volatility of V , its only role is determine the standard deviation ofln S at time T .
à Bond options
In addition to trading in the OTC market, bond options are frequently embedded in standard bonds in order tomake them more attractive to the issuer or potential purchasers. Examples include callable bonds, puttable fondsand conventional mortages with early payment privileges. Most OTC bond options and some embedded bondoptions are European.
In using Black's model to value European bond options, it is assumed that the bond price ST at the maturity ofthe option is lognormal with the standard deviation of ln ST = s
è!!!!T . The forward price of the bond is
F0 =S0 - CÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL
where S0 is the current price and C is the present value of the coupons that will be paid during the life of theoption. The value of a call option is given by (2) and the value of a put option by (3).
The spot and forward prices in this formula are cash ("dirty") prices, and so the strike price should also be thecash price. In practice, the strike price is often the quoted ("clean") price applicable when the option is exercised,in which case K should be set equal to the strike price plus accrued interest at the expiration date of the option.
The volatility parameter s is the volatility of the forward price F. Volatilities are usually quoted in terms ofyield volatilities. Practitioners typically use bond duration to convert yield volatilities sy to price volatilities by
s = D y0 sy.
where D is modified duration and y0 is the forward yield.
2 Black'sModel.nb
The Black model assumes that volatility (variance) of the underlying increases linearly with time to maturity.However, the price of a bond as it approaches maturity must be equal to the principal plus coupon ("pull to pareffect"). Therefore, the volatility of a bond over its life first increases then decreases to zero. Pricing of Euro-pean bond options using the Black model should therefore be limited to options of short maturity compared tothe maturity of the bond. A rule of thumb used by some traders is that the time to maturity of the option shouldbe no more than one-fifth of the maturity of the underlying bond.
à Interest rate cap
Recall that an interest rate cap is a portfolio of interest rate options. The caplet corresponding to the rate rk
observed at time tk provides a payoff at time tk+1 of
L Dt maxHrk - cap, 0Lwhere is L is the notional principal. This is effectively a call option on rk with the payoff made at time tk+1. Inusing Black's model to value caps and floors, it is assumed that the interest rate rk is lognormal with volatilitysk. Applying equation (1), the expected payoff at time tk+1 is
E@L Dt maxHrk - cap, 0LD = L Dt @ fk NHd1L - cap NHd2LDwhere fk is the forward rate between time tk and tk+1, and
d1 =lnH fk ê capL + sk
2 tk ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
sk è!!!!tk
, d2 = d1 - s è!!!!tk
The present value of the caplet is
vcaplet = PH0, t + 1L L Dt @ fk NHd1L - cap NHd2LDThe value of an interest rate cap is the sum of the values of its constituent caplets, each valued according theprevious equation. Similarly, the value of an interest rate floor is the sum of its constituent floorlets, each ofwhich is analagous to a put option
vfloorlet = PH0, t + 1L L Dt @ floor NH-d2L - fk NH-d1L DWhere the same volatility is used to price each caplet, these are known as flat volatilities. In contrast, spotvolatilities are specific to each caplet. While many traders like to use spot volatilities, cap prices are usuallyquoted in the market in terms of flat implied volatilities.
à Swaption
The payoff from a payer swaption in each period is
L Dt maxHrT - rS , 0Lwhere L is the notional principal of the swap, rT is the actual swap rate at the maturity of the option, and rS is thefixed rate payable under the option. To apply Black's model to value the swaption, it is assumed that rT islognormal with volatility (of log rT ) equal to s. The expected value of the cash flow received at time Ti is
Black'sModel.nb 3
v = PH0, Ti L L Dt @r0 NHd1L - rS NHd2LDwhere r0 is the forward swap rate and
d1 =lnHr0 ê rSL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
s è!!!!T
, d2 = d1 - s è!!!!T
The total value of the swaption is
vpayer = ‚i=1
m n
PH0, Ti L L Dt @r0 NHd1L - rS NHd2LDwhere m = 1 ê Dt is the compounding frequency, n is the length of option (in years), and Ti = T + i ê m. Define
A =1
ÅÅÅÅÅÅÅm
‚i=1
m n
PH0, Ti LThe value of a payer swaption is
vpayer = A L @r0 NHd1L - rS NHd2LDSimilarly, the payoff of a receiver swaption in each period is
L Dt maxHrS - rT , 0Lwhich is analogous to a put option on rT . The value of the receiver swaption
vreceiver = A L @rS NH-d2L - r0 NH-d1L DBrokers provide tables of implied volatilities for European swap options.
à Conclusion
Practitioners typically use an adaptation of Black's model for valuing the most popular interest rate derivatives -bond options, interest rate caps and swaptions. Each model is based on the assumption that a key variable islognormal. These assumptions are inconsistent with one another. For example, if future bond prices are lognor-mal, future zero rates and swap rates are not. Similarly, if future zero rates are lognormal, future bond prices andswap rates are not. However, the great popularity of this model for pricing both caps and swaptions indicatesthat this inconsistency is not significant economically.
4 Black'sModel.nb
Determining price volatility from yield volatility
First consider an option maturing at t on a discount bond which matures at T . The initial price of the bond is
S0 = ã-rT´T
At maturity of the option, the price of the bond is
St = ãrt´1 S0 = ã
-rT-t´HT-tL
where rT-t is a random variable.
Differentiating
¶St
¶rT-t
= -HT - tL ã-rT-t´HT-tL
= -HT - tL St
so that for small changes
D St
St
= -HT - tL D rT-t = -HT - tL´ rT-t ´
D rT-t
rT-t
and so the price volatility is
VolD St
St
= HT - tL´ rT-t ´VolD rT-t
rT-t
The first term T - t is the duration of the discount bond at the maturity of the option.
Analogously, for a coupon bond, we have
VolD St
St
= D´ rT-t ´VolD rT-t
rT-t
where D is the modified duration of the bond at maturity of the option.
The Black-Derman-Toy modelThe single factor model of Black-Derman-Toy stands out as a very attractive tool for pricinginterest rate options in India. This model is lattice based, incorporates mean reversion, assumeslevel independent volatility and is calibrated through an exogenously specified yield curve.
Jayant Varma, 1996
Michael CarterThe fundamental assumption underlying Black-Scholes and most other financial models is that asset pricesfollow a particular stochastic process called geometric Brownian motion (GBM)
(1)dS = m S dt + s S dz
or
dSÅÅÅÅÅÅÅÅÅS
= m dt + s dz
This implies that the logarithm of the stock price follows
(2)d ln S = n dt + s dz
where n = m - 1ÅÅÅÅ2 s2. The derivation of (2) from (1) is an application of Ito's lemma (Hull: 2003:232-233,Luenberger 1998:313). This implies that the asset price is lognormal, that is
lnHStL = lnHSt-1L + n Dt + s et è!!!!!
Dt
or
St = ‰n Dt + s et è!!!!!!
Dt St-1
The basic binomial model sets n = 0, so that
St = u St-1 or St = d St-1
with
u = ‰ s è!!!!!!Dt and d = ‰- s è!!!!!!Dt
The Black-Derman-Toy model assumes that short rates follow the stochastic process
d ln r = nHtL dt + sHtL dz
which implies that the short rate is lognormal
lnHrtL = lnHrt-1L + nHtL Dt + sHtL et è!!!!!
Dt
or
rt = ‰nHtL Dt + sHtL et è!!!!!!Dt rt-1
This can be approximated by a binomial model
rt = u rt-1 or rt = d rt-1
with
u = ‰ nt Dt + stè!!!!!!Dt and d = ‰nt Dt - st
è!!!!!!Dt
where nt and st are chosen to match the actual term structure of interest rates and volatilities.
The single asset and interest rate models are compared in the following table.
Lognormal lnHStL = lnHSt-1L + n Dt + s et è!!!!!
Dt lnHrtL = lnHrt-1L + nHtL Dt + sHtL et è!!!!!
Dtor or
or St = ‰n Dt + s et è!!!!!!Dt St-1 rt = ‰nHtL Dt + sHtL et è!!!!!!Dt rt-1
∞ ∞
Binomial St = u St-1 or St = d St-1 rt = u rt-1 or rt = d rt-1
with with
u = ‰ s è!!!!!!Dt d = ‰- s è!!!!!!Dt u = ‰ nt Dt + stè!!!!!!Dt d = ‰nt Dt - st
è!!!!!!Dt
2 BlackDermanToy.nb
Kolmogorov�Smirnov One�Sided Test
n ��� ���� ����� ���� �����
� ������ ������ ������ ������ ������
� ��� ������ ����� ������ �����
����� ����� ������ ����� �����
� ������ ������ ����� ���� �����
� ������ ������ ���� ������ �����
� ������ ����� ����� ������ ������
� ���� ����� ���� ���� �����
��� ������ ����� ������ �����
� ���� ���� ����� ������ ����
�� ����� ���� ������ ������ ����
�� ��� ����� ����� ����� ������
�� ����� ��� ����� ������ ������
� ����� ����� ����� ����� �����
�� ����� ����� ���� ���� ������
�� ������ ����� ���� ����� ������
�� ����� ������ ���� ����� �����
�� ������ ���� ���� ���� ����
� ����� ����� ����� ����� �����
�� ���� ������ ����� ���� �����
�� ����� ������ ������ ���� �����
�� ������ ����� ����� ����� ����
�� ������ ����� ����� ���� ����
� ������ ������ ������ ���� �����
�� ������ ������ ����� ����� �����
�� ������ ����� ������ ������ �����
�� ������ ���� ������ ����� �����
�� ����� ������ ������ ����� �����
� ����� ������ ������ ������ ������
�� ����� ������ ������ ������ ������
� ����� ������ ������ ������ �����
� ���� ������ ����� ������ ����
� ����� ����� ����� ������ �����
����� ������ ���� ����� �����
� ������ ������ ������ ����� �����
� ������ ����� ������ ������ ������
� ������ ������ ������ ����� �����
� ������ ������ ���� ������ �����
������ ����� ������ ������ �����
� ������ ������ ������ ����� ������
�� ������ ����� ������ ����� ������
� �� �����pn �����
pn ����
pn �����
pn ����
pn