Notes on Stochastic FinanceChapter20 PricingandHedginginJumpModels In this chapter we consider option pricing and hedging in jump-diﬀusion models

Chapter 20Pricing and Hedging in Jump Models

In this chapter we consider option pricing and hedging in jump-diffusionmodels. In comparison with the continuous case the situation is further com-plicated by the existence of multiple risk-neutral probability measures. Asa consequence, perfect replicating hedging strategies cannot be computed ingeneral.

20.1 Market Returns vs Gaussian and Power Tails . . . . 67920.2 Risk-Neutral Probability Measures . . . . . . . . . . . . . . 68420.3 Pricing in Jump Models . . . . . . . . . . . . . . . . . . . . . . . . 68520.4 Black-Scholes PDE with Jumps . . . . . . . . . . . . . . . . . 68720.5 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68920.6 Self-Financing Hedging with Jumps . . . . . . . . . . . . . 693Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

20.1 Market Returns vs Gaussian and Power Tails

The modeling of risky asset by stochastic processes with continuous paths,based on Brownian motions, suffers from several defects. First, the path con-tinuity assumption does not seem reasonable in view of the possibility ofsudden price variations (jumps) resulting of market crashes, gaps or openingjumps, see e.g. Chapter 1 of Cont and Tankov (2004). Secondly, the modelingof risky asset prices by Brownian motion relies on the use of the Gaussiandistribution which tends to underestimate the probabilities of extreme events.

The R package Quantmod is installed through the command:

1 install.packages("quantmod")

" 679

This version: January 15, 2020https://www.ntu.edu.sg/home/nprivault/indext.html

https://www.ntu.edu.sg/home/nprivault/indext.html

N. Privault

The following scripts allow us to fetch DJI and STI index data using Quant-mod.1 library(quantmod)2 getSymbols("^STI",from="1990-01-03",to="2015-02-01",src="yahoo")3 stock.rtn=diff(log(Ad(`STI`)));returns <- as.vector(stock.rtn)4 m=mean(returns,na.rm=TRUE);s=sd(returns,na.rm=TRUE)5 times=index(stock.rtn)6 n = sum(is.na(returns))+sum(!is.na(returns))7 x=seq(1,n);y=rnorm(n,mean=m,sd=s)8 plot(times,returns,pch=19,cex=0.03,xaxs="i",col="blue", ylab="X", xlab="n", main = '')9 segments(x0 = times, x1 = times, y0 = 0, y1 = returns,col="blue")

10 points(times,y,pch=19,cex=0.3,col="red", ylab="X", xlab="n", main = '')11 pnorm(3*s,0,s)-pnorm(-3*s,0,s)

1 getSymbols("^DJI",from="1990-01-03",to="2015-02-01",src="yahoo")2 stock.rtn=diff(log(Ad(`DJI`)));returns <- as.vector(stock.rtn)3 m=mean(returns,na.rm=TRUE);s=sd(returns,na.rm=TRUE)4 times=index(stock.rtn)5 n = sum(is.na(returns))+sum(!is.na(returns))6 x=seq(1,n);y=rnorm(n,mean=m,sd=s)7 plot(times,returns,pch=19,xaxs="i",cex=0.03,col="blue", ylab="X", xlab="n", main = '')8 segments(x0 = times, x1 = times, y0 = 0, y1 = returns,col="blue")9 points(times,y,pch=19,cex=0.3,col="red", ylab="X", xlab="n", main = '')

10 abline(h = m+3*s, col="black", lwd =1)11 abline(h = m, col="black", lwd =1)12 abline(h = m+-3*s, col="black", lwd =1)

The

next figures illustrate the mismatch between the distributional properties ofmarket vs Gaussian returns based on the above code.

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Fig. 20.1: Market returns vs normalized Gaussian returns.

680


"


Pricing and Hedging in Jump Models

1 stock.ecdf=ecdf(as.vector(stock.rtn))2 x <- seq(-0.25, 0.25, length=100)3 px <- pnorm((x-m)/s)4 plot(stock.ecdf, xlab = 'Sample Quantiles', col="blue",ylab = '', main = 'Empirical

Cumulative Distribution')5 lines(x, px, type="l", lty=2, col="red",xlab="x value",ylab="Density", main="Gaussian cdf")6 legend("topleft", legend=c("Empirical cdf", "Gaussian cdf"),col=c("blue", "red"), lty=1:2,

cex=0.8)

−0.10 −0.05 0.00 0.05 0.10

0.00.2

0.40.6

0.81.0

Empirical Cumulative Distribution

Sample Quantiles

Empirical cdfGaussian cdf

Fig. 20.2: Empirical vs Gaussian CDF.

The following Quantile-Quantile plot is plotting the normalized empiricalquantiles against the standard Gaussian quantiles, and is obtained with theqqnorm(returns) command.

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−3 −2 −1 0 1 2 3

−0.05

0.00

0.05

0.10

Normal Q−Q Plot

Theoretical Quantiles

Sample

Quantile

s

Fig. 20.3: Quantile-Quantile plot.

1 ks.test(y,"pnorm",mean=m,sd=s)2 ks.test(returns,"pnorm",mean=m,sd=s)

The Kolmogorov-Smirnov test clearly rejects the null (normality) hypothesis.

" 681



N. Privault

One-sample Kolmogorov-Smirnov test

data: returnsD = 0.075577, p-value < 2.2e-16alternative hypothesis: two-sided

1 x <- seq(-0.25, 0.25, length=100)2 qx <- dnorm(x,mean=m,sd=s)3 stock.dens=density(returns,na.rm=TRUE)4 plot(stock.dens, xlab = 'Sample Quantiles', lwd=2, col="red",ylab = '', main = 'Empirical

Cumulative Distribution',panel.first = abline(h = 0, col='grey', lwd =0.2))5 lines(x, qx, type="l", lty=2, lwd=2, col="blue",xlab="x value",ylab="Density",

main="Gaussian cdf")6 legend("topleft", legend=c("Empirical cdf", "Gaussian cdf"),col=c("red", "blue"), lty=1:2,

cex=0.8)

−0.05 0.00 0.05 0.10

010

2030

4050

Empirical Cumulative Distribution

Sample Quantiles

Empirical cdfGaussian cdf

Fig. 20.4: Empirical density vs normalized Gaussian density.

On the other hand, power tail densities can provide a better fit of empiricaldensities, as shown in Figure 20.5.

682


"



−0.10 −0.05 0.00 0.05 0.10

010

2030

4050

Empirical densitypower density

Fig. 20.5: Empirical density vs power density.

The above fitting of empirical density is using a power probability densitydefined by a rational fraction obtained by the following R script.

1 install.packages("pracma")2 x <- seq(-0.25, 0.25, length=1000)3 stock.dens=density(returns,na.rm=TRUE, from = -0.1, to = 0.1, n = 1000)4 library(pracma)5 a<-rationalfit(stock.dens$x, stock.dens$y, d1=2, d2=2)6 plot(stock.dens$x,stock.dens$y, lwd=2, type = "l",xlab = '', col="red",ylab = '', main = '',

panel.first = abline(h = 0, col='grey', lwd =0.2))7 lines(x,

(a$p1[3]+a$p1[2]*x+a$p1[1]*x^2)/(a$p2[3]+a$p2[2]*x+a$p2[1]*x^2),type="l",lty=2,col="blue",xlab="xvalue",lwd=2, ylab="Density",main="")

8 legend("topleft", legend=c("Empirical density", "power density"),col=c("red", "blue"),lty=1:2, cex=0.8)

The output of the rationalfit command is

$p1[1] -0.184717249 -0.001591433 0.001385017

$p2[1] 1.000000e+00 -6.460948e-04 1.314672e-05

which yields a rational fraction of the form

x 7−→ 0.001385017− 0.001591433× x− 0.184717249× x2

1.314672 10−5 − 6.460948 10−4 × x+ x2

' −0.184717249− 0.001591433x

+0.001385017

x2 ,

which approximates the empirical density of DJI returns in the least squaressense.

A solution to this tail problem is to use stochastic processes with jumps, thatwill account for sudden variations of the asset prices. On the other hand,

" 683



N. Privault

such jump models are generally based on the Poisson distribution which hasa slower tail decay than the Gaussian distribution. This allows one to assignhigher probabilities to extreme events, resulting in a more realistic modelingof asset prices. Stable distributions with parameter α ∈ (0, 2) provide typicalexamples of probability laws with power tails, as their probability densityfunctions behave asymptotically as x 7→ Cα/|x|1+α when x→ ±∞.

20.2 Risk-Neutral Probability Measures

Consider an asset price modeled by the equation,

dSt = µStdt+ σStdBt + St−dYt, (20.1)

where (Yt)t∈R+ is the compound Poisson process defined in Section 19.2, withjump size distribution ν(dx) under Pν . The equation (20.1) has for solution

St = S0 exp(µt+ σBt −

σ2

2 t

) Nt∏k=1

(1 + Zk), (20.2)

t ∈ R+. An important issue for non-abitrage pricing is to determine a risk-neutral probability measure (or martingale measure) P∗ under which the dis-counted process ( e−rtSt)t∈R+ is a martingale, and this goal can be achievedusing the Girsanov Theorem for jump processes, cf. Section 19.6.

We have

d( e−rtSt) = −r e−rtStdt+ e−rtdSt= (µ− r) e−rtStdt+ σ e−rtStdBt + e−rtSt−dYt= (µ− r+ λ IEν [Z]) e−rtStdt+ σ e−rtStdBt + e−rtSt−(dYt − λ IEν [Z]dt),

which yields a martingale under P provided that

µ− r+ λ IEν [Z] = 0,

however that condition may not be satisfied by the market parameters µ, r,λ and IEν [Z].

In this case, a change of measure might be needed. The discounted assetprice process

S̃t := e−rtSt, t ∈ R+,

satisfies the equation

dS̃t = (µ− r)S̃tdt+ σS̃tdBt + e−rtS̃t−dYt= (σu− λ̃ IEν̃ [Z])S̃tdt+ σS̃tdBt + S̃t−dYt

684


"



= σS̃t(dBt + udt) + S̃t−(dYt − λ̃ IEν̃ [Z]dt)= σS̃t

(dBt + udt+ dYt − λ̃ IEν̃ [Z]dt

),

where u ∈ R. When the drift parameter u, the intensity λ̃ > 0 and a jumpsize distribution ν̃ are chosen to satisfy

µ− r = σu− λ̃ IEν̃ [Z], (20.3)

and σu+ r− µ > 0, then

λ̃ =σu+ r− µIEν̃ [Z]

> 0,

and the Girsanov Theorem 19.18 for jump processes then shows that

dBt + udt+ dYt − λ̃ IEν̃ [Z]dt

is a martingale under the probability measure Pu,λ̃,ν̃ defined in (19.31). As aconsequence, the discounted price process (S̃t)t∈R+ = ( e−rtSt)t∈R+ becomesa martingale under Pu,λ̃,ν̃ .

In this setting, the non-uniqueness of the risk-neutral probability measureis apparent since additional degrees of freedom are involved in the choicesof u, λ and the measure ν̃, whereas in the continuous case the choice ofu = (µ− r)/σ in (7.10) was unique.

20.3 Pricing in Jump Models

Recall that a market is without arbitrage if and only it admits at least onerisk-neutral probability measure.

Consider the probability measure Pu,λ̃,ν̃ built in the previous section, un-der which the discounted asset price

d( e−rtSt) = e−rtSt−(dYt − λ̃ IEν [Z]dt) + σ e−rtStdB̂t,

is a martingale, and B̂t = Bt + udt is a standard Brownian motion underPu,λ̃,ν̃ .

Then the arbitrage price of a claim with payoff C is given by

e−(T−t)r IEu,λ̃,ν̃ [C | Ft] (20.4)

under Pu,λ̃,ν̃ .

" 685



N. Privault

Clearly the price (20.4) of C is no longer unique in the presence of jumpsdue to the infinity of choices satisfying the martingale condition (20.3), andsuch a market is not complete, except if either λ̃ = λ = 0, or (σ = 0 andν̃ = ν = δ1).

Various techniques can be used for the selection of a risk-neutral proba-bility measure, such as the determination of a minimal entropy risk-neutralprobability measure P∗

u,λ̃,ν̃ that minimizes the Kullback-Leibler relative en-tropy

Q 7−→ I(Q, P) := IE[dQ

dPlog dQ

dP

]among the probability measures Q equivalent to P.

Pricing Vanilla Options

The price of a vanilla option with payoff of the form φ(ST ) on the underlyingasset ST can be written from (20.4) as

e−(T−t)r IEu,λ̃,ν̃ [φ(ST ) | Ft], (20.5)

where the expectation can be computed as

IEu,λ̃,ν̃[φ(ST ) | Ft

]= IEu,λ̃,ν̃

[φ

(S0 exp

(µT + σBT −

σ2

2 T

) NT∏k=1

(1 + Zk)

) ∣∣∣Ft]

= IEu,λ̃,ν̃

[φ

(St exp

((T − t)µ+ (BT −Bt)σ−

σ2

2 (T − t)) NT∏k=Nt+1

(1 + Zk)

) ∣∣∣Ft]

= IEu,λ̃,ν̃

[φ

(x exp

((T − t)µ+ (BT −Bt)σ−

σ2

2 (T − t)) NT∏k=Nt+1

(1 + Zk)

)]x=St

=∑n>0

Pu,λ̃,ν̃ (NT −Nt = n)

IEu,λ̃,ν̃

[φ

(x e(T−t)µ+(BT−Bt)σ−(T−t)σ2/2

NT∏k=Nt+1

(1 + Zk)

) ∣∣∣NT −Nt = n

]x=St

= e−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!

× IEu,λ̃,ν̃

[φ


n∏k=1

(1 + Zk)

)]x=St

686


"



= e−λ̃(T−t)∑n>0

(λ̃(T − t))n

n!

w∞−∞· · ·

w∞−∞︸︷︷︸

n times

IEu,λ̃,ν̃

[φ


n∏k=1

(1 + zk)

)]x=St

ν̃(dz1) · · · ν̃(dzn),

hence the price of the vanilla option with payoff φ(ST ) is given by

e−(T−t)r IEu,λ̃,ν̃ [φ(ST ) | Ft]

=1√

2(T − t)πe−(r+λ̃)(T−t)

∑n>0

(λ̃(T − t))n

n!

w∞−∞· · ·

w∞−∞︸︷︷︸

n+1 times

φ

(St e(T−t)µ+σx−(T−t)σ2/2

n∏k=1

(1 + zk)

)e−(x+(T−t)u)2/(2(T−t))ν̃(dz1) · · · ν̃(dzn)dx.

20.4 Black-Scholes PDE with Jumps

Recall that by the Markov property of (St)t∈R+ the price (20.5) at time t ofthe option with payoff φ(ST ) can be written as a function f(t,St) of t andSt, i.e.

f(t,St) = e−(T−t)r IEu,λ̃,ν̃ [φ(ST ) | Ft], (20.6)

with the terminal condition f(T ,x) = φ(x). In addition, the process

t 7−→ e(T−t)rf(t,St)

is a martingale under Pu,λ̃,ν̃ by the same argument as in (7.1).

In this section we derive a Partial Integro-Differential Equation (PIDE)for the function (t,x) 7−→ f(t,x). We have

dSt = rStdt+ σStdB̂t + St−(dYt − λ̃ IEν̃ [Z]dt), (20.7)

where B̂t = Bt + ut is a standard Brownian motion under Pu,λ̃,ν̃ . Next, bythe Itô formula with jumps (19.19), we have

df(t,St)

=∂f

∂t(t,St)dt+ rSt

∂f

∂x(t,St)dt+ σSt

∂f

∂x(t,St)dB̂t +

σ2

2 S2t∂2f

∂x2 (t,St)dt

− λ̃ IEν̃ [Z]St∂f

∂x(t,St)dt+ (f(t,St−(1 + ZNt))− f(t,St−))dNt

= σSt∂f

∂x(t,St)dB̂t + (f(t,St−(1 + ZNt))− f(t,St−))dNt

" 687



N. Privault

− λ̃ IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=Stdt

+

(∂f

∂t(t,St) + rSt

∂f

∂x(t,St) +

σ2

2 S2t∂2f

∂x2 (t,St))dt

+

(λ̃ IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=St − λ̃ IEν̃ [Z]St

∂f

∂x(t,St)

)dt.

Based on the discounted portfolio value differential

d( e−rtf(t,St))

= e−rtσSt∂f

∂x(t,St)dB̂t

+ e−rt(f(t,St−(1 + ZNt))− f(t,St−))dNt − λ̃ IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=Stdt

)+ e−rt

(−rf(t,St) +

∂f

∂t(t,St) + rSt

∂f

∂x(t,St) +

σ2

2 S2t∂2f

∂x2 (t,St))dt

(20.8)

+ e−rt(λ̃ IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=St − λ̃ IEν̃ [Z]St

∂f

∂x(t,St)

)dt,

(20.9)

obtained from the Itô Table 19.1 with jumps, and the facts that

• the Brownian motion (B̂t)t∈R+ is a martingale under Pu,λ̃,ν̃ ,

• by the smoothing formula Proposition 19.9 the process given by the dif-ferential

(f(t,St−(1+ZNt))−f(t,St−))dNt− λ̃ IEν̃ [(f(t,x(1+Z))−f(t,x))]x=Stdt,

is a martingale under Pu,λ̃,ν̃ ,

• the discounted portfolio value process t 7−→ e−rtf(t,St), is also a martin-gale under the risk-neutral probability measure Pu,λ̃,ν̃ ,

we conclude to the vanishing of the terms (20.8)-(20.9) above, i.e.

−rf(t,St) +∂f

∂t(t,St) + rSt

∂f

∂x(t,St) +

σ2

2 S2t∂2f

∂x2 (t,St)

+λ̃ IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=St − λ̃ IEν̃ [Z]St∂f

∂x(t,St) = 0,

or

∂f

∂t(t,x) + rx

∂f

∂x(t,x) + σ2

2 x2 ∂2f

∂x2 (t,x)

+λ̃w∞−∞

(f(t,x(1 + y))− f(t,x))ν̃(dy)− λ̃x∂f∂x

(t,x)w∞−∞

yν̃(dy) = rf(t,x),

688


"



which leads to the Partial Integro-Differential Equation (PIDE)

rf(t,x) = ∂f

∂t(t,x) + rx

∂f

∂x(t,x) + σ2

2 x2 ∂2f

∂x2 (t,x)

+λ̃w∞−∞

(f(t,x(1 + y))− f(t,x)− yx∂f

∂x(t,x)

)ν̃(dy),

(20.10)

under the terminal condition f(T ,x) = φ(x).

A major technical difficulty when solving the PIDE (20.10) numerically isthat the operator

f 7−→w∞−∞

(f(t,x(1 + y))− f(t,x)− yx∂f

∂x(t,x)

)ν̃(dy)

is nonlocal, therefore adding significant difficulties to the application of stan-dard discretization schemes, cf. e.g. Section 21.2.

In addition we have found that the change df(t,St) in the portfolio value(20.6) is given by

df(t,St) = σSt∂f

∂x(t,St)dB̂t + rf(t,St)dt (20.11)

+(f(t,St−(1 + ZNt))− f(t,St−))dNt − λ̃ IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=Stdt.

In the case of Poisson jumps with fixed size a, i.e. when Yt = aNt andν(dx) = δa(dx), the PIDE (20.10) reads

rf(t,x) = ∂f

∂t(t,x) + rx

∂f

∂x(t,x) + σ2

2 x2 ∂2f

∂x2 (t,x)

+λ̃

(f(t,x(1 + a))− f(t,x)− ax∂f

∂x(t,x)

),

and we have

df(t,St) = σSt∂f

∂x(t,St)dB̂t + rf(t,St)dt

+(f(t,St−(1 + a))− f(t,St−))dNt − λ̃(f(t,St(1 + a))− f(t,St))dt.

20.5 Exponential Models

Instead of modeling the asset price (St)t∈R+ through a stochastic exponential(20.2) solution of the stochastic differential equation with jumps of the form

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N. Privault

(20.1), we may consider an exponential price process of the form

St := S0 eµt+σBt+Yt

= S0 exp(µt+ σBt +

Nt∑k=1

Zk

)

= S0 eµt+σBtNt∏k=1

eZk

= S0 eµt+σBt∏

06s6te∆Yt , t ∈ R+,

from Relation (19.8), i.e. ∆Yt = ZNt∆Nt. The process (St)t∈R+ is equiva-lently given by the log-return dynamics

d logSt = µdt+ σdBt + dYt, t ∈ R+.

In this exponential model we also have

St = S0 e(µ+σ2/2)t+σBt−σ2t/2+Yt

and the process St satisfies the stochastic differential equation

dSt =

(µ+

σ2

2

)Stdt+ σStdBt + St−( e∆Yt − 1)dNt

=

(µ+

σ2

2

)Stdt+ σStdBt + St−( eZNt − 1)dNt,

hence the process St has jumps of size ST−k( eZk − 1), k > 1, and (20.3) reads

µ+σ2

2 − r = σu− λ̃ IEν̃ [ eZ − 1].

Under this condition we can choose a risk-neutral probability measure Pu,λ̃,ν̃under which ( e−rtSt)t∈R+ is a martingale, and the expected value

e−(T−t)r IEu,λ̃,ν̃ [φ(ST ) | Ft]

represents a (non-unique) arbitrage price at time t ∈ [0,T ] for the contingentclaim with payoff φ(ST ).

This arbitrage price can be expressed as

e−(T−t)r IEu,λ̃,ν̃[φ(ST )

∣∣Ft] = e−(T−t)r IEu,λ̃,ν̃[φ(S0 eµT+σBT+YT )

∣∣Ft]= e−(T−t)r IEu,λ̃,ν̃

[φ(St e(T−t)µ+(BT−Bt)σ+YT−Yt)

∣∣Ft]690


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= e−(T−t)r IEu,λ̃,ν̃[φ(x e(T−t)µ+(BT−Bt)σ+YT−Yt)

]x=St

= e−(T−t)r IEu,λ̃,ν̃

φx exp

(T − t)µ+ (BT −Bt)σ+

NT∑k=Nt+1

Zk

x=St

= e−(T−t)r−(T−t)λ̃

×∑n>0

(λ̃(T − t))n

n!IEu,λ̃,ν̃

[φ

(x e(T−t)µ+(BT−Bt)σ exp

(n∑k=1

Zk

))]x=St

.

The Merton Model

We assume that (Zk)k>1 is a family of independent identically distributedGaussian N (δ, η2) random variables under Pu,λ̃,ν̃ with

µ+σ2

2 − r = σu− λ̃ IEν̃ [ eZ − 1] = σu− λ̃( eδ+η2/2 − 1),

as in (20.3), hence by the Girsanov Theorem 19.18 for jump processes, Bt +ut+ Yt − λ̃ IEν̃ [ eZ − 1]t is a martingale and Bt + ut is a standard Brownianmotion under Pu,λ̃,ν̃ . For simplicity we choose u = 0, which yields

µ = r− σ2

2 − λ̃( eδ+η2/2 − 1),

and we have

e−(T−t)r IEλ̃,ν̃ [φ(ST ) | Ft]

= e−(T−t)r−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!

× IEλ̃,ν̃

[φ

(x e(T−t)µ+(BT−Bt)σ exp

(n∑k=1

Zk

))]x=St

= e−(T−t)r−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!IE[φ(x e(T−t)µ+nδ+Xn)

]x=St

= e−(T−t)r−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!

w∞−∞

φ(St e(T−t)µ+nδ+y) e−y2/(2((T−t)σ2+nη2))√4((T − t)σ2 + nη2)π

dy,

where

Xn := (BT −Bt)σ+n∑k=1

(Zk − δ) ' N (0, (T − t)σ2 + nη2), n > 0,

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N. Privault

is a centered Gaussian random variable with variance

v2n := (T − t)σ2 +

n∑k=1

VarZk = (T − t)σ2 + nη2.

Hence when φ(x) = (x−κ)+ is the payoff function of a European call option,using the relation

Bl(x,κ, v2

n/τ , r, τ)= e−rτ IE

[(x eXn−v2

n/2+rτ −K)+]

we get

e−(T−t)r−(T−t)λ̃ IEλ̃,ν̃ [(ST − κ)+ | Ft]

= e−(T−t)r−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!IE[(x e(T−t)µ+nδ+Xn − κ

)+]x=St

= e−(T−t)r−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!

× IE[(x e(r−σ2/2−λ̃( eδ+η2/2−1))(T−t)+nδ+Xn − κ

)+]x=St

= e−(T−t)r−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!

× IE[(x enδ+nη2/2−λ̃( eδ+η2/2−1)(T−t)+Xn−v2)n/2+(T−t)r − κ

)+]x=St

= e−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!

×Bl(St enδ+nη2/2−λ̃( eδ+η2/2−1)(T−t),κ,σ2 + nη2/(T − t), r,T − t).

We may also write

e−(T−t)r−(T−t)λ̃ IEλ̃,ν̃ [(ST − κ)+ | Ft]

= e−(T−t)λ̃∑n>0

((T − t)λ̃)n

n!enδ+nη2/2−λ̃( eδ+η2/2−1)(T−t)

×Bl(St,κ e−nδ−nη2/2+λ̃( eδ+η2/2−1)(T−t),σ2 + nη2/(T − t), r,T − t

)= e−λ̃ eδ+η2/2(T−t)

∑n>0

(λ̃ eδ+nη2/2(T − t))n

n!

×Bl(St,κ,σ2 + nη2/(T − t), r+ n

δ + η2/2T − t

− λ̃( eδ+η2/2 − 1),T − t)

.

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20.6 Self-Financing Hedging with Jumps

Consider a portfolio valued

Vt := ηtAt + ξtSt = ηt ert + ξtSt

at time t ∈ R+, and satisfying the self-financing condition (5.3), i.e.

dVt = ηtdAt + ξtdSt = rηt ertdt+ ξtdSt.

Assuming that the portfolio value takes the form Vt = f(t,St) for all timest ∈ [0,T ], by (20.7) we have

dVt = df(t,St)= rηt ertdt+ ξtdSt

= rηt ertdt+ ξt(rStdt+ σStdB̂t + St−(dYt − λ̃ IEν̃ [Z]dt))= rVtdt+ σξtStdB̂t + ξtSt−(dYt − λ̃ IEν̃ [Z]dt)= rf(t,St)dt+ σξtStdB̂t + ξtSt−(dYt − λ̃ IEν̃ [Z]dt), (20.12)

has to match

df(t,St) = rf(t,St)dt+ σSt∂f

∂x(t,St)dB̂t (20.13)

+ (f(t,St−(1 + ZNt))− f(t,St−))dNt − λ̃ IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=Stdt,

which is obtained from (20.11).

In such a situation we say that the claim payoff C can be exactly replicated.

Exact replication is possible in essentially only two situations:

(i) Continuous market, λ = λ̃ = 0. In this case we find the usual Black-Scholes Delta:

ξt =∂f

∂x(t,St). (20.14)

(ii) Poisson jump market, σ = 0 and Yt = aNt, ν(dx) = δa(dx). In thiscase we find

ξt =1

aSt−(f(t,St−(1 + a))− f(t,St−)). (20.15)

Note that in the limit a→ 0 this expression recovers the Black-ScholesDelta formula (20.14).

When Conditions (i) or (ii) above are not satisfied, exact replication is notpossible and this results into an hedging error given from (20.12) and (20.13)

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N. Privault

by

VT − φ(ST ) = VT − f(T ,ST )

= V0 +w T

0dVt − f(0,S0)−

w T

0df(t,St)

= V0 − f(0,S0) + σw T

0St

(ξt −

∂f

∂x(t,St)

)dB̂t

+w T

0ξtSt−(ZNtdNt − λ̃ IEν̃ [Z]dt)

−w T

0(f(t,St−(1 + ZNt))− f(t,St−))dNt

+λ̃w T

0IEν̃ [(f(t,x(1 + Z))− f(t,x))]x=Stdt.

Assuming for simplicity that Yt = aNt, i.e. ν(dx) = δa(dx), we get

VT − f(T ,ST ) = V0 − f(0,S0) + σw T

0St−

(ξt −

∂f

∂x(t,St−)

)dB̂t

−w T

0(f(t,St−(1 + a))− f(t,St−)− aξtSt−)(dNt − λ̃dt),

hence the mean square hedging error is given by

IEu,λ̃[(VT − f(T ,ST )

)2]= (V0 − f(0,S0))

2 + σ2 IEu,λ̃

[(w T

0St−

(ξt −

∂f

∂x(t,St−)

)dB̂t

)2]

+ IEu,λ̃

[( w T

0(f(t,St−(1 + a))− f(t,St−)− aξtSt−)(dNt − λ̃dt)

)2]= (V0 − f(0,S0))

2 + σ2 IEu,λ̃

[w T

0S2t−

(ξt −

∂f

∂x(t,St−)

)2dt

]

+λ̃ IEu,λ̃

[w T

0

((f(t,St−(1 + a))− f(t,St−)− aξtSt−)

)2dt

],

where we applied the Itô isometry (19.17). Clearly, the initial portfolio valueV0 that minimizes the above quantity is

V0 = f(0,S0) = e−rT IEu,λ̃,ν̃ [φ(ST )].

When hedging only the risk generated by the Brownian part we let

ξt =∂f

∂x(t,St−)

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as in the Black-Scholes model, and in this case the hedging error due to thepresence of jumps becomes

IEu,λ̃[(VT −f(T ,ST )

)2]= λ̃ IEu,λ̃

[w T

0

((f(t,St−(1 + a))− f(t,St−)− aξtSt−)

)2dt

].

Next, let us find the optimal strategy (ξt)t∈R+ that minimizes the remaininghedging error

IEu,λ̃

[w T

0

(σ2S2

t−

(ξt −

∂f

∂x(t,St−)

)2+ λ̃

((f(t,St−(1 + a))− f(t,St−)− aξtSt−)

)2)dt

].

For all t ∈ [0,T ], the almost-sure minimum of

ξt 7−→ σ2S2t−

(ξt −

∂f

∂x(t,St−)

)2+ λ̃((f(t,St−(1+a))−f(t,St−)−aξtSt−)

)2is given by differentiation with respect to ξt, as the solution of

2σ2S2t−

(ξt −

∂f

∂x(t,St−)

)−2aλ̃St− ((f(t,St−(1 + a))− f(t,St−)− aξtSt−)) = 0,

i.e.

ξt =σ2

σ2 + a2λ̃

∂f

∂x(t,St−) +

a2λ̃

σ2 + a2λ̃× f(t,St−(1 + a))− f(t,St−)

aSt−,

(20.16)

t ∈ (0,T ]. We note that the optimal strategy (20.16) is a weighted average ofthe Brownian and jump hedging strategies (20.14) and (20.15) according tothe respective variance parameters σ2 and a2λ̃ of the continuous and jumpcomponents.

Clearly, if aλ̃ = 0 we get

ξt =∂f

∂x(t,St−), t ∈ [0,T ],

which is the Black-Scholes perfect replication strategy, and when σ = 0 werecover

ξt =f(t, (1 + a)St−)− f(t,St−)

aSt−, t ∈ [0,T ].

which is (20.15).

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N. Privault

Note that the fact that perfect replication is not possible in a jump-diffusion model can be interpreted as a more realistic feature of the model,as perfect replication is not possible in the real world.

See Jeanblanc and Privault (2002) for an example of a complete marketmodel with jumps, in which continuous and jump noise are mutually exclud-ing each other over time.

In the following table we summarize the properties of geometric Brownianmotion vs jump-diffusion models in terms of asset price and market behaviors.

PropertiesModel Geometric Brownian motion Jump-diffusion model Real world

Discontinuous asset prices 7 X XFat tailed market returns 7 X X

Complete market X 7 7

Unique prices and risk-neutral measure X 7 7

Table 20.1: Market models and properties.

Exercises

Exercise 20.1 Consider a standard Poisson process (Nt)t∈R+ with inten-sity λ > 0 under a probability measure P. Let (St)t∈R+ be defined by thestochastic differential equation

dSt = rStdt+ ηSt−(dNt − αdt),

where η > 0.

a) Find the value of α ∈ R such that the discounted process ( e−rtSt)t∈R+

is a martingale under P.b) Compute the price at time t ∈ [0,T ] of a power option with payoff |ST |2

at maturity T .

Exercise 20.2 Consider a long forward contract with payoff ST −K on ajump diffusion risky asset (St)t∈R+ given by

dSt = µStdt+ σStdBt + St−dYt.

a) Show that the forward claim admits a unique arbitrage price to be com-puted in a market with risk-free rate r > 0.

b) Show that the forward claim admits an exact replicating portfolio strategybased on the two assets St and ert.

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c) Recover portfolio strategy of Question (b) using the optimal portfoliostrategy formula (20.16).

Exercise 20.3 Consider (Bt)t∈R+ a standard Brownian motion and (Nt)t∈R+

a standard Poisson process with intensity λ > 0, independent of (Bt)t∈R+ ,under a probability measure P∗. Let (St)t∈R+ be defined by the stochasticdifferential equation

dSt = µStdt+ ηSt−dNt + σStdBt. (20.17)

a) Solve the equation (20.17).b) We assume that µ, η and the risk-free rate r > 0 are chosen such that the

discounted process ( e−rtSt)t∈R+ is a martingale under P∗. What relationdoes this impose on µ, η, λ and r?

c) Under the relation of Question (b), compute the price at time t ∈ [0,T ]of a European call option on ST with strike price κ and maturity T , usinga series expansion of Black-Scholes functions.

Exercise 20.4 Consider (Nt)t∈R+ a standard Poisson process with inten-sity λ > 0 under a probability measure P. Let (St)t∈R+ be defined by thestochastic differential equation

dSt = rStdt+ YNtSt−dNt,

where (Yk)k>1 is an i.i.d. sequence of uniformly distributed random variableson [−1, 1].

a) Show that the discounted process ( e−rtSt)t∈R+ is a martingale under P.b) Compute the price at time 0 of a European call option on ST with strike

price κ and maturity T , using a series of multiple integrals.

Exercise 20.5 Consider a standard Poisson process (Nt)t∈R+ with inten-sity λ > 0 under a probability measure P. Let (St)t∈R+ be defined by thestochastic differential equation

dSt = rStdt+ YNtSt−(dNt − αdt),

where (Yk)k>1 is an i.i.d. sequence of uniformly distributed random variableson [0, 1].

a) Find the value of α ∈ R such that the discounted process ( e−rtSt)t∈R+

is a martingale under P.b) Compute the price at time t ∈ [0,T ] of the long forward contract with

maturity T and payoff ST − κ.

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N. Privault

Exercise 20.6 Consider (Nt)t∈R+ a standard Poisson process with intensityλ > 0 under a risk-neutral probability measure P∗. Let (St)t∈R+ be definedby the stochastic differential equation

dSt = rStdt+ αSt−(dNt − λdt), (20.18)

where α > 0. Consider a portfolio with value

Vt = ηt ert + ξtSt

at time t ∈ [0,T ], and satisfying the self-financing condition

dVt = rηt ertdt+ ξtdSt.

We assume that the portfolio hedges a claim payoff C = φ(ST ), and that theportfolio value can be written as a function Vt = f(t,St) of t and St for alltimes t ∈ [0,T ].

a) Solve the stochastic differential equation (20.18). (10 marks)b) Price the claim C = φ(ST ) at time t ∈ [0,T ] using a series expansion. (10

marks)c) Show that under self-financing, the variation dVt of the portfolio value Vt

satisfiesdVt = rf(t,St)dt+ αξtSt−(dNt − λdt). (20.19)

d) Show that the claim payoff C = φ(ST ) can be exactly replicated by thehedging strategy

ξt =1

αSt−(f(t,St−(1 + α))− f(t,St−)).

Exercise 20.7 Pricing by the Esscher transform (Gerber and Shiu (1994)).Consider a compound Poisson process (Yt)t∈[0,T ] with IE

[e(Yt−Ys)θ

]=

e(t−s)m(θ), θ ∈ R, 0 6 s 6 t, and the asset price process St := ert+Yt ,t ∈ [0,T ]. Given θ ∈ R, let

Nt :=eθYt

IE[eθYt

] = eθYt−tm(θ) = Sθt e−rθt−tm(θ),

and consider the probability measure Pθ defined as

dPθ|Ft

dP|Ft:=

NTNt

= e(YT−Yt)θ−(T−t)m(θ), 0 6 t 6 T .

a) Check that (Nt)t∈R+ is a martingale under P.698


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b) Find a condition on θ such that the discounted price process ( e−rtSt)t∈[0,T ] =(eYt)t∈[0,T ] is a martingale under Pθ.

c) Price the European call option with payoff (ST −K)+ by taking Pθ asrisk-neutral measure.

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Documents

Notes on Stochastic FinanceChapter20 PricingandHedginginJumpModels In this chapter we consider option pricing and hedging in jump-diﬀusion models