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160 7 Poisson Jumps
with
f
JSimple
1
(
φ
)=exp
−
1
2
(1+i
φ
)
λ
JT +
λ
T exp[(1 + i
φ
)ln(1+J)] −
λ
T
. (7.16)
In a similar fashion, we obtain
f
2
(
φ
)=E[exp(i
φ
x(T))]
= E
exp(i
φ
rT + i
φ
x
0
−i
φ
1
2
v
2
+
λ
J
T + i
φ
ln(1+J)
T
0
dY(t)
= exp
i
φ
rT + i
φ
x
0
−i
φ
1
2
v
2
+
λ
J
T +
1
2
(1+i
φ
)
2
v
2
T
+
λ
T exp[i
φ
ln(1+J)] −
λ
T)
= exp(i
φ
(x
0
+ rT)) f
BS
2
(
φ
) f
JSimple
2
(
φ
) (7.17)
with
f
JSimple
2
(
φ
)=exp
−
1
2
i
φλ
JT +
λ
T exp[i
φ
ln(1+J)] −
λ
T
. (7.18)
The most appealing attribute to f
j
(
φ
) is that they are continuous functions of
φ
although the Poisson distribution is not continuous. The closed-form pricing for-
mula for the European call options is then given in the standard way where the
probabilities F
j
are expressed by the Fourier inversion. If we omit the diffusion parts
from these two CFs, that is, v = 0 and r−
λ
J = q, we obtain the same option pricing
formula as given in (7.12). Obviously, this new formula is more easily computable
and unified with other pricing formulas in a same Fourier transform framework. Due
to the deterministic specification of jump size J, the simple jump model presented
above finds only a limited application in practice, and does not have large flexibility
to capture smile features of implied volatilities. In the following two subsections,
we discuss two more complicated and realistic jump models.
7.3 Lognormal Jumps
In the line with Merton (1976), we go back to the process (7.7), the so-called jump-
diffusion model, and further assume that jump size J is no longer deterministic
but lognormally distributed. The Brownian motion w
1
(t), the Poisson term Y(t)
and the jump size J are mutually stochastically independent. Formally, we have the
following specification,
10
ln(1+J) ∼ N
ln(1+
μ
J
) −
1
2
σ
2
J
,
σ
2
J
,
μ
J
−1. (7.19)
10
Compare with Scott (1997) and BCC (1997).
7.3 Lognormal Jumps 161
This means that J has a mean of
μ
J
and ln(1 + J) has a standard deviation of
σ
J
.
Because of the normal distribution of the random variable ln(1+ J), this jump itself
is log normally distributed, and is referred to as lognormal jump. Lognormal jump
model is the most popular jump model in financial literature, and is widely used to
examine the empirical properties of stock prices and the smile features of implied
volatilities.
We obtain immediately
dx(t)=[r(t) −
λμ
J
−
1
2
v
2
(t)]dt + v(t)dW
1
+ ln(1+ J)dY (t), (7.20)
where it is clear that the abnormal stock returns governed by the term ln(1+J)dY(t)
have a normally distributed jump size. Hence, the abnormal stock returns could be
both positive and negative, and are modeled more realistically than in a simple jump
model. Merton (1976) gave a pricing formula in terms of an infinite sum,
C =
∞
∑
n=0
exp[−
λ
T(1 +
μ
J
)](
λ
T(1 +
μ
J
))
n
n!
C
n
, (7.21)
with C
n
as the Black-Scholes price with an instantaneous variance v
2
+
n
T
σ
2
J
and the
risk-free rate r −
λμ
J
+
n
T
ln(1 +
μ
J
). This formula is obviously not a closed-form
solution since it includes an infinite number of Black-Scholes formulas. Again, we
can express formula (7.21) in an alternative way with CF, and for convenience, let
the interest rate and volatility be constant for the time being.
We perform the calculation of the first CF as follows:
f
1
(
φ
)=E
S(T)
S
0
e
rT
exp(i
φ
x(T))
= E
exp
i
φ
(rT + x
0
) −(1 + i
φ
)
λ
T
μ
J
−(1+ i
φ
)
1
2
v
2
T
+
1
2
(1+i
φ
)
2
v
2
T +(1 + i
φ
)ln(1+J)
T
0
dY(t)
= E[exp(
λ
T exp((1 + i
φ
)ln(1+J)) −
λ
T)]
× exp
i
φ
(rT + x
0
) −(1 + i
φ
)
λ
T
μ
J
−(1+ i
φ
)
1
2
v
2
T +
1
2
(1+i
φ
)
2
v
2
T
= exp
i
φ
(rT + x
0
) −(1 + i
φ
)
λ
T
μ
J
−(1+ i
φ
)
1
2
v
2
T
+
1
2
(1+i
φ
)
2
v
2
T +
λ
T ·E[exp((1 + i
φ
)ln(1+J))] −
λ
T
162 7 Poisson Jumps
= exp
i
φ
(rT + x
0
) −(1 + i
φ
)
λ
T
μ
J
+
1
2
i
φ
(1+i
φ
)v
2
T
+
λ
T[(1 +
μ
J
)
(1+i
φ
)
e
1
2
i
φ
(1+i
φ
)
σ
2
J
−1]
,
=(i
φ
(x
0
+ rT)) f
BS
1
(
φ
) f
LogN
1
(
φ
) (7.22)
with
f
LogN
1
(
φ
)=exp
−(1+ i
φ
)
λ
T
μ
J
+
λ
T[(1 +
μ
J
)
(1+i
φ
)
e
1
2
i
φ
(1+i
φ
)
σ
2
J
−1]
, (7.23)
where, since J and dY (t) are independent of each other,
11
we can set the expectation
operator directly in the front of exp[(1 + i
φ
)ln(1+J)].
In a similar manner, we may calculate
f
2
(
φ
)=E[exp(i
φ
x(T))]
= exp
i
φ
(rT + lnS
0
) −i
φλ
T
μ
J
+
1
2
i
φ
(1+i
φ
)v
2
T
+
λ
T[(1 +
μ
J
)
i
φ
e
1
2
i
φ
(i
φ
−1)
σ
2
J
−1]
=(i
φ
(x
0
+ rT)) f
BS
2
(
φ
) f
LogN
2
(
φ
) (7.24)
with
f
LogN
2
(
φ
)=exp
−i
φλ
T
μ
J
+
λ
T[(1 +
μ
J
)
i
φ
e
1
2
i
φ
(i
φ
−1)
σ
2
J
−1]
. (7.25)
The closed-form pricing formula is then given in the usual way by applying the
the Fourier inversion. Additionally, we can easily do comparative statics with re-
spect to the jump parameters
λ
and
μ
J
while it is rather cumbersome to do this with
Merton’s formula. If
σ
J
is approaching zero, then the jump size will converge to the
deterministic one and the lognormal jump becomes a simple jump which is exactly
the case addressed above.
11
The detailed calculation is as follows:
E[exp((1 + i
φ
)ln(1 + J))]
= exp
(1+ i
φ
)ln(1 +
μ
J
) −
1
2
(1+ i
φ
)
σ
2
J
+
1
2
(1+ i
φ
)
2
σ
2
J
=(1 +
μ
J
)
(1+i
φ
)
exp
1
2
i
φ
(1+ i
φ
)
σ
2
J
.
7.4 Pareto Jumps 163
7.4 Pareto Jumps
An other way to enhance the variants of jumps is to specify the jump size by an
alternative distribution. In contrast to the above section, the logarithm of the random
jump size may be suggested to be exponentially distributed with a mean of
μ
as
follows:
ln(1+J) ∼ Ex[
μ
], (7.26)
where the probability density function (pdf) of an exponential distribution takes the
following form
pd f
Ex
(y;
μ
)=
1
μ
exp
−
y
μ
, 0 < y < ∞, (7.27)
that is defined only in positive real line.
Duffie, Pan and Singleton (1999) applied this type of distribution to model the
jumps occurring in a volatility process. The mean and standard deviation of the
exponentially distributed variable are identical and are given by
E[y]=
μ
, Std[y]=
μ
. (7.28)
Therefore, an exponential distribution is entirely characterized by a single parameter
μ
. We can immediately derive from (7.27) that J has a probability density function
of the following form
pd f
J
(y)=
1
μ
(1+y)
exp
−
ln(1+y)
μ
=
1
μ
(1+y)
−
1
μ
−1
, 0 < y < ∞, (7.29)
which means that the jump size J is Pareto distributed.
12
Hence, we call this a Pareto
jump. The interchangeability between an exponential distribution and its Pareto
counterpart is similar to the interchangeability between a normal distribution and
its lognormal counterpart. One of the important implications of the Pareto distribu-
tion and the associated exponential distribution is that the random jump size J and its
logarithm ln(1+ J) can only be positive and have positive means. Given 0 <
μ
< 1,
the mean of J exists and is equal to
E[J]=
μ
1 −
μ
. (7.30)
Therefore, if we specify the abnormalities in the dynamic movements of stock
prices using a single positive Pareto jump, the jumps can only be positive. To in-
corporate possible negative jumps into stock price process, we may introduce two
Parato jumps with different expected jump sizes and jump intensities, one is re-
12
For more details see Stuard and Ord (1994), or Johnson, Kotz and Balakrishnan (1994).
164 7 Poisson Jumps
sponsible for positive surprises, the other for negative surprises. More precisely, we
specify the following dynamics for stock returns:
dx(t)=[r −
μ
−
1
2
v
2
]dt +vdW
1
+ ln(1 + J
1
)dY
1
(t) −ln(1+ J
2
)dY
2
(t),
=[r −
μ
−
1
2
v
2
]dt +vdW
1
+ y
1
dY
1
(t) −y
2
dY
2
(t), (7.31)
with
μ
=
λ
1
E[J
1
] −
λ
1
E[J
2
]=
λ
1
μ
J
1
1 −
μ
J
1
−
λ
2
μ
J
2
1 −
μ
J
2
,
ln(1+J
1
) ∼
1
μ
J
1
exp
−
ln(1+J
1
)
μ
J
1
, 0 < J
1
< ∞,
ln(1+J
2
) ∼
1
μ
J
2
exp
−
ln(1+J
2
)
μ
J
2
, 0 < J
2
< ∞.
The jump Y
1
(t) with jump intensity
λ
1
characterizes the positive abnormal move-
ments in stock prices while Y
2
(t) with jump intensity
λ
2
characterizes the negative
abnormal movements. Their joint impact on the drift of the risk-neutral process is
then
μ
=
λ
1
E[J
1
] −
λ
2
E[J
2
]. Together we use 4 parameters to specify two jumps.
Note, if y is exponentially distributed with a mean
μ
, for any real or complex
number k, the moment-generating function of y is given by
E[e
ky
]=
1
1 −k
μ
.
Based on the risk-neutralized process, applying the moment-generating function of
y, we can calculate the two CFs that are relevant for option pricing as follows:
f
1
(
φ
)
= E
S(T)
S
0
e
rT
exp(i
φ
x(T))
= E
exp
i
φ
(rT + x
0
) −(1 + i
φ
)
μ
T −(1+i
φ
)
1
2
v
2
T +
1
2
(1+i
φ
)
2
v
2
T
+(1 +i
φ
)ln(1+J
1
)
T
0
dY
1
(t) −(1+ i
φ
)ln(1+ J
2
)
T
0
dY
2
(t)
= exp
i
φ
(rT + x
0
) −(1 + i
φ
)
μ
T −(1+i
φ
)
1
2
v
2
T +
1
2
(1+i
φ
)
2
v
2
T
× E[exp[(
λ
1
T exp((1 + i
φ
)ln(1+J
1
)) −
λ
1
T)
− (
λ
2
T exp((1 + i
φ
)ln(1+J
2
)) −
λ
2
T)]].
7.5 The Kou Model: An Equivalence to Pareto Jumps 165
Computing the expected value under exponential distribution yields
f
1
(
φ
)=exp
i
φ
(rT + x
0
) −(1 + i
φ
)
μ
T −(1+i
φ
)
1
2
v
2
T +
1
2
(1+i
φ
)
2
v
2
T
+
λ
1
T ·E[exp((1 + i
φ
)ln(1+J
1
))] −
λ
1
T
−
λ
2
T ·E[exp((1 + i
φ
)ln(1+J
2
))] −
λ
2
T)
= exp
i
φ
(rT + x
0
) −(1 + i
φ
)
μ
T +
1
2
i
φ
(1+i
φ
)v
2
T
+
λ
1
T[(1 −(1+ i
φ
)
μ
J
1
)
−1
−1]−
λ
2
T[(1 −(1+ i
φ
)
μ
J
2
)
−1
−1]
=(i
φ
(x
0
+ rT)) f
BS
1
(
φ
) f
Paret o
1
(
φ
) (7.32)
with
f
Paret o
1
(
φ
)=exp
−(1+ i
φ
)
μ
T +
λ
1
T[(1 −(1+ i
φ
)
μ
J
1
)
−1
−1]
−
λ
2
T[(1 −(1+ i
φ
)
μ
J
2
)
−1
−1]
. (7.33)
We have the similar result for the second CF,
f
2
(
φ
)=E[exp(i
φ
x(T))]
= exp
i
φ
(rT + x
0
) −i
φμ
T +
1
2
i
φ
(1+i
φ
)v
2
T
+
λ
1
T[(1 −i
φμ
J
1
)
−1
−1]−
λ
2
T[(1 −i
φμ
J
2
)
−1
−1]
=(i
φ
(x
0
+ rT)) f
BS
2
(
φ
) f
Paret o
2
(
φ
) (7.34)
with
f
Paret o
2
(
φ
)=exp
−i
φμ
T +
λ
1
T[(1 −i
φμ
J
1
)
−1
−1]
−
λ
2
T[(1 −i
φμ
J
2
)
−1
−1]
. (7.35)
Given these two CFs, the option pricing formula with Pareto jumps may be cor-
respondingly derived.
7.5 The Kou Model: An Equivalence to Pareto Jumps
Kou (2002) proposed a jump-diffusion model with jumps that may be regarded as
two weighted Pareto jumps. Under the risk-neutral measure, Kou’s model reads
dS(t)
S(t)
=(r −
λ
E[U −1])dt + vdW (t)+d
Y(t)
∑
i=1
(U
i
−1)
=(r −
λ
E[J])dt + vdW(t)+JdY(t), (7.36)
166 7 Poisson Jumps
where it is assumed that u = lnU has an asymmetric double-exponential distribution
with a density function
pd f
u
(z)=p
η
1
e
−
η
1
z
1
(z0)
+ q
η
2
e
η
2
z
1
(z<0)
,
with
η
1
> 1 and
η
2
> 0. The parameters p and q with p+q = 1 are considered as the
probabilities for positive and negative abnormalities, that correspond to the positive
and negative abnormalities respectively in the above discussed Pareto jump model.
In other words, we have a weighted jump,
u = pu
+
+ q(−u
−
)=pu
+
−qu
−
,
with
E[u]=p
1
η
1
−q
1
η
2
,
E[U]=pE[e
u
+
]+qE[e
−u
−
]=p
η
1
η
1
−1
+ q
η
2
η
2
+ 1
,
where u
+
and u
−
are two exponential random variables with means 1/
η
1
and 1/
η
2
respectively, and are weighted according to the Bernoulli law.
To show that the Kou model is equivalent to the Pareto jump model, we denote
μ
1
=
1
η
1
,
μ
2
=
1
η
2
, z
−
= −z1
(z<0)
,
then z
−
> 0 is exponentially distributed with mean
μ
2
=
1
η
2
. With these notations,
the expected value of U can be rewritten as
E[U]=p
1
1 −
μ
1
+ q
1
1 +
μ
2
.
It follows immediately,
pd f
u
(z)=p
1
μ
1
exp
−
z
μ
1
1
(z0)
+ q
1
μ
2
exp
−
z
−
μ
1
1
(z
−
>0)
. (7.37)
Therefore, the stock price process S(t) can be restated as
dS(t)
S(t)
=(r −
λ
E[J])dt + vdW(t)
+ pJ
+
dY(t)+q(−J
−
)dY(t), (7.38)
with J
+
+ 1 = e
u
+
and J
−
+ 1 = e
−u
−
. This implies that u
+
= ln(J
+
+ 1) and u
−
=
−ln(J
−
+ 1) are exponentially distributed. The process for stock return x(t) takes
the following form,
7.5 The Kou Model: An Equivalence to Pareto Jumps 167
dx(t)=
r −
λ
E[J] −
1
2
v
2
dt + vdW(t)
+ pln(J
+
+ 1)dY(t) −qln(J
−
+ 1)dY(t)
=
r −
λ
E[J] −
1
2
v
2
dt + vdW(t)
+ ln(J
+
+ 1)dY
1
(t) −ln(J
−
+ 1)dY
2
(t), (7.39)
where two new Poisson processes are defined by
dY
1
(t)=pdY (t), dY
2
(t)=qdY (t)
with two arrival intensity parameters
λ
1
= p
λ
and
λ
2
= q
λ
. By a comparison with
the processes (7.31) and (7.39), the new formulation of Kou’s jump model is obvi-
ously equivalent to the Pareto jump model discussed above. Finally we can verify
E[U]=pE[e
u
+
]+qE[e
−u
−
]=pE[1+J
+
]+qE[1+J
−
]
= pE[J
+
]+qE[J
−
]+1 = E[J]+1, (7.40)
from which it follows
E[J]=E[U] −1 = p
μ
1
1 −
μ
1
+ q
μ
2
1 +
μ
2
.
To emphasis that a negative exponentially distributed random variable is incorpo-
rated into this jump model, we set
μ
∗
2
= −
μ
2
, therefore we have
μ
=
λ
E[J]=
λ
1
μ
1
1 −
μ
1
−
λ
2
μ
∗
2
1 −
μ
∗
2
, (7.41)
that is identical to the jump correction term in the Pareto jump model. The Kou
model can be re-formulated in the following log-normal form,
dx(t)=
r −
μ
−
1
2
v
2
dt +vdW(t)
+ ln(J
+
+ 1)dY
1
(t) −ln(J
−
+ 1)dY
2
(t)
=
r −
μ
−
1
2
v
2
dt +vdW(t)
+ u
+
dY
1
(t) −u
−
dY
2
(t), (7.42)
that is mathematically identical to the process given by (7.31) in the Pareto jump
model.
In spite of the clear similarity of the Pareto jump model and the Kou model, the
interpretations of these two jump models are different and subtle. While the Pareto
jump model is equipped with positive and negative jumps, both of which can happen
simultaneously, and overlap each other, the positive and negative jumps in Kou’s
model are weighted according to Bernoulli’s law, and can not occur simultaneously.
Although we may define two Poisson processes Y
1
(t) and Y
2
(t) in the Kou model,
168 7 Poisson Jumps
but they are driven by a same randomness. As a result, conditional on a jump event
occurring, either a positive or a negative surprise can happen in the Kou model.
Summing up, by setting
μ
1
=
1
η
1
,
μ
∗
2
= −
1
η
2
,
λ
1
= p
λ
,
λ
2
= q
λ
, (7.43)
we have established an equivalence between the Pareto jump model and Kou’s
model. Kou’s original option pricing formula is rather complicated and is expressed
in terms of a sum of Hh functions, a special function of mathematical physics.
The numerical computation of European-style option prices is then very extensive.
In contrast, as an equivalence of Kou’s model to the Pareto jumps is established,
a pricing formula for Kou’s model similar to (7.33) and (7.35) is available, and
shares the same computation method as other jump models. The application of the
Fourier transform in Kou’s model then gains many advantages in terms of efficiency,
tractability and simplicity.
7.6 Affine Jump-Diffusions
In this chapter, we have actually discussed jumps in a jump-diffusion model where
the underlying stock prices S(t) are driven jointly by a diffusion, i.e., a Brown-
ian motion W(t), and a jump component JdY(t). In fact, the jump-diffusion as an
enhanced setting may be applied to model any financial quantity. Duffie, Pan and
Singleton (DPS, 2000) used affine structures of a multi-dimensional jump-diffusion
X(t) as background-driving factors to model the dynamics of all pricing-relevant
rates, such as stock returns, interest rates, volatilities, and foreign exchanges. As in
Bakshi and Madan (1999), Duffie, Pan and Singleton considered a discounted CF
(see Section 2.2.2) as follows
f
∗
(
φ
,T −t)=E
exp
−
T
t
r(X(s),s)ds
e
i
φ
X(T )
|F
t
, (7.44)
where interest rate r(X(t),t) is affine in X(t). By setting
φ
= 0, we obtain the expres-
sion for zero-coupon bond. An important insight of DPS for affine jump-diffusion
is that the discounted CF f
∗
(
φ
) under some technical regularity conditions admits a
solution of exponential affine form,
f
∗
(
φ
,T −t)=e
α
(t)+
β
(t)X(t)
, (7.45)
where
α
(t) and
β
(t) are determined by two ODEs, and have the closed-form solu-
tions for many particular processes.
In the line of the jump-diffusion model discussed above, we propose that X(t) ∈
R
n
is governed by the following n-dimensional process under the risk-neutral mea-
sure,
7.6 Affine Jump-Diffusions 169
dX(t)=
μ
(X)dt + v(X)dW(t)+qdY(t), (7.46)
where the interest rate r(X) ∈ R, the drift
μ
∈ R
n
, the variance v(X)v(X)
∈ R
n×n
and the intensity
λ
(X) ∈ R
n
all take an affine functional form of X. In details,
•
μ
(x)=A
0
+ A
1
·x, A =(A
0
,A
1
) ∈ R
n
×R
n×n
.
• (v(x)v(x)
)
ij
=(B
0
)
ij
+(B
1
)
ij
·x, B =(B
0
,B
1
) ∈ R
n×n
×R
n×n×n
.
•
λ
(x)=c
0
+C
1
·x, C =(c
0
,C
1
) ∈ R×R
n
.
• r(x)=d
0
+ D
1
·x, D =(d
0
,D
1
) ∈ R×R
n
.
Therefore, based on the factor process X(t), the coefficients (A,B,C,D), called the
characteristics of affine jump-diffusion, completely determine all dynamics of the
underlying rates that are relevant for pricing. Because the drift of X(t) is speci-
fied explicitly, r(X) is only responsible for discounting. Additionally, the counting
mechanism for jumps of X(t) has an stochastic intensity
λ
(x).
13
Now we denote f
Y
(z;q) as an “extended” Fourier transform or a moment-
generating function of the jump component qdY(t),
14
f
Y
(z;q)=E
e
zqY (1)
= exp
λ
(E[e
zq
] −1)
= exp
λ
f
q
(z) −1)
(7.47)
where f
q
(z) may be again regarded as an “extended” Fourier transform of q. With
the special affine structure characterized by (A,B,C,D), the functions
α
(t) and
β
(t)
of the exponential affine solution of the discounted CF f
∗
(
φ
,t) satisfy two complex-
valued ODEs,
β
(t)=D
1
−A
1
β
(t) −
1
2
B
1
v
2
(t) −C
1
[ f
q
(
β
(t)) −1], (7.48)
α
(t)=d
0
−A
0
β
(t) −
1
2
B
0
v
2
(t) −c
0
[ f
q
(
β
(t)) −1], (7.49)
with boundary conditions
β
(T)=i
φ
,
α
(T)=0.
These two ODEs may be considered as an extended version of the Feynman-Kac
theorem at the presence of Poisson jumps. In fact, the above two ODEs are the
results of the PIDE given in (7.9) under the risk-neutral measure. Note that the
ODE for
β
(t) ∈ R
n
in (7.48) is an equation system consisting of n one-dimensional
ODEs, while the ODE for
α
(t) in (7.49) is only one-dimensional. If the closed-form
solutions for these ODEs are not available for some special specifications of X(t),
we would solve them numerically. To ensure the solutions
α
(t) and
β
(t) solvable,
the coefficients (A,B,C, D) are required to be well-behaved, which means in turn
that the following integrability conditions must be satisfied:
13
A poisson process with stochastic intensity is also called Cox process.
14
As discussed in Chapter 8, “extended” Fourier transform essentially is the generalized Fourier
transform.