Chap 11. Introduction to Jump Process
Stochastic Calculus for Finance II
Steven E. Shreve
財研二 范育誠
AGENDA
 11.5 Stochastic Calculus for Jump Process
 11.5.1 Ito-Doeblin Formula for One Jump Process
 11.5.2 Ito-Doeblin Formula for Multiple Jump Process
 11.6 Change of Measure
 11.7 Pricing a European Call in Jump Model
Ito-Doeblin Formula for Continuous-Path Process
 For a continuous-path process, the Ito-Doeblin formula is the
following. Let
X
c
 t   X  0  0   s  dW  s   0   s  ds
c
t
t
In differential notation, we write
dX c  t     s  dW  s     s  ds
 Let
f  x   C 2 where C 2 : 1st and 2nd derivatives are defined and continuous.
Then
df  X c  s    f   X c  s   dX c  s  
1
f   X c  s   dX c  s  dX c  s 
2
1
 f   X c  s     s  dW  s   f   X c  s     s  ds  f   X c  s    2  s  ds
2
Write in integral form as
f  X c  t    f  X c  0     f   X c  s     s  dW  s    f   X c  s     s  ds 
t
t
0
0
1 t
f   X c  s    2  s  ds

2 0
Ito-Doeblin Formula for One Jump Process
 Add a right-continuous pure jump term J
X  t   X  0  I  t   R t   J t 
where
 Define
 I  t  : Ito integral term

 R  t  : Riemann integral term

 J  t  : Pure jump term
X c  t   X  0  I  t   R  t 
 Between jumps of J
1
df  X  s    f   X  s   dX  s   f   X  s   dX  s  dX  s 
2
 f   X  s     s  dW  s   f   X  s     s  ds 
 f   X  s   dX c  s  
1
f   X  s    2  s  ds
2
1
f   X  s   dX c  s  dX c  s 
2
Ito-Doeblin Formula for One Jump Process
 Theorem 11.5.1
Let
X t 
be a jump process and f  x   C 2 . Then
f  X  t    f  X  0     f   X  s   dX c  s  
t
0

  f  X  s    f  X  s   
1 t
c
c

f
X
s
dX
s
dX




s


2 0
0 s t
 PROOF :
Fix   , which fixes the path of X , and let 0  1   2     n1  t
be the jump times in 0,t  of this path of the process X .
We set  0  0 is not a jump time, and  n  t, which may or
may not be a jump time.
PROOF (con.)
Whenever u  v , u, v   j , j 1 
f  X  v    f  X  u     f   X c  s   dX c  s  
v
u
1 v
c
c

f
X
s
dX
s
dX




s


2 u
Letting u   j , v   j 1 and using the right-continuity of
We conclude that
f  X      f  X   
j 1

 j 1
j
j
f   X  s   dX c  s  
1  j1
f   X  s   dX c  s  dX c  s 

2 j
Now add the jump in f  x  at time  j 1

 
f X  j 1   f X  j 

 j 1
j

1  j1
f   X  s   dX  s    f   X  s   dX c  s  dX c  s 
2 j
c

 
 f X  j 1   f X  j 1  

X
PROOF (con.)
Summing over j  0,1, , n  1
f  X t   f  X  0
n 1

 

   f X  j 1   f X  j  


j 0

t
0
1 t
f   X  s   dX  s    f   X  s   dX c  s  dX c  s 
2 0
c
n 1

 

   f X  j 1   f X  j 1   


j 0
Example (Geometric Poisson Process)
 Geometric Poisson Process
S  t   S  0  exp  N  t  log   1   t  S  0  e   t   1
 We may write S  t   S  0 f  X  t  
f  x   ex
where
X  t    t  N  t  log   1
0
t
S  0
 1

S  0  0u t
t
S  0  0
S  u   du 
 S u   S u     S u  
If there is no jump at time u
1 t
f   X  u   dX c  u  dX c  u     f  X  u    f  X  u    

2 0
0u t
S u 
1
 du  

  S  u   S  u  
0 S 0
S  0  0u t 
 
 t
1
 1
S
u
du



  S  u   N  u 
0
 1 
S  u   S  u    1
We have
S  u   S  u     S u   N u 
S t 
 f  X t 
S  0
t
If there is a jump at time u
S u   S u   0
X c  t    t
 f  X  0     f   X  u   dX c  u  
N t 
1 t
 S  u   dN  u 
S  0  0
Example (con.)
S  t   S  0    S  u   du    S  u   dN  u 
t
t
0
0
 M u   N u   u 
t
 S  0     S  u   dM  u 
0
 In this case, the Ito-Doeblin formula has a differential form
dS  t    S  t   dM  t    S  t  dt   S  t   dN t 
Independence Property
 Corollary 11.5.3

W  t  : Brownian motion


 N  t  : Poisson process ,   0
defined on the same space  , F,   and relative to the same filtration F t 
 W  t  and N  t  are independent
 PROOF :
1


Y  t   exp u1W  t   u2 N  t   u12t    eu2  1 t 
2


1
Define f  x   e x , X  s   u1W  s   u2 N  s   u12 s    eu2  1 s
2
1
X c  s   u1W  s   u12 s    eu2  1 s
2
1
dX c  s   u1dW  s   u12 ds    eu2  1 ds
2
dX c  s  dX c  s   u12 ds
PROOF (con.)
 If Y has a jump at time s, then
1


Y  s   exp u1W  s   u2  N  s    1  u12 s    eu2  1 s   Y  s   eu2
2


u2
Y  s   Y  s     e  1 Y  s  
Therefore,
Y  s   Y  s     eu2  1 Y  s   N  s 
 According to Ito-Doeblin formula
Y t   f  X t 
 f  X  0     f   X  s   dX c  s  
1 t
f   X  s   dX c  s  dX c  s     f  X  s    f  X  s    

0
2 0
0  s t
t
t
t
t
1
1
 1  u1  Y  s  dW  s   u12  Y  s  ds    eu2  1  Y  s  ds  u12  Y  s  ds   Y  s   Y  s   
0
0
0
0
2
2
0  s t
t
 1  u1  Y  s  dW  s     eu2  1  Y  s   ds   eu2  1  Y  s   dN  s 
t
t
t
0
0
0
 1  u1  Y  s  dW  s    eu2  1  Y  s   dM  s 
t
t
0
0
PROOF (con)
 Y is a martingale and Y  0  1, EY  t   1 for all t.
 In other words
1


 exp u1W  t   u2 N  t   u12t    eu2  1 t   1
2


 e
u1W  t  u2 N  t 
e

e

1 2
u1 t  eu2 1 t
2
 The corollary asserts more than the independence between
N(t) and W(t) for fixed time t, saying that the process N and
W are independent. For example, max0st W  s  is
t
independent of 0 N  s  ds
AGENDA
 11.5 Stochastic Calculus for Jump Process
 11.5.1 Ito-Doeblin Formula for One Jump Process
 11.5.2 Ito-Doeblin Formula for Multiple Jump Process
 11.6 Change of Measure
 11.7 Pricing a European Call in Jump Model
Ito-Doeblin Formula for Multiple Jump Process
 Theorem 11.5.4 (Two-dimensional Ito-Doeblin formula)
f  C 2 , X 1 , X 2 are jump processes
Continuous Part
Jump Part
Ito’s Product Rule for Jump Process
 Corollary 11.5.5
 PROOF :
f  x1 , x2   x1 x2
f x1  x2 , f x2  x1 , f x1 x1  f x2 x2  0 , f x1 x2  f x2 x1  1
cross variation
PROOF (con.)
X 1  0  X 2  0    X 2  s  dX 1c  s    X 1  s  dX 2c  s 
  X 1c , X 2c   t  
t
t
0
0
  X  s  X  s   X  s   X  s  
0  s t
1
2
1
2
 X1  t   X1c t   J1 t  , X 2  t   X 2c t   J 2 t  are pure jump parts
of
X1  t 
and X 2  t , respectively.
PROOF (con.)
 Show the last sum in the previous slide is the same as
  X  s  X  s   X  s   X  s  
0  s t
1
2
1
2
Doleans-Dade Exponential
 Corollary 11.5.6
Let X  t  be a jump process.
The Doleans-Dade exponential of X is defined to be the process
1


Z X  t   exp  X c  t    X c , X c   t    1  X  s  
2

 0  s t
It is the solution to the S.D.E.
dZ X  Z X  t   dX  t  with Z X  0   1
integral form is
Z X  t   1   Z X  s   dX  s 
t
0
Comparison (Girsanov ' s THM ) :
1 t
 t

Z  t   exp     s  dW  s     2  s  ds 
2 0
 0

1


 exp  X c  s    X c , X c   s  
2


dZ  t   Z  t  dX c  t 
PROOF
 PROOF of Corollary 11.5.6 :
X  t   X c  t   J t  where X c  t      s  dW  s      s  ds
t
t
0
0
1


Define Y  t   exp  X c  t    X c , X c   t  
2


t
1 t 2
 t

 exp     s  dW  s      s  ds     s  ds 
0
2 0
 0

From the Ito-Doeblin formula
dY  t   Y t  dX c t   Y t   dX c t 
Note : We define K(0)=1
 1  X  s  
K  t   K  t   1  X t  
K  t   K  t   K t    K t   X t 
Define K  t  
0 s t
PROOF (con.)
 Use Ito’s product rule for jump process to obtain
[Y,K](t)=0
dZ X  Z X  t   dX t  with Z X  0  1
AGENDA
 11.5 Stochastic Calculus for Jump Process
 11.6 Change of Measure
 11.6.1 Change of Measure for a Poisson Process
 11.6.2 Change of Measure for a Compound Poisson Process
 11.6.3 Change of Measure for a Compound Poisson Process
and a Brownian Motion
 11.7 Pricing a European Call in Jump Model
Change of Measure for a Poisson Process
    t   
Z t   e
 

 Define
N t 
 Lemma 11.6.1
 
Z  t   dM  t 

Z  t  is a martingale under P and Z  t   1 for all t.
dZ  t  
 PROOF :
 
M t 
M(t)=N(t)-λt

 
X c t      t , J t  
N t 

Define X  t  


Then  X c , X c   t   0, and if there is a jump at time t, then X  t  

so 1  X  t  

 

PROOF (con.)
 Z(t) may be written as
1


Z  t   exp  X c  t    X c , X c   t    1  X  s  
2

 0 s t
 We can get the result by corollary 11.5.6
Let X  t  be a jump process.
The Doleans-Dade exponential of X is defined to be the process
1


Z X  t   exp  X c  t    X c , X c   t    1  X  s  
2

 0  s t
It is the solution to the S.D.E.
dZ X  Z X  t   dX  t  with Z X  0   1
integral form is
Z X  t   1   Z X  s   dX  s 
t
0
Change of Poisson Intensity
 Theorem 11.6.2
Under the probability measure P , the process N  t  , 0  t  T
is Poisson with intensity 
 PROOF :
Example (Geometric Poisson Process)
 Geometric Poisson Process
S  t   S  0  exp  t  N  t  log   1   t
  1  
 t
 e S t  is a martingale under P-measure, and hence S  t 
has mean rate of return α.
dS  t    S  t  dt   S t   dM t 
11.6.4 
 M t   N t   t, N t  is a Poisson process with intensity  under P 
 We would like to change measure such that
dS  t   rS  t  dt   S  t   dM  t 
11.6.5
 M t   N t   t, N t  is a Poisson process with intensity  under P 
 S  0  e
   t
N t
Example (con.)
 The “dt” term in (11.6.4) is
    S t  dt
11.6.6 
The “dt” term in (11.6.5) is
11.6.7 
 r    S t  dt
 Set (11.6.6) and (11.6.7) are equal, we can obtain
 
 r

We then change to the risk-neutral measure by
 To make the change of measure, we must have
 0 
 r

    t   
Z t   e
 

N t 
If the inequality doesn’t hold, then there must be an arbitrage.
Example (con.)
  0
 r

If   0 , then S  t   S  0  ert   1 N t   S  0  ert
Borrowing money S(0) at the interest rate r to invest in the
stock is an arbitrage.
AGENDA
 11.5 Stochastic Calculus for Jump Process
 11.6 Change of Measure
 11.6.1 Change of Measure for a Poisson Process
 11.6.2 Change of Measure for a Compound Poisson Process
 11.6.3 Change of Measure for a Compound Poisson Process
and a Brownian Motion
 11.7 Pricing a European Call in Jump Model
Definition
 N  t  is a Poisson process with intensity 
Y1 , Y2 ,... are i.i.d. random variables defined on a probability space   ,F,P 
Assume that Yi is independent of N  t 
 Compound Poisson Process
N t 
Q  t    Yi
i 1
 If N   jumps at time t, then Q  jumps at time t and
Q  t   YN t 
Jump-Size R.V. have a Discrete Distribution
 Yi takes one of finitely many possible nonzero values y1 , y2 , ... , yM
p  ym   P Yi  ym  , m  1, 2, ... ,M
M
 p y  1
m 1
m
 According to Corollary 11.3.4
M
N t    Nm t 
m 1
Nm  t  is the number of jumps in Q  t  of size ym up to and including time t
N1 , N2 , ... ,NM are independent and each Nm has intensity m   p  ym 
N t 
M
i 1
m 1
 Q  t    Yi   ym N m  t 
Jump-Size R.V. have a Discrete Distribution
 Let 1 , 2 , ... ,M be given positive numbers, and set
 Lemma 11.6.4
The process Z  t  is a martingale.
In particular, EZ  t   1 for all t.
PROOF of Lemma 11.6.4
 From Lemma 11.6.1, we have
 Z m is a martingale.
For m  n, Nm and Nn have no simultaneous jumps   Zm , Zn   0
 By Ito’s product rule
Z1 , Z 2 are martingales and the integrands are left-continuous
 Z1 Z 2 is a martingale
 In the same way, we can conclude that
Z  t   Z1  t  Z2 t 
Z M t 
is a martingale.
Jump-Size R.V. have a Discrete Distribution
 Because Z T   0 almost surely and EZ T   1 , we can use Z T 
to change the measure, defining
P  A   Z T  dP
A
for all Z  F
 Theorem 11.6.5 (Change of compound Poisson intensity
and jump distribution for finitely many jump sizes)
Under P , Q Mt  is a compound Poisson process with
intensity    m , and Y1 , Y2 , are i.i.d. R.V. with
m 1
m
P Yi  ym   p  ym  

Zm t   e
 m m t  m 
 
 m 
M
Z t    Zm t 
m 1
Nm  t 
PROOF of Theorem 11.6.5
Jump-Size R.V. have a Continuous Distribution
 The Radon-Nykodym derivative process Z(t) may be written as
 We could change the measure so that Q  t  has intensity  and
have a different density f  y  by using the
M
Radon-Nykodym derivative process
Y1 , Y2 ,...
 N t  x
m 1
m
 Notice that, we assume that f  y   0 whenever f  y   0
m
N t 
  Xi
i 1
AGENDA
 11.5 Stochastic Calculus for Jump Process
 11.6 Change of Measure
 11.6.1 Change of Measure for a Poisson Process
 11.6.2 Change of Measure for a Compound Poisson Process
 11.6.3 Change of Measure for a Compound Poisson Process
and a Brownian Motion
 11.7 Pricing a European Call in Jump Model
Definition
 Compound Poisson Process
 Let   0 , f  y   0 whenever f  y   0 ,
  t  is an adaptive process
 Lemma 11.6.8
The process Z  t  of (11.6.33) is a martingale. In particular,
EZ  t   1 for all t  0
 PROOF :
Z1  t  is continuous


   Z1 , Z 2   t   0
Z 2  t  has no Ito integral part 

By Ito’s product rule,
Z1(s-),Z2(s-)
are left-continuous
Z1(s),Z
2(s) are martingales
Z1(t)Z2(t) is a martingale
 Theorem 11.6.9
Under the probability measure P , the process
is a Brownian motion, Q  t  is a compound Poisson process
with intensity  and i.i.d. jump sizes having density f  y ,
and the processes W  t  and Q  t  are independent.
 PROOF :
The key step in the proof is to show

Y  u    euy f  y  dy

？
Is Θ independent with Q (or Z2) ?
PROOF (con.)
 Define
 We want to show that X1 t  Z1 t  , X 2 t  Z2 t  , X1 t  Z1 t  X 2 t  Z2 t 
are martingales under P.
No drift term.
So X1(t)Z1(t) is a martingale
PROOF (con.)
 The proof of theorem 11.6.7 showed that X2(t)Z2(t) is a
martingale.
 Finally, because X1(t)Z1(t) is continuous and X2(t)Z2(t) has
no Ito integral part, [X1Z1,X2Z2](t)=0. Therefore, Ito’s
product rule implies
X2(s)Z2(s), X1(s)Z1(s) are
martingales.
X1(s-)Z1(s-), X2(s-)Z2(s-)
are left-continuous.
 Theorem 11.4.5 implies that X1(t)Z1(t)X2(t)Z2(t) is a
martingale. It follows that
 Theorem 11.6.10 (Discrete type)
Under the probability measure P , the process
is a Brownian motion, Q  t  is a compound Poisson process
with intensity  and i.i.d. jump sizes satisfying PYi  ym  p  ym 
for all i and m  1, 2,..., M , and the processes W  t  and Q  t 
are independent.
AGENDA
 11.5 Stochastic Calculus for Jump Process
 11.6 Change of Measure
 11.7 Pricing a European Call in Jump Model
 11.7.2 Asset Driven by Brownian Motion and Compound Poisson
Process
 Definition
Q(t)-λβt is a martingale.
 Theorem 11.7.3
The solution to
is
PROOF of Theorem 11.7.3
 Let
 We show that
is a solution to the SDE.
X is continuous and J
is a pure jump process
→ [ X,J ](t)=0
PROOF (con.)
 The equation in differential form is