Chap 11. Introduction to Jump Process - Tian
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Transcript Chap 11. Introduction to Jump Process - Tian
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 C2 where C2 : 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 n1 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 j1
c
c
f
X
s
dX
s
dX
s
2 j
Now add the jump in f x at time j 1
f X f X
j 1
j 1
j
j
f X s dX c s
1 j1
f X s dX c s dX c s
2 j
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 e
where
x
X t t N t log 1
0
t
S 0
1
S 0 0u 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 0u 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 e u2 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 u12ds
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
1 2
u1 t eu2 1 t
2
e
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, max0st 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 , X1, X 2 are jump processes
Continuous Part
Jump Part
Ito’s Product Rule for Jump Process
Corollary 11.5.5
PROOF :
f x1, x2 x1x2
f x1 x2 , f x2 x1 , f x1x1 f x2 x2 0 , f x1x2 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 0 s dW s 0 s ds
t
t
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.)
r
0
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 , Z2 are martingales and the integrands are left-continuous
Z1Z2 is a martingale
In the same way, we can conclude that
Z t Z1 t Z2 t
ZM 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
P Yi ym p ym m
m m t m
Zm t e
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 PYi 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