212
20.2 Up and out European call.
Let 0
K L be given. The payoff at time T is
S T , K + 1fS T Lg;
where
S T = 0maxT S t:
t
To simplify notation, assume that IP is already the riskneutral measure, so the value at time zero of
the option is
h
i
v 0; S 0 =
CHAPTER 22. Summary of Arbitrage Pricing Theory
231
Example 22.4 (Continuation of Example 22.3)
Xt = exp , Z t ru; Y u du vt; St; Y t
t 
0
z
1= t
e
Xt 1
d t = t ,rt; Y tvt; St; Y t dt
is a martingale under IP . We have
+ vt dt + vx dS + vy dY
+ vxx dS
CHAPTER 22. Summary of Arbitrage Pricing Theory
229
it has the same probabilityzero sets as the original measure;
it makes all the discounted asset prices be martingales.
To see if the riskneutral measure is unique, compute the differential of all disco
230
Then
Xt
= ert
e e,rT hST F t
IE
= vt; St; Y t
and
e
is a martingale under IP .
Xt = e,rt vt; St; Y t
t
Example 22.3 (Homework problem 4.2)
e
dSt = rt; Y t Stdt + t; Y tSt dBt;
e
dY t = t; Y t dt + t; Y t dBt;
V = hST:
Take St and Y t to be the state v
Chapter 23
Recognizing a Brownian Motion
Theorem 0.62 (Levy) Let B t; 0
F t; 0 t T , such that:
t T; be a process on ; F ; P, adapted to a ltration
1. the paths of B t are continuous,
2.
3.
B is a martingale,
hBit = t; 0 t T , (i.e., informally dBt dBt =
234
Rt
Now 0 ueuB vdB v is a martingale (by condition 2), and so
IE
Z t
ueuBv dBv F s
=,
s
Zs
= 0:
0
Z t
ueuBv dBv + IE
0
ueuB vdBv F s
It follows that
1
IE euBt F s = euB s + 2 u2
We dene
'v = IE
Zt
IE euBv F s dv:
s
euBv F s
;
so that
's = euBs
and
1
'
236
23.2 Reversing the process
Suppose we are given that
dS1 = r dt + dW ;
1 1
S1
dS2 = r dt + dW ;
2 2
S
2
where W1 and W2 are Brownian motions with correlation coefcient . We want to nd
=
so that
0 =
=
=
"
"
"
11
21
12
22
"
"
11
21
12
22
11
12
21
22
2 +
CHAPTER 23. Recognizing a Brownian Motion
237
so both B1 and B2 are Brownian motions. Furthermore,
dB1 dB2 = p 1
1,
=p 1
1,
2 dW1 dW2 ,
2
dW1 dW1
dt , dt = 0:
We can now apply an Extension of Levys Theorem that says that Brownian motions with zero
cross
CHAPTER 24. An outside barrier option
241
Recall that the barrier process is
dY = r dt +
Y
so
e
1 dB1 ;
o
n
1
e
Y t = Y 0 exp rt + 1 B1 t , 2 2 t :
1
Set
b
= r= 1 , 1=2;
b e
b
B t = t + B1 t;
c
b
M T = 0maxT B t:
t
Then
b
Y t = Y 0 expf 1B tg;
c
Y T = Y
Chapter 24
An outside barrier option
Barrier process:
dY t = dt +
Y t
Stock process:
dS t = dt +
S t
0; 2 0; ,1
; F ; P. The option pays off:
where 1
1
dB1 t:
q
2 dB1 t + 1 ,
2 2 dB2 t;
1, and B1 and B2 are independent Brownian motions on some
S T , K +
240
so that
dY = r dt + dB
1 e1
Y
= r dt + 1 1 dt + 1 dB1 ;
dS = r dt + dB + q1 , 2 dB
2 e1
2 e2
S
q
= r dt + 2 1 dt + 1 , 2 2 2 dt
q
+ 2 dB1 + 1 , 2 2 dB2 :
We must have
= r + 1 1 ;
q
= r + 21 + 1 ,
(0.1)
2 2 2 :
(0.2)
We solve to get
1 = , r ;
1
p r ,
CHAPTER 24. An outside barrier option
243
If we have an option whose payoff depends only on
original equation for S ,
dS = dt +
S
2 dB1 +
S , then Y
q
1,
is superuous. Returning to the
2 2 dB2 ;
q
we should set
dW = dB1 + 1 , 2dB2 ;
so W is a Brownian mot
228
1.
V
is given,
2. Dene X t; 0 t T , by (3.3) (not by (3.1) or (3.2), because we do not yet have t).
3. Construct t so that (3.2) (or equivalently, (3.1) is satised by the
dened in step 2.
X t; 0 t T ,
To carry out step 3, we rst use the tower property
CHAPTER 22. Summary of Arbitrage Pricing Theory
and IP is the riskneutral measure. If a different choice of
227
is made, we have
S t = S 0 expf t + Btg;
dS t = + z1 2 S t dt + S t dBt:
 2
= rS t dt +
e
B has the same paths as B.
i
h ,r
dt + dBt :

z
e
214
give the result for the other three. The exponent in the rst integrand is
x2
x , 2T + x , 1 2 T
2
= , 21 x , T , T 2 + 1 2T + T
2
T
2
T
= , 21 x , rT , 2 + rT:
T
In the rst integral we make the change of variable
p
p
y = x , rT= , T=2= T; dy = dx= T;
CHAPTER 20. Pricing Exotic Options
215
v(t,L) = 0
L
v(T,x) = (x  K)+
v(t,0) = 0
T
Figure 20.4: Initial and boundary conditions.
If we let L!1 we obtain the classical BlackScholes formula
"
~
p
p !
b ,r T , T
v0; S 0 = S 0 1 , N p
2
T
"
~
p
p !
b
T
,rT
216
Let S 0
0 be given and dene the stopping time
= minft 0; S t = Lg:
Theorem 2.61 The process
e,rt^ v t ^ ; S t ^ ; 0 t T;
is a martingale.
Proof: First note that
S T L T:
Let ! 2 be given, and choose t 2 0; T . If ! t, then
IE e,rT S T , K +1fS T Lg F
CHAPTER 20. Pricing Exotic Options
217
For 0 t T , we compute the differential
1
d e,rt vt; S t = e,rt,rv + vt + rSvx + 2 2 S 2vxx dt + e,rt Svx dB:
Integrate from 0 to t ^ :
e,rt^ v t ^ ; S t ^ = v0; S 0
Z t^
+
e,ru ,rv + vt + rSvx + 1 2 S 2vxx du
2
0
Z
218
v(T, x)
0
K
L
x
L
x
v(t, x)
0
K
Figure 20.5: Practial issue.
20.3 A practical issue
For t
T but t near T , vt; x has the form shown in the bottom part of Fig. 20.5.
In particular, the hedging portfolio
t = vx t; S t
can become very negative near the k
Chapter 21
Asian Options
Stock:
dS t = rS t dt + S t dB t:
Payoff:
V =h
Value of the payoff at time zero:
X 0 = IE
Z T
0
"
e,rT h
!
S t dt
Z T
0
!
S t dt :
Introduce an auxiliary process Y t by specifying
dY t = S t dt:
With the initial conditions
S t = x
220
21.1 FeynmanKac Theorem
The function u satises the PDE
ut + rxux + 1 2x2 uxx + xuy = 0; 0 t T; x 0; y 2 IR;
2
the terminal condition
uT; x; y = hy ; x 0; y 2 IR;
and the boundary condition
ut; 0; y = hy ; 0 t T; y 2 IR:
One can solve this equation. T
CHAPTER 21. Asian Options
221
The differential of the value of the option is
dv t; S t;
Zt
0
S u du = vtdt + vx dS + vy S dt + 1 vxx dS dS
2
= vt + rSvx + Svy + 1 2 S 2vxx dt + Svx dB
2
= rv t; S t dt + vx t; S t S t dB t: (From Eq. 1.1)
Compare this with
Chapter 22
Summary of Arbitrage Pricing Theory
A simple European derivative security makes a random payment at a time xed in advance. The
value at time t of such a security is the amount of wealth needed at time t in order to replicate the
security by tra
224
2
3
7
1 6 r
X1 !1 = 1 + r 6 1 + , , d X2 !z ; H + u , 1 + r X2!z1; T 7 ;
4 u d  1
5
u,d 
V ! ;H
V ! ;T
1 + r , d
u , 1 + r X T ;
1
X0 = 1 + r u , d X1H + u , d 1
! T
1 !1 = X2 !1 ; H , X2! 1;T ;
S2 !1; H , S2 1
X1 H , X1T :
0 = S H , S T
1
1
1
1
CHAPTER 22. Summary of Arbitrage Pricing Theory
225
k
f
P
If we introduce a probability measure I under which Sk is a martingale, then
martingale, regardless of the portfolio used. Indeed,
Xk
k will also be a
f
f
IE Xk+1 F k = IE Xk + k Sk+1 , Sk F k
k+
226
B t; 0 t T , be a Brownian motion dened on a probability space ; F ; P.
2 IR, the paths of
t + Bt
Let
For any
accumulate quadratic variation at rate 2 per unit time. We want to dene
S t = S 0 expf t + Btg;
so that the paths of
log S t = log S 0 + t +
242
24.1 Computing the option value
i
fh
v 0; S 0; Y 0 = IE e,rT S T , K + 1fY T Lg
1
f
= e,rT IE S 0 exp r , 2 2 ,
2
:1fY 0exp M T Lg
b
b
2 T +
q
b
2 B T + 1 ,
2 2 B2 T
e
,K
1
b c
e
We know the joint density of B T ; M T . The density of B2 T is
b b b
CHAPTER 1. Introduction to Probability Theory
15
of stock, investing V0 , 0S0 dollars in the money market to nance this. At time 1, the agent has
a portfolio (excluding the short position in the option) valued at
X1 = 0S1 + 1 + rV0 , 0 S0:
(1.10)
Although
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