Unformatted text preview: 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 K 1 , N p , r T +
,e
2
T
p
p !
1
0
= S 0N p log SK + r T + 2 T
T
p
p !
1
T :
,rT KN p log S 0 + r T ,
,e
K
2
T If we replace T by T , t and replace S 0 by x in the formula for v 0; S 0, we obtain a formula
for v t; x, the value of the option at the time t if S t = x. We have actually derived the formula
under the assumption x K L, but a similar albeit longer formula can also be derived for
K x L. We consider the function h
i
vt; x = IE t;x e,rT ,tS T , K + 1fS T Lg ; 0 t T; 0 x L: This function satisﬁes the terminal condition vT; x = x , K + ; 0 x L
and the boundary conditions vt; 0 = 0; 0 t T;
vt; L = 0; 0 t T:
We show that v satisﬁes the BlackScholes equation 1
,rv + vt + rxvx + 2 2x2vxx; 0 t T; 0 x L: ...
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 Fall '15
 mr.somebody
 Real options analysis, Myron Scholes, Boundary conditions, 0 L, classical BlackScholes formula

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