stochastic.216

# stochastic.216 - CHAPTER 20 Pricing Exotic Options 215...

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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 Black-Scholes formula "  ~ p p ! b ,r T , T v0; 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 0N 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 vt; x = IE t;x e,rT ,tS T  , K + 1fS T  Lg ; 0  t  T; 0  x  L: This function satisﬁes the terminal condition vT; x = x , K + ; 0  x L and the boundary conditions vt; 0 = 0; 0  t  T; vt; L = 0; 0  t  T: We show that v satisﬁes the Black-Scholes equation 1 ,rv + vt + rxvx + 2 2x2vxx; 0  t T; 0  x  L: ...
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