stochastic.214

# stochastic.214 - CHAPTER 20 Pricing Exotic Options 213 ~...

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Unformatted text preview: CHAPTER 20. Pricing Exotic Options 213 ~ M(T) y x=y ~ m (B(T), M(T)) lies in here ~ b x ~ B(T) e f Figure 20.3: Possible values of B T ; M T . We consider only the case S 0  K L; so 0  ~ m: b ~ The other case, K S 0  L leads to ~ 0  m and the analysis is similar. b ~ R m R m : : :dy dx: ~ ~ We compute ~ x b Z mZ m ~ ~   , 2 p S 0 expf xg , K  22y , x exp , 2y2T x + x , 1 2 T dy dx 2 ~ x b T 2T  y=m Zm ~ 2 ~ 1 = ,e,rT ~ S 0 expf xg , K  p 1 exp , 2y , x + x , 2 2 T dx 2T b 2T y=x "  2  Z~ ,rT m S 0 expf xg , K  p 1 exp , x + x , 1 2 T =e 2 ~ 2T b 2T   2m , x2 + x , 1 2 T dx ~ , exp , 2T 2   ~ 1 e,rT S 0 Z m exp x , x2 + x , 1 2 T dx =p 2 ~ 2T 2T b   ~ 1 e,rT K Z m exp , x2 + x , 1 2 T dx ,p 2 ~ 2T 2T b   ~ 1 e,rT S 0 Z m exp x , 2m , x2 + x , 1 2 T dx ~ ,p 2 ~ 2T 2T b   ~ 1 e,rT K Z m exp , 2m , x2 + x , 1 2 T dx: ~ +p 2 ~ 2T 2T b v 0; S 0 = e,rT The standard method for all these integrals is to complete the square in the exponent and then recognize a cumulative normal distribution. We carry out the details for the ﬁrst integral and just ...
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