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Unformatted text preview: 202 3 Exotic Options Therefore according to these relations above, for the integral given we have e r Z S Z S 1 T S 1 T ( S 1 T , S 2 T ; S 1 , S 2 , t ) dS 2 T dS 1 T = S * 1 1 2 det P 1 2 ZZ 1 S 1 T S 2 T e (  1 Pe 1 ) T P 1 (  1 Pe 1 ) / 2 dS 2 T dS 1 T = S * 1 1 2 p 1 2 201 ZZ * e [ 2 1 ( z 21 ) 2 201 1 ( z 21 ) 2 ( z 01 )+ 2 2 ( z 01 )] / 2(1 2 201 ) dz 21 dz 01 = S * 1 1 2 p 1 2 201 Z Z e [ 2 1 ( z 21 ) 2 201 1 ( z 21 ) 2 ( z 01 )+ 2 2 ( z 01 )] / 2(1 2 201 ) dz 21 dz 01 = S * 1 1 2 p 1 2 201 Z 2 (0) Z 1 (0) e ( 2 1 2 201 1 2 + 2 2 ) / 2(1 2 201 ) d 1 d 2 = S * 1 N 2 ln S * 1 S * 2 + 2 12 2 12 , ln S * 1 S * + 2 1 2 1 ; 1 12 2 12 . 26. Suppose that S 1 and S 2 are the prices of two assets A and B respectively. The random variables S 1 and S 2 satisfy dS 1 = 1 S 1 dt + 1 S 1 dX 1 , dS 2 = 2 S 2 dt + 2 S 2 dX 2 , where 1 , 2 , 1 and 2 are constants, and dX 1 and dX 2 are two Wieners processes with E[dX 1 dX 2 ] = 12 dt . Also, suppose that the two assets pay dividends continuously, and that the dividend yields of the assets A and B are D 01 and D 02 respectively. Consider a European option on the minimum of S 1 , S 2 and S , i.e., its payoff function is min( S , S 1 , S 2 ) , where S is a constant. Let V min ( S 1 , S 2 , t ) be the price of the option. Show that 3 Exotic Options 203 V min ( S 1 , S 2 , t ) = S * N 2 ln S * 1 S * 2 1 2 1 , ln S * 2 S * 2 2 2 2 ; 12 + S * 1 N 2 ln S * 2 S * 1 2 12 2 12 , ln S * S * 1 2 1 2 1 ; 1 12 2 12 + S * 2 N 2 ln S * S * 2 2 2 2 2 , ln S * 1 S * 2 2 12 2 12 ; 2 12 1 12 , where S * = S e r , S * 1 = S 1 e D 01 , and S * 2 = S 2 e D 02 , denoting T t . Solution : The price of such an option is the solution of the following problem: V t + 1 2 2 1 S 2 1 2 V S 2 1 + 12 1 2 S 1 S 2 2 V S 1 S 2 + 1 2 2 2 S 2 2 2 V S 2 2 +( r D 01 ) S 1 V S 1 + ( r D 02 ) S 2 V S 2 rV = 0 , S 1 , S 2 , t T, V ( S 1 , S 2 , T ) = min( S , S 1 , S 2 ) , S 1 , S 2 ....
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This note was uploaded on 02/11/2010 for the course MATH 6203 taught by Professor Zhu during the Spring '10 term at University of North Carolina Wilmington.
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