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hw10-1 - 202 3 Exotic Options Therefore according to these...

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202 3 Exotic Options Therefore according to these relations above, for the integral given we have e - Z S 0 Z S 1 T 0 S 1 T ψ ( S 1 T , S 2 T ; S 1 , S 2 , t ) dS 2 T dS 1 T = S * 1 1 2 πτ det P σ 1 σ 2 · ZZ Ω 1 S 1 T S 2 T e - ( η - σ 1 τ Pe 1 ) T P - 1 ( η - σ 1 τ Pe 1 ) / 2 dS 2 T dS 1 T = S * 1 1 2 π p 1 - ρ 2 201 · ZZ Ω * e - [ ζ 2 1 ( z 21 ) - 2 ρ 201 ζ 1 ( z 21 ) ζ 2 ( z 01 )+ ζ 2 2 ( z 01 )] / 2(1 - ρ 2 201 ) dz 21 dz 01 = S * 1 1 2 π p 1 - ρ 2 201 · Z 0 -∞ Z 0 -∞ e - [ ζ 2 1 ( z 21 ) - 2 ρ 201 ζ 1 ( z 21 ) ζ 2 ( z 01 )+ ζ 2 2 ( z 01 )] / 2(1 - ρ 2 201 ) dz 21 dz 01 = S * 1 1 2 π p 1 - ρ 2 201 · Z ζ 2 (0) -∞ Z ζ 1 (0) -∞ e - ( ζ 2 1 - 2 ρ 201 ζ 1 ζ 2 + ζ 2 2 ) / 2(1 - ρ 2 201 ) 1 2 = S * 1 N 2 ln S * 1 S * 2 + σ 2 12 2 τ σ 12 τ , ln S * 1 S * 0 + σ 2 1 2 τ σ 1 τ ; σ 1 - ρ 12 σ 2 σ 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 = μ 1 S 1 dt + σ 1 S 1 dX 1 , dS 2 = μ 2 S 2 dt + σ 2 S 2 dX 2 , where μ 1 , μ 2 , σ 1 and σ 2 are constants, and dX 1 and dX 2 are two Wiener’s processes with E[dX 1 dX 2 ] = ρ 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 0 , i.e., its payoff function is min( S 0 , S 1 , S 2 ) , where S 0 is a constant. Let V min ( S 1 , S 2 , t ) be the price of the option. Show that

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3 Exotic Options 203 V min ( S 1 , S 2 , t ) = S * 0 N 2 ln S * 1 S * 0 - σ 2 1 2 τ σ 1 τ , ln S * 2 S * 0 - σ 2 2 2 τ σ 2 τ ; ρ 12 + S * 1 N 2 ln S * 2 S * 1 - σ 2 12 2 τ σ 12 τ , ln S * 0 S * 1 - σ 2 1 2 τ σ 1 τ ; σ 1 - ρ 12 σ 2 σ 12 + S * 2 N 2 ln S * 0 S * 2 - σ 2 2 2 τ σ 2 τ , ln S * 1 S * 2 - σ 2 12 2 τ σ 12 τ ; σ 2 - ρ 12 σ 1 σ 12 , where S * 0 = S 0 e - , S * 1 = S 1 e - D 01 τ , and S * 2 = S 2 e - D 02 τ , τ denoting T - t . Solution : The price of such an option is the solution of the following problem: ∂V ∂t + 1 2 σ 2 1 S 2 1 2 V ∂S 2 1 + ρ 12 σ 1 σ 2 S 1 S 2 2 V ∂S 1 ∂S 2 + 1 2 σ 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 , 0 S 1 , 0 S 2 , 0 t T, V ( S 1 , S 2 , T ) = min( S 0 , S 1 , S 2 ) , 0 S 1 , 0 S 2 . Let ξ 10 = S * 1 S * 0 = S 1 e - D 01 ( T - t ) S 0 e - r ( T - t ) , ξ 20 = S * 2 S * 0 = S 2 e - D 02 ( T - t ) S 0 e - r ( T - t ) , and V 0 = V S * 0 = V S 0 e - r ( T - t ) . Using Itˆ o’s lemma, we know that ξ 10 and ξ 20 also are lognormal variables. Let σ 10 and σ 20 be the volatilities of ξ 10 and ξ 20 , respectively. Also let ρ 120 denote the correlation coefficient between
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hw10-1 - 202 3 Exotic Options Therefore according to these...

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