Exercise 7.5 (Black-Scholes-Merton equation for lookback option). We wish to verify by direct computation that the function v(t, x, y) of (7.4.35) satisfies the Black-Scholes-Merton equation (7.4.6). As we saw in Subsection 7.4.3, this is equivalent to showing that the function u defined by (7.4.36) satisfies the Black-Scholes-Merton equation (7.4.18). We verify that u(t, z) satisfies (7.4.18) in the following steps. Let 0 < t < T be given, and define T=T-t. (i) Use (7.8.1) to compute u₁(t, z), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that ut(t, z) = re¯ -TT 1 'N ( − 8_ (7, ²)) — —½±0 ²e¯¯¯ 2¹¯³½³N ( − 8_ (7, z¯¹³)) στ =N′ (8+ (7, 2)). (7.8.18) (ii) Use (7.8.2) to compute uz(t, z), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that uz(t, z) = (: (1+227) N (8+ (7, 2)) + 2r 1 - 2r )e¯z¯N( - 8- (7, 2¯¹)) - 1. (7.8.19) (iii) Use (7.8.19) and (7.8.2) to compute uz(t, z), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that 2r -TT - Uzz(t, z) = (1-2) - e¯π 2 - 3 4 - 1 N ( − 8 - (7, 2¯¹³)) + 272 √7 N' (8+ (7,2)). T, (7.8.20) (iv) Verify that u(t, z) satisfies the Black-Scholes-Merton equation (7.4.18). (v) Verify that u(t, z) satisfies the boundary condition (7.4.20).

Exercise 7.5 (Black-Scholes-Merton equation for lookback option). We wish to verify by direct computation that the function v(t, x, y) of (7.4.35) satisfies the Black-Scholes-Merton equation (7.4.6). As we saw in Subsection 7.4.3, this is equivalent to showing that the function u defined by (7.4.36) satisfies the Black-Scholes-Merton equation (7.4.18). We verify that u(t, z) satisfies (7.4.18) in the following steps. Let 0 < t < T be given, and define T=T-t. (i) Use (7.8.1) to compute u₁(t, z), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that ut(t, z) = re¯ -TT 1 'N ( − 8_ (7, ²)) — —½±0 ²e¯¯¯ 2¹¯³½³N ( − 8_ (7, z¯¹³)) στ =N′ (8+ (7, 2)). (7.8.18) (ii) Use (7.8.2) to compute uz(t, z), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that uz(t, z) = (: (1+227) N (8+ (7, 2)) + 2r 1 - 2r )e¯z¯N( - 8- (7, 2¯¹)) - 1. (7.8.19) (iii) Use (7.8.19) and (7.8.2) to compute uz(t, z), and use (7.8.3) and (7.8.4) to simplify the result, thereby showing that 2r -TT - Uzz(t, z) = (1-2) - e¯π 2 - 3 4 - 1 N ( − 8 - (7, 2¯¹³)) + 272 √7 N' (8+ (7,2)). T, (7.8.20) (iv) Verify that u(t, z) satisfies the Black-Scholes-Merton equation (7.4.18). (v) Verify that u(t, z) satisfies the boundary condition (7.4.20).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 51E

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