HW2_LIMO_withName - 642:623 Computational Finance Report of...

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642:623 Computational Finance: Report of Homework 2 Due on January 31, 2012 Tues 6:30PM Mo Li
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Mo Li 642:623 Computational Finance : Report of Homework 2 [Algorithm] Algorithm 1. Closed-Form formula for up-and-out barrier European option. Use closed-form formulae by Black Scholes Model to compute the price of European up-and-out barrier call option with dividend yield. Recall the formula for European call option c ( t, x ) = xe - N ( d + ( τ, x )) - e - KN ( d - ( τ, x )) We have the closed-form formula for up-and-in call option[Hull, p.551] when barrier H is higher than Strike K . c ui = S 0 N ( x i ) e - qT - Ke - rT N ( x 1 - σ T ) - S 0 e - qT ( H/S 0 ) 2 λ [ N ( - y ) - N ( - y 1 )] + Ke - rT ( H/S 0 ) 2 λ - 2 [ N ( - y + σ T ) - N ( - y 1 + σ T )] and c uo = c - c ui where λ = r - q + σ 2 / 2 σ 2 y = ln H 2 / ( S 0 K ) σ T + λσ T x 1 = ln ( S 0 /H ) σ T + λσ T y 1 = ln ( H/S 0 ) σ T + λσ T Implemented by C++ functions of double BSBarrierCallClosedForm( double Expiry, double Spot, double Vol, double r, 5 double Strike, double Barrier, double Dividend ); 2. Use Monte Carlo Simulation to compute the price for a European-style call with an up-and-out barrier Based on the Geometric Brownian Motion dS ( t ) = S ( t ) rdt + S ( t ) σdW ( t ) and its finite difference form S ( t i +1 ) = S ( t i ) + S ( t i )[ + σ τZ i ] with Z i be iid N (0 , 1) random variables. Recording the maximum value of underlying [Algorithm] continued on next page. . . Page 1 of 4
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Mo Li 642:623 Computational Finance : Report of Homework 2 [Algorithm] (continued) asset price M ( T ) M ( t ) = max 0 τ t {
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