Hong, Zheng HW7

# Hong, Zheng HW7 - CNETzhengh HW7 Zheng Hong Contents 1...

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CNETzhengh HW7 Zheng Hong November 16, 2009 Contents 1 Simulate the Normal-Uniform Hybrid Mark 2 2 correlated diﬀusion diﬀerentials 2 3 the Black-Sholes PDE problem 3 4 The Greeks (Sensitivity Coeﬃcients) 3 5 Black-Scholes European Option Pricing 4 1

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1 Simulate the Normal-Uniform Hybrid Mark -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0 2 4 6 8 10 12 14 16 18 Figure 1: Q(q) mean(Q)=13.0114, standard deviation(Q)=3.9836, skewness(Q)=-0.8365, kurtosis(Q)=2.5464 Matlab Codes: clc; clf; a = -0.0947;b = 0.1096; mu = 2.448e-4;sigma=1.121e-2;pu=0.6; k=1; qq=zeros(1,5000); q=normrnd(mu,sigma,1,10000); for i=1:10000 if q ( i ) < = b N q ( i ) > = a N k < = 5000 qq(k)=q(i) k=k+1 end end c = numeric ( int ( exp ( - t 2 / (2 * sigma 2 )) ,a - mu,b - mu )); phiQ = pu/ ( b - a ) + (1 - pu ) * exp ( - ( qq - mu ) . 2 / (2 * sigma 2 )) /c ; bar(qq,phiQ) 2 correlated diﬀusion diﬀerentials (a) dW b ( t ) dW s ( t ) dt = ρ ( t ) dt ; dW p ( t ) dW s ( t ) dt = 0; dW b ( t ) dW s ( t ) = ( α ( t ) dW s ( t ) + β ( t ) dW p ( t )) dW s ( t ) dt = α ( t )( dW s ( t )) 2 = α ( t ) dt ; α ( t ) = ρ ( t ); 2
(b) Since dW p ( t ) is uncorrelated with dW s ( t ), Δ W m p ( t W n s ( t ) = 0 E [(Δ W b ) 2 ( t )(Δ W s ) 2 ( t )] = E [( αdW s + βdW p ) 2 ( t )(Δ W s ) ( t )] = E [ α 2 ( t W 4 s ( t )] = ρ 2 ( t ) Z -∝ 1 2 π Δ t e - t

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Hong, Zheng HW7 - CNETzhengh HW7 Zheng Hong Contents 1...

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