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HW9

HW9. 100071021 楊佑濬. Convertible Bonds. function [ price, lattice] = CBprice ( S, I, Q, r, T, sigma, N) deltaT = T/N; u = exp ( sigma * sqrt ( deltaT ) ); d = 1/u; p = ( exp( r * deltaT ) - d ) / ( u - d ); for i = 0 : N

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HW9

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  1. HW9 100071021 楊佑濬

  2. Convertible Bonds function [ price, lattice] = CBprice( S, I, Q, r, T, sigma, N) deltaT = T/N; u = exp( sigma * sqrt(deltaT) ); d = 1/u; p = ( exp( r * deltaT ) - d ) / ( u - d ); for i = 0 : N CB(i+1) = max( I/Q, S * u^(N-i) * d^(i) ); end for i = N : -1 : 1 for j = 0 : i-1 CB(j+1) = exp(-r*deltaT) * ( CB(j+1)*p + CB(j+2)*(1-p) ); S_p = S * u^(i-1-j) * d^(j) ; CB(j+1) = max( CB(j+1) , S_p ); end lattice(1:i,i) = CB(1:i); end price = CB(1); end

  3. S = 35; I = 100000; Q = 2500; r = 0.02; T = 5; sigma = 0.3; N = 100; >> CBprice(S,I,Q,r,T,sigma,N) ans = 44.959 >>

  4. Demo Code S = 35; I = 100000; Q = 2500; r = 0.02; T = 5; sigma = 0.3; N = 100; for ST = 1 : 100 P(ST) = CBprice(ST,I,Q,r,T,sigma,N); end B = I/Q * exp(-r*T); figure; plot(ones(100)*B); hold on; plot(P); title('S0 to CB price'); for i = 1 : 100 P(i) = CBprice(S,I,Q,r,T,i*0.002,N); end figure; plot(ones(100)*B); hold on; plot(P); title('Sigma to CB price');

  5. Monte Carlo of Normal CDF function [ p ] = MCnormcdf( X, N ) if(nargin<2) N = 10000000; end for i = 1:length(X) p(i) = 1/sqrt(2*pi) * mean( exp(-1* ((rand(1,N)*X(i)).^2) / 2 ) ) * X(i) + 0.5; end end

  6. Demo Code x = linspace(-40,40,1000); tic; plot(x,MCnormcdf(x),x,normcdf(x)); toc;

  7. Outcome • i7-4770 3.40GHz 8 cores / 8G memory >> MCdemo Elapsed time is 128.180229 seconds. >> • i5-2520M 2.50GHz 4 cores / 8G memory >> MCdemo Elapsed time is 302.060766 seconds. >>

  8. Demo Code x = linspace(-3,3,1000); tic; plot(x,MCnormcdf(x),x,normcdf(x)); toc; for i = 1:10000 MC(i) = MCnormcdf(1.96,i); end plot(1:10000,MC,1:10000,normcdf(1.96));

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