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Option Pricing

Option Pricing. Downloads. Today’s work is in: matlab_lec08.m Functions we need today: pricebinomial.m, pricederiv.m. Derivatives. A derivative is any security the payout of which fully depends on another security Underlying is the security on which a derivative’s value depends

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Option Pricing

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  1. Option Pricing

  2. Downloads • Today’s work is in: matlab_lec08.m • Functions we need today: pricebinomial.m, pricederiv.m

  3. Derivatives • A derivative is any security the payout of which fully depends on another security • Underlying is the security on which a derivative’s value depends • European Call gives owner the option to buy the underlying at expiry for the strike price • European Put gives owner the option to sell the underlying at expiry for the strike price

  4. Binomial Tree

  5. Arbitrage Pricing (1 period) • Lets make a portfolio that exactly replicates underlying payoff, buy Δ shares of stock, and B dollars of bond • CH = B*Rf+ΔPS(1+σ) CL = B*Rf+ΔPS(1-σ) • Solve for B and Δ: • Δ=(CH-CL)/(2σPS) and B=(CH-ΔPS(1+σ))/Rf • PU= ΔPS+B

  6. pricebinomial.m function out=pricebinomial(pS,Rf,sigma,Ch,Cl); D=(Ch-Cl)/(pS*2*sigma); B=(Ch-(1+sigma)*pS*D)/Rf; pC=B+pS*D; out=[pC D];

  7. Price Call • Suppose the price of the underlying is 100 and the volatility is 10%; suppose the risk free rate is 2% • The payoff of a call with strike 100 is 10 in the good state and 0 in the bad state: C=max(P-X,0) • What is the price of this call option? >>pS=100; Rf=1.02; sigma=.1; Ch=10; Cl=0; >>pricebinomial(pS,Rf,sigma,Ch,Cl) Price=5.88, Δ=.5

  8. Larger Trees • The assumption that the world only has two states is unrealistic • However its not unrealistic to assume that the price in one minute can only take on two values • This would imply that in one day, week, year, etc. there are many possible prices, as in the real world • In fact, at the limit, the binomial assumption implies a log-normal distribution of prices at expiry

  9. Binomial Tree (multiperiod)

  10. Tree as matrix 100.0000 108.0000 116.6400 125.9712 0 92.0000 99.3600 107.3088 0 0 99.3600 107.3088 0 0 84.6400 91.4112 0 0 0 107.3088 0 0 0 91.4112 0 0 0 91.4112 0 0 0 77.8688

  11. Prices of Underlying Recursively define prices forward >>N=3; P=zeros(2^N,N+1); %create a price grid for underlying >>P(1,1)=pS; for i=1:N; for j=1:2^(i-1); P((j-1)*2+1,i+1)=P(j,i)*(1+sigma); P((j-1)*2+2,i+1)=P(j,i)*(1-sigma); %disp([i j i+1 (j-1)*2+1 (j-1)*2+2]); end; end;

  12. Indexing disp([i j i+1 (j-1)*2+1 (j-1)*2+2]); 1 1 2 1 2 2 1 3 1 2 2 2 3 3 4 3 1 4 1 2 3 2 4 3 4 3 3 4 5 6 3 4 4 7 8

  13. Payout at Expiry • Payout of derivative at expiry is a function of the underlying • European Call: C(:,N+1)=max(P(:,N+1)-X,0); • European Put: C(:,N+1)=max(X-P(:,N+1),0); • This procedure can price any derivative, as long as we can define its payout at expiry as a function of the underlying • For example C(:,N+1)=abs(P(:,N+1)-X); would be a type of volatility hedge

  14. Payout at Expiry

  15. Prices of Derivative Recursively define prices backwards >>X=100; C(:,N+1)=max(P(:,N+1)-X,0); %call option >>for k=1:N; i=N+1-k; for j=1:2^(i-1); Ch=C((j-1)*2+1,i+1); Cl=C((j-1)*2+2,i+1); pStemp=P(j,i); out=pricebinomial(pStemp,Rf,sigma,Ch,Cl); C(j,i)=out(1); %disp([i j i+1 (j-1)*2+1 (j-1)*2+2]); end; end;

  16. Indexing >>disp([i j i+1 (j-1)*2+1 (j-1)*2+2]); 3 1 4 1 2 3 2 4 3 4 3 3 4 5 6 3 4 4 7 8 2 1 3 1 2 2 2 3 3 4 1 1 2 1 2

  17. pricederiv.m function out=pricederiv(pS,Rf,sigmaAgg,X,N) sigma=sigmaAgg/sqrt(N); Rf=Rf^(1/N); %define sigma, Rf for shorter period C=zeros(2^N,N+1); P=zeros(2^N,N+1); %initialize price vectors P(1,1)=pS; for i=1:N; %create price grid for underlying for j=1:2^(i-1); P((j-1)*2+1,i+1)=P(j,i)*(1+sigma); P((j-1)*2+2,i+1)=P(j,i)*(1-sigma); end; end; C(:,N+1)=max(P(:,N+1)-X,0); %a european call for k=1:N; %create price grid for option i=N+1-k; for j=1:2^(i-1); Ch=C((j-1)*2+1,i+1); Cl=C((j-1)*2+2,i+1); pStemp=P(j,i); x=pricebinomial(pStemp,Rf,sigma,Ch,Cl); C(j,i)=x(1); end; end; out=C(1,1);

  18. Investigating N >>pS=100; Rf=1.02; sigmaAgg=.3; X=100; • B-S value of this call is 12.8 http://www.blobek.com/black-scholes.html >>for N=1:15; out(N,1)=N; out(N,2)=pricederiv(pS,Rf,sigmaAgg,X,N); end; >>plot(out(:,1),out(:,2)); • This converges to B-S as N grows!

  19. Convergence to B-S Price

  20. Investigating Strike Price >>pS=100; Rf=1.02; sigmaAgg=.3; N=10; >>for i=1:50; X=40+120*(i-1)/(50-1); out(i,1)=X; out(i,2)=pricederiv(pS,Rf,sigmaAgg,X,N); end; >>plot(out(:,1),out(:,2)); >>xlabel('Strike'); ylabel('Call');

  21. InvestigatingUnderlying Price >>X=100; Rf=1.02; sigmaAgg=.3; N=10; >>for i=1:50; pS=40+120*(i-1)/(50-1); out(i,1)=pS; out(i,2)=pricederiv(pS,Rf,sigmaAgg,X,N); end; >>plot(out(:,1),out(:,2)); >>xlabel('Price'); ylabel('Call');

  22. Investigating sigma >>X=100; Rf=1.02; pS=100; N=10; >>for i=1:50; sigmaAgg=.01+.8*(i-1)/(50-1); out(i,1)=sigmaAgg; out(i,2)=pricederiv(pS,Rf,sigmaAgg,X,N); end; >>plot(out(:,1),out(:,2)); >>xlabel('Sigma'); ylabel('Call');

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