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  C H = B*Rf+ΔP S (1+σ) C L = B*Rf+ΔP S (1-σ)  Solve for B and Δ:  Δ=(C H -C L )/(2σP S ) and B=(C H - ΔP S (1+σ))/Rf  P U = ΔP S +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 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;

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;

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; 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; 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');

22. Investigating Underlying 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');

24. 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');