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Option Pricing

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Option Pricing

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

- 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

- 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

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];

- 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

- 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

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

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;

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

- 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

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;

>>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

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);

>>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!

>>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');

>>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');

>>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');