Simulate a plain Call- and Put Option

Hi

run a Monte Carlo Simulation on a plain Call option based on the Black-Scholes Model. This is as far as i came:

Call Option:

clear all
randn('state',3)
K=100;
r=0.1;
sigma=0.4;
T=1;
s0=80;
n=50;
h=1/n;
M=1000;
%Simultaneous Creation of the Wiender-process with M tracks
dW=sqrt(h)*randn(n,M);

%Simultaneous Calculation of the equity Market for all M tracks
S=zeros (n+1,M);
S(1,:)=s0; %initial value
for i=1:n
    S(i+1,:)=S(i,:).*(1+r*h+sigma*dW(i,:));
end

%Simultaneous Calculation of a payoff function 
payoff=max(0,S(n+1,:)-K);

%Simultaneous Calculation of the option price estimator
V=exp(-r*T)*(cumsum(payoff)./(1:M));

%Graphical output
Vexakt=call(s0,0,K,r,sigma,T);
plot(abs(V-Vexakt*ones(1,M)/Vexakt)

However it always results:

??? Undefined function or method 'call'
    for input arguments of type 'char'.
    Why is this?

And is the rest of the code correct?

Answer 1

[Matt Tearle]

It looks like there should be a function call that takes a bunch of inputs and returns the call price. Unless that function is somewhere on your path, MATLAB can't find it, so returns an error. There doesn't seem to be a call function in any of the financial MATLAB toolboxes, so this is probably supposed to be a user-defined function.

EDIT TO ADD: Here's what I have that successfully creates a plot (right or wrong, I don't know).

randn('state',3)
K=100;
r=0.1;
sigma=0.4;
T=1;
S0=80;
n=50;
h=1/n;
M=10000;
%Simulataneous Creation of the Wiener-Process for M Path
dW=sqrt(h)*randn(n,M);
%Simultaneous Calculation of the Equity Prices for all M Path
S=zeros(n+1,M);
S(1,:)=S0; %Initial Values
for i=1:n
S(i+1,:)=S(i,:).*(1+r*h+sigma*dW(i,:));
end
%Simultaneous Calculation of the Payoff
payoff= max(0,S(n+1,:)-K);
%Simultaneous Calculationof the Estimator and the Option Prices
V=exp(-r*T)*(cumsum(payoff)./(1:M));
%Grafical Output
Vexakt=call(S,0,K,r,sigma,T);
plot (abs(bsxfun(@minus,V,Vexakt))./Vexakt)

And in call.m:

function result = call(S,t,K,r,sigma,T)

d1=(log(S/K)+(r+0.5*sigma^2)*(T-t))/(sigma*sqrt(T-t));
d2 = d1-sigma*sqrt(T-t);
n1=0.5*(1+erf(d1/sqrt(2)));
n2=0.5*(1+erf(d2/sqrt(2)));
result=S.*n1-K*exp(-r*(T-t))*n2;

The biggest change was the last line of the script (the plot command). I'm not sure what the intent is there, but Vexakt is (n+1)-by-M, so matrix multiplication with a 1-by-M won't work. Multiplication by M-by-1 works, but will be a different size to V, so the subtraction won't work. Elementwise multiplication also has issues, so I just went with something that at least makes that line run, but could be completely different in intent.

Also note that t wasn't defined, so I just hard-coded in a value (0).

Answer 2

[Nathalie Frischknecht]

Dear mister Matt Tearle Thank you for your help. I tried to create a call function. So mi new code is. function result = call(S,t,K,r,sigma,T)

d1=(log(S/K)+(r+0.5*sigma^2)*(T-t))/(sigma*sqrt(T-t));

d1 = d1-sigma*sqrt(T-t);

n1=0.5*(1+erf(d1/sqrt(2)));

n2=0.5*(1+erf(d2/sqrt(2)));

result=S.*n1-K*exp(-r*(T-t))*n2;

randn('state',3)

K=100;

r=0.1;

sigma=0.4;

T=1;

S0=80;

n=50;

h=1/n;

M=10000;

%Simulataneous Creation of the Wiener-Process for M Path

dW=sqrt(h)*randn(n,M);

%Simultaneous Calculation of the Equity Prices for all M Path

S=zeros(n+1,M);

S(1,:)=S0; %Initial Values

for i=1:n

S(i+1,:)=S(i,:).*(1+r*h+sigma*dW(i,:));

end

%Simultaneous Calculation of the Payoff

payoff= max(0,S(n+1,:)-K);

%Simultaneous Calculationof the Estimator and the Option Prices

V=exp(-r*T)*(cumsum(payoff)./(1:M));

%Grafical Output

Vexakt=call(S,t,K,r,sigma,T;
)
plot (abs(V-Vexakt*ones(1,M))/Vexakt)

However, it does not creat the call function properly since always the following message results: function creation is not allowed in this context. What can I do about this? Whank youIin advance.

[Matt Tearle]

That error usually indicates that you're trying to put function code within a script. Functions need to be defined in their own file (in this case, call.m). There are one or two other bugs in your code, but I was able to make something work -- no idea if it's correct. See edit to my answer (above) for code.