Matlab → Simulations → Monte Carlo →  Antithetic Technique → Asian Option

The following Matlab codes calculate Asian Option Prices using Monte Carlo Simulation Method in Matlab. In the first code we used the for loop to calculate the arithmetic Asian call option price. Later, we used the powerful cumprod command to simplify the Matlab codes. In the end, the for loop is used to calculate the geometric Asian call option.

Asian call option using Monte Carlo option pricing method
function Asian = AsianMonteCarlo(so,k,r,v,t,m,n)
dt = t/m; AsianPayoffSum=0;
for i=1:n
s = so;
stSum = so;
at = so;
for j=1:m
st = s*exp(((r-v^2/2)*dt)+(v*randn*sqrt(dt)));
stSum = stSum + st; % For geometric: STSum = STSum*ST;
at = stSum/(j+1); % For geometric: AT = (STSum)^(1/(J+1));
s = st;
end
AsianPayoff = max(at-k,0);
AsianPayoffSum = AsianPayoffSum + AsianPayoff;
end
AsianCall = exp(-r*t)*(AsianPayoffSum/n)

We can also calculate the Asian call and put option prices using the cumprod function instead of using the for loop. For example,

function Asian = AsianMonteCarlo(s,k,r,v,t,m,n)
dt = t/m;
st = s*[ones(n,1),cumprod(exp((r-0.5*v^2)*dt+v*sqrt(dt)*randn(n,m)),2)];
averagePrice = mean(st,2);
payoffCall = max(averagePrice-k,0);
payoffPut = max(k-averagePrice,0);
AsianCall = exp(-r*t)*mean(payoffCall)
AsianPut = exp(-r*t)*mean(payoffPut)

And, to calculate geometric Asian option prices using Monte Carlo method, we need to change mean by geomean in line 4 of the above Matlab code. Like,

function Asian = AsianMonteCarlo(s,k,r,v,t,m,n)
dt = t/m;
st = s*[ones(n,1),cumprod(exp((r-0.5*v^2)*dt+v*sqrt(dt)*randn(n,m)),2)];
averagePrice = geomean(st,2);
payoffCall = max(averagePrice-k,0);
payoffPut = max(k-averagePrice,0);
AsianCall = exp(-r*t)*mean(payoffCall)
AsianPut = exp(-r*t)*mean(payoffPut)

In the next example, Both Asian call and put option prices are calculated using Monte Carlo simulation method for option pricing. Here we used the antithetic variable technique for variance reduction.

function Asian = AsianMonteCarlo(s,k,r,v,t,m,n)
dt = t/m;
st1 = s*[ones(n,1),cumprod(exp((r-0.5*v^2)*dt+v*sqrt(dt)*randn(n,m)),2)];
st2 = s*[ones(n,1),cumprod(exp((r-0.5*v^2)*dt+v*sqrt(dt)*(-randn(n,m))),2)];
averagePrice1 = mean(st1,2);
averagePrice2 = mean(st2,2);
payoffCall1 = max(averagePrice1-k,0);
payoffCall2 = max(averagePrice2-k,0);
payoffCall = 0.5*(payoffCall1+payoffCall2);
payoffPut1 = max(k-averagePrice1,0);
payoffPut2 = max(k-averagePrice2,0);
payoffPut = 0.5*(payoffPut1+payoffPut2);
AsianCall = exp(-r*t)*mean(payoffCall)
AsianPut = exp(-r*t)*mean(payoffPut)

In the next example geometric Asian call option price is calculate when we have used the for loop.

function Asian = AsianMonteCarlo(so,k,r,v,t,m,n)
dt = t/m;; AsianPayoffSum=0;
for i=1:n
s = so;
stSum = so;
at = so;
for j=1:m
st = s*exp(((r-v^2/2)*dt)+(v*randn*sqrt(dt)));
stSum = stSum*st;
at = (stSum)^(1/(j+1));
s = st;
end
AsianPayoff = max(at-k,0);
AsianPayoffSum = AsianPayoffSum + AsianPayoff;
end
AsianCall = exp(-r*t)*(AsianPayoffSum/n)