From ThetaWiki

Description

At expiry an Asian option pays the difference between the strike and an average of the underlying price achieved during a specified averaging period in the option's term. Many different Asian options can be defined. Specifically, it is important whether to include the first and the last option price observation or not. An example where neither the first nor the last option price is included is given below:

 
%% The model Asian_call computes of the price "P" of 
%% Asian call option   
model Asian_call
    import S    "Stock"
    import K    "Strike"
    import EUR  "Numeraire"
    import T    "Maturity time"
    export P    "Option Value"
 
    P = V!
    n = 125
 
    % no observation at time t=0
    I = 0
    loop n-1
        Theta T/n
        I = I + S
    end     
    % no observation at time t = T
    V = max(1/(n-1)*I-K,0)*EUR
end

The value of an option with volatility sigma=25%, risk-free rate r=5%, strike K=100 and a maturity of T=0.5 is 4.646.

Thetagram

Floating Strike Asian Option

A floating strike Asian option has a payoff based on the difference between the underlying at the expiration T and the average I of the underlying prior to the expiration.

ThetaScript

%% The model Asian_lookback computes the price "P" of 
%% an Asian lookback option, which pays the difference between
%% the average price of the underlying minus the terminal 
%% asset price.
Model Asian_lookback  
import S   "Underlying stock"            
import EUR "Numeraire" 
import T   "Maturity time"
export V   "Option value"            
 
    I=0
    n=250       
    loop n
        Theta 1/n         
        I=I+S
    end 
    V= max(S - I/n,0)*EUR
end

Numerical convergence

A graph of the numerical convergence of a floating strike Asian option is shown below. The graph presents the expected Monte Carlo error versus the number of simulations n.

Number of Monte Carlo simulations: 1000 
random_seed: varied from 1..100