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# Moving Window Asian Option

## Overview

A moving window Asian option (MWAO) is computationally challenging to evaluate. The description of a MWAO is simple: similar to an American option that pays the difference between the current underlying price and a fixed strike, the MWAO pays the difference between the current stock price and a moving average.

The computational difficulty of computing the price of a MWAO comes from the underlying strike being a discretely-sampled moving average of stock prices. This leads to very high dimensionality in the equation for the optimal early exercise strategy. Theta Suite can deal with this high-dimensional problem easily since it has a built-in function E that uses a sparse-grid technique that efficiently solves the "curse of dimensionality" problem.

## Implementation in ThetaML

The model mwao implemented in ThetaML efficiently prices a Moving Window Asian Option. The high-dimensionality problem of the early exercise strategy is solved with the ThetaML function E. The underlying stock price process 'S' is simulated externally in Geometric Brownian Motion, the discount factor process 'EUR' uses the constant discount curve simulated in Discounting.

model mwao
import S	"Stock prices"
import EUR	"Discount numeraire"
import m	"Window length"
import n	"Number of observations"
import T	"Option maturity time"
export V	"Option process"
export P	"Option prices"
export A	"Current average"

%at current time, set the option prices to have the same distribution as the variable 'V'
%the ThetaML future operator '!' accompanying the variable 'V' acts like a function on 'V,
%such that the values of 'V' at current time remains to be determined at a later instance
%when 'V' is assigned some values
P = V!
%assuming daily exercise interval
deltaT = 1/252

%an array used to store the last m stock prices
C = 0
%initialize the moving average to 0
A = 0
%computing the moving averages 'A'
%the ThetaML command 'fork' virtually parallels the processes defined
%inside the 'fork...end' statement and the processes defined outside it
fork
%initialize 'index' to 'm'
index = m
%loop 'n' times
loop n
%add the averages of stock prices in the moving window;
%at the end of the window 'm', subtract the stock value
%at the beginning of the window divided by 'm'
A = A + S/m - C[index]/m
%store the stock prices in the array 'C'
C[index] = S
%update the 'index'
index= mod(index, m)+1
%the ThetaML command 'Theta' passes time by 'deltaT' time interval
Theta deltaT
end
end

%the ThetaML command 'Theta' passes time by 'deltaT*(m-1)'
%since the window length is 'm', the first early-exercise evaluation is
%at time 'deltaT*(m-1)'
Theta deltaT*(m-1)
%loop 'n - m + 1' times
loop n - m + 1
%only compute values for in-the-money paths
if (A > S)
%compute optimal early exercise strategy
if (A - S)*EUR > E(V!)
%exercise option payoffs
V = max( A - S, 0)*EUR
end
end
%the ThetaML command 'Theta' passes time by 'deltaT'
Theta deltaT
end
%option payoffs at maturity
V = max(A - S,0)*EUR
end

## Numerical Example

An option with the following parameters has a value of about 7.65.

 Parameter Symbol Value Option type Moving Window Asian Option Maturity T 0.4 years Risk free rate r 5 % p.a. Volatility sigma 40 % p.a. Daily observations Delta_t 1/250 years Early exercise at each observation with $t\geq 10/250$ years Length of observation m 10 days Exercise value $P(S, t_i) =\max\left( \frac{1}{M} \left( \sum \limits_{j=i-M+1}^i S_{t_j} \right) - S_{t_i},0\right)$