## Pricing Asian options in the Heston model

Introduction

The following stochastic volatility model for the stock price dynamic in an incomplete market was introduced by Heston in 1993 [1]. Under a Risk-Neutral probability , it writes:

where and where are such that . Here and are two standard Brownian motions under the probability measure . Consider the Asian Call option of maturity and strike for which there is no explicit formula:

As a first step, we project onto , so that where is a standard Brownian motion independent of under the probability measure . Then writes

where .

Consider now and two functional quantizers of the Brownian motion.

where tabs andt are available here.

For and , we numerically solve the following ordinary differential equations.

(Here, we used a Runge Kutta IV method.) The option price is now approximated by

Numerical test

Here, we propose a method to compute the Asian call option price based on the functional quantization of the Brownian motion as described in Article [2] (Section 8).

Here et are the sizes of optimal functional quantizers of the standard Brownian motion ( “points”, i.e. paths, for et for ). Parameter stands for the number of time-steps used to solve the ordinary differential equation written above. (We used a fourth order Runge Kutta scheme). Each grid couple yields a price approximation. The final result is a Romberg extrapolation between both prices, based on a rate of convergence for the quadrature error.

The following program was developed with the C programming language, and interfaced with Ruby. You can contact the authors for the source code.

 Spot Constant interest rate Constant dividend rate Initial variance Mean reversion Asymptotic variance Volatility of volatility Correlation Maturity Strike
This can take up to one minute of computation before displaying the results.

The site team would like to thank David Delavennat (CNRS ingeneer at LAMA-UMR 8050 Univ. MLV - Paris 12 from 2003 to 2007) for his advice on the Ruby interface.

### References

1. Steven L. Heston, The Review of Financial Studies, vol. 6, issue 2, pp. 327-343, 1993.
2. Gilles Pagès, and Jacques Printems, Monte Carlo Methods and Appl., vol. 11, no. 11, pp. 407-446, 2005.