**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.

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

- Steven L. Heston,
"A closed-form solution for options with stochastic volatility with an application to bond and currency options",
*The Review of Financial Studies*, vol. 6, issue 2, pp. 327-343, 1993. - Gilles Pagès, and Jacques Printems,
"Functional quantization for numerics with an application to option pricing",
*Monte Carlo Methods and Appl.*, vol. 11, no. 11, pp. 407-446, 2005.