A Gaussian-Legendre quadrature method is combined with analytical formulas for moments of the cumulative return under GARCH for pricing American option. To enhance the convergence speed for pricing American GARCH options, a modified Richardson extrapolation technique was employed. We show that the Gaussian-Legendre Quadrature (GLQ) GARCH option model performs superior, in both accuracy and computational time, than the alternative tree methods.

A Gaussian-Legendre quadrature method is combined with analytical formulas for moments of the cumulative return under GARCH for pricing American option. To enhance the convergence speed for pricing American GARCH options, a modified Richardson extrapolation technique was employed. We show that the Gaussian-Legendre Quadrature (GLQ) GARCH option model performs superior, in both accuracy and computational time, than the alternative tree methods.

Contents
1. Introduction……………………………………….…………….….1
2. Pricing option under GARCH of Duan, Gauthier & Simonato (1999)……………………………………………………………….6
3. Pricing the numerical option……………………………………....8
3.1 Simple and Gaussian Legendre quadrature method………...8
3.2 Apply the Gaussian-Legendre quadrature rule to European option in the GARCH framework……………………...……10
3.3 Apply the Gaussian-Legendre quadrature rule to American option in the GARCH framework…………………………...14
4. Numerical analysis…..…………………………………………….17
4.1 Option prices………………………………………………….17
4.2 Hedge ratios…………………………………………………..21
5. Conclusions…..…………………………………………………....24
6. Appendix…………..………………………………………………25
7. References…………………………………………………………29
Lists of Figures and Tables
Figure
1. The histogram and an approximating distribution of …………..31
2. A left-tail histogram magnified Figure 1 and an approximating distribution of …………………………………………………...32
Table
1. Comparing ranges for approximating a European put option price
by the Gaussian Legendre quadrature rule………………………...33
2. Comparing methods for approximating a European call option price………………………………………………………………..34
3. (a)(b)(c)The performances of the European call option in the GARCH framework………………………………………..35, 36, 37
4. At-the-money European call option prices and their time
(comparing with Ritchken & Trevor (1999))………………...38
5. (a) The moments of the cumulative return…………………………39
(b) The European put option……………………………………….39
6. American put option in the GARCH framework comparing with R&T………………………………………………………………..40
7. American put option in the GARCH framework comparing with different maturity………………………..……………………..…..41
8. The European call option’s Delta…………………………………..42
9. The Amreican put option’s Delta…………………………………..43

Referneces
Abramowitz, M., and Stegun, I. A. (1964), Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series 53, U.S. Government Pricing Office, Washington, D. C.
Barraquand, J. and Martineau, D. (1995). Numerical Valuation of High Dimensional Multivariate American Securities. Journal of Financial and Quantitative Anaysis, 30, 383-338.
Black, F., and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economics, 81, 637-659.
Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31, 307-327.
Broadie, M. and Glasserman, P. (1997). Pricing American-Style Securities Using Simulation. Journal of Economic Dynamics and Contral 21, 1323-1352.
Chang, C. C., Chung, S. L., and Stapleton, R. C. (2001). Richardson Extrapolation Techniques for Pricing American-style Options.
Davis, P. J., and Rabinowitz, P. (1984), Methods of Numerical Integration ( ed.), Academic Press, Orlando, Fla.
Duan, J. C. (1995). The GARCH option pricing model. Mathematical Finance, 5, 13-32.
Duan, J. C. (1996a). Cracking the Smile. Risk, 9(12), 55-59.
Duan, J. C. (1997). Augmented GARCH(p,q) process and its diffusion limit. Journal of Econometrics, 79, 97-127.
Duan, J. C., Gauthier, C., Sasseville, C., and Simonato, J. G. (2001). An analytical approximations for the GJR-GARCH and EGARCH option pricing models. Working paper, HEC Montreal.
Duan, J. C., Gauthier, C., and Simonato, J. G. (1999). An analytical approximating for the GARCH option pricing model. Journal of Computational Finance, 2, 75-116.
Duan, J. C. and Simonato, J. C. (1998). Empirical martingale simulation for asset prices. Management Science, 44, 1218-1233.
Duan, J. C. and Simonato, J. G. (2001). American option pricing under GARCH by a Markov chain approximation. Journal of Economic Dynamics and Contral, 25, 1689-1718.

Engle, R. (1982). Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation. Econometrica, 50, 987-1008.
Eric, J. and Michael, R. (2001). Gram-Charlier densities, Journal of Economic Dynamics & Contral, 25, 1457-1483.
Engle, R. and Ng, V. (1993). Measuring and testing of the impact of news on volatility. Journal of Finance, 48, 1749-1778.
Heston, S., and Nandi, S. (2000). A closed-form GARCH option pricing model. Review of Financial Studies, 13, 586-625.
Jarrow, R., and Rudd, A. (1982). Approximate option valuation for arbitrary stochastic processes. Journal of Financial Economics, 10, 347-369.
Nelson, D. (1991). Conditional Heteroskedasticity in Asset returns: A New Approach. Econometrica, 59, 347-370.
Tilley, J. A. (1993). Valuing American Options in a Path Simulation Model. Transactions of the Society of Actuaries, 45, 83-104.
Omberg, E. (1988). Efficient discrete time jump process models in option pricing. Journal of Financial and Quantitative Analysis, 161-174.
Ritchken, P., and Trevor, R. (1999). Pricing options under generalized GARCH and stochastic volatility processes. Journal of Finance, 54(1), 377-402.
Sullivan, M. A., (2000), Valuing American Put Options Using Gaussian Quadrature, Review of Financial Studies Vol. 13, No.1, 75-94.