We have developed a practical and elegant closed-form option pricing formula for general GARCH models using a risk-neutral argument. To estimate the parameters, we propose a procedure and utilize Monte Carlo simulation to calculate the prices. Our formula has been successfully applied to S&P 500 index options and Chinese SSE 50 ETF options, providing empirical evidence that it outperforms the Black-Scholes formula with constant volatility in both the U.S. and Chinese financial markets. While there may be other equivalent martingale measures in this setting, our formula serves as a useful reference for pricing options.
References
[1]
Engle, R.F. (2002) New Frontiers for ARCH Models. Journal of Applied Econometrics, 17, 425-446. https://doi.org/10.1002/jae.683
[2]
Black, F., and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-659. https://doi.org/10.1086/260062
[3]
Merton, R.C. (1973) Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4, 141-183. https://doi.org/10.2307/3003143
[4]
Cox, J., Ross, S. and Rubinstein, M. (1979) Option Pricing: A Simplified Approach. Journal of Financial Economics, 7, 229-263. https://doi.org/10.1016/0304-405X(79)90015-1
[5]
Engle, R.F. (1982) Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation. Econometrica, 50, 987-1008. https://doi.org/10.2307/1912773
[6]
Bollerslev, T. (1986) Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31, 307-327. https://doi.org/10.1016/0304-4076(86)90063-1
[7]
Nelson, D.B. (1991) Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica, 59, 347-370. https://doi.org/10.2307/2938260
[8]
Francq, C. and Zakoian, J.-M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley, Hoboken. https://doi.org/10.1002/9780470670057
[9]
Engle, R.F., Lilien, D.M. and Robins, R.P. (1987) Estimating the Time Varying Risk Premia in the Term Structure: The ARCH-M Model. Econometrica, 55, 391-407. https://doi.org/10.2307/1913242
[10]
Hull, J. and White, A. (1987) The Pricing of Options on Assets with Stochastic Volatilities. Journal of Finance, 42, 281-300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x
[11]
Bollerslev, T. and Mikkelsen, H. (1996) Modeling and Pricing Long Memory in Stock Market Volatility. Journal of Econometrics, 73, 151-184. https://doi.org/10.1016/0304-4076(95)01736-4
Zumbach, G. and Fernández, L. (2013) Fast and Realistic European ARCH Option Pricing and Hedging. Quantitative Finance, 13, 713-728. https://doi.org/10.1080/14697688.2012.750009
[14]
Zumbach, G. and Fernández, L. (2014) Option Pricing with Realistic ARCH Processes. Quantitative Finance, 14, 143-170. https://doi.org/10.1080/14697688.2013.816437
[15]
Christensen, B., Dahl, C. and Iglesias, E. (2012) Semiparametric Inference in a GARCH-in-Mean Model. Journal of Econometrics, 167, 458-472. https://doi.org/10.1016/j.jeconom.2011.09.028
[16]
Conrad, C. and Mammen, E. (2016) Asymptotics for Parametric GARCH-in-Mean Models. Journal of Econometrics, 194, 319-329. https://doi.org/10.1016/j.jeconom.2016.05.010
[17]
Fiorentini, G., Sentana, E. and Shephard, N. (2004) Likelihood-Based Estimation of Latent Generalized ARCH Structures. Econometrica, 72, 1481-1517. https://doi.org/10.1111/j.1468-0262.2004.00541.x
[18]
Aït-Sahalia Y. and Jacod, J. (2014) High-Frequency Financial Econometrics. Princeton University Press, Princeton. https://doi.org/10.23943/princeton/9780691161433.001.0001
[19]
Noureldin, D., Shephard, N. and Sheppard, K. (2012) Multivariate High-Frequency-Based Volatility (HEAVY) Models. Journal of Applied Econometrics, 27, 907-933. https://doi.org/10.1002/jae.1260
[20]
Shephard, N. and Sheppard, K. (2010) Realising the Future: Forecasting with High-Frequency-Based Volatility (HEAVY) Models. Journal of Applied Econometrics, 25, 197-231. https://doi.org/10.1002/jae.1158
