International Journal of Financial Management

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An Analytical Approximation for Option Price under the Affine GARCH Model–A Comparison with the Closed-Form Solution of Heston-Nandi


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1 BESTMOD, Higher Institute of Management, Tunis, Tunisia
     

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In the option pricing theory, two important approaches have been developed to evaluate the prices of a European option. The first approach develops an almost closed-form option pricing formula under a specific GARCH process (Heston&Nandi, 2000). The second approach develops an analytical approximation for computing European option prices with more widespread NGARCH models (Duan, Gauthier&Simonato, 1999). The analytical approximation was also developed under GJR-GARCH and EGARCH models by Duan, Gauthier, Sasseville&Simonato (2006). However, no empirical work was performed to study the comparative performance of these two formulas (closed-form solution and analytical approximation). Also, it is possible to develop an analytical approximation under the specific GARCH model of Heston&Nandi (2000). In this paper, we have filled up those gaps. We started with the development of an analytical approximation, for computing European option prices, under Heston-Nandi's GARCH model. In the second step, we carried out a comparative analysis of the three formulas using CAC 40 index returns from 31 December 1987 to 31 December 2013.

Keywords

GARCH, Option, Pricing, Approximation, Performance, Hedging.
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  • Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. Journal of Finance, 52, 2003-2049.

  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-654.

  • Bollerslev, T. (1986). Generalized auto regressive conditional heteroskedasticity. Journal of Econometrics, 31, 307-327.

  • Christoffersen, P., Dorion, C., Jacobs, K., & Wang, K., (2010). Volatility components: Affine restrictions and non-normal innovations. Journal of Business and Economic Statistics, 28, 483-502.

  • Christoffersen, P., Jacobs, K., & Ornthanalai, C., (2013). GARCH Option Valuation: Theory and Evidence. Journal of Derivatives, 21, 8-41.

  • Christoffersen, P., Jacobs, K., Ornthanalai, C., & Wang, Y. (2008). Option valuation with long-run and short-run volatility components. Journal of Financial Economics, 90, 272-297.

  • Duan, J. C. (1995). The GARCH option pricing models. Mathematical Finance, 5, 13-32.

  • Duan, J. C., Gauthier, G., & Simonato, J. G. (1999). An analytical approximation for the GARCH option pricing model. Journal of Computational Finance, 2(4), 75-116.

  • Duan, J. C., Gauthier, G., Sasseville, C., & Simonato, J. G. (2006). Approximating the JGR-GARCH and EGARCH option pricing models analytically. Journal of Computational Finance, 9(3).

  • Duan, J. C., Ritchken, P., & Sun, Z. (2007). Jump Starting GARCH: Pricing Options with Jumps in Returns and Volatilities, Working Paper.

  • Duan, J. C., & Simonato, J. G. (1999). American option pricing under GARCH by a Markov chain approximation, working paper, Hong Kong University of Science and Technology.

  • Engle, R. (1982). Auto regressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica, 50, 987-1008.

  • Engle, R., & Ng, V. (1993). Measuring and testing the impact of news on volatility. Journal of Finance, 48, 1749-1778.

  • Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779-1801.

  • Hanke, M. (1997). Neutral network approximation of option pricing formulas for analytically intractable option pricing models. Journal of Computational Intelligence in Finance, 5, 20-27.

  • Heston S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327-343.

  • Heston, S., & Nandi, S. (2000). A closed-form GARCH option pricing model. Review of Financial Studies, 13, 585-626.

  • Jarrow, R. A., & Rudd, A. (1982). Approximate option valuation for arbitrary stochastic processes. Journal of Financial Economics, 10, 347-369.

  • Nandi, S. (1996). Pricing and hedging index options under stochastic volatility: An empirical examination. Working paper, Federal Reserve of Atlanta.

  • Nelson, D. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347-370.

  • Ritchken, P., & Trevor R. (1999). Pricing options under generalized GARCH and stochastic volatility processes. Journal of Finance, 54, 377-402.


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  • An Analytical Approximation for Option Price under the Affine GARCH Model–A Comparison with the Closed-Form Solution of Heston-Nandi

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Authors

Noureddine Lahouel
BESTMOD, Higher Institute of Management, Tunis, Tunisia
Slaheddine Hellara
BESTMOD, Higher Institute of Management, Tunis, Tunisia

Abstract


In the option pricing theory, two important approaches have been developed to evaluate the prices of a European option. The first approach develops an almost closed-form option pricing formula under a specific GARCH process (Heston&Nandi, 2000). The second approach develops an analytical approximation for computing European option prices with more widespread NGARCH models (Duan, Gauthier&Simonato, 1999). The analytical approximation was also developed under GJR-GARCH and EGARCH models by Duan, Gauthier, Sasseville&Simonato (2006). However, no empirical work was performed to study the comparative performance of these two formulas (closed-form solution and analytical approximation). Also, it is possible to develop an analytical approximation under the specific GARCH model of Heston&Nandi (2000). In this paper, we have filled up those gaps. We started with the development of an analytical approximation, for computing European option prices, under Heston-Nandi's GARCH model. In the second step, we carried out a comparative analysis of the three formulas using CAC 40 index returns from 31 December 1987 to 31 December 2013.

Keywords


GARCH, Option, Pricing, Approximation, Performance, Hedging.

References