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Estimating option greeks under the stochastic volatility using simulation

Author

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  • Shafi, Khuram
  • Latif, Natasha
  • Shad, Shafqat Ali
  • Idrees, Zahra
  • Gulzar, Saqib

Abstract

As the Black–Scholes (BS) equation being widely used to price options, which is based on a hypothesis that the underlying (bonds & stocks) volatility is constant. Many scholars proposed the extended version of this formula to predict the behavior of the volatility. So stochastic volatility model is the improved version in which fixed volatility is replaced. The purpose of this study is to adopt one of the famous stochastic volatility models, Heston Model (1993), to price European call options. Put option values can easily obtained by call–put parity if it is needed. Simulation has proved to be a valuable tool for estimating options price derivatives i.e. “Greeks”. This paper proposes the method for the simulation of stock prices and variance under the Heston stochastic volatility model. We consider three different models based on the Heston model. We present two direct methods a Path-wise method and Likelihood ratio method for estimating the derivatives of Options. Then we compare it with Black–Scholes equation, and make a sensitivity analysis for its parameters by using estimator’s approaches.

Suggested Citation

  • Shafi, Khuram & Latif, Natasha & Shad, Shafqat Ali & Idrees, Zahra & Gulzar, Saqib, 2018. "Estimating option greeks under the stochastic volatility using simulation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 1288-1296.
    • Handle: RePEc:eee:phsmap:v:503:y:2018:i:c:p:1288-1296
    DOI: 10.1016/j.physa.2018.08.032
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    References listed on IDEAS

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    1. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    2. Devreese, J.P.A. & Lemmens, D. & Tempere, J., 2010. "Path integral approach to Asian options in the Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(4), pages 780-788.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. Lin, Lisha & Li, Yaqiong & Wu, Jing, 2018. "The pricing of European options on two underlying assets with delays," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 495(C), pages 143-151.
    5. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Wang, Guanying & Wang, Xingchun & Zhou, Ke, 2017. "Pricing vulnerable options with stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 485(C), pages 91-103.
    8. Contreras, Mauricio & Hojman, Sergio A., 2014. "Option pricing, stochastic volatility, singular dynamics and constrained path integrals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 393(C), pages 391-403.
    9. Boyle, Phelim P., 1977. "Options: A Monte Carlo approach," Journal of Financial Economics, Elsevier, vol. 4(3), pages 323-338, May.
    10. Huang, Guanghui & Wan, Jianping, 2008. "A nonparametric approach for European option valuation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(10), pages 2306-2316.
    11. Cassagnes, Aurelien & Chen, Yu & Ohashi, Hirotada, 2014. "Path integral pricing of outside barrier Asian options," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 394(C), pages 266-276.
    12. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    13. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    14. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    15. Walter, J.-C. & Barkema, G.T., 2015. "An introduction to Monte Carlo methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 418(C), pages 78-87.
    16. Ser-Huang Poon & Clive W.J. Granger, 2003. "Forecasting Volatility in Financial Markets: A Review," Journal of Economic Literature, American Economic Association, vol. 41(2), pages 478-539, June.
    17. Sun, Lin, 2013. "Pricing currency options in the mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(16), pages 3441-3458.
    18. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
    19. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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