Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-28T18:11:48.423Z Has data issue: false hasContentIssue false
  • English
  • Français

A closed-form approximation formula for pricing European options under a three-factor model

Published online by Cambridge University Press:  18 August 2021

Hye-mee Kil  and
Jeong-Hoon Kim Open the ORCID record for Jeong-Hoon Kim [Opens in a new window]
Hye-mee Kil
Affiliation:
School of Mathematics & Computing, Yonsei University, Seoul, Republic of Korea. E-mail: hyemee@yonsei.ac.kr, jhkim96@yonsei.ac.kr
Jeong-Hoon Kim
Affiliation:
School of Mathematics & Computing, Yonsei University, Seoul, Republic of Korea. E-mail: hyemee@yonsei.ac.kr, jhkim96@yonsei.ac.kr
Get access Rights & Permissions [Opens in a new window]

Abstract

The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

Keywords

CalibrationDouble-mean-reverting modelEuropean optionImplied volatility
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 1214 - 1240
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bayer, C., Gatheral, J., & Karlsmark, M. (2013). Fast Ninomiya-Victoir calibration of the double-mean-reverting model. Quantitative Finance 13: 1813–1829.CrossRefGoogle Scholar
Black, F. & Scholes, M.S. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654.CrossRefGoogle Scholar
Cox, J.C. & Ross, S.A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics 3: 145–166.CrossRefGoogle Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R., & Sølna, K. (2011). Multiscale stochastic volatility for equity, interest rate, and credit derivatives. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Fouque, J.-P., Lorig, M., & Sircar, R. (2016). Second order multiscale stochastic volatility asymptotics: Stochastic terminal layer analysis and calibration. Finance and Stochastics 20: 543–588.CrossRefGoogle Scholar
Gatheral, J. (2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society, London, July 18.Google Scholar
Gatheral, J. (2011). The volatility surface: A practitioner's guide, vol. 357. New York: John Wiley & Sons.Google Scholar
Hagan, P., Kumar, D., Lesniewski, A., & Woodward, D. (2002). Managing smile risk. Wilmott Magazine 1: 84–108.Google Scholar
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.CrossRefGoogle Scholar
Huh, J., Jeon, J., & Kim, J.-H. (2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515.CrossRefGoogle Scholar
Øksendal, B. (2003). Stochastic differential equations: An introduction with applications. Universitext (1979). Berlin: Springer.CrossRefGoogle Scholar
Ramm, A.G. (2001). A simple proof of the Fredholm alternative and a characterization of the Fredholm operators. The American Mathematical Monthly 108: 855–860.CrossRefGoogle Scholar