Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T16:33:01.142Z Has data issue: false hasContentIssue false
  • English
  • Français

Pricing VIX derivatives using a stochastic volatility model with a flexible jump structure

Published online by Cambridge University Press:  27 January 2022

Wuyi Ye ,
Bin Wu  and
Pengzhan Chen Open the ORCID record for Pengzhan Chen [Opens in a new window]
Wuyi Ye
Affiliation:
International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230026, China. E-mail: cpz@ustc.edu.cn
Bin Wu
Affiliation:
International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230026, China. E-mail: cpz@ustc.edu.cn
Pengzhan Chen*
Affiliation:
International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230026, China. E-mail: cpz@ustc.edu.cn
*
*Corresponding author. E-mail: cpz@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proposes a novel stochastic volatility model with a flexible jump structure. This model allows both contemporaneous and independent arrival of jumps in return and volatility. Moreover, time-varying jump intensities are used to capture jump clustering. In the proposed framework, we provide a semi-analytical solution for the pricing problem of VIX futures and options. Through numerical experiments, we verify the accuracy of our pricing formula and explore the impact of the jump structure on the pricing of VIX derivatives. We find that the correct identification of the market jump structure is crucial for pricing VIX derivatives, and misspecified model setting can yield large errors in pricing.

Keywords

Hawkes processImplied volatilityJump intensityVIX futuresVIX options
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 1 , January 2023 , pp. 245 - 274
Copyright
Copyright © The Author(s), 2022. 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.)

Footnotes

†

Wuyi Ye and Bin Wu contributed equally to this work and should be considered co-first authors.

References

Aït-Sahalia, Y., Cacho-Diaz, J., & Laeven, R.J. (2015). Modeling financial contagion using mutually exciting jump processes. Journal of Financial Economics 117(3): 585–606.CrossRefGoogle Scholar
An, Y. & Li, C. (2015). Approximating local volatility functions of stochastic volatility models: a closed-form expansion approach. Probability in the Engineering and Informational Sciences 29(4): 547–563.CrossRefGoogle Scholar
Bandi, F.M. & Reno, R. (2016). Price and volatility co-jumps. Journal of Financial Economics 119(1): 107–146.CrossRefGoogle Scholar
Bardgett, C., Gourier, E., & Leippold, M. (2019). Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets. Journal of Financial Economics 131(3): 593–618.CrossRefGoogle Scholar
Bates, D.S. (1996). Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options. The Review of Financial Studies 9(1): 69–107.CrossRefGoogle Scholar
Bates, D.S. (2000). Post-’87 crash fears in the S&P 500 futures option market. Journal of Econometrics 94(1–2): 181–238.CrossRefGoogle Scholar
Bates, D.S. (2006). Maximum likelihood estimation of latent affine processes. The Review of Financial Studies 19(3): 909–965.CrossRefGoogle Scholar
Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81(3): 637–654.CrossRefGoogle Scholar
Chen, P. & Ye, W. (2021). Stochastic volatility model with correlated jump sizes and independent arrivals. Probability in the Engineering and Informational Sciences 35(3): 513–531.CrossRefGoogle Scholar
Cheng, J., Ibraimi, M., Leippold, M., & Zhang, J.E. (2012). A remark on Lin and Chang's paper consistent modeling of S&P 500 and VIX derivatives. Journal of Economic Dynamics and Control 36(5): 708–715.CrossRefGoogle Scholar
Da Fonseca, J. & Ignatieva, K. (2019). Jump activity analysis for affine jump-diffusion models: evidence from the commodity market. Journal of Banking & Finance 99: 45–62.CrossRefGoogle Scholar
Duan, J.-C. & Yeh, C.-Y. (2010). Jump and volatility risk premiums implied by VIX. Journal of Economic Dynamics and Control 34(11): 2232–2244.CrossRefGoogle Scholar
Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6): 1343–1376.CrossRefGoogle Scholar
Eraker, B. (2004). Do stock prices and volatility jump? Reconciling evidence from spot and option prices. The Journal of Finance 59(3): 1367–1403.CrossRefGoogle Scholar
Eraker, B. & Wu, Y. (2017). Explaining the negative returns to volatility claims: an equilibrium approach. Journal of Financial Economics 125(1): 72–98.CrossRefGoogle Scholar
Eraker, B. & Yang, A. (2020). The price of higher order catastrophe insurance: the case of VIX options. Available at SSRN 3520256.CrossRefGoogle Scholar
Eraker, B., Johannes, M., & Polson, N. (2003). The impact of jumps in volatility and returns. The Journal of Finance 58(3): 1269–1300.CrossRefGoogle Scholar
Han, X. (2019). Valuation of vulnerable options under the double exponential jump model with stochastic volatility. Probability in the Engineering and Informational Sciences 33(1): 81–104.CrossRefGoogle Scholar
Jing, B., Li, S., & Ma, Y. (2020). Pricing VIX options with volatility clustering. Journal of Futures Markets 40(6): 928–944.CrossRefGoogle Scholar
Lewis, A.L. (2000). Option valuation under stochastic volatility. Newport Beach: Finance Press.Google Scholar
Lian, G.-H. & Zhu, S.-P. (2013). Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance 36(1): 71–88.CrossRefGoogle Scholar
Lin, Y.-N. (2007). Pricing VIX futures: evidence from integrated physical and risk-neutral probability measures. Journal of Futures Markets: Futures, Options, and Other Derivative Products 27(12): 1175–1217.CrossRefGoogle Scholar
Luo, X., Zhang, J.E., & Zhang, W. (2019). Instantaneous squared VIX and VIX derivatives. Journal of Futures Markets 39(10): 1193–1213.CrossRefGoogle Scholar
Pan, J. (2002). The jump-risk premia implicit in options: evidence from an integrated time-series study. Journal of Financial Economics 63(1): 3–50.CrossRefGoogle Scholar
Pasricha, P. & Goel, A. (2020). A closed-form pricing formula for European exchange options with stochastic volatility. Probability in the Engineering and Informational Sciences, 1–10, in press.Google Scholar
Poularikas, A.D. (2006). Transforms and applications handbook. Boca Raton: CRC Press.Google Scholar
Psychoyios, D., Dotsis, G., & Markellos, R.N. (2010). A jump diffusion model for VIX volatility options and futures. Review of Quantitative Finance and Accounting 35(3): 245–269.CrossRefGoogle Scholar
Schürger, K. (2002). Laplace transforms and suprema of stochastic processes. In Sandmann, K., & Schönbucher, P.J. (Eds.), Advances in Finance and Stochastics, pp. 285–294. Springer.CrossRefGoogle Scholar
Sepp, A. (2008). VIX option pricing in a jump-diffusion model. Risk Magazine, pp. 84–89.Google Scholar
Sepp, A. (2008). Pricing options on realized variance in the Heston model with jumps in returns and volatility. Journal of Computational Finance 11(4): 33–70.CrossRefGoogle Scholar
Wu, B., Chen, P., & Ye, W. (2021). Jump activity analysis of the equity index and the corresponding volatility: evidence from the Chinese market. Journal of Futures Markets 41(7): 1055–1073.CrossRefGoogle Scholar
Zhu, S.-P. & Lian, G.-H. (2012). An analytical formula for VIX futures and its applications. Journal of Futures Markets 32(2): 166–190.CrossRefGoogle Scholar