Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T11:11:55.116Z Has data issue: false hasContentIssue false
  • English
  • Français

VOLATILITY SWAPS VALUATION UNDER A MODIFIED RISK-NEUTRALIZED HESTON MODEL WITH A STOCHASTIC LONG-RUN VARIANCE LEVEL

Part of: Mathematical finance

Published online by Cambridge University Press:  26 September 2022

XIN-JIANG HE Open the ORCID record for XIN-JIANG HE [Opens in a new window]  and
SHA LIN Open the ORCID record for SHA LIN [Opens in a new window]
XIN-JIANG HE
Affiliation:
School of Economics, Zhejiang University of Technology, Hangzhou, China; e-mail: [email protected]
SHA LIN*
Affiliation:
School of Finance, Zhejiang Gongshang University, Hangzhou, China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the pricing of discretely sampled volatility swaps under a modified Heston model, whose risk-neutralized volatility process contains a stochastic long-run variance level. We derive an analytical forward characteristic function under this model, which has never been presented in the literature before. Based on this, we further obtain an analytical pricing formula for volatility swaps which can guarantee the computational accuracy and efficiency. We also demonstrate the significant impact of the introduced stochastic long-run variance level on volatility swap prices with synthetic as well as calibrated parameters.

Keywords

volatility swapsmodified Heston modelrisk-neutralized volatilitystochastic long-run varianceanalytical solution

MSC classification

Primary: 91G20: Derivative securities
Type
Research Article
Information
The ANZIAM Journal , Volume 64 , Issue 3 , July 2022 , pp. 250 - 263
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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

Andersen, T. G., Benzoni, L. and Lund, J., “An empirical investigation of continuous-time equity return models”, J. Finance 57 (2002) 1239–1284; doi:10.1111/1540-6261.00460.CrossRefGoogle Scholar
Bakshi, G., Ju, N. and Ou-Yang, H., “Estimation of continuous-time models with an application to equity volatility dynamics”, J. Financ. Econ. 82 (2006) 227–249; doi:10.1016/j.jfineco.2005.09.005.CrossRefGoogle Scholar
Cao, J., Kim, J.-H. and Zhang, W., “Pricing variance swaps under hybrid CEV and stochastic volatility”, J. Comput. Appl. Math. 386 (2021) Article Id 113220; doi:10.1016/j.cam.2020.113220.CrossRefGoogle Scholar
Cao, J., Lian, G. and Roslan, T. R. N., “Pricing variance swaps under stochastic volatility and stochastic interest rate”, Appl. Math. Comput. 277 (2016) 72–81; doi:10.1016/j.amc.2015.12.027.CrossRefGoogle Scholar
Cao, J., Roslan, T. R. N. and Zhang, W., “Pricing variance swaps in a hybrid model of stochastic volatility and interest rate with regime-switching”, Methodol. Comput. Appl. Probab. 20 (2018) 1359–1379; doi:10.1007/s11009-018-9624-5.CrossRefGoogle Scholar
Cao, J., Roslan, T. R. N. and Zhang, W., “The valuation of variance swaps under stochastic volatility, stochastic interest rate and full correlation structure”, J. Korean Math. Soc. 57 (2020) 1167–1186; doi:10.4134/JKMS.j190616.Google Scholar
Carr, P. and Lee, R., “Realized volatility and variance: options via swaps”, Risk 20 (2007) 76–83; https://www.risk.net/derivatives/equity-derivatives/1500285/realised-volatility-and-variance-options-swaps.Google Scholar
Carr, P. and Lee, R., “Robust replication of volatility derivatives”, In PRMIA award for Best Paper in Derivatives, MFA 2008 Annual Meeting, http://faculty.baruch.cuny.edu/lwu/890/CarrLee2004.pdf.CrossRefGoogle Scholar
Christoffersen, P., Heston, S. and Jacobs, K., “The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well”, Manage. Sci. 55 (2009) 1914–1932; doi:10.1287/mnsc.1090.1065.CrossRefGoogle Scholar
Cui, Z., Kirkby, J. L. and Nguyen, D., “A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps”, European J. Oper. Res. 262 (2017) 381–400; doi:10.1016/j.ejor.2017.04.007.CrossRefGoogle Scholar
Elliott, R. J. and Lian, G.-H., “Pricing variance and volatility swaps in a stochastic volatility model with regime switching: discrete observations case”, Quant. Finance 13 (2013) 687–698; doi:10.1080/14697688.2012.676208.CrossRefGoogle Scholar
Elliott, R. J., Siu, T. K. and Chan, L., “Pricing volatility swaps under Heston’s stochastic volatility model with regime switching”, Appl. Math. Finance 14 (2007) 41–62; doi:10.1080/13504860600659222.CrossRefGoogle Scholar
Gil-Pelaez, J., “Note on the inversion theorem”, Biometrika 38 (1951) 481–482; doi:10.2307/2332598.CrossRefGoogle Scholar
Grünbichler, A. and Longstaff, F. A., “Valuing futures and options on volatility”, J. Bank. Financ. 20 (1996) 985–1001; doi:10.1016/0378-4266(95)00034-8.CrossRefGoogle Scholar
He, X.-J. and Chen, W., “A closed-form pricing formula for European options under a new stochastic volatility model with a stochastic long-term mean”, Math. Financ. Econ. 15 (2021) 381–396; doi:10.1007/s11579-020-00281-y.CrossRefGoogle Scholar
He, X.-J. and Zhu, S.-P., “A series-form solution for pricing variance and volatility swaps with stochastic volatility and stochastic interest rate”, Comput. Math. Appl. 76 (2018) 2223–2234; doi:10.1016/j.camwa.2018.08.022.Google Scholar
He, X.-J. and Zhu, S.-P., “Variance and volatility swaps under a two-factor stochastic volatility model with regime switching”, Int. J. Theor. Appl. Finance 22 (2019) Article Id 1950009; doi:10.1142/S0219024919500092.CrossRefGoogle Scholar
Heston, S. and Nandi, S., “Derivatives on volatility: some simple solutions based on observables” (Federal Reserve Bank of Atlanta WP No. 2000-20, November 2000); doi:10.2139/ssrn.249173.CrossRefGoogle Scholar
Heston, S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Financ. Stud. 6 (1993) 327–343; doi:10.1093/rfs/6.2.327.CrossRefGoogle Scholar
Howison, S., Rafailidis, A. and Rasmussen, H., “On the pricing and hedging of volatility derivatives”, Appl. Math. Finance 11 (2004) 317–346; doi:10.1080/1350486042000254024.CrossRefGoogle Scholar
Issaka, A., “Variance swaps, volatility swaps, hedging and bounds under multi-factor Heston stochastic volatility model”, Stoch. Anal. Appl. 38 (2020) 856–874; doi:10.1080/07362994.2020.1730903.CrossRefGoogle Scholar
Kim, S.-W. and Kim, J.-H., “Variance swaps with double exponential Ornstein–Uhlenbeck stochastic volatility”, North Am. J. Econ. Finance 48 (2019) 149–169; doi:10.1016/j.najef.2019.01.018.CrossRefGoogle Scholar
Lee, M.-K., Kim, S.-W. and Kim, J.-H., “Variance swaps under multiscale stochastic volatility of volatility”, Methodol. Comput. Appl. Probab. 24 (2022) 39–64; doi:10.1007/s11009-020-09834-6.CrossRefGoogle Scholar
Little, T. and Pant, V., “A finite difference method for the valuation of variance swaps”, J. Comput. Finance 5 (2001) 81–101; doi:10.21314/JCF.2001.057.CrossRefGoogle Scholar
Liu, W. and Zhu, S.-P., “Pricing variance swaps under the Hawkes jump-diffusion process”, J. Futures Markets 39 (2019) 635–655; doi:10.1002/fut.21997.CrossRefGoogle Scholar
Pan, J., “The jump-risk premia implicit in options: evidence from an integrated time-series study”, J. Financ. Econ. 63 (2002) 3–50; doi:10.1016/S0304-405X(01)00088-5.CrossRefGoogle Scholar
Pun, C. S., Chung, S. F. and Wong, H. Y., “Variance swap with mean reversion, multifactor stochastic volatility and jumps”, European J. Oper. Res. 245 (2015) 571–580; doi:10.1016/j.ejor.2015.03.026.CrossRefGoogle Scholar
Wu, H., Jia, Z., Yang, S. and Liu, C., “Pricing variance swaps under double Heston stochastic volatility model with stochastic interest rate”, Probab. Engrg. Inform. Sci. 36 (2022) 564–580; doi:10.1017/S0269964820000662.CrossRefGoogle Scholar
Zhu, S.-P. and Lian, G.-H., “Analytically pricing volatility swaps under stochastic volatility”, J. Comput. Appl. Math. 288 (2015) 332–340; doi:10.1016/j.cam.2015.04.036.CrossRefGoogle Scholar