[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v114y2018icp130-144.html
   My bibliography  Save this article

Analytic solutions for variance swaps with double-mean-reverting volatility

Author

Listed:
  • Kim, See-Woo
  • Kim, Jeong-Hoon
Abstract
A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston’s stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.

Suggested Citation

  • Kim, See-Woo & Kim, Jeong-Hoon, 2018. "Analytic solutions for variance swaps with double-mean-reverting volatility," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 130-144.
    • Handle: RePEc:eee:chsofr:v:114:y:2018:i:c:p:130-144
    DOI: 10.1016/j.chaos.2018.06.024
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077918304685
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2018.06.024?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Adrian Dragulescu & Victor Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 443-453.
    2. Carole Bernard & Zhenyu Cui, 2014. "Prices and Asymptotics for Discrete Variance Swaps," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(2), pages 140-173, April.
    3. Peter Christoffersen & Steven Heston & Kris Jacobs, 2009. "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well," Management Science, INFORMS, vol. 55(12), pages 1914-1932, December.
    4. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Cao, Jiling & Lian, Guanghua & Roslan, Teh Raihana Nazirah, 2016. "Pricing variance swaps under stochastic volatility and stochastic interest rate," Applied Mathematics and Computation, Elsevier, vol. 277(C), pages 72-81.
    7. Pun, Chi Seng & Chung, Shing Fung & Wong, Hoi Ying, 2015. "Variance swap with mean reversion, multifactor stochastic volatility and jumps," European Journal of Operational Research, Elsevier, vol. 245(2), pages 571-580.
    8. Bara Kim & In-Suk Wee, 2014. "Pricing of geometric Asian options under Heston's stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1795-1809, October.
    9. Christian Bayer & Jim Gatheral & Morten Karlsmark, 2013. "Fast Ninomiya--Victoir calibration of the double-mean-reverting model," Quantitative Finance, Taylor & Francis Journals, vol. 13(11), pages 1813-1829, November.
    10. Long Teng & Matthias Ehrhardt & Michael Günther, 2016. "On The Heston Model With Stochastic Correlation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(06), pages 1-25, September.
    11. Tinne Haentjens & Karel J. in 't Hout, 2015. "ADI Schemes for Pricing American Options under the Heston Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(3), pages 207-237, July.
    12. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    13. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    14. Ballestra, Luca Vincenzo & Cecere, Liliana, 2016. "A numerical method to estimate the parameters of the CEV model implied by American option prices: Evidence from NYSE," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 100-106.
    15. Lee, Min-Ku & Yang, Sung-Jin & Kim, Jeong-Hoon, 2016. "A closed form solution for vulnerable options with Heston’s stochastic volatility," Chaos, Solitons & Fractals, Elsevier, vol. 86(C), pages 23-27.
    16. Wendong Zheng & Yue Kuen Kwok, 2014. "Closed Form Pricing Formulas For Discretely Sampled Generalized Variance Swaps," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 855-881, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kim, Seong-Tae & Kim, Jeong-Hoon, 2020. "Stochastic elasticity of vol-of-vol and pricing of variance swaps," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 420-440.
    2. Kim, See-Woo & Kim, Jeong-Hoon, 2019. "Variance swaps with double exponential Ornstein-Uhlenbeck stochastic volatility," The North American Journal of Economics and Finance, Elsevier, vol. 48(C), pages 149-169.
    3. Min-Ku Lee & See-Woo Kim & Jeong-Hoon Kim, 2022. "Variance Swaps Under Multiscale Stochastic Volatility of Volatility," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 39-64, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kim, See-Woo & Kim, Jeong-Hoon, 2019. "Variance swaps with double exponential Ornstein-Uhlenbeck stochastic volatility," The North American Journal of Economics and Finance, Elsevier, vol. 48(C), pages 149-169.
    2. Ah-Reum Han & Jeong-Hoon Kim & See-Woo Kim, 2021. "Variance Swaps with Deterministic and Stochastic Correlations," Computational Economics, Springer;Society for Computational Economics, vol. 57(4), pages 1059-1092, April.
    3. Alexander Lipton, 2024. "Hydrodynamics of Markets:Hidden Links Between Physics and Finance," Papers 2403.09761, arXiv.org.
    4. Kim, See-Woo & Kim, Jeong-Hoon, 2020. "Pricing generalized variance swaps under the Heston model with stochastic interest rates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 168(C), pages 1-27.
    5. Wu, Bin & Chen, Pengzhan & Ye, Wuyi, 2024. "Variance swaps with mean reversion and multi-factor variance," European Journal of Operational Research, Elsevier, vol. 315(1), pages 191-212.
    6. Yang, Ben-Zhang & Yue, Jia & Wang, Ming-Hui & Huang, Nan-Jing, 2019. "Volatility swaps valuation under stochastic volatility with jumps and stochastic intensity," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 73-84.
    7. Tong, Zhigang & Liu, Allen, 2022. "Pricing variance swaps under subordinated Jacobi stochastic volatility models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 593(C).
    8. Coqueret, Guillaume & Tavin, Bertrand, 2016. "An investigation of model risk in a market with jumps and stochastic volatility," European Journal of Operational Research, Elsevier, vol. 253(3), pages 648-658.
    9. Rehez Ahlip & Laurence A. F. Park & Ante Prodan, 2017. "Pricing currency options in the Heston/CIR double exponential jump-diffusion model," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-30, March.
    10. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.
    11. Jiling Cao & Teh Raihana Nazirah Roslan & Wenjun Zhang, 2018. "Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1359-1379, December.
    12. Teh Raihana Nazirah Roslan & Wenjun Zhang & Jiling Cao, 2016. "Pricing variance swaps with stochastic volatility and stochastic interest rate under full correlation structure," Papers 1610.09714, arXiv.org, revised Apr 2020.
    13. Ben-zhang Yang & Jia Yue & Nan-jing Huang, 2017. "Variance swaps under L\'{e}vy process with stochastic volatility and stochastic interest rate in incomplete markets," Papers 1712.10105, arXiv.org, revised Mar 2018.
    14. Li, Shaoyu & Zhang, Yuanyuan & Zhu, Chunhui, 2021. "A closed-form exact solution for pricing fixed-income variance swaps with affine-jump model," The North American Journal of Economics and Finance, Elsevier, vol. 58(C).
    15. Dong-Mei Zhu & Jiejun Lu & Wai-Ki Ching & Tak-Kuen Siu, 2019. "Option Pricing Under a Stochastic Interest Rate and Volatility Model with Hidden Markovian Regime-Switching," Computational Economics, Springer;Society for Computational Economics, vol. 53(2), pages 555-586, February.
    16. Yanhong Zhong & Guohe Deng, 2019. "Geometric Asian Options Pricing under the Double Heston Stochastic Volatility Model with Stochastic Interest Rate," Complexity, Hindawi, vol. 2019, pages 1-13, January.
    17. Recchioni, M.C. & Sun, Y., 2016. "An explicitly solvable Heston model with stochastic interest rate," European Journal of Operational Research, Elsevier, vol. 249(1), pages 359-377.
    18. Zhu, Song-Ping & Lian, Guang-Hua, 2015. "Pricing forward-start variance swaps with stochastic volatility," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 920-933.
    19. repec:uts:finphd:41 is not listed on IDEAS
    20. Long Teng & Matthias Ehrhardt & Michael Günther, 2016. "On The Heston Model With Stochastic Correlation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(06), pages 1-25, September.
    21. Youngin Yoon & Jeong-Hoon Kim, 2023. "A Closed Form Solution for Pricing Variance Swaps Under the Rescaled Double Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 61(1), pages 429-450, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:114:y:2018:i:c:p:130-144. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.