IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v11y2023i9p162-d1236051.html
   My bibliography  Save this article

Pricing of Averaged Variance, Volatility, Covariance and Correlation Swaps with Semi-Markov Volatilities

Author

Listed:
  • Anatoliy Swishchuk

    (Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada
    These authors contributed equally to this work.)

  • Sebastian Franco

    (Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada
    These authors contributed equally to this work.)

Abstract

In this paper, we consider the problem of pricing variance, volatility, covariance and correlation swaps for financial markets with semi-Markov volatilities. The paper’s motivation derives from the fact that in many financial markets, the inter-arrival times between book events are not independent or exponentially distributed but instead have an arbitrary distribution, which means they can be accurately modelled using a semi-Markov process. Through the results of the paper, we hope to answer the following question: Is it possible to calculate averaged swap prices for financial markets with semi-Markov volatilities? This question has not been considered in the existing literature, which makes the paper’s results novel and significant, especially when one considers the increasing popularity of derivative securities such as swaps, futures and options written on the volatility index VIX. Within this paper, we model financial markets featuring semi-Markov volatilities and price-averaged variance, volatility, covariance and correlation swaps for these markets. Formulas used for the numerical evaluation of averaged variance, volatility, covariance and correlation swaps with semi-Markov volatilities are presented as well. The formulas that are detailed within the paper are innovative because they provide a new, simplified method to price averaged swaps, which has not been presented in the existing literature. A numerical example involving the pricing of averaged variance, volatility, covariance and correlation swaps in a market with a two-state semi-Markov process is presented, providing a detailed overview of how the model developed in the paper can be used with real-life data. The novelty of the paper lies in the closed-form formulas provided for the pricing of variance, volatility, covariance and correlation swaps with semi-Markov volatilities, as they can be directly applied by derivative practitioners and others in the financial industry to price variance, volatility, covariance and correlation swaps.

Suggested Citation

  • Anatoliy Swishchuk & Sebastian Franco, 2023. "Pricing of Averaged Variance, Volatility, Covariance and Correlation Swaps with Semi-Markov Volatilities," Risks, MDPI, vol. 11(9), pages 1-22, September.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:9:p:162-:d:1236051
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/11/9/162/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/11/9/162/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Windcliff, H. & Forsyth, P.A. & Vetzal, K.R., 2006. "Pricing methods and hedging strategies for volatility derivatives," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 409-431, February.
    2. Giovanni Salvi & Anatoliy V. Swishchuk, 2012. "Modeling and Pricing of Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities," Papers 1205.5565, arXiv.org.
    3. Peter Carr & Hélyette Geman & Dilip Madan & Marc Yor, 2005. "Pricing options on realized variance," Finance and Stochastics, Springer, vol. 9(4), pages 453-475, October.
    4. Mark Broadie & Ashish Jain, 2008. "The Effect Of Jumps And Discrete Sampling On Volatility And Variance Swaps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(08), pages 761-797.
    5. Fred Espen Benth & Martin Groth & Rodwell Kufakunesu, 2007. "Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(4), pages 347-363.
    6. Joost Driessen & Pascal J. Maenhout & Grigory Vilkov, 2009. "The Price of Correlation Risk: Evidence from Equity Options," Journal of Finance, American Finance Association, vol. 64(3), pages 1377-1406, June.
    7. Robert Elliott & Tak Kuen Siu & Leunglung Chan, 2007. "Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(1), pages 41-62.
    8. Giovanni Salvi & Anatoliy V. Swishchuk, 2014. "Covariance And Correlation Swaps For Financial Markets With Markov-Modulated Volatilities," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(01), pages 1-23.
    Full references (including those not matched with items on IDEAS)

    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. Anatoliy Swishchuk, 2013. "Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8660.
    2. Giovanni Salvi & Anatoliy V. Swishchuk, 2012. "Modeling and Pricing of Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities," Papers 1205.5565, arXiv.org.
    3. 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.
    4. 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.
    5. 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.
    6. Song-Ping Zhu & Guang-Hua Lian, 2018. "On the Convexity Correction Approximation in Pricing Volatility Swaps and VIX Futures," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 14(03), pages 383-401, November.
    7. Semere Habtemicael & Indranil SenGupta, 2016. "Pricing variance and volatility swaps for Barndorff-Nielsen and Shephard process driven financial markets," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-35, December.
    8. Wendong Zheng & Chi Hung Yuen & Yue Kuen Kwok, 2016. "Recursive Algorithms For Pricing Discrete Variance Options And Volatility Swaps Under Time-Changed Lévy Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-29, March.
    9. Gong, Yaxian, 2020. "Credit default swap and two-sided moral hazard," Finance Research Letters, Elsevier, vol. 34(C).
    10. 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.
    11. Lian, Guanghua & Chiarella, Carl & Kalev, Petko S., 2014. "Volatility swaps and volatility options on discretely sampled realized variance," Journal of Economic Dynamics and Control, Elsevier, vol. 47(C), pages 239-262.
    12. Fred Espen Benth & Martin Groth & Rodwell Kufakunesu, 2007. "Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(4), pages 347-363.
    13. Carole Bernard & Zhenyu Cui, 2013. "Prices and Asymptotics for Discrete Variance Swaps," Papers 1305.7092, arXiv.org.
    14. Anqi Zou & Jiajie Wang & Chiye Wu, 2023. "Pricing Variance Swaps under MRG Model with Regime-Switching: Discrete Observations Case," Mathematics, MDPI, vol. 11(12), pages 1-30, June.
    15. Weiyi Liu & Song‐Ping Zhu, 2019. "Pricing variance swaps under the Hawkes jump‐diffusion process," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(6), pages 635-655, June.
    16. Peter Carr & Roger Lee & Liuren Wu, 2012. "Variance swaps on time-changed Lévy processes," Finance and Stochastics, Springer, vol. 16(2), pages 335-355, April.
    17. M.E. Mancino & S. Scotti & G. Toscano, 2020. "Is the Variance Swap Rate Affine in the Spot Variance? Evidence from S&P500 Data," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(4), pages 288-316, July.
    18. Subhojit Biswas & Diganta Mukherjee & Indranil SenGupta, 2020. "Multi-asset Generalised Variance Swaps in Barndorff-Nielsen and Shephard model," Papers 2011.13474, arXiv.org.
    19. Nicolas Merener, 2012. "Swap rate variance swaps," Quantitative Finance, Taylor & Francis Journals, vol. 12(2), pages 249-261, May.
    20. Martin Keller-Ressel & Johannes Muhle-Karbe, 2013. "Asymptotic and exact pricing of options on variance," Finance and Stochastics, Springer, vol. 17(1), pages 107-133, 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:gam:jrisks:v:11:y:2023:i:9:p:162-:d:1236051. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.