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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
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    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.
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    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.
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    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.
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