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Pricing Derivatives in a Regime Switching Market with Time Inhomogeneous Volatility

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  • Milan Kumar Das
  • Anindya Goswami
  • Tanmay S. Patankar

Abstract

This paper studies pricing derivatives in an age-dependent semi-Markov modulated market. We consider a financial market where the asset price dynamics follow a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local B-S-M type PDE. We establish existence and uniqueness of a classical solution of the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. Finally, we analyze the PDE to establish continuous dependence of the solution on the instantaneous transition rates of semi-Markov processes. An explicit expression of quadratic residual risk is also obtained.

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  • Milan Kumar Das & Anindya Goswami & Tanmay S. Patankar, 2016. "Pricing Derivatives in a Regime Switching Market with Time Inhomogeneous Volatility," Papers 1611.02026, arXiv.org.
    • Handle: RePEc:arx:papers:1611.02026
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    References listed on IDEAS

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    1. Tanmay S. Patankar, 2016. "Asset Pricing in a Semi-Markov Modulated Market with Time-dependent Volatility," Papers 1609.04907, arXiv.org.
    2. Robert J. Elliott & Leunglung Chan & Tak Kuen Siu, 2005. "Option pricing and Esscher transform under regime switching," Annals of Finance, Springer, vol. 1(4), pages 423-432, October.
    3. Anindya Goswami & Jeeten Patel & Poorva Shevgaonkar, 2015. "A system of non-local parabolic PDE and application to option pricing," Papers 1506.01467, arXiv.org, revised May 2016.
    4. John Buffington & Robert J. Elliott, 2002. "American Options With Regime Switching," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(05), pages 497-514.
    5. Anindya Goswami & Ravi Kant Saini, 2014. "Volterra equation for pricing and hedging in a regime switching market," Cogent Economics & Finance, Taylor & Francis Journals, vol. 2(1), pages 1-11, December.
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    Cited by:

    1. Milan Kumar Das & Anindya Goswami, 2019. "Testing of binary regime switching models using squeeze duration analysis," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(01), pages 1-20, March.
    2. Milan Kumar Das & Anindya Goswami & Sharan Rajani, 2019. "Inference of Binary Regime Models with Jump Discontinuities," Papers 1910.10606, arXiv.org, revised Mar 2022.
    3. Milan Kumar Das & Anindya Goswami, 2018. "Testing of Binary Regime Switching Models using Squeeze Duration Analysis," Papers 1807.04393, arXiv.org, revised Aug 2018.
    4. Anindya Goswami & Omkar Manjarekar & Anjana R, 2018. "Option Pricing in a Regime Switching Jump Diffusion Model," Papers 1811.11379, arXiv.org, revised Oct 2019.
    5. Anindya Goswami & Kedar Nath Mukherjee & Irvine Homi Patalwala & Sanjay N. S, 2022. "Regime recovery using implied volatility in Markov modulated market model," Papers 2201.10304, arXiv.org, revised Mar 2022.

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