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Option Pricing with Stochastic Volatility, Equity Premium, and Interest Rates

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  • Nicole Hao
  • Echo Li
  • Diep Luong-Le

Abstract

This paper presents a new model for options pricing. The Black-Scholes-Merton (BSM) model plays an important role in financial options pricing. However, the BSM model assumes that the risk-free interest rate, volatility, and equity premium are constant, which is unrealistic in the real market. To address this, our paper considers the time-varying characteristics of those parameters. Our model integrates elements of the BSM model, the Heston (1993) model for stochastic variance, the Vasicek model (1977) for stochastic interest rates, and the Campbell and Viceira model (1999, 2001) for stochastic equity premium. We derive a linear second-order parabolic PDE and extend our model to encompass fixed-strike Asian options, yielding a new PDE. In the absence of closed-form solutions for any options from our new model, we utilize finite difference methods to approximate prices for European call and up-and-out barrier options, and outline the numerical implementation for fixed-strike Asian call options.

Suggested Citation

  • Nicole Hao & Echo Li & Diep Luong-Le, 2024. "Option Pricing with Stochastic Volatility, Equity Premium, and Interest Rates," Papers 2408.15416, arXiv.org.
    • Handle: RePEc:arx:papers:2408.15416
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    References listed on IDEAS

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    1. John Y. Campbell & Luis M. Viceira, 1999. "Consumption and Portfolio Decisions when Expected Returns are Time Varying," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(2), pages 433-495.
    2. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    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. Lech A. Grzelak & Cornelis W. Oosterlee, 2012. "On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(1), pages 1-35, February.
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