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Valuation of European Options under an Uncertain Market Price of Volatility Risk

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  • Bartosz Jaroszkowski
  • Max Jensen

Abstract

We propose a model to quantify the effect of parameter uncertainty on the option price in the Heston model. More precisely, we present a Hamilton-Jacobi-Bellman framework which allows us to evaluate best and worst case scenarios under an uncertain market price of volatility risk. For the numerical approximation the Hamilton--Jacobi--Bellman equation is reformulated to enable the solution with a finite element method. A case study with butterfly options exhibits how the dependence of Delta on the magnitude of the uncertainty is nonlinear and highly varied across the parameter regime. Keywords: Uncertain market price, Volatility risk, Hamilton-Jacobi-Bellman equation, Finite element method, Uncertainty quantification

Suggested Citation

  • Bartosz Jaroszkowski & Max Jensen, 2021. "Valuation of European Options under an Uncertain Market Price of Volatility Risk," Papers 2105.09581, arXiv.org.
    • Handle: RePEc:arx:papers:2105.09581
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    References listed on IDEAS

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    3. 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.
    4. Gurdip Bakshi & Nikunj Kapadia, 2003. "Delta-Hedged Gains and the Negative Market Volatility Risk Premium," The Review of Financial Studies, Society for Financial Studies, vol. 16(2), pages 527-566.
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    6. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
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    Keywords

    uncertain market price; volatility risk; hamilton-jacobi-bellman equation; finite element method; uncertainty quantification;
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