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Two Stochastic Volatility Processes - American Option Pricing

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Abstract

In this paper we consider the pricing of an American call option whose underlying asset dynamics evolve under the influence of two independent stochastic volatility processes of the Heston (1993) type. We derive the associated partial differential equation (PDE) of the option price using hedging arguments and Ito's lemma. An integral expression for the general solution of the PDE is presented by using Duhamel's principle and this is expressed in terms of the joint transition density function for the driving stochastic processes. We solve the Kolmogorov PDE for the joint transition density function by first transforming it to a corresponding system of characteristic PDEs using a combination of Fourier and Laplace transforms. The characteristic PDE system is solved by using the method of characteristics. With the full price representation in place, numerical results are presented by first approximating the early exercise surface with a bivariate log linear function. We perform numerical comparisons with results generated by the method of lines algorithm and note that our approach is very competitive in terms of accuracy.

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  • Carl Chiarella & Jonathan Ziveyi, 2011. "Two Stochastic Volatility Processes - American Option Pricing," Research Paper Series 292, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:292
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    Cited by:

    1. Ziveyi, Jonathan & Blackburn, Craig & Sherris, Michael, 2013. "Pricing European options on deferred annuities," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 300-311.
    2. Blessing Taruvinga & Boda Kang & Christina Sklibosios Nikitopoulos, 2018. "Pricing American Options with Jumps in Asset and Volatility," Research Paper Series 394, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Belssing Taruvinga, 2019. "Solving Selected Problems on American Option Pricing with the Method of Lines," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2019, January-A.

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    More about this item

    Keywords

    American options; Fourier transform; Laplace transform; method of characteristics;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory

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