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A Monte Carlo Method using PDE Expansions for a Diversifed Equity Index Model

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Abstract

This paper considers a new class of Monte Carlo methods that are combined with PDE expansions for the pricing and hedging of derivative securities for multidimensional diffusion models. The proposed method combines the advantages of both PDE and Monte Carlo methods and can be directly applied to models with more than two state variables. The pricing procedure is illustrated using a three-component index model that captures some of the key features of a diversified stock index over long time periods. The method is widely applicable and is demonstrated here in the general setting of the benchmark approach, where spatial boundary limiting conditions for the PDE need to be appropriately chosen and approximated. The PDE expansion is based on a Taylor series approximation for the underlying three-component PDE. A Monte Carlo method with variance reduction is then formulated to approximate the true solution. Almost exact simulation schemes are described for the given state variables in the model. Numerical results are presented that demonstrate the effectiveness and tractability of the proposed pricing and hedging methodology.

Suggested Citation

  • David Heath & Eckhard Platen, 2014. "A Monte Carlo Method using PDE Expansions for a Diversifed Equity Index Model," Research Paper Series 350, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:350
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    File URL: https://www.uts.edu.au/sites/default/files/rp350.pdf
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    References listed on IDEAS

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    1. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    2. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    3. Platen, Eckhard, 2000. "A minimal financial market model," SFB 373 Discussion Papers 2000,91, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    4. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    5. David Heath & Eckhard Platen, 2002. "A variance reduction technique based on integral representations," Quantitative Finance, Taylor & Francis Journals, vol. 2(5), pages 362-369.
    6. Eckhard Platen & Renata Rendek, 2012. "The Affine Nature of Aggregate Wealth Dynamics," Research Paper Series 322, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Boyle, Phelim P., 1977. "Options: A Monte Carlo approach," Journal of Financial Economics, Elsevier, vol. 4(3), pages 323-338, May.
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    Cited by:

    1. Coskun Sema & Korn Ralf, 2018. "Pricing barrier options in the Heston model using the Heath–Platen estimator," Monte Carlo Methods and Applications, De Gruyter, vol. 24(1), pages 29-41, March.

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    More about this item

    Keywords

    Multi-factor diffusion; Monte Carlo methods; diversified equity index; pricing PDE; exact simulation; variance reduction; benchmark approach;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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