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The Evaluation of American Compound Option Prices Under Stochastic Volatility Using the Sparse Grid Approach

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Abstract

A compound option (the mother option) gives the holder the right, but not obligation to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we demonstrate a partial differential equation (PDE) approach to pricing American-type compound options where the underlying dynamics follow Heston’s stochastic volatility model. This price is formulated as the solution to a two-pass free boundary PDE problem. A modified sparse grid approach is implemented to solve the PDEs, which is shown to be accurate and efficient compared with the results from Monte Carlo simulation combined with the Method of Lines.

Suggested Citation

  • Carl Chiarella & Boda Kang, 2009. "The Evaluation of American Compound Option Prices Under Stochastic Volatility Using the Sparse Grid Approach," Research Paper Series 245, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:245
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-02/QFR-rp245.pdf
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    1. Thomas Adolfsson & Carl Chiarella & Andrew Ziogas & Jonathan Ziveyi, 2013. "Representation and Numerical Approximation of American Option Prices under Heston Stochastic Volatility Dynamics," Research Paper Series 327, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. 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.
    3. Geske, Robert, 1979. "The valuation of compound options," Journal of Financial Economics, Elsevier, vol. 7(1), pages 63-81, March.
    4. Geske, Robert & Johnson, Herb E, 1984. "The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-1524, December.
    5. Carl Chiarella & Boda Kang & Gunter H. Meyer & Andrew Ziogas, 2009. "The Evaluation Of American Option Prices Under Stochastic Volatility And Jump-Diffusion Dynamics Using The Method Of Lines," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 393-425.
    6. 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.
    7. Brenner, Menachem & Ou, Ernest Y. & Zhang, Jin E., 2006. "Hedging volatility risk," Journal of Banking & Finance, Elsevier, vol. 30(3), pages 811-821, March.
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    Citations

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    Cited by:

    1. Ming-Chi Chang & Yuan-Chung Sheu & Ming-Yao Tsai, 2015. "Pricing Perpetual American Compound Options under a Matrix-Exponential Jump-Diffusion Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(6), pages 553-575, December.
    2. Andrey Itkin, 2015. "LSV models with stochastic interest rates and correlated jumps," Papers 1511.01460, arXiv.org, revised Nov 2016.
    3. 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.
    4. Ingo Beyna, 2013. "Interest Rate Derivatives," Lecture Notes in Economics and Mathematical Systems, Springer, edition 127, number 978-3-642-34925-6, October.
    5. Cao, Jiling & Lian, Guanghua & Roslan, Teh Raihana Nazirah, 2016. "Pricing variance swaps under stochastic volatility and stochastic interest rate," Applied Mathematics and Computation, Elsevier, vol. 277(C), pages 72-81.
    6. 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.
    7. Pavel V. Gapeev, 2022. "Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 749-788, June.
    8. Mahayni, Antje & Schoenmakers, John G.M., 2011. "Minimum return guarantees with fund switching rights—An optimal stopping problem," Journal of Economic Dynamics and Control, Elsevier, vol. 35(11), pages 1880-1897.
    9. Pavel V. Gapeev & Peter M. Kort & Maria N. Lavrutich & Jacco J. J. Thijssen, 2022. "Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 789-813, June.
    10. Susanne Griebsch, 2013. "The evaluation of European compound option prices under stochastic volatility using Fourier transform techniques," Review of Derivatives Research, Springer, vol. 16(2), pages 135-165, July.
    11. Carl Chiarella & Boda Kang & Gunter H. Meyer, 2010. "The Evaluation Of Barrier Option Prices Under Stochastic Volatility," Research Paper Series 266, Quantitative Finance Research Centre, University of Technology, Sydney.
    12. Wang, Xiandong & He, Jianmin, 2017. "A simple method for generalized sequential compound options pricing," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 85-91.

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    More about this item

    Keywords

    American compound option; stochastic volatility; free boundary problem; sparse grid; combination technique; Monte Carlo simulation; method of lines;
    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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