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Polynomial chaos for simulating random volatilities

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  • Pulch, Roland
  • van Emmerich, Cathrin

Abstract

In financial mathematics, the fair price of options can be achieved by solutions of parabolic differential equations. The volatility usually enters the model as a constant parameter. However, since this constant has to be estimated with respect to the underlying market, it makes sense to replace the volatility by an according random variable. Consequently, a differential equation with stochastic input occurs, whose solution determines the fair price in the refined model. Corresponding expected values and variances can be computed approximately via a Monte Carlo method. Alternatively, the generalised polynomial chaos yields an efficient approach for calculating the required data. Based on a parabolic equation modelling the fair price of Asian options, the technique is developed and corresponding numerical simulations are presented.

Suggested Citation

  • Pulch, Roland & van Emmerich, Cathrin, 2009. "Polynomial chaos for simulating random volatilities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(2), pages 245-255.
    • Handle: RePEc:eee:matcom:v:80:y:2009:i:2:p:245-255
    DOI: 10.1016/j.matcom.2009.05.008
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    References listed on IDEAS

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    1. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
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    3. 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.
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    5. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    6. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    Cited by:

    1. Kathrin Hellmuth & Christian Klingenberg, 2022. "Computing Black Scholes with Uncertain Volatility-A Machine Learning Approach," Papers 2202.07378, arXiv.org.
    2. Lin, Y.-T. & Shih, Y.-T. & Chien, C.-S. & Sheng, Q., 2021. "A note on stochastic polynomial chaos expansions for uncertain volatility and Asian option pricing," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    3. Ledermann, Daniel & Alexander, Carol, 2012. "Further properties of random orthogonal matrix simulation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 83(C), pages 56-79.
    4. Pulch, Roland, 2011. "Modelling and simulation of autonomous oscillators with random parameters," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(6), pages 1128-1143.

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