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A 2 Dimensional Pde For Discrete Asian Options

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  • Eric Benhamou

    (London School of Economics)

  • Alexandre Duguet

    (London School of Economics)

Abstract

This paper presents a new method for the pricing of discrete Asian options when assuming a deterministic volatility as specified in Dupire (1993). Using a homogeneity property, we show how to reduce an n+1 dimensional problem to a 2 dimensional one. Previous research has been intensively focussing on continuous time Asian options using Black Scholes assumptions. However, traded Asian options are based on a discrete time sampling and can exhibit a pronounced volatility smile. Previous works which have tried to find approached closed forms have the major drawback to be not extendable to more complex volatility model, like the Dupire model, as well as to American type options: Vorst (1992), Geman and Yor (1993), Turnbull and Wakeman (1991), Levy (1992), Jacques (1995), Zhang (1997) and Milevsky and Posner (1998). Works which have focussed at numerical methods do not take into account volatility smile and focus at continuous Asian options: Kemma and Vorst (1990), Hull and White (1993), Caverhill and Clewlow (1990), Benhamou (1999), Roger and Shi (1995), He and Takahashi (1996), Alziary et al. (1997) and Zvan et al. (1998). Our paper oñers a solution to the pricing of discrete Asian options with volatility smile. As suggested by Dupire (1993), we assume a volatility function of time and the underlying so as to take the volatility smile into account. Assuming furthermore that our volatility function can be written as a function of the normalized underlying, we prove that the option price is homogeneous in the underlying price and the strike. By means of this property, we show how to reduce the numerical problem of a discrete Asian options with n fixing dates, which normally should be computed as an n +1 dimensional problem (n fixings and the time) to a 2 dimensional problem. This is of considerable interest for the efficient computation of discrete Asian options. This generalizes to discrete Asian options the dimension reduction found for continuous Asian options by Rogers and Shi (1995). We derive a PDE for the computation of the Asian option and solve it with the standard Crank Nicholson method. Because of the dimension reduction, we need to interpolate our conditional price at each fixing dates. Diñerent methods of interpolation are examined. The rest of the article focuses at numerical specification of our finite difference (grid boundaries, time and space steps) as well as an extension to the case of non proportional discrete dividends, using a jump condition. We compare our result with a Quasi Monte Carlo simulation based on Sobol sequences. Our simulations show us interesting results about the delta hedge and its jump over fixing dates.

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Bibliographic Info

Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 2000 with number 33.

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Date of creation: 05 Jul 2000
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Handle: RePEc:sce:scecf0:33

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Postal: CEF 2000, Departament d'Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fargas, 25,27, 08005, Barcelona, Spain
Fax: +34 93 542 17 46
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Web page: http://enginy.upf.es/SCE/
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  1. Marc Potters & Rama Cont & Jean-Philippe Bouchaud, 1996. "Financial markets as adaptative systems," Science & Finance (CFM) working paper archive 500037, Science & Finance, Capital Fund Management.
  2. Vorst, Ton, 1992. "Prices and hedge ratios of average exchange rate options," International Review of Financial Analysis, Elsevier, vol. 1(3), pages 179-193.
  3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
  4. Alziary, Benedicte & Decamps, Jean-Paul & Koehl, Pierre-Francois, 1997. "A P.D.E. approach to Asian options: analytical and numerical evidence," Journal of Banking & Finance, Elsevier, vol. 21(5), pages 613-640, May.
  5. Peter A. Abken & Dilip B. Madan & Sailesh Ramamurtie, 1996. "Estimation of risk-neutral and statistical densities by Hermite polynomial approximation: with an application to Eurodollar futures options," Working Paper 96-5, Federal Reserve Bank of Atlanta.
  6. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
  7. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
  8. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
  9. 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 637-54, May-June.
  10. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
  11. 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.
  12. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-51, July.
  13. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
  14. Turnbull, Stuart M. & Wakeman, Lee Macdonald, 1991. "A Quick Algorithm for Pricing European Average Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(03), pages 377-389, September.
  15. Knut K. Aase, 1993. "A Jump/Diffusion Consumption-Based Capital Asset Pricing Model and the Equity Premium Puzzle," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 65-84.
  16. Kemna, A. G. Z. & Vorst, A. C. F., 1990. "A pricing method for options based on average asset values," Journal of Banking & Finance, Elsevier, vol. 14(1), pages 113-129, March.
  17. Jarrow, Robert & Rudd, Andrew, 1982. "Approximate option valuation for arbitrary stochastic processes," Journal of Financial Economics, Elsevier, vol. 10(3), pages 347-369, November.
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