Application of high-precision computing for pricing arithmetic asian options

ABSTRACT

Asian options are notoriously hard to price. Even though they have been the focus of much attention in recent years, there is no single technique which would widely be accepted to price Asian options for all choices of market parameters. In practice, estimation of price sensitivities is often as important as evaluation of the prices themselves since price sensitives are important measures of risk. The main goal of this paper is to demonstrate how high-precision methods can be used to efficiently price and estimate the market sensitivities of an arithmetic Asian option.

REFERENCES

• 1: Abate, J., and Valko, P.P., Multi-precision Laplace transform inversion, Int. J. Numer. Meth. Engng, 60, 2004, pp. 979--993.
• 2: Abate, J., and Whitt, W., Numerical Inversion of Laplace Transforms of Probability Distributions. ORSA Journal on Computing, 7, 1995, pp. 36--43.
• 3: Barrieu, P., Rouault, A., and Yor, M., A study of the Hartman-Watson distribution motivated by numerical problems related tothe pricing of asian options, Journal of Applied Probability, 41, 2004, pp. 1049--1058.
• 4: Michael Curran, Valuing Asian and portfolio options by conditioning on the geometric mean price, Management Science, v.40 n.12, p.1705-1711, Dec. 1994 [doi>10.1287/mnsc.40.12.1705]
• 5: Dufresne, D., The Laguerre Series for Asian and Other Options, Mathematical Finance, 10, 2000, pp. 407--428.
• 6: Davies, B., Integral Transforms and Their Appplications, 3-rd ed., Springer, New York, 2002.
• 7: Erdelyi, A., Higher Transcendental Functions, MacGraw-Hill, New York, 1981.
• 8: Fu, M., Madan, D., and Wang, T., Pricing Continuous Asian Options: A Comparison of Monte Carlo and Laplace Transform Inversion Methods,, Journal of Computational Finance, 2, 1999, pp. 49--74.
• 9: Geman, H., and Eydeland, A., Domino Effect, Risk, 4 (8), 1995, pp. 65--67.
• 10: Geman, H., and Yor, M., Bessel Processes, Asian Options and Perpetuities, Mathematical Finance, 3, 1993, pp. 349--75.
• 11: Y. d'Halluin , P. A. Forsyth , G. Labahn, A Semi-Lagrangian Approach for American Asian Options under Jump Diffusion, SIAM Journal on Scientific Computing, v.27 n.1, p.315-345, 2005 [doi>10.1137/030602630]
• 12: Hoogland, J., and Neumann, D., Local Scale Invariance and Contingent Claim Pricing, International Journal of Theoretical and Applied Science, 4, 2001, pp. 1--21.
• 13: Hartman, P., and Watson, G.S., Normal distribution on spheres and the modified Bessel functions, Annals of Probability, 2, 1974, pp. 593--607.
• 14: Hull, J., and White, A., Efficient Procedures for Valuing European and American Path-Dependent Options, Journal of Derivatives, 1, 1993, 21--31.
• 15: Klassen, T., Simple, fast and flexible pricing of Asian options, Journal of Computational Finance, 4, 2001, 89--124.
• 16: Ju, N., Pricing Asian and basket options via Taylor expansion, Journal of Computational Finance, 5, 2002, pp. 79--103.
• 17: Kemna, A., and Vorst, A., A pricing method for options based on average asset values, Journal of Banking and Finance, 14, 1990, 113--130.
• 18: Levy, E., Pricing European average rate currency options, Journal of Internaltional Money and Finance, 11, pp. 474--491.
• 19: Vadim Linetsky, Spectral Expansions for Asian (Average Price) Options, Operations Research, v.52 n.6, p.856-867, November 2004 [doi>10.1287/opre.1040.0113]
• 20: Milevsky, M., and Posner, S., Asian options, the sum of lognormals, and the reciprocal gamma distribution, Journal of Financial and Quantitative Analysis, 33, 1998, pp. 409--422.
• 21: Milevsky, M., and Posner, S., A closed-form approximation for valuing basket options, Journal of Derivatives, 5, 1998, pp. 54--61.
• 22: Milevsky, M., and Posner, S., Valuing exotic options by approximating the SPD with higher moments, Journal of Financial Engineering, 7, 1998, pp. 109--125.
• 23: Rogers, L., and Shi, Z., The value of an Asian option, Journal of Applied Probability, 32, 1995, pp. 1077--1088.
• 24: William T. Shaw, Modelling Financial Derivatives with Mathematica, Press Syndicate of the University of Cambridge, 1998
• 25: W. Shaw, Pricing Asian Options by Contour Integration, Including Asymptotic Methods for Low Volatility, Working Paper, 2002.
• 26: Shreve, S., Večer, J., Options on a traded account: Vacation calls, vacation puts and passport options, Finance and Stochastics, 2000, pp. 255--274.
• 27: Yi Su , Michael C. Fu, Simulation in financial engineering: importance sampling in derivative securities pricing, Proceedings of the 32nd conference on Winter simulation, December 10-13, 2000, Orlando, Florida
• 28: Thompson, G.W.P., Fast narrow bounds on the value of Asian options, Working paper, Centre for Financial Research, Judge Institute of Management Science, University of Cambridge, 2000.
• 29: Turnbull, S., and Wakeman, L., A quick algorithm for pricing European average options, Journal of Financeal and Quantitative Analysis, 26, 1991, pp. 377--389.
• 30: Valko, P.P., and Abate, J., Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion, Computational Mathematics and Applications, 2004.
• 31: Večer, J., A new PDE approach for pricing arithmetic average Asian optons, Journal of Computational Finance, 4, 2001, pp. 105--113.
• 32: Yor, M., Loi de l'indice du lacet Brownien et distribution de Hartman-Watson, Z. Wahrscheinlichkeitstheorie, 53, 1980, pp. 71--95.
• 33: Yor, M., On some exponential functionals of Brownian motion, Advances in Applied Probability, 24, 1992, pp. 509--531.
• 34: Zhang, J.E., A semi-analytical method for pricing and hedging continuously sampled arithmetic average rate options, Journal of Computational Finance, 5, 2001, pp. 59--79.

INDEX TERMS

Primary Classification:
  H. Information Systems
  H.0 GENERAL

General Terms:
Algorithms, Economics