Abstract
The pricing of discretely monitored barrier options is a difficult problem. In general, there is no known closed form solution for pricing such options. A path integral approach to the evaluation of barrier options is developed. This leads to a backward recursion functional equation linking the pricing functions at successive barrier points. This functional equation is solved by expanding the pricing functions in Fourier-Hermite series. The backward recursion functional equation then becomes the backward recurrence relation for the coefficients in the Fourier-Hermite expansion of the pricing functions. A very efficient and accurate method for generating the pricing function at any barrier point is thus obtained. A number of numerical experiments with the method are performed in order to gain some understanding of the nature of convergence. Results for various volatility values and different numbers of basis functions in the Fourier-Hermite expansion are presented. Comparisons are given between pricing of discrete barrier option in the path integral framework and by use of finite difference methods.
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References
Boyle, P. and Tian, Y.: 1997, An explicit finite difference approach to the pricing of barrier options, Working paper, University of Waterloo and University of Cinicinnati.
Broadie, M., Glasserman, P. and Kou, S.: 1997a, Connecting discrete and continuous path-dependent options, Working paper, Columbia Business School and University of Michigan.
Broadie, M., Glasserman, P. and Kou, S.: 1997b, A continuity correction for discrete barrier options, Mathematical Finance 7(4), 328–349.
Chance, D.M.: 1994, The pricing and hedging of limited exercise caps and spreads, Journal of Financial Res. 17, 561–584.
Chiarella, C. and El-Hassan, N.: 1997, Evaluation of derivative security prices in the Heath-Jarrow-Morton framework as path integrals using Fast Fourier Transform techniques, Journal of Financial Engineering 6(2), 121–147.
Chiarella, C., El-Hassan, N. and Kucera, A.: 1999, Evaluation of American option prices in a path integral framework using Fourier-Hermite series expansions, Journal of Economic and Dynamic Control 23, 1387–1424.
Dash, J.: 1988, Path Integrals and Options — Part I, CRNS Preprints.
Eydeland, A.: 1994, A fast algorithm for computing integrals in function spaces: Financial applications, Computational Economics 7(4), 277–285.
Figlewski, S. and Gao, B.: 1997, The adaptive mesh model: A new approach to efficient option pricing, Working Paper S-97-5, Salomon Centre, NYU.
Fusai, G., Abrahams, I.D. and Sgarra, C.: 2006, An exact analytical solution for discrete barrier options, Finance and Stochastic 10, 1–26.
Heynen, R. and Kat, H.M.: 1996, Discrete partial barrier options with a moving barrier, Journal of Financial Engineering 5(3), 199–209.
Kat, H.M.: 1995, Pricing lookback options using binomial trees: An evaluation, Journal of Financial Engineering 4, 375–397.
Levy, E. and Mantion, F.: 1997, Discrete by nature, Risk 10(1), 8–22.
Linetsky, V.: 1997, The path integral approach to financial modeling and option pricing, Computational Economics 11(1), 129–163.
Makici, M.S.: 1995, Path integral Monte Carlo method for valuation of derivative securities: Algorithm and parallel implementation, Technical Report SCCS 650, NPAC.
Merton, R.: 1973, The theory of rational option pricing, Bell Journal of Economics and Management Science 4, 141–183.
Rubinstein, M. and Reiner, E.: 1991, Breaking down the barriers, Risk 4(4), 28–35.
Sullivan, M.A.: 2000, Pricing discretely monitored barrier options, Journal of Computational Finance 3(4), 35–54.
Tavella, D.: 2002, Quantitative Methods in Derivatives Pricing, Wiley Finance.
Tian, Y.: 1996, Breaking the barrier with a two-stage binomial model, Working paper, University of Cincicinnati.
Wei, J.Z.: 1998, Valuation of discrete barrier options by interpolations, Journal of Derivatives 6(1), 51–73.
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Chiarella, C., El-Hassan, N., Kucera, A. (2008). The Evaluation of Discrete Barrier Options in a Path Integral Framework. In: Kontoghiorghes, E.J., Rustem, B., Winker, P. (eds) Computational Methods in Financial Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77958-2_7
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DOI: https://doi.org/10.1007/978-3-540-77958-2_7
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