On constructive complex analysis in finance: Explicit formulas for Asian options
Author:
Michael Schröder
Journal:
Quart. Appl. Math. 66 (2008), 633-658
MSC (2000):
Primary 33Cxx, 91B28; Secondary 33F05, 41A58, 41A60, 44A10
DOI:
https://doi.org/10.1090/S0033-569X-08-01107-X
Published electronically:
October 7, 2008
MathSciNet review:
2465139
Full-text PDF Free Access
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Abstract: This paper develops a two-stage approach to the explicit valuation of Asian options which stresses the rôle of complex analytic techniques in general, and special functions in particular. First, integral representations for their Black-Scholes values in terms of Hermite functions are obtained; this is based on a Laplace transform arising in the work of Yor, and proceeds by analytical inversion. Explicit formulas for these integrals are then derived as a second step; this is done by combining series and asymptotic expansions which have as terms at worst special functions such as the error integral. Numerical examples then illustrate how the results enable a benchmark valuation of these options.
References
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
- Richard Beals, Advanced mathematical analysis. Periodic functions and distributions, complex analysis, Laplace transform and applications, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, No. 12. MR 0530403
- P. Barrieu, A. Rouault, and M. Yor, A study of the Hartman-Watson distribution motivated by numerical problems related to the pricing of Asian options, J. Appl. Probab. 41 (2004), no. 4, 1049–1058. MR 2122799, DOI https://doi.org/10.1017/s0021900200020817
- P. Carr, Personal communication, November 1999.
- P. Carr and M. Schröder, On the valuation of arithmetic-average Asian options: The Geman-Yor Laplace transform revisited, manuscript 24pp., 2000.
- P. Carr and M. Schröder, Bessel processes, the integral of geometric Brownian motion, and Asian options, Teor. Veroyatnost. i Primenen. 48 (2003), no. 3, 503–533 (English, with Russian summary); English transl., Theory Probab. Appl. 48 (2004), no. 3, 400–425. MR 2141348, DOI https://doi.org/10.1137/S0040585X97980543
- Gustav Doetsch, Handbuch der Laplace-Transformation. Band I: Theorie der Laplace-Transformation, Birkhäuser Verlag, Basel-Stuttgart, 1971 (German). Verbesserter Nachdruck der ersten Auflage 1950; Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, Band 14. MR 0344807
- Daniel Dufresne, Laguerre series for Asian and other options, Math. Finance 10 (2000), no. 4, 407–428. MR 1785163, DOI https://doi.org/10.1111/1467-9965.00101
- Daniel Dufresne, The integral of geometric Brownian motion, Adv. in Appl. Probab. 33 (2001), no. 1, 223–241. MR 1825324, DOI https://doi.org/10.1239/aap/999187905
- Michèle Breton and Hatem Ben-Ameur (eds.), Numerical methods in finance, GERAD 25th Anniversary Series, vol. 9, Springer, New York, 2005. MR 2180792
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- H. Geman and M. Yor, Bessel processes, Asian options, and perpetuities, Math. Finance 3 (1993), 349–375.
- Damien Lamberton and Bernard Lapeyre, Introduction to stochastic calculus applied to finance, Chapman & Hall, London, 1996. Translated from the 1991 French original by Nicolas Rabeau and François Mantion. MR 1422250
- N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
- Vadim Linetsky, Spectral expansions for Asian (average price) options, Oper. Res. 52 (2004), no. 6, 856–867. MR 2104142, DOI https://doi.org/10.1287/opre.1040.0113
- A. Lipton-Lifschitz, Similarities via self-similarities, RISK (September 1999), 101–105.
- Henry P. McKean Jr., Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc. 82 (1956), 519–548. MR 87012, DOI https://doi.org/10.1090/S0002-9947-1956-0087012-3
- F. W. J. Olver, Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders, J. Res. Nat. Bur. Standards Sect. B 63B (1959), 131–169. MR 109898
- F. W. J. Olver, Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms, SIAM J. Math. Anal. 22 (1991), no. 5, 1475–1489. MR 1112521, DOI https://doi.org/10.1137/0522095
- Frank W. J. Olver, Asymptotics and special functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR 1429619
- L. C. G. Rogers and Z. Shi, The value of an Asian option, J. Appl. Probab. 32 (1995), no. 4, 1077–1088. MR 1363350, DOI https://doi.org/10.1017/s0021900200103559
- M. Schröder, Mathematical ramifications of option valuation: The case of the Asian option, Habilitationsschrift, Universität Mannheim, April 2002.
- Michael Schröder, On the integral of geometric Brownian motion, Adv. in Appl. Probab. 35 (2003), no. 1, 159–183. In honor of Joseph Mecke. MR 1975509, DOI https://doi.org/10.1239/aap/1046366104
- Michael Schröder, Laguerre series in contingent claim valuation, with applications to Asian options, Math. Finance 15 (2005), no. 3, 491–531. MR 2147159, DOI https://doi.org/10.1111/j.1467-9965.2005.00230.x
- M. Schröder, A constructive Hartman-Watson approach to stochastic functionals, with applications to finance, 2005.
- Michael Schröder, On ladder height densities and Laguerre series in the study of stochastic functionals. II. Exponential functionals of Brownian motion and Asian option values, Adv. in Appl. Probab. 38 (2006), no. 4, 995–1027. MR 2285691, DOI https://doi.org/10.1239/aap/1165414589
- J. Michael Steele, Stochastic calculus and financial applications, Applications of Mathematics (New York), vol. 45, Springer-Verlag, New York, 2001. MR 1783083
- Nico M. Temme, Numerical and asymptotic aspects of parabolic cylinder functions, J. Comput. Appl. Math. 121 (2000), no. 1-2, 221–246. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780050, DOI https://doi.org/10.1016/S0377-0427%2800%2900347-2
- Nico M. Temme and Raimundas Vidunas, Parabolic cylinder functions: examples of error bounds for asymptotic expansions, Anal. Appl. (Singap.) 1 (2003), no. 3, 265–288. MR 1993339, DOI https://doi.org/10.1142/S0219530503000168
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
- E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Cambridge University Press, New York, 1962. Fourth edition. Reprinted. MR 0178117
- Marc Yor, Loi de l’indice du lacet brownien, et distribution de Hartman-Watson, Z. Wahrsch. Verw. Gebiete 53 (1980), no. 1, 71–95 (French). MR 576898, DOI https://doi.org/10.1007/BF00531612
- Marc Yor, On some exponential functionals of Brownian motion, Adv. in Appl. Probab. 24 (1992), no. 3, 509–531. MR 1174378, DOI https://doi.org/10.2307/1427477
- Marc Yor (ed.), Exponential functionals and principal values related to Brownian motion, Biblioteca de la Revista Matemática Iberoamericana. [Library of the Revista Matemática Iberoamericana], Revista Matemática Iberoamericana, Madrid, 1997. A collection of research papers. MR 1648653
- Marc Yor, Exponential functionals of Brownian motion and related processes, Springer Finance, Springer-Verlag, Berlin, 2001. With an introductory chapter by Hélyette Geman; Chapters 1, 3, 4, 8 translated from the French by Stephen S. Wilson. MR 1854494
References
- M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, New York, 1964. MR 0167642 (29:4914)
- R. Beals, Advanced mathematical analysis, Springer, New York, 1973. MR 0530403 (58:26624)
- P. Barrieu, A. Rouault, and M. Yor, A study of the Hartman-Watson distribution motivated by numerical problems related to the pricing of Asian options, J. Appl. Prob. 41 (2003), 1049–1058. MR 2122799 (2006a:60032)
- P. Carr, Personal communication, November 1999.
- P. Carr and M. Schröder, On the valuation of arithmetic-average Asian options: The Geman-Yor Laplace transform revisited, manuscript 24pp., 2000.
- P. Carr and M. Schröder, Bessel processes, the integral of geometric Brownian motion, and Asian options, Theorija verojatnostej i ee primenenija 48, 503–533 (2003); English version in, Theory of probability and its applications 48, 400–425 (2004). MR 2141348 (2006e:60119)
- G. Doetsch, Handbuch der Laplace-Transformation. Band I, Birkhäuser Verlag, Basel, 1971. MR 0344807 (49:9546)
- D. Dufresne, Laguerre series for Asian and other options, Math. Finance 10 (2000), 407–428. MR 1785163 (2001i:91058)
- D. Dufresne, The integral of geometric Brownian motion, Adv. Appl. Prob. 33 (2001), 223–241. MR 1825324 (2002c:60132)
- D. Dufresne, Bessel processes and Asian options, Numerical Methods in Finance (M. Breton and H. Ben-Ameur, eds.), Springer, New York, 2005, pp. 35–57. MR 2180792 (2006e:91005)
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher transcendental functions, R.E. Krieger Publ., Malabar, 1953, reprinted 1981. MR 0058756 (15:419i)
- H. Geman and M. Yor, Bessel processes, Asian options, and perpetuities, Math. Finance 3 (1993), 349–375.
- D. Lamberton and B. Lapeyre, Introduction to stochastic calculus applied to finance, Chapman and Hall, London, 1996. MR 1422250 (98b:90018)
- N.N. Lebedev, Special functions and their applications, Dover Publications, New York, 1971. MR 0350075 (50:2568)
- V. Linetsky, Spectral expansions for Asian options, Oper. Res. 52 (2004), 856–867. MR 2104142 (2005i:60156)
- A. Lipton-Lifschitz, Similarities via self-similarities, RISK (September 1999), 101–105.
- H.P. McKean, Elementary solutions of certain parabolic differential equations, Transactions AMS 82 (1956), 519–548. MR 0087012 (19:285b)
- F.W.J. Olver, Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders, J. Res. Nat. Bur. Standards 63B (1959), 131–169. MR 0109898 (22:781)
- F.W.J. Olver, Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms, SIAM J. Math. Anal. 22 (1991), 1475–1489. MR 1112521 (92g:41035)
- F.W.J. Olver, Asymptotics and special functions, Reprint of the 1974 original, A.K. Peters, Wellesley, MA, 1997. MR 1429619 (97i:41001)
- L.C.G. Rogers and Z. Shi, The value of an Asian option, J. Appl. Prob. 32 (1995), 1077-1088. MR 1363350 (96j:90017)
- M. Schröder, Mathematical ramifications of option valuation: The case of the Asian option, Habilitationsschrift, Universität Mannheim, April 2002.
- M. Schröder, On the integral of geometric Brownian motion, Adv. Appl. Prob. 35 (2003), 159–183. MR 1975509 (2004b:60202)
- M. Schröder, Laguerre series in contingent claim valuation, with applications to Asian options, Mathematical Finance 15 (2005), 491–531. MR 2147159 (2006h:91084)
- M. Schröder, A constructive Hartman-Watson approach to stochastic functionals, with applications to finance, 2005.
- M. Schröder, On ladder height densities and Laguerre series in the study of stochastic functionals. II: Exponential functionals of Brownian motion and Asian option values, Adv. Appl. Prob. 38 (2006), 995–1027. MR 2285691
- J.M. Steele, Stochastic calculus and financial applications, Springer, New York, 2000. MR 1783083 (2001i:60080)
- N. Temme, Numerical and asymptotic aspects of parabolic cylinder functions, J. Comput. Appl. Math. 121 (2000), 221–246. MR 1780050 (2002a:33007)
- R. Vidunas and N. Temme, Parabolic cylinder functions: Examples of error bounds for asymptotic expansions, CWI Report MAS-R0225, 2002. MR 1993339 (2004e:41040)
- G.N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 2nd rev ed. 1944, reprinted 1995. MR 0010746 (6:64a)
- E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge University Press, Cambridge, 1927; 4th ed. 1962. MR 0178117 (31:2375)
- M. Yor, Loi de l’indice du lacet Brownien, et distribution de Hartman-Watson, Z. Wahrscheinlichkeitstheorie 53 (1980), 71–95. MR 576898 (82a:60120)
- M. Yor, On some exponential functionals of Brownian motion, Adv. Appl. Prob. 24 (1992), 509–531. MR 1174378 (94b:60095)
- M. Yor, Exponential functionals and principal values related to Brownian motion, Biblioteca de la revista mathemática iberoamericana., Madrid, 1997. MR 1648653 (99d:60003)
- M. Yor, Exponential functionals of Brownian motion and related processes, Springer, Heidelberg, 2001. MR 1854494 (2003f:60147)
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Additional Information
Michael Schröder
Affiliation:
Keplerstraße 30, D-69469 Weinheim (Bergstraße), Germany
Keywords:
Complex analytic methods and special functions in finance,
Asian options,
asymptotic expansions and series for Asian option valuation,
parabolic cylinder functions,
hypergeometric and confluent hypergeometric functions
Received by editor(s):
March 1, 2007
Published electronically:
October 7, 2008
Article copyright:
© Copyright 2008
Michael Schröder
