Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T14:30:36.273Z Has data issue: false hasContentIssue false
  • English
  • Français

Differential equations and asymptotic solutions for arithmetic Asian options: ‘Black–Scholes formulae’ for Asian rate calls

Published online by Cambridge University Press:  01 August 2008

J. N. DEWYNNE  and
W. T. SHAW
J. N. DEWYNNE
Affiliation:
OCIAM, The Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK email: [email protected]
W. T. SHAW
Affiliation:
Mathematics Department, King's College, London WC2R 2LS, UK email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we present a simplified means of pricing Asian options using partial differential equations (PDEs). We first provide a concise derivation of the well-known similarity reduction and exact Laplace transform solution. We then analyse the problem afresh as a power series in the volatility-scaled contract duration, with a view to obtaining an asymptotic solution for the low-volatility limit, a limit which presents difficulties in the context of the general Laplace transform solution. The problem is approached anew from the point of view of asymptotic expansions and the results are compared with direct, high precision, inversion of the Laplace transform and with numerical results obtained by V. Linetsky and J. Vecer. Our asymptotic formulae are little more complicated than the standard Black–Scholes formulae and, working to third order in the volatility-scaled expiry, are accurate to at least four significant figures for standard test problems. In the case of zero risk-neutral drift, we have the solution to fifth order and, for practical purposes, the results are effectively exact. We also provide comparisons with the hybrid analytic and finite-difference method of Zhang.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 4 , August 2008 , pp. 353 - 391
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abramowitz, M. & Stegun, I. A. (1970) Handbook of Mathematical Functions, Dover Edition, New York.Google Scholar
[2]Fu, M., Madan, D. & Wang, T. (1998) Pricing continuous time Asian options: A comparison of Monte Carlo and Laplace transform inversion methods. J. Comp. Fin. 2, 49–74.Google Scholar
[3]Geman, H. & Eydeland, A. (1995) Domino effect: Inverting the Laplace transform. RISK Magazine, March 1995.Google Scholar
[4]Geman, H. & Yor, M. (1993) Bessel processes, Asian options and perpetuities. Math. Finance 3, 349–375.CrossRefGoogle Scholar
[5]Gradshteyn, I. S. & Ryzhik, I. M. (1980) Table of Integrals, Series and Products, Corrected and enlarged edition, Academic Press, New York.Google Scholar
[6]Hinch, E. J. (1991) Perturbation Methods, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[7]Howison, S. D. (2005) Matched asymptotic expansions in financial engineering. J. Engrg. Math. 53, 385–406.Google Scholar
[8]Howison, S. D. (2007) A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 2: Bermudan options. Appl. Math. Finance 14, 91–104.CrossRefGoogle Scholar
[9]Howison, S. D. & Steinberg, M. (2007) A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 1: Barrier options. Appl. Math. Finance 14, 63–89.CrossRefGoogle Scholar
[10]Ingersoll, J. E. (1987) Theory of Financial Decision Making, Rowman & Littlefield Studies in Financial Economics, Rowman and Littlefield, Savage, Maryland, USA.Google Scholar
[11]Lewis, A. (2002) Asian connections. In: Algorithms, Wilmott Magazine, September 2002, pp. 57–63.Google Scholar
[12]Linetsky, V. (2004) Spectral expansions for Asian (average price) options. Oper. Res. 52, 856.CrossRefGoogle Scholar
[13]Rogers, L. C. G. & Shi, Z. (1995) The value of an Asian option. J. Appl. Probab. 32, 1077–1088.Google Scholar
[14]Shaw, W. T. (1998) Modelling Financial Derivatives with Mathematica, Cambridge University Press, Cambridge, UK.Google Scholar
[15]Shaw, W. T. (1998) A Reply to “Pricing continuous Asian options: A comparison of Monte Carlo and Laplace transform inversion methods” by Fu, Madan and Wang. J. Comp. Fin. 2, 49–74. Working Paper, 2000. URL: http://www.mth.kcl.ac.uk/~shaww/web_page/papers/JCFReply.pdf, http://www.mth.kcl.ac.uk/~shaww/web_page/papers/JC-FReply.nb.Google Scholar
[16]Shaw, W. T.Transform Calculus, Conformal Mapping and Applications to Mathematical Finance, OCIAM, Oxford (2003) and Judge Business School (2004) presentations (seminar working paper based on work with MacDonald, A. and Dewynne, J.). URL: http://www.mth.kcl.ac.uk/~shaww/web_page/talks/conformal.pdf.Google Scholar
[17]Shaw, W. T.Mathematica implementations of Asian option valuation by contour integration, Mathematica Note Book (2007 version). URL: http://www.mth.kcl.ac.uk/~shaww/web_page/papers/asianmathematica.nb.Google Scholar
[18]Shaw, W. T. (2003) Pricing Asian Options by Contour Integration, Including Asymptotic Methods for Low Volatility, Working Paper, Oxford Mathematical Institute. URL: http://www.mth.kcl.ac.uk/~shaww/web_page/papers/lowvol.pdf, http://www.mth.kcl.ac.uk/~shaww/web_page/papers/lowvol.nb.Google Scholar
[19]Van Dyke, M. (1978) Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, CA.Google Scholar
[20]Vecer, J. (2001) A new PDE approach for pricing arithmetic Asian options. J. Comp. Fin. 4, 105–113.Google Scholar
[21]Vecer, J. (2002) Unified Asian pricing. RISK 15 (6), 113–116.Google Scholar
[22]Wilmott, P., Dewynne, J. & Howison, S. (1993) Option Pricing, Mathematical Models and Computation, Oxford Financial Press, Oxford.Google Scholar
[23]Wilmott, P., Howison, S. & Dewynne, J. (1995) Mathematics of Financial Derivatives, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[24]Wong, E. (1964) The construction of a class of stationary Markoff processes. In: Proceedings of Symposia in Applied Mathematics, Vol. XVI, Stochastic Processes in Mathematical Physics and Engineering, American Mathematical Society, Providence, RI, pp. 265–276.CrossRefGoogle Scholar
[25]Yor, M. (2001) Exponential Functionals of Brownian Motion and Related Processes, Springer Finance, Berlin.Google Scholar
[26]Zhang, J. E. (2001) A semi-analytical method for pricing and hedging continuously sampled arithmetic average rate options. J. Comp. Fin. 5, 59–79.Google Scholar
[27]Zhang, J. E. (2003) Pricing continuously sampled Asian options with perturbation method. J. Futures Mkts 23 (6), 535–560.CrossRefGoogle Scholar