Skip to main content

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

  • J. N. DEWYNNE (a1) and W. T. SHAW (a2)

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.

Hide All
[1]Abramowitz, M. & Stegun, I. A. (1970) Handbook of Mathematical Functions, Dover Edition, New York.
[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, 4974.
[3]Geman, H. & Eydeland, A. (1995) Domino effect: Inverting the Laplace transform. RISK Magazine, March 1995.
[4]Geman, H. & Yor, M. (1993) Bessel processes, Asian options and perpetuities. Math. Finance 3, 349375.
[5]Gradshteyn, I. S. & Ryzhik, I. M. (1980) Table of Integrals, Series and Products, Corrected and enlarged edition, Academic Press, New York.
[6]Hinch, E. J. (1991) Perturbation Methods, Cambridge University Press, Cambridge.
[7]Howison, S. D. (2005) Matched asymptotic expansions in financial engineering. J. Engrg. Math. 53, 385406.
[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, 91104.
[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, 6389.
[10]Ingersoll, J. E. (1987) Theory of Financial Decision Making, Rowman & Littlefield Studies in Financial Economics, Rowman and Littlefield, Savage, Maryland, USA.
[11]Lewis, A. (2002) Asian connections. In: Algorithms, Wilmott Magazine, September 2002, pp. 57–63.
[12]Linetsky, V. (2004) Spectral expansions for Asian (average price) options. Oper. Res. 52, 856.
[13]Rogers, L. C. G. & Shi, Z. (1995) The value of an Asian option. J. Appl. Probab. 32, 10771088.
[14]Shaw, W. T. (1998) Modelling Financial Derivatives with Mathematica, Cambridge University Press, Cambridge, UK.
[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, 4974. Working Paper, 2000. URL:,
[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:
[17]Shaw, W. T.Mathematica implementations of Asian option valuation by contour integration, Mathematica Note Book (2007 version). URL:
[18]Shaw, W. T. (2003) Pricing Asian Options by Contour Integration, Including Asymptotic Methods for Low Volatility, Working Paper, Oxford Mathematical Institute. URL:,
[19]Van Dyke, M. (1978) Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, CA.
[20]Vecer, J. (2001) A new PDE approach for pricing arithmetic Asian options. J. Comp. Fin. 4, 105113.
[21]Vecer, J. (2002) Unified Asian pricing. RISK 15 (6), 113116.
[22]Wilmott, P., Dewynne, J. & Howison, S. (1993) Option Pricing, Mathematical Models and Computation, Oxford Financial Press, Oxford.
[23]Wilmott, P., Howison, S. & Dewynne, J. (1995) Mathematics of Financial Derivatives, Cambridge University Press, Cambridge.
[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.
[25]Yor, M. (2001) Exponential Functionals of Brownian Motion and Related Processes, Springer Finance, Berlin.
[26]Zhang, J. E. (2001) A semi-analytical method for pricing and hedging continuously sampled arithmetic average rate options. J. Comp. Fin. 5, 5979.
[27]Zhang, J. E. (2003) Pricing continuously sampled Asian options with perturbation method. J. Futures Mkts 23 (6), 535560.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 24 *
Loading metrics...

Abstract views

Total abstract views: 222 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th March 2018. This data will be updated every 24 hours.