Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T01:59:14.069Z Has data issue: false hasContentIssue false
  • English
  • Français

Essentially exact asymptotic solutions for Asian derivatives

Published online by Cambridge University Press:  10 January 2012

S. SIYANKO
S. SIYANKO*
Affiliation:
University College London, Gower Street, London WC1E 6BT, UK email: sergei.siyanko.10@ucl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we will show how to obtain asymptotic solutions for the problem of pricing Asian options. Under the assumption that the underlying follows geometric Brownian motion, we will derive Taylor expansion series for the fixed and floating strike Asian options. While there will be no analytical formulae for calculating expansion coefficients, we will provide relatively simple algorithms for calculating them. The methodology is particularly effective for the case of continuously sampled fixed-strike Asian calls where it takes only seconds to obtain constants for the Taylor expansion series that can converge beyond 10 significant digits. It is needless to say that we need to calculate Taylor expansion constants only once and the option price would be an analytical expression constructed from a cumulative normal distribution function, an exponential function and finite sums.

Keywords

Asian optionsoption pricingmatched asymptotic expansionsfinancial derivatives
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 3 , June 2012 , pp. 395 - 415
Copyright
Copyright © Cambridge University Press 2012

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]Dewynne, J. N. & Shaw, W. T. (2008) Differential equations and asymptotic solutions for arithmetic Asian option: “Black–Scholes formulae” for Asian rate calls. Eur. J. Appl. Math. 19, 353–391.CrossRefGoogle Scholar
[2]Geman, H. & Yor, M. (1993) Bessel processes, Asian options and perpetuities. Math. Fin. 3, 349–375.CrossRefGoogle Scholar
[3]Henderson, V. & Wojakowski, R. (2001) On the equivalence of floating and fixed-strike. J. Appl. Probab. 39, 391–394.CrossRefGoogle Scholar
[4]Howison, S. D. (2005) Matched asymptotic expansions in financial engineering. J. Eng. Math. 53 (3–4), 385–406.CrossRefGoogle Scholar
[5]Howison, S. D. & Steinberg, M. (2007) A matched asymptotic expansions approach to continuity corrections for discretely sampled options. I. Barrier options. Appl. Math. Fin. 14 (1), 63–89.CrossRefGoogle Scholar
[6]Linetsky, V. (2004) Spectral expansions for Asian (average price) options. Oper. Res. 52 (6), 856–867.CrossRefGoogle Scholar
[7]Rogers, L. & Shi, Z. (1995) The value of an Asian option. J. Appl. Probab. 32, 1077–1088.CrossRefGoogle Scholar
[8]Shaw, W. T. (1998) Modelling Financial Derivatives with Mathematica, Cambridge University Press, Cambridge, UK.Google Scholar
[9]Vecer, J. (2002) A new PDE approach for pricing arithmetic Asian options. J. Comp. Fin. 4, 105–113.CrossRefGoogle Scholar
[10]Wolfram Research, Inc. (2007) Mathematica, Version 6.0, Champaign, IL.Google Scholar
[11]Zhang, J. E. (2001) A semi-analytical method for pricing and hedging continuously sampled arithmetic average rate options, J. Comp. Fin. 5, 59–79.CrossRefGoogle Scholar
[12]Zhang, J. E. (2003) Pricing continuously sampled Asian options with perturbation method, J. Futures Markets 23, 535–560.CrossRefGoogle Scholar
[13]Yor, M. (1992) On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24, 509–531, 1992a.CrossRefGoogle Scholar