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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.

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[2] M. Fu , D. Madan & T. Wang (1998) Pricing continuous time Asian options: A comparison of Monte Carlo and Laplace transform inversion methods. J. Comp. Fin. 2, 4974.

[4] H. Geman & M. Yor (1993) Bessel processes, Asian options and perpetuities. Math. Finance 3, 349375.

[5] I. S. Gradshteyn & I. M. Ryzhik (1980) Table of Integrals, Series and Products, Corrected and enlarged edition, Academic Press, New York.

[7] S. D. Howison (2005) Matched asymptotic expansions in financial engineering. J. Engrg. Math. 53, 385406.

[8] S. D. Howison (2007) A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 2: Bermudan options. Appl. Math. Finance 14, 91104.

[12] V. Linetsky (2004) Spectral expansions for Asian (average price) options. Oper. Res. 52, 856.

[13] L. C. G. Rogers & Z. Shi (1995) The value of an Asian option. J. Appl. Probab. 32, 10771088.

[20] J. Vecer (2001) A new PDE approach for pricing arithmetic Asian options. J. Comp. Fin. 4, 105113.

[25] M. Yor (2001) Exponential Functionals of Brownian Motion and Related Processes, Springer Finance, Berlin.

[26] J. E. Zhang (2001) A semi-analytical method for pricing and hedging continuously sampled arithmetic average rate options. J. Comp. Fin. 5, 5979.

[27] J. E. Zhang (2003) Pricing continuously sampled Asian options with perturbation method. J. Futures Mkts 23 (6), 535560.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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