Articlegreen2005

THE BLACK‐SCHOLES EQUATION REVISITED: ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONS

Martin Widdicks · Peter W. Duck · Ari D. Andricopoulos · David Newton

49 citations0/0 files storedMathematical Finance

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Stochastic processes and financial applications Financial Markets and Investment Strategies Differential Equations and Numerical Methods

Abstract

In this paper, novel singular perturbation techniques are applied to price European, American, and barrier options. Employment of these methods leads to a significant simplification of the problem in all cases, by reducing the number of parameters. For American options, the valuation problem is reduced to a procedure that may be performed on a rudimentary handheld calculator. The method also sheds light on the evolution of option prices for all of the cases considered, the results being particularly illuminating for American and barrier options.

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References

1973The Pricing of Options and Corporate Liabilities 1997An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs 2002American options on assets with dividends near expiry 1998Optimal exercise boundary for an American put option 1999THE EARLY EXERCISE BOUNDARY FOR THE AMERICAN PUT NEAR EXPIRY: NUMERICAL APPROXIMATION

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