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

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.     

Extending quadrature methods to value multi-asset and complex path dependent options

The exposition of the quadrature (QUAD) method (Andricopoulos, Widdicks, Duck, and Newton, 2003. Universal option valuation using quadrature methods. Journal of Financial Economics 67, 447–471 (see also Corrigendum, Journal of Financial Economics 73, 603 (2004)) is significantly extended to cover notably more complex and difficult problems in option valuations involving one or more underlyings. Trials comparing several techniques in the literature, adapted from standard lattice, grid and Monte Carlo methods to tackle particular types of problem, show that QUAD offers far greater flexibility, superior convergence, and hence, increased accuracy and considerably reduced computational times. The speed advantage of QUAD means that, even under the curse of dimensionality, it is not necessary to resort to Monte Carlo methods (certainly for options involving up to five underlying assets). Given the universality and flexibility of the method, it should be the method of choice for pricing options involving multiple underlying assets, in the presence of many features, such as early exercise or path dependency.     

On the enhanced convergence of standard lattice methods for option pricing

For derivative securities that must be valued by numerical techniques, the trade‐off between accuracy and computation time can be a severe limitation. For standard lattice methods, improvements are achievable by modifying the underlying structure of these lattices; however, convergence usually remains non‐monotonic. In an alternative approach of general application, it is shown how to use standard methods, such as Cox, Ross, and Rubinstein (CRR), trinomial trees, or finite differences, to produce uniformly converging numerical results suitable for straightforward extrapolation. The concept of Λ, a normalized distance between the strike price and the node above, is introduced, which has wide ranging significance. Accuracy is improved enormously with computation times reduced, often by orders of magnitude. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:315–338, 2002