We explore theoretical and computational issues for the problem of early exercise (or American) options in the classical continuous time Black-Scholes framework.

The early exercise boundary is derived in three distinct ways:

1. solution of a special saltus problem (Kolodner-1956);
2. Fourier transform method (McKean-1965); and
3. solution of an equivalent inhomogeneous pde (Jamshidian-1990).

The behaviour of the free-boundary at expiry is discussed. It is known that in the zero dividend yield case, it has sqrt(t|log t|) behaviour. We report on what is known about this limit when the dividend yield is non-zero.

We develop a numerical algorithm to solve the integral equation for the early exercise boundary. This algorithm represents the boundary as a cubic spline in the variable sqrt(t). While this is not the precise behaviour near t = 0 (expiry), it at least has a singularity in slope there. Since the early exercise boundary is always smooth and monotonic, only a few spline knots are needed for high-order accuracy, even for long term (>5yrs) options.

We first apply the algorithm to a toy Stefan-type problem with a known exact solution in order to test its accuracy. We then apply it to american options and compare with finite difference, binomial tree, Method of Lines and published semi-analytical algorithms.

We dispute the claim made in ref.[4] below that the early exercise boundary can be determined exactly by a certain closed-form implicit functional equation.

References

• Kolodner, I. (1956), Free boundary problems for the heat equation with applications to problems of change of phase, Comm. on Pure and Applied Maths., {9}, 1-31.
• McKean, H.P. (1965), A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, {6}, 32-39.
• Jamshidian, F. (1990), An analysis of American options, Merrill Lynch Capital Markets, working paper.
• Barone-Adesi, G. & Elliott, R. (1991), Approximations for the value of American options, Stochastic Anal. & Applics., {9}, 115-131.

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