A Canonical Optimal Stopping Problem for American Options
This Demonstration shows a recursive integral method from [1, 2] for approximating the early exercise boundary of American options. A European financial option is an instrument that allows its holder the right to buy or sell an equity at a future maturity date for a fixed price called the "strike price". An American option allows its holder to exercise the contract at any time up to the maturity date, and because of this, it is worth more than the European option by an amount called the "early exercise premium". For the holder of an American put, the early exercise becomes optimal when the underlying asset price falls below a critical boundary , where the intrinsic value of the option becomes greater than its holding value.
According to the Kim method in , the valuation of the American option derives from an integral expression of the early exercise premium as a function of the critical boundary plus the value of the European option. In , Huang, Subrahmanyam and Yu propose the use of a piecewise step function to approximate the critical boundary, assuming that it remains constant within each time subinterval. In order to accelerate the option's value approximation, they apply Richardson extrapolation over three crude option estimates , , , deriving from uniform steps, respectively: . In , Ju proposes to approximate the critical boundary by a piecewise exponential function within each time subinterval, using a closed expression for Kim's integral. In [1, 2], AitSahlia and Lai apply a transformation to approximate the critical boundary by a continuous linear spline (in the canonical scale).
This Demonstration expands the method in [1, 2] by enabling the use of a nonuniform time mesh guided by the regularized incomplete beta function . The uniform mesh derives as a special case when (the two lines coincide). The plot shows the critical boundary approximation, with the grid lines indicating the time subintervals of the adjusted time mesh.
The symbols have the following meanings in this Demonstration:
is the current time;
is the maturity date;
is the stock price at time ;
is the strike price;
is the stock dividend yield;
is the risk-free interest rate;
is the stock volatility;
is the cumulative distribution function of the standard normal distribution;
is the moving free boundary;
is the critical boundary.
and be a non-negative continuous function of time. Consider a contract whose value at time is
where denotes the value of a European put option on at time with strike price and maturity . The critical boundary for the American put option is obtained by solving the "value matching condition":
for for all .
The value of the American put option is then given by .
Subject to the value matching condition, the method in  proposes to numerically approximate the critical asset price at time by a recursive procedure. This method requires solving integral equations, where is the number of time steps.
The method in  proposes to evaluate analytically the integrals, assuming that the remains constant within each time subinterval, instead of employing a numerical technique (e.g. the composite Simpson's rule) to approximate the integrals. The method in  further expands this idea by assuming that is an exponential function within each time subinterval. The method in [1, 2] applies a transformation by introducing the following change of variables:
Under this change of variables, becomes in a new coordinate system, where
and the exponential function in  transforms into a linear function. This transformation allows the use of a continuous linear spline (in the canonical scale) to approximate the critical boundary, while the method in  obtains a discontinuous boundary.
Furthermore, this Demonstration allows the use of a nonuniform time mesh to improve the approximation of the critical boundary, especially near expiry. Following the regularized incomplete beta function, the temporal point of the nonuniform time mesh is obtained by
where , and .
 F. AitSahlia and T. L. Lai, "A Canonical Optimal Stopping Problem for American Options and Its Numerical Solution," Journal of Computational Finance, 3(2), 2000 pp. 33–52. doi:10.21314/JCF.1999.039.
 F. AitSahlia and T. L. Lai, "Exercise Boundaries and Efficient Approximations to American Option Prices and Hedge Parameters," Journal of Computational Finance, 4(4), 2001 pp. 85–104. doi:10.21314/JCF.2001.063.
 I. J. Kim, "The Analytic Valuation of American Options," The Review of Financial Studies, 3(4), 1990 pp. 547–572. www.jstor.org/stable/2962115.
 J. Huang, M. G. Subrahmanyam and G. G. Yu, "Pricing and Hedging American Options: A Recursive Integration Method," The Review of Financial Studies, 9(1), 1996 pp. 277–300. www.jstor.org/stable/2962372.
 N. Ju, "Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function," The Review of Financial Studies, 11(3), 1998 pp. 627–646. www.jstor.org/stable/2646012.