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# A Recursive Integration Method for Options Pricing

This Demonstration shows a recursive integral method from [2] for pricing 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 [1], 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 [2] 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: .
This Demonstration expands the method in [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 plot shows the critical boundary approach by the step function, while the grid lines indicate the time subintervals. The table shows the American put approximation depending on the time steps, and the time mesh parameters , .

### DETAILS

In this detailed description, the symbols have the following meanings:
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
Let
,
,
and be a non-negative continuous function of time. Consider a contract whose value at time is given by:
,
where denotes the value at time of a European put option on 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 [1] 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 [2] 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 [3] further expands this idea by assuming that is an exponential function within each time subinterval. In this Demonstration, the method in [2] derives as a special case of the method in [3], where the exponential function's exponent is set to zero.
Following the regularized incomplete beta function, the temporal point of the nonuniform time mesh is obtained by , where , and .
References
[1] 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.
[2] 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.
[3] 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.

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