# A Recursive Integration Method for Options Pricing

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.

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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: =(-8+9)/2.

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n=1,2,3

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This Demonstration expands the method in [2] by enabling the use of a nonuniform time mesh guided by the regularized incomplete beta function (a,β). 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 , .

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a=β=1

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