# Applications in Finance: American Put Options

## Abstract

The homotopy analysis method (HAM) is successfully combined with the Laplace transform to solve the famous American put option equation in finance. Unlike asymptotic and/or perturbation formulas that are often valid only a couple of days or weeks prior to expiry, our homotopy approximation of the optimal exercise boundary *B*(ι) in polynomials of \(\sqrt \tau \) to *o*(ι^{M})
may be valid a couple of dozen years, or even a half century, as long as *M* is large enough. It is found that the homotopyapproximation of *B*(ι) in polynomial of \(\sqrt \tau \) to *o*(ι^{48}) is often valid in so many years that the well-known theoretical perpetual optimal exercise price is accurate enough thereafter, so that the combination of them can be regarded as an analytic formula valid in the whole time interval 0≤ι<+∞. A practical Mathematica code APOh is provided in the Appendix 13.2 for businessmen to gain accurate enough optimal exercise price of American put option at large expiration-time by a laptop only in a few seconds, which is free available (Accessed 25 Nov 2011, will be updated in the future) at http://numericaltank.sjtu.edu.cn/HAM.htm

## Keywords

Homotopy Analysis Method Exercise Price Optimal Exercise Convertible Bond Perturbation Formula## Preview

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