Applications in Finance: American Put Options

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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq % aHepaDaSqabaaaaa!37D4! $$\sqrt \tau $$ to o(ιM) 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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq % aHepaDaSqabaaaaa!37D4! $$\sqrt \tau $$ to o(ι48) 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