MELLIN TRANSFORM METHOD FOR THE VALUATION OF AMERICAN POWER PUT OPTION

Abstract

American Power Put Option (APPO) is a financial contract with a nonlinear payoff that can be applied at any time on or before its expiration date and offers flexibility to investors. Analytical approximations and numerical techniques have been proposed for the valuation of Plain American Put Option (PAPO) but there is no known closed-form solution for the price of APPO. Mellin transform is a useful method for dealing with unstable mathematical systems. This study was designed to derive a closed-form solution for APPO by means of the Mellin transform method that enables option equations to be solved directly in terms of market prices and to investigate the efficiency and robustness of the method. The Ito’s lemma under the geometric Brownian motion was used to derive a non-homogeneous Partial Differential Equation (PDE) for the price of APPO. The Mellin transform with its shifting and derivative properties were used to solve the non-homogeneous PDE. The Mellin inversion formula and the value-matching condition were used to recover the integral representations for the price and the free boundary of APPO, respectively. The convolution theorem for the Mellin transform was used to prove the equivalence of the integral representation for the price of APPO, for n = 1. The integral representation was transformed into a form that permits the use of the Gauss-Laguerre quadrature method to obtain the closed-form solution for the price of APPO. By varying the volatility (σ), strike price (K) and time to expiry (T), numerical experiments were performed to compare the results of the Mellin transform method for the price of APPO for n = 1 with accelerated binomial model, binomial model, finite difference and recursive methods. A non-homogeneous Black-Scholes-Merton-like PDE for the price of APPO was obtained. The integral representations for the price and the free boundary of APPO were obtained respectively as: A n p (S n t , t) = 1 2πi Z c+i∞ c−i∞ Kω+1 ω(ω + 1)e 1 2 n 2σ 2 (ω 2+α ∗ 1ω−α2)(T −t) (S n t ) −ω dω + rK 2πi Z c+i∞ c−i∞ Z T t (S n t ) −ω (S¯n y ) ω ω e 1 2 n 2σ 2 (ω 2+α ∗ 1ω−α2)(y−t) dydω − q 2πi Z c+i∞ c−i∞ Z T t (S n t ) −ω (S¯n y ) ω+1 ω + 1 e 1 2 n 2σ 2 (ω 2+α ∗ 1ω−α2)(y−t) dydω and K − S¯n t = 1 2πi Z c+i∞ c−i∞ Kω+1 ω(ω + 1)e 1 2 n 2σ 2 (ω 2+α ∗ 1ω−α2)(T −t) (S¯n t ) −ω dω + rK 2πi Z c+i∞ c−i∞ Z T t (S¯n t ) −ω (S¯n y ) ω ω e 1 2 n 2σ 2 (ω 2+α ∗ 1ω−α2)(y−t) dydω − q 2πi Z c+i∞ c−i∞ Z T t (S¯n t ) −ω (S¯n y ) ω+1 ω + 1 e 1 2 n 2σ 2 (ω 2+α ∗ 1ω−α2)(y−t) dydω for (S n t , t) ∈ {(0,∞) × [0, T]}, c ∈ (0, ∞), {ω ∈ C|0 <