An Application of Mellin Transform Techniques to a Black–scholes Equation Problem

Abstract

In this article, we use a Mellin transform approach to prove the existence and uniqueness of the price of a European option under the framework of a Black-Scholes model with time-dependent coefficients. The formal solution is rigorously shown to be a classical solution under quite general European contingent claims. Specifically, these include claims that are bounded and continuous, and claims whose difference with some given but arbitrary polynomial is bounded and continuous. We derive a maximum principle and use it to prove uniqueness of the option price. An extension of the put-call parity which relates the aforementioned two classes of claims is also given.

Approximate ordinary differential equations for the optimal exercise boundaries of American put and call options

“…In particular, it was shown in [26] that the Mellin transform of the Black-Scholes kernel K (·, t, u) with respect to x iŝ In particular, it was shown in [26] that the Mellin transform of the Black-Scholes kernel K (·, t, u) with respect to x iŝ…”

“…It was rigorously shown in [26] that if φ is continuous and bounded, then (2.10) is the unique classical solution of (2.9). It was rigorously shown in [26] that if φ is continuous and bounded, then (2.10) is the unique classical solution of (2.9).…”

“…[26], where an exact valuation formula for a European option with a general payoff was obtained using the Mellin transform). We give an alternative derivation of valuation formulas for the American put and call option prices that depend on their respective optimal exercise boundaries.…”

“…For the convenience of the reader, we give the formal derivation of the valuation formula for a European option with a general payoff given in [26]. In Section 2 we derive some properties of the Black-Scholes kernel (to be defined later) and recall some properties of the Mellin transform.…”

A New Representation of the Local Volatility Surface

“…[17,20]). In particular, if q, r, and σ are constants, then z 1 = d 1 , z 2 = d 2 and we recover the BSM formula (2.1)-(2.3).…”