A Mellin transform approach to barrier option pricing

Abstract

A barrier option is an exotic path-dependent option contract that, depending on terms, automatically expires or can be exercised only if the underlying asset ever reaches a predetermined barrier price. Using a partial differential equation approach, we provide an integral representation of the barrier option price via the Mellin transform. In the case of knock-out barrier options, we obtain a decomposition of the barrier option price into the corresponding European option value minus a barrier premium. The integral representation formula can be expressed in terms of the solution to a system of coupled Volterra integral equations of the first kind. Moreover, we suggest some possible numerical approaches to the problem of barrier option pricing.

A Unified Analytical Approach to Fixed and Moving Boundary Problems for the Heat Equation

“…where g ± = g ± (t) and u 0 = u 0 (x) are given. We wish to determine u = u(x, t) and s = s(t) that satisfy (27) and also the interface condition…”

Closed-form pricing formula for foreign equity option with credit risk

“…This transformation replaces (u 1 , u 2 , u 3 ) with (x 1 , x 2 , x 3 ). To solve the C 1 (t, s, v, y), we apply (26) to (25). If we define C 10 (t, s, v, y) as…”

“…Yoon and Kim [22] first used the Mellin transforms to obtain vulnerable European option prices. Recently, many studies showed that the Mellin transforms are useful to solve the PDE for various types of financial derivatives with credit risk (Asian option [23], exchange option [24], path-dependent option [25,26], dynamic fund protection [27] European option with early credit risk [28], lookback option [29]). This paper deals with the valuation of foreign equity option price with credit risk based on the PDE approach and provides a closed-form pricing formula of the options using the Mellin transforms.…”

A Mellin Transform Approach to the Pricing of Options with Default Risk

“…( 2017 ) and Guardasoni et al. ( 2020 ). In this study, we first employ singular perturbation technique to obtain PDEs for the vulnerable option price and then exploit the double Mellin transform introduced by Krapivsky and Ben-Naim ( 1994 ) to derive an analytic solution for each PDE.…”

“…In general, the Mellin transform technique, if it is plausible for the pricing of a given option, would not require the complexity of the calculation as appeared in the probabilistic approaches. The Mellin transform technique has been used for pricing financial derivatives in Panini and Srivastav (2004), Srivastav (2005), Brychkov (1992), Frontczak andSchöbel (2010), Frontczak (2013), Jeon et al (2016), Jeon et al (2017) and Guardasoni et al (2020). In this study, we first employ singular perturbation technique to obtain PDEs for the vulnerable option price and then exploit the double Mellin transform introduced by Krapivsky and Ben-Naim (1994) to derive an analytic solution for each PDE.…”