On approximating deep in‐the‐money Asian options under exponential Lévy processes

Abstract

This note proposes a new approach of valuing deep in-the-money fixed strike and discretely monitoring arithmetic Asian options. This new approach prices Asian options whose underlying asset price evolves according to the exponential of a Lévy process as a weighted sum of European options. Numerical results from experimenting on three different types of Lévy processes-a diffusion process, a jump diffusion process, and a pure jump process-illustrate the accuracy of the approach.

Comment: “On Approximating Deep in‐the‐money Asian Options Under Exponential Lévy Processes”

“…We would like to point out that the reasoning described in Equations (11) and (12) of Tchuindjo () cannot provide the first equality in Equation (13). Invoking the convexity of the function , one can argue that this equality should be a “less than or equal,” providing an upper bound for the approximating price in Equation (10) of an Asian option.…”

“…

The approximation in Tchuindjo (2012) for the value of the in-the-money arithmetic Asian options in the exponential Lévy setting is shown to be an upper bound, which cannot be smaller than the optimal upper bound derived in Albrecher et al (2005). Consequently, some of the results in Table VIII of Tchuindjo (2012) are inaccurate.

…”

“…This is based on the well-established comonotonicity theory, see, e.g., Dhaene et al (2002a), and, in particular for applications to Asian options, Simon et al (2000), Dhaene et al (2002b), Vanmaele et al (2006), Chen et al (2008Chen et al ( , 2015. However, the numerical applications in Tchuindjo (2012) are not consistent with this theoretical conclusion.…”

“…Tchuindjo () proposed an easy method to approximate the value of a deep in‐the‐money Asian option with a weighted sum of the prices of European options matured on the monitoring dates of the Asian option. The strikes of these European options differ from the strikes in Albrecher et al () which are computed in terms of the cumululative distribution function of the underlying asset price at the monitoring dates.…”

“…The superhedging strategy with the strikes proposed in Albrecher et al (2005) provides in the exponential Lévy setting an optimal upper bound for the Asian option which should not be higher than the upper bounds with other strikes such as those in Tchuindjo (2012). This is based on the well-established comonotonicity theory, see, e.g., Dhaene et al (2002a), and, in particular for applications to Asian options, Simon et al (2000), Dhaene et al (2002b), Vanmaele et al (2006), Chen et al (2008Chen et al ( , 2015.…”