Pricing Perpetual American Put Options with Asset-Dependent Discounting

Jonas Al-Hadad,Zbigniew Palmowski

doi:10.3390/jrfm14030130

Abstract

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.

Highlights

In this paper, we consider a perpetual American put option with asset-dependent discounting
The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting with the value function defined in (1) and the asset price process St given in (2)
We have presented a novel approach to pricing the perpetual American put options with asset-dependent discounting

Summary

Introduction

We consider a perpetual American put option with asset-dependent discounting. We consider a standard stochastic background for this problem, i.e., we define a complete filtered risk-neutral probability space (Ω, F , {Ft }t≥0 , P), on which we define the asset price process St. Ft is a natural filtration of St satisfying the usual conditions and P is a risk-neutral measure under which the discounted (with respect to the risk-free interest rate r > 0) asset price process e−rt St is a local martingale. A family of Ft -stopping times is denoted by T while Es denotes the expectation with respect to P when S0 = s = e x. The value function of the perpetual American put option with asset-dependent discounting can be represented by h Rτ i.

Objectives

Methods

Conclusion