Valuation and parities for exchange options

TL;DR

This paper develops valuation formulas and parity relations for European and American exchange options within a comprehensive financial model that incorporates jumps, default risk, and asset price bubbles, extending classical results.

Contribution

It introduces generalized valuation and parity formulas for exchange options in a model with jumps, default risk, and bubbles, modifying traditional folklore results.

Findings

Derived valuation formulas using change-of-numeraire technique

Extended parity relations to models with jumps and bubbles

Discussed modifications to Merton's no-early-exercise theorem

Abstract

Valuation and parity formulas for both European-style and American-style exchange options are presented in a general financial model allowing for jumps, possibility of default and "bubbles" in asset prices. The formulas are given via expectations of auxiliary probabilities using the change-of-numeraire technique. Extensive discussion is provided regarding the way that folklore results such as Merton's no-early-exercise theorem and traditional parity relations have to be altered in this more versatile framework.