Bubbles, convexity and the Black--Scholes equation

TL;DR

This paper investigates the effects of asset bubbles on the Black--Scholes equation, establishing existence, uniqueness, and convexity properties of option prices in markets with strict local martingale assets.

Contribution

It provides new theoretical results on the Black--Scholes equation and convexity properties in markets with bubbles, where standard assumptions do not hold.

Findings

Existence and uniqueness of solutions to the Black--Scholes equation in bubble markets

American options preserve convexity, European options preserve concavity for general payoffs

Convexity of European options holds only for bounded payoffs

Abstract

A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black--Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general payoffs and convexity only for bounded contracts.