BUBBLES, CONVEXITY AND THE BLACK–SCHOLES EQUATION

Erik Ekström · Johan Tysk

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Stochastic processes and financial applications Nonlinear Partial Differential Equations Economic theories and models

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. 1. Introduction. Recently

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References

1999Continuous Martingales and Brownian Motion 2008Stochastic differential equations and applications 1976Stochastic Differential Equations and Applications 2006Moment explosions in stochastic volatility models 1998Robustness of the Black and Scholes Formula 2010ASSET PRICE BUBBLES IN INCOMPLETE MARKETS 2005Local martingales, bubbles and option prices 2002Option valuation under stochastic volatility: with mathematica code

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