Bubbles, convexity and the Black–Scholes equation

By Erik Ekström, Johan Tysk. 2009

62 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/0b57a1be561b7e4bfc6ed61bc60a562a16ba14ae

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Citations in the corpus

On the martingale property of certain local martingales (2009, background): “Bubbles have recently attracted attention in the mathematical finance literature; see e.g. Cox and Hobson [9], Ekström and Tysk [13], Heston, Loewenstein, and Willard [19], Jarrow, Protter, and Shimbo [22], Madan and Yor [32], Pal and Protter [36].”
Distribution of the time to explosion for one-dimensional diffusions (2013, methodology): “…that X o ( · ) is a strict local martingale; and the connection with the Bessel process allows the computation of the distribution of X o ( T ) as in Ekstr¨om and Tysk (2009), namely On the other hand, with the choices of (6.1), the equation of (2.1) becomes whereas the functions of Subsection…”
The Black–Scholes equation in stochastic volatility models (2010, background): “This situation resembles the one with bubbles treated in [8], and uniqueness of solutions may be lost for general contracts.”
Analysis of continuous strict local martingales via h-transforms (2007, result): “Before we end, let us mention that similar results can be obtained from the semimartingale decomposition formulas of Madan and Yor [27] and by PDE methods, as in Ekström and Tysk [8].”
HEDGING UNDER ARBITRAGE (2010, unknown): “In this context, we provide sufficient conditions to ensure the differentiabilityof the hedging price, generalizing results by Heath and Schweizer (2000), Janson and Tysk (2006), and Ekström and Tysk (2009).”
Optimal arbitrage under model uncertainty. (2011, background): “…from the theory of stochastic flows [Kunita (1990), Protter (2004)] and from parabolic partial differential equations [Janson and Tysk (2006), Ekström and Tysk (2009)], Theorem 2 in Ruf (2011) shows that the function Γ(·, ·) is then of class C1,2 locally on (0,∞) × (0,∞)n, and solves there…”