Bubbles, convexity and the Black-Scholes equation

DOI10.1214/08-AAP579zbMath1219.91138arXiv0908.4468MaRDI QIDQ835063

Erik Ekström, Johan Tysk

Publication date: 27 August 2009

Published in: The Annals of Applied Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0908.4468

zbMATH Keywords

parabolic equationsstochastic representationlocal martingalespreservation of convexity

Mathematics Subject Classification ID

Degenerate parabolic equations (35K65) Stopping times; optimal stopping problems; gambling theory (60G40) Martingales with continuous parameter (60G44) Derivative securities (option pricing, hedging, etc.) (91G20)

Related Items (24)

Uniqueness in Cauchy problems for diffusive real-valued strict local martingales ⋮ Distribution of the time to explosion for one-dimensional diffusions ⋮ Diffusion transformations, Black-Scholes equation and optimal stopping ⋮ On the multiplicity of option prices under CEV with positive elasticity of variance ⋮ Numerical option pricing in the presence of bubbles ⋮ Bubbles in discrete-time models ⋮ Hedging for the long run ⋮ On backward stochastic differential equations and strict local martingales ⋮ Can time-homogeneous diffusions produce any distribution? ⋮ A note on options and bubbles under the CEV model: implications for pricing and hedging ⋮ Outperforming the market portfolio with a given probability ⋮ Optimal arbitrage under model uncertainty ⋮ On the martingale property of certain local martingales ⋮ On the uniqueness of classical solutions of Cauchy problems ⋮ DUPIRE'S EQUATION FOR BUBBLES ⋮ The stochastic solution to a Cauchy problem for degenerate parabolic equations ⋮ HEDGING UNDER ARBITRAGE ⋮ Strict local martingale deflators and valuing American call-type options ⋮ The Black-Scholes equation in stochastic volatility models ⋮ Analysis of continuous strict local martingales via \(h\)-transforms ⋮ Approximating functionals of local martingales under lack of uniqueness of the Black–Scholes PDE solution ⋮ A Mathematical Theory of Financial Bubbles ⋮ THE EFFECT OF TRADING FUTURES ON SHORT SALE CONSTRAINTS ⋮ A liquidity-based model for asset price bubbles

Uses Software

• Mathematica

Cites Work

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• On exponential local martingales associated with strong Markov continuous local martingales
• Parabolic interior Schauder estimates by the maximum principle
• Volatility misspecification, option pricing and superreplication via coupling
• Volatility time and properties of option prices
• Comparison results for stochastic volatility models via coupling
• Moment explosions in stochastic volatility models
• Properties of American option prices
• Local martingales, bubbles and option prices
• Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle
• Robustness of the Black and Scholes Formula
• ARBITRAGE AND FREE LUNCH WITH BOUNDED RISK FOR UNBOUNDED CONTINUOUS PROCESSES
• Complications with stochastic volatility models
• OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS
• FEYNMAN–KAC FORMULAS FOR BLACK–SCHOLES-TYPE OPERATORS
• Class 𝐷 supermartingales

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