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Strict local martingales and bubbles. (English) Zbl 1336.91076

The goal of this paper is to determine the influence of asset price bubbles on the pricing of derivatives. When an asset price bubble exists, the market price of the asset is higher than its fundamental value. From a mathematical point of view, this is the case when the stock price process is modeled by a positive strict local martingale under the equivalent local martingale measure. To deal with the local martingales instead of martingales, especially in the case of call options, the authors prove some technical results and then obtain decomposition formulas for some classes of European path-dependent options under the NFLVR (no free lunch with vanishing risk) condition. Furthermore, they express the risk-neutral price of an exchange option in the presence of asset price bubbles as an expectation involving the last passage time at the strike level under the new measure. Even the case where the candidate density process for the risk-neutral measure is only a strict local martingale and the NFLVR condition is not fulfilled and risk-neutral valuation fails, is considered. In this case the real-world measure is applied.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60G30 Continuity and singularity of induced measures
60G44 Martingales with continuous parameter
60G48 Generalizations of martingales
91G99 Actuarial science and mathematical finance
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References:

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