Diffusion-based models for financial markets without martingale measures. (English) Zbl 1306.91125
Biagini, Francesca (ed.) et al., Risk measures and attitudes. In part based on a conference, Munich, Germany, December 2010. London: Springer (ISBN 978-1-4471-4925-5/pbk; 978-1-4471-4926-2/ebook). EAA Series, 45-81 (2013).
Summary: We consider a general class of diffusion-based models and show that, even in the absence of an Equivalent Local Martingale Measure, the financial market may still be viable, in the sense that strong forms of arbitrage are excluded and portfolio optimisation problems can be meaningfully solved. Relying partly on the recent literature, we provide necessary and sufficient conditions for market viability in terms of the market price of risk process and martingale deflators. Regardless of the existence of a martingale measure, we show that the financial market may still be complete and contingent claims can be valued under the original (real-world) probability measure, provided that we use as numéraire the Growth-Optimal Portfolio.
For the entire collection see [Zbl 1258.60005].
For the entire collection see [Zbl 1258.60005].
MSC:
| 91G10 | Portfolio theory |
| 60G44 | Martingales with continuous parameter |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
Keywords:
arbitrage; hedging; contingent claim valuation; market price of risk; martingale deflator; growth-optimal portfolio; numéraire portfolio; market completeness; utility indifference valuation; benchmark approachReferences:
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