Brody, DC; Hughston, LP; Mackie, E; (2012) General Theory of Geometric Lévy Models for Dynamic Asset Pricing. **Proceedings of the Royal Society of London A** , 468 1778 - 1798. 10.1098/rspa.2011.0670.

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## Abstract

The geometric Lévy model (GLM) is a natural generalization of the geometric Brownian motion (GBM) model used in the derivation of the Black–Scholes formula. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying Lévy process has been specified, the GLM has four parameters: the initial price, the interest rate, the volatility and the risk aversion. The pricing kernel is the product of a discount factor and a risk aversion martingale. For GBM, the risk aversion parameter is the market price of risk. For a GLM, this interpretation is not valid: the excess rate of return is a nonlinear function of the volatility and the risk aversion. It is shown that for positive volatility and risk aversion, the excess rate of return above the interest rate is positive, and is increasing with respect to these variables. In the case of foreign exchange, Siegel's paradox implies that one can construct foreign exchange models for which the excess rate of return is positive for both the exchange rate and the inverse exchange rate. This condition is shown to hold for any geometric Lévy model for foreign exchange in which volatility exceeds risk aversion.

Type: | Article |
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Title: | General Theory of Geometric Lévy Models for Dynamic Asset Pricing |

DOI: | 10.1098/rspa.2011.0670 |

Keywords: | Lévy processes, asset pricing, risk premium, risk aversion, Siegel's paradox |

UCL classification: | UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics |

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