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General Theory of Geometric L\'evy Models for Dynamic Asset Pricing

  • Dorje C. Brody
  • Lane P. Hughston
  • Ewan Mackie
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    The geometric L\'evy model (GLM) is a natural generalisation of the geometric Brownian motion model (GBM) 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 L\'evy 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 both for the exchange rate and the inverse exchange rate. This condition is shown to hold for any geometric L\'evy model for foreign exchange in which volatility exceeds risk aversion.

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    Paper provided by in its series Papers with number 1111.2169.

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    Date of creation: Nov 2011
    Date of revision: Jan 2012
    Publication status: Published in Proc. R. Soc. A June 8, 2012 468 1778-1798
    Handle: RePEc:arx:papers:1111.2169
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