The theory of L\'evy models for asset pricing simplifies considerably if one takes a pricing kernel approach, which enables one to bypass market incompleteness issues. The special case of a geometric L\'evy model (GLM) with constant parameters can be regarded as a natural generalisation of the standard geometric Brownian motion model  used in the  Black-Scholes theory . In one dimension, once the underlying L\'evy process has been specified, the GLM is characterised by four parameters: the initial asset price, the interest rate, the volatility, and a risk aversionfactor . The pricing kernel is given by the product of a discount factor and the Esscher martingale associated with the risk aversion parameter. The model is fixed by the requirement that for each asset the product of the asset price and the pricing kernel should be a martingale. In the GBMcase , the risk aversion factor is the so-called market price of risk. In the GLMcase , this interpretation is no longer valid; instead, the excess rate of return is given by a nonlinear function of the volatility and the risk aversion. It is shown that for positive values of the volatility and the risk aversion the excess rate of return above the interest rate is positive, and is monotonically increasing with respect to these variables. In the case of foreign exchange, Siegel's paradox implies that it is possible to construct foreign exchange models for which the excess rate of return (above the interest rate differential) 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 the volatility exceeds the risk aversion.
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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