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 aversion factor. 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 GBM case, the risk aversion factor is the so-called market
price of risk. In the GLM case, this interpretation is no longer valid;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.