Abstract
We present an overview of the broad class of financial models in which the prices of assets are Lévy-Ito processes driven by an n-dimensional Brownian motion and an independent Poisson random measure. The Poisson random measure is associated with an n-dimensional Lévy process. Each model consists of a pricing kernel, a money market account, and one or more risky assets. We show how the excess rate of return above the interest rate can be calculated for risky assets in such models, thus showing the relationship between risk and return when asset prices have jumps. The framework is applied to a variety of asset classes, allowing one to construct new models as well as interesting generalizations of familiar models.
Highlights
Pricing models driven by Levy processes have been considered by many authors [1, 16, 19, 22, 25, 26, 28, 35, 41, 56, 61, 75]
The need for a systematic theory of Levy-Ito models in finance is plain, for if an asset price is driven by a Levy process, the price process of an option or other derivative based on that asset cannot itself in general be represented by a Levy model, but it can typically be represented by a Levy-Ito model, provided that the payoff is reasonably well behaved; and as we know well [9, 64], most securities and other financial assets, both corporate and sovereign, can be viewed as complex options based on the cash flows associated with one or more simpler underlying assets
We comment on the nature of the excess rate of return above the short rate of interest in a Levy-Ito setting, and in equation (3.38) we show that the excess rate of return per unit of jump intensity can be expressed as the product of a random volatility and a random market-price-of-risk for each admissible jump vector of the Levy process associated with the Poisson random measure
Summary
Pricing models driven by Levy processes have been considered by many authors [1, 16, 19, 22, 25, 26, 28, 35, 41, 56, 61, 75]. We are concerned here with a broader family of pricing models, namely, the so-called Levy-Ito models Such models are driven both by a Brownian motion and a Poisson random measure, where the Poisson random measure is associated with an underlying Levy process. In the case of a two-currency model driven by a single Brownian motion this result is known as Siegel’s paradox [8, 76], and here we have shown that the Siegel condition can be satisfied for each exchange rate pair in a multi-currency Brownian market model. This leads us to conjecture that the Siegel condition can be satisfied in any multi-currency Levy-Ito model with an appropriate choice of the risk-aversion functions
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