Changes of Probability Measures in Finance and Insurance: A Synthesis

Changes of probability measures now pave a very efficient way towards the study of many problems in Finance and Insurance. To quote only a few, this approach has proven to be useful in portfolio selection, option pricing and premium valuation in insurance. Many techniques have been developed more or less independently: changes of numéraire, pricing kernels (equivalents of probability measure changes) , Esscher transforms. The aim of this paper is to give a survey on these approaches and illustrate them with the Black and Scholes model taken as a benchmark; and then to see to what extent these changes of measure can be adapted to a more general context than the standard diffusion processes extensively assumed in the financial literature for modelling price processes. More specifically, we explore these techniques when price movements are modelled as Lévy processes. In sections two, three, four, five of this article we concentrate on diffusions – our aim being to exhibit the fundamental ideas behind the commonly used tools. From section six, we move on to the case of jump-diffusions, and in particular to the case of assets following geometric Lévy motions.