Lévy Processes in Finance and Insurance
Ernesto Mordecki
In this short course we consider
(i) a model of a financial market with two stocks, one deterministic and the other driven by a Lévy process, we call this model a Lévy Market and, simultaneously,
(ii) a model of the capital of an insurance company described by a Lévy process, i.e. a Lévy risk process.
In the first case corresponding to the financial modelling, we obtain a generalization of the classical Black-Scholes model, obtained when the Lévy process has no jumps, and reduces to a Brownian Motion. In what concerns actuarial models, our proposal generalizes the classical Crámer-Lundberg model, where the risk process is a Lévy process with no Gaussan component, no positive jumps, and finite intensity negative jumps, i.e. a compound Poisson process.
In the first situation (finance) we address the problem of option pricing, discussing Eurpean, American, and Perpetual American options, with stress in this last particular case, where closed solutions can be obtained. In the second, we address the explicit computation of ruin probabilities.
Both results (pricing of american perpetual options and computation of ruin probabilities) happen to be connected, more precisely, the optimal strategy and option price in a Lévy Market can be explicitely obtained if the ruin probability for an associated Lévy risk process can be computed.
We plan to devote special attention to the introduction of Lévy process, to the consideration of particular examples and explain how the obtained results apply in practice.