Encyclopedia of Quantitative Finance, IV Volume Set

Large Deviations

Modern large deviations theory, pioneered by Donsker and Varadhan [11], concerns the study of rare events, and it has become a common tool for the analysis of stochastic systems in a variety of scientific disciplines and in engineering. The theory developed by Donsker and Varadhan is a generalization of Laplace’s principle and Cramér’s theorem. Here we concentrate on applications to finance and risk management.

A wealth of monographs discuss the large deviations theory in detail; see, for instance, [8–10]. Here, we shall discuss applications to mathematical finance, including option pricing (see Option Pricing: General Principles), risk management (see Risk Exposures; Risk Management: Historical Perspectives), and portfolio optimization (see Risk–Return Analysis; Merton Problem). The reader may also consult [21], which includes details behind many of the topics touched upon here. Before discussing these applications, however, we provide a brief introduction to some basic concepts underlying the theory of large deviations.

Definition 1 A sequence of random objects (Z_n : n ≥ 0) taking values on some topological space S satisfies a, large deviations principle with rate function (J (z) : z ∈ S ), if for each Borel measurable set A ∈ S

and J (·) is nonnegative and upper ...