Change of measure techniques are ubiquitous in many areas of applied probability including in model selection in finance and in efficient simulation of rare events.
Broadly, this technique amounts to changing the underlying probability measure (the prior) to a new one (the posterior), so that the new probability measure may satisfy certain desirable properties for the problem at hand. For example the posterior measure may be satisfying some constraints while being close in a specified sense to given probability measure, or it may be efficient to simulate from the posterior probability distribution to ascertain probability of some rare event by simulation-estimation. In this thesis we develop change of measure in two broad areas: Model selection for financial applications and simulation-estimation of rare event probabilities.
The first abstracted problem corresponds to finding a probability measure that minimizes the relative entropy (also called I-divergence) with respect to a known measure while it satisfies certain moment constraints on functions of underlying assets. We show that under I-divergence, the optimal solution may not exist when the underlying assets have fat tailed distributions, popular in financial practice. We note that this drawback may be corrected if `polynomial-divergence' is used. This divergence can be seen to be equivalent to the well known (relative) Tsallis or (relative) Renyi entropy. We discuss existence and uniqueness issues related to this new optimization problem. We also identify the optimal solution structure under I-divergence as well as polynomial-divergence when the associated constraints include those on marginal distribution of functions of underlying assets. These results are applied to a simple problem of model calibration to options prices as well as to portfolio modeling in Markowitz framework, where we note that a reasonable view that a particular portfolio of assets has heavy tailed losses may lead to fatter and more reasonable tail distributions of all assets.
The second problem is further classified as estimation of large deviation probabilities and estimation of probability that sum of few random variables exceeds a large threshold, the latter is applied to pricing deep out of the money financial options. We exploit the saddle-point based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. We note that these representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Further, the integrands have enough structure so that it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and prove that asymptotically they possess strong efficiency properties.
