Large Deviations and Asymptotic Efficiencies

Asymptotic Arbitrage and Large Deviations

Typical models of mathematical finance admit equivalent martingale measures up to any finite time horizon but not globally, and this means that arbitrage opportunities arise in the long run. In this paper, we derive explicit estimates for asymptotic arbitrage, and we show how they are related to large deviation estimates for the market price of risk. As a case study we consider a geometric Orns...

Large Deviations

Large Deviations

This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. 1. Large deviations for sums. The role of " large deviations " is best understood through an example. Suppose that X random variables, for instance, normally distributed with mean zero and variance 1. Then, E e θ(X 1 +···+X n) = E[e θX ...

Large Deviations for Large Capacity

In this paper, we study large deviations of large capacity loss networks with xed routing. We use two-level modelling for the loss networks: the call level and the cell level. At the call level, a call request is accepted if it succeeds an admission test. The test is based on a polyhedral set of the number of calls in progress when a new call arrives. After being accepted, a call then transmits...

Large Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models

Let σt(x) denote the implied volatility at maturity t for a strike K = S0e , where x ∈ R and S0 is the current value of the underlying. We show that σt(x) has a uniform (in x) limit as maturity t tends to infinity, given by the formula σ∞(x) = √ 2 ( h(x) + (h(x)− x) ) , for x in some compact neighbourhood of zero in the class of affine stochastic volatility models. The function h∗ is the convex...