Volume 25, Issue 4 - 4

In recent years importance has become the standard tool for estimation of probabilities of rare events. Of special interest is efficient importance sampling which allows a substantial reduction of the computational burden. Efficiency of importance sampling has been proved (see Sadowsky and Bucklew [19]) under rather strong assumptions, which often cannot be verified for particular test statistics and estimators. In this paper we show that efficient importance sampling correctly works for calculation of moderate deviation probabilities of statistics having influence functions.

We introduce and study no-good-deal valuation bounds defined in terms of expected utility. A utility-based good deal is a payoff whose expected utility is too high in comparison to the utility of its price. Forbidding good deals induces, via duality, restrictions on pricing kernels and thereby gives tighter valuation bounds on payoffs than absence of arbitrage alone. Our approach extends earlier work by Černý (2003) in several directions: We give rigorous results for a general probability space instead of finite Ω; we systematically use duality results to provide a streamlined approach with simple arguments; we do all this rigorously for both static and dynamic situations; and we give a systematic comparison between local and global (conditional) pricing kernel restrictions for the temporally dynamic setting. For the dynamic case, we show in a Lévy framework that defining no-good-deal valuation measures by imposing local conditional restrictions on their instantaneous market prices of risk gives valuation bounds having very good dynamic properties as processes over time. We also show that global restrictions cannot yield such results in general.

Using a backward stochastic differential equation (BSDE) approach in a Brownian motion setting, we determine in an incomplete market an initial price Y 0 for a non-attainable claim ξ ∈ L p , 1 < p < ∞, that takes the hedging risk into account. Y 0 is chosen to be the best price such that the minimal replication error of the optimal hedging strategy with initial capital Y 0 has vanishing risk, the risk being measured with a certain coherent risk measure. This risk measure allows an interpretation as measuring total risk by accumulating instantaneously measured risks. Intertemporal consistency properties of best prices and uniqueness of optimal hedging strategies with vanishing risk are shown. The result can be interpreted as a robust predictable martingale representation property of Brownian motion w.r.t. the intersection of the scenario set defining the risk measure and the set of equivalent martingale measures for the assets traded in the market.

In this paper, we address the problem of testing the goodness of fit of a copula density function in the context of a d -dimensional i.i.d. sample by using a L 2 -type statistic. In extension to Fermanian [4], we propose to use a bootstrap approximation of the distribution of the test statistic. We show the validity of the bootstrap and perform a simulation study. Stochastic independence can be characterised in terms of the underlying copula and so we use the test as a test of independence.