Risk measures with the CxLS property

In the present contribution, we characterise law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (Math. Finance 16:419–441, 2006), we show that these risk measures can be identified with a class of generalised shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (Math. Finance, 2014, http://onlinelibrary.wiley.com/doi/10.1111/mafi.12080/abstract) and Bellini and Bignozzi (Quant. Finance 15:725–733, 2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterisation of robustness for convex risk measures in terms of a weak notion of mixture continuity.     

Representation of the penalty term of dynamic concave utilities

In the context of a Brownian filtration and with a fixed finite time horizon, we provide a representation of the penalty term of general dynamic concave utilities (hence of dynamic convex risk measures) by applying the theory of g-expectations.     

An Interest Rate Model with Upper and Lower Bounds

We propose a new interest rate dynamicsmodel where the interest rates fluctuate in a bounded region. The model ischaracterised by five parameters which are sufficiently flexible to reflect theprediction of the future interest rates distribution. The interest rate convergesin law to a Beta distribution and has transition probabilities which arerepresented by a series of Jacobi polynomials. We derive the moment evaluationformula of the interest rate. We also derive the arbitrage free pure discountbond price formula by a weighted series of Jacobi polynomials. Furthermore wegive simple lower and upper bounds for the arbitrage free discount bond pricewhich are tight for the narrow interest rates region case. Finally we show thatthe numerical evaluation procedure converges to the exact value in the limitand evaluate the accuracy of the approximation formulas for the discount bondprices.     

Weighted norm inequalities and hedging in incomplete markets

Let be an -valued special semimartingale on a probability space with canonical decomposition . Denote by the space of all random variables , where is a predictable -integrable process such that the stochastic integral is in the space of semimartingales. We investigate under which conditions on the semimartingale the space is closed in , a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of in . Most of these conditions deal with BMO-martingales and reverse H?lder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the F?llmer-Schweizer decomposition.     

A Note on Option Pricing for the Constant Elasticity of Variance Model

We study the arbitrage free optionpricing problem for the constant elasticity of variance (CEV) model. To treatthestochastic aspect of the CEV model, we direct attention to the relationship between the CEV modeland squared Bessel processes. Then we show the existence of a unique equivalentmartingale measure and derive the Cox's arbitrage free option pricing formulathrough the properties of squared Bessel processes. Finally we show that the CEVmodel admits arbitrage opportunities when it is conditioned to be strictlypositive.     

A note on the no arbitrage condition for international financial markets

We consider an international financial market model that consists ofN currencies. The purpose is to derive a no arbitrage condition which is not affected by the choice of numéraire between theN currencies. As a result, we show that a finiteness condition for an arbitrary chosen currency and the no arbitrage condition for the basket currency are necessary and sufficient for the no arbitrage property of all theN currencies. Research supported in part by Nomura Foundation for Social Science and by the European Community Stimulation Plan for Economic Science contract Number SPES-CT91-0089. The authors thank an anonymous FEJM referee for helpful comments.     

Partitioning uncertain workloads

We present a method for determining the ratio of the tasks when breaking any complex workload in such a way that once the outputs from all tasks are joined, their full completion takes less time and exhibit smaller variance than when running on the undivided workload. To do that, we have to infer the capabilities of the processing unit executing the divided workloads or tasks. We propose a Bayesian Inference algorithm to infer the amount of time each task takes in a way that does not require prior knowledge on the processing unit capability. We demonstrate the effectiveness of this method in two different scenarios; the optimization of a convex function and the transmission of a large computer file over the Internet. Then we show that the Bayesian inference algorithm correctly estimates the amount of time each task takes when executed in one of the processing units.     

A Simple Counterexample to Several Problems in the Theory of Asset Pricing

We give an easy example of two strictly positive local martingales that fail to be uniformly integrable, but such that their product is a uniformly integrable martingale. The example simplifies an earlier example given by the second author. We give applications in mathematical finance and we show that the phenomenon is present in many incomplete markets.