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The presence of any friction in financial markets qualitatively changes the nature of the optimization problem faced by an investor. It requires one to either act or do nothing, an issue which, of course, does not arise in frictionless situations. The investor considered here accumulates wealth without consuming until some terminal point in time when he consumes all. His objective is to maximize the expected utility derived from that terminal consumption. We postpone the terminal point far into the future to obtain a stationary portfolio rule. The portfolio policy is in the form of two control barriers between which portfolio proportions are allowed to fluctuate. We show how to calculate them.

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to represent mortality risk. This paper represents a …rst attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. We extend to couples the Cox processes set up, namely the idea that mortality is driven by a jump process whose intensity is itself a stochastic process, proper of a particular generation within each gender. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. We also provide a methodology for …tting the joint survival function by working separately on the (analytical) copula and the (analytical) margins. First, we calibrate and select the best …t copula according to the methodology of Wang and Wells (2000b) for censored data. Then, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, a¢ ne process: this gives the marginal survival functions. By coupling the best …t copula with the calibrated margins we obtain a joint survival function which incorporates the stochastic nature of mortality improvements. Several measures of time dependent association can be computed out of it.We apply the methodology to a well known insurance dataset, using a sample generation. The best …t copula turns out to be a Nelsen one, which implies not only positive dependency, but dependency increasing with age.JEL Classi…cation: G22

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We discuss a Levy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behaviour of a series of stocks or indexes and to study a multi-firm, value-based default model. Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a stochastic time-change technique and provide the details for a Gamma change. The main feature of the model is the fact that—opposite to other, non-jointly Gaussian settings—its risk-neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit.Levy processes, Multivariate asset modelling, Copulas, Risk neutral dependence,

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The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Frechet bounds. Applications provided include prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, no-arbitrage pricing bounds, as well as values consistent with the independence of the underlying assets are provided. As a final reference value, a copula function calibrated on historical data is used.Bivariate Option Pricing, Copula Functions, Pricing Kernel, Applications,