PRICING AND HEDGING OF DERIVATIVES BASED ON NONTRADABLE UNDERLYINGS

This paper is concerned with the study of insurance related derivatives on financial markets that are based on nontradable underlyings, but are correlated with tradable assets. We calculate exponential utility‐based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward‐backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical “delta hedge” in complete markets.     

Additional utility of insiders with imperfect dynamical information

In this paper we consider a market driven by a Wiener process where there is an insider and a regular trader. The insider has privileged information which has been deformed by an independent noise vanishing as the revelation time approaches. At this time, the information of every trader is the same. We obtain the semimartingale decomposition of the original Wiener process under dynamical enlargement of the filtration, and we prove that if the rate at which the additional noise in the insiders information vanishes is slow enough then there is no arbitrage and the additional utility of the insider is finite.Received: 1 October 2003, Mathematics Subject Classification: 60G48, 90A09, 60H07, 90A60JEL Classification: D82, G11, G14     

Malliavin's Calculus in Insider Models: Additional Utility and Free Lunches

We consider simple models of financial markets with regular traders and insiders possessing some extra information hidden in a random variable that is accessible to the regular trader only at the end of the trading interval. The problems we focus on are the calculation of the additional utility of the insider and a study of his free lunch possibilities. The information drift—that is, the drift to eliminate in order to preserve the martingale property in the insider's filtration—turns out to be the crucial quantity needed to answer these questions. It is most elegantly described by the logarithmic Malliavin trace of the conditional laws of the insider information with respect to the filtration of the regular trader. Several examples are given to illustrate additional utility and free lunch possibilities. In particular, if the insider has advance knowledge of the maximal stock price process, given by a regular diffusion, arbitrage opportunities exist.