| The technique of minimum relative entropy (MRE) calibration provides a simple and computationally friendly way to perform moment matching of probability distributions. Its application to problems in financial modelling, and option pricing models in particular, is our area of study. In this work, we present techniques which can be used in concert with two existing applications. Thinking of current asset prices as discounted expectations of future cash flours, we first consider the calibration of probability densities of future asset prices to correctly price liquidly traded options. Using this method, the shape of the calibrated density may be quite irregular and is dependent on the initial, or prior, distribution used as a starting point for calibration. We present an iterative technique for increasing the regularity of the calibrated distribution and removing the dependence on the starting point. We study empirically the rate of convergence of this iteration to a fixed-point, and present a characterization of the limit as the solution of an eigenvalue problem. Second, we consider the problem of MRE calibration applied to monte carlo simulation. In this method, non-uniform weights are assigned to the generated paths in order to price correctly a benchmark set of instruments. One drawback of weighted monte carlo is that re-weighting the paths can destroy the martingale property of the simulated process, important in financial problems to ensure the absence of "model arbitrage", i.e. of the possibility that the model signals spurious arbitrage opportunities. We introduce a class of synthetic instruments and accompanying price constraints that can eliminate arbitrage opportunities and improve the pricing of path dependent options (e.g. forward starting options) without adversely affecting the pricing of the original benchmark instruments. Additionally; using these synthetic instruments, we characterize the minimum entropy martingale measure of a stochastic process as an infinite set of moment constraints. We describe a hierarchy of minimization problems whose solution in the limit converges to the minimum entropy martingale measure. In both cases, we combine theoretical discussion and numerical experimentation using real financial data, to demonstrate our methods. |
