A BENCHMARK APPROACH TO RISK MANAGEMENT

Eckhard Platen

Abstract

This paper describes an approach that is based on a diversified world index in a general financial market setting with the growth optimal portfolio (GOP) as reference unit or benchmark. The diversified world index is shown to approximate the GOP under general assumptions when the number of securities of the market is sufficiently large. It is demonstrated that the trend of the GOP is a key financial quantity. It turns out that it can be directly observed since the expected increments of the GOP equal four times those of the quadratic variation of its square root. This holds without any major modeling assumptions. Using a world stock index as proxy for the GOP it is shown that, in reality, the trend of the discounted index does not vary greatly in the long term. This leads for the index to a natural model, where the index is a transformed square root process. The squared index volatility appears then as the inverse of the square root process. This feature explains most of the stylized empirical facts on index volatility. Furthermore, it suggests that the Radon-Nikodym derivative of the candidate risk neutral measure is in reality a strict supermartingale. This indicates that deviations from the standard risk neutral pricing theory have to be expected when pricing and hedging long term derivatives on indices and other securities. Furthermore, interesting consequences for the integrated management of market risk arise from the proposed benchmark approach that uses the GOP as benchmark.