Alternative to beta coefficients in the context of diffusions

Abstract

We develop an alternative to the beta coefficient of the CAPM theory. We show the link between this notion and the Wiener chaos expansion of the underlying processes. In the setting of Markov diffusions, we define the drift-neutral beta, which is the quantity of benchmark such that the resulting portfolio is immune to an infinitesimal change of drift on the Brownian motion driving the benchmark. Our approach yields a coefficient which in many practical cases depends on the initial values of both the portfolio and its benchmark. It can also be used to take into account extreme risks and not only the variance. We study several classical diffusion processes and give a full analysis in the case of Jacobi processes. Examples with credit indices show the efficiency of the method in hedging a portfolio.

Keywords:

• Beta coefficient
• CAPM theory
• Risk management
• Wiener chaos decomposition
• Jacobi processes
• Credit assets allocation

AMS Subject Classifications:

• C31
• G11

Acknowledgements

The opinions and views expressed in this document are those of the authors and do not necessarily reflect those of Natixis Asset Management. We are grateful to the referees for their valuable comments on the first version of this work. We wish to thank Flavien Bellocq for his help in the numerical applications.

Notes

No potential conflict of interest was reported by the authors.