On Bayesian Value at Risk: From Linear to Non-linear Portfolios

This paper proposes the use of Bayesian approach to implement Value at Risk (VaR) model for both linear and non-linear portfolios. The Bayesian approach provides risk traders with the flexibility of adjusting their VaR models according to their subjective views. First, we deal with the case of linear portfolios. By imposing the conjugate-prior assumptions, a closed-form expression for the Bayesian VaR is obtained. The Bayesian VaR model can also be adjusted in order to deal with the ageing effect of the past data. By adopting Gerber-Shiu’s option-pricing model, our Bayesian VaR model can also be applied to deal with non-linear portfolios of derivatives. We obtain an exact formula for the Bayesian VaR in the case of a single European call option. We adopt the method of back-testing to compare the non-adjusted and adjusted ∗Tak Kuen Siu, Ph.D., is a Lecturer in the Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, United Kingdom. E-mail: [email protected] †Howell Tong, Hons. F.I.A., Ph.D., is the Chair Professor of the Department of Statistics, London School of Economics, U.K., [email protected] ‡Hailiang Yang, A.S.A., Ph.D., is an Associate Professor in the Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, e-mail: [email protected].