Normalized Exponential Tilting: Pricing and Measuring Multivariate Risks

This paper discusses methods of risk neutralizing statistical distributions by applying exponential tilting of the probability density of a risk X, with respect to a reference risk Y. It proposes a normalization procedure based on percentile matching to convert the reference risk Y to a standard normal variable Z. The resulting normalized exponential tilting extends classic theories of pricing risks, including CAPM and the Black- Merton-Scholes. The paper then extends normalized exponential tilting to multivariate risks, establishing a link to multivariate probability distortions. The paper provides efficient routines for computing risk-neutralized multivariate probability distributions and illustrative examples of pricing contingent claims on multiple risks.