A Multivariate Extension of Equilibrium Pricing Transforms: The Multivariate Esscher and Wang Transforms for Pricing Financial and Insurance Risks

Abstract

This paper proposes a multivariate extension of the equilibrium pricing transforms for pricing general financial and insurance risks. The multivariate Esscher and Wang transforms are derived from Bühlmann's equilibrium pricing model (1980) under some assumptions on the aggregate risk. It is shown that the Esscher and Wang transforms coincide with each other when the underlying risks are normally distributed.

Pricing Mortality Securities With Correlated Mortality Indexes

“…Exponential tilting is an incomplete market pricing method that neutralizes statistical distributions, which is consistent with the literature on nonarbitrage pricing of contingent claims (see Buhlmann, ; Gerber and Shiu, ; Madan and Unal, ; Kijima, ; Wang, ; and others). It can be applied in pricing risks embedded in loan defaults, mortgage refinancing, electricity trading, weather derivatives, catastrophic insurance, and insurance‐linked securities (Duffie, ; Karatzas and Shreve, ; Heston, ; Gerber and Shiu, ; Cox, Lin, and Wang, ; Milidonis, Lin, and Cox, ).…”

“…It can be applied in pricing risks embedded in loan defaults, mortgage refinancing, electricity trading, weather derivatives, catastrophic insurance, and insurance‐linked securities (Duffie, ; Karatzas and Shreve, ; Heston, ; Gerber and Shiu, ; Cox, Lin, and Wang, ; Milidonis, Lin, and Cox, ). Kijima () and Wang () extend univariate exponential tilting to multivariate cases. The need for changing multivariate probability measures arises from pricing contingent claims on multiple underlying assets or liabilities.…”

Multivariate Exponential Tilting and Pricing Implications for Mortality Securitization

“…Wang (2006) shows that normalized exponential tilting of the probability density function (PDF) of X (with respect to Z ) is equivalent to applying the Wang transform to the cumulative distribution of X , and is an extension of the capital asset pricing model to risks with general‐shaped distributions. By using a multiperiod equilibrium model, Kijima (2006) also reaches the same conclusions as those of multivariate exponential tilting (Wang, 2006). Our article proposes to use multivariate exponential titling to price mortality securities.…”

“… Y 1 , Y 2 , … , Y k is equivalent to applying Wang transforms to X i with The correlation matrix between X 1 , X 2 , … , X n is unchanged after the normalized multivariate exponential tilting , Σ* xx =Σ xx . Kijima ( 2006 ) reaches the same conclusion as by using a multiperiod equilibrium argument.…”

Pricing Survivor Derivatives With Cohort Mortality Dependence Under the Lee–Carter Framework

“…Dowd et al () offer the first pricing model for a survivor swap based on the Wang transform (Wang 2000). To capture cohort mortality dependence between ages, we apply the multivariate extension of the Wang transformation (Cox, Lin, and Wang, ; Kijima, ) and obtain the mortality rate under the Wang risk measure Q , based on our proposed Lee–Carter model with cohort mortality dependence. For a multivariate setting, assume that follows a Gaussian copula with a correlation matrix .…”