Inverse Problems to Estimate Market Price of Risk in Catastrophe Bonds

Abstract

This research focuses on evaluating the market price of risk for catastrophe bonds (CAT bonds). Our approach involves constructing a model for CAT bonds that incorporates stochastic process interest rates and losses, followed by numerical methods. Recognizing the inherent challenge of directly obtaining the market price of risk from the market, we utilize inverse problems to derive it. Our assumptions include the CIR stochastic process model for the interest rates and the jump-diffusion stochastic process model for the loss. Through the analysis of a risk-free portfolio, we illustrate the alignment of CAT bonds with partial integral differential equations (PIDE). Employing inverse problems, we then estimate the market price of risk by solving the PIDE. Specifically, we implement Tikhonov regularization and propose a systematic method for determining the market price of risk.