This paper develops a unit-consistent actuarial framework for pricing capped cumulative temperature-index insurance under long-range dependence and stochastic variability. Daily temperature anomalies are modeled as increments of fractional Brownian motion evaluated at an operational time generated by the integral of a stationary normalized Cox--Ingersoll--Ross process. We show that the stochastic time change preserves stationarity and the long-memory covariance decay of the increments while introducing additional variability through the random operational clock. The cumulative temperature index admits a conditionally Gaussian representation, which leads to an exact conditional exponential kernel for capped stop-loss contracts and ensures existence of the entropic premium for every positive risk-aversion parameter. Consequently, valuation reduces to an outer Monte Carlo expectation over the accumulated CIR time, avoiding fractional Brownian path simulation and covariance-matrix construction. We further establish monotonicity properties of the premium with respect to risk aversion and conditional volatility. An empirical illustration based on Chicago temperature data shows that both long memory and stochastic time change can materially affect insurance premiums relative to conventional Brownian and fractional Brownian benchmarks, with the Hurst parameter playing an important role in valuation uncertainty. The proposed framework therefore provides a tractable approach for incorporating persistent dependence, stochastic variability, and bounded insurance losses into climate-index pricing.
This paper investigates the optimal surplus management problem of an insurance company operating in a financial market with stochastic interest rates and jump-driven liabilities. The insurer dynamically allocates its surplus between a risky stock and a risk-free zero-coupon bond while facing insurance claims modeled by a compound Poisson process with exponentially distributed claim sizes. The short term interest rate follows a Cox-Ingersoll-Ross (CIR) process, which captures mean-reverting dynamics commonly observed in term structure models. The insurer maximizes the expected exponential utility of terminal surplus. Using stochastic control techniques, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation. Although the exponential utility structure suggests an exponential affine representation, the interaction between the interest rate hedge and the surplus state generates quadratic surplus terms in the HJB equation. To obtain a tractable formulation, we adopt a normalized surplus projection method, which provides an approximate reduction of the full three-dimensional problem to a nonlinear system of partial differential equations (which is subsequently numerically validated). The optimal investment policy admits an economically meaningful decomposition consisting of a myopic demand component and an interest rate hedging component. Numerical experiments illustrate how the optimal strategy and the surplus distribution depend on interest rate volatility, claim intensity, and risk aversion. The results highlight the importance of jointly modeling stochastic interest rates and insurance liability risk when designing optimal investment policies for insurance companies.
Nader Karimi, F. Shokrollahi, Masoumeh Shahmoradi· 1 citation