Mathematical reserves of unit-linked insurance policies with fractional volatility

Abstract

We study models of mathematical reserves of unit-linked insurance policies with fractional volatility. Mathematical reserves are specified amounts of capital that an insurance company is legally obligated to allot to cover its expected claims in a given period. Unit-linked insurance policies differ from regular policies in that the benefits are connected to an asset from the financial market. The risk associated to an insurance policy, coupled with the risk from the financial asset, presents the need for accurate predictions of mathematical reserves in the unlikely event that uncommonly large claims are made. Our principle objectives are to provide a new financial volatility model based on familiar ones and to apply it in the mentioned insurance setting through realistic examples. The assumption that the volatility is deterministic often suffers from being too unrealistic. Our volatility model is based on a fractional Ornstein-Uhlenbeck process, which is driven by a fractional Brownian motion (fBM). In particular, we will apply a Hurst parameter H less than 1/2, yielding fBM processes with negatively correlated increments, effectively mimicking the unstable nature of financial volatility. The employment of our suggested volatility model in the mentioned insurance setting has been successful. Using computer simulation, our main findings show that the proposed volatility leads to increased mathematical reserves of unit-linked insurance policies, compared to the classical case of a deterministic volatility. Specifically, our model suggests that the larger the value H less than 1/2, the larger the reserves. The effects of each parameter in the model are studied. Finally, methods for simulating and visualizing mathematical reserves under our volatility model are provided.