A numerical approach to ruin probability in finite time for fitted models with investment

Abstract

In this paper we present a numerical method for solving a partial integro-differential equation (PIDE) associated with ruin probability, when the surplus is continuously invested in stochastic assets. The method uses precalculated Gaussian quadrature rules for the numerical integration. Except for the numerical integration part, the method is based largely on the finite differences method used in Halluin et al. (2005) for a PIDE associated with a more general option pricing problem. In our numerical examples we use historical data for inflation and returns on U.S. Treasury bills, U.S. Treasury bonds and American stocks. The log-returns of the investments are adjusted for an assumed constant force of inflation. We consider four different strategies for continuous investment: (a) U.S. Treasury bills with a constant maturity of 3 months, (b) U.S. Treasury bonds with a constant maturity of 10 years, and (c) the Standard and Poor 500 index and (d) another index of American stocks. For each of these strategies a geometric Brownian motion process is fitted to the aforementioned historical data. The results suggest that the ruin probabilities obtained can vary substantially, depending on whether the models are fitted to data for the last decade or for a longer time period. We also discuss numerical solution of investment models with jumps.