Worst-case scenario investment for insurers

We consider the investment problem of an insurance company who is facing a risk process from its own business and can additionally invest money into a stock index. This index is threatened by a possible market crash but otherwise is assumed to follow a geometric Brownian motion. Building up on work by Browne [Browne, S., 1995. Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Math. Operations Res. 29, 937–957] and Korn and Wilmott [Korn, R., Wilmott, P., 2002. Optimal portfolios under the threat of a crash. Int. J. Theor. Appl. Finance 5, 171–187] or Korn and Menkens [Korn, R., Menkens, O., 2002. Worst-case scenario portfolio optimization: a new stochastic control approach. Working paper] for optimal worst-case investment in a pure stock and bond market we determine equilibrium strategies for an insurer maximizing exponential utility. These strategies are not constant ones – as in the crash-free setting of Browne (1995) – and are optimal in a worst-case maximizing sense.