In this paper we consider the optimal trading strategy for an investor with an exponentially distributed horizon who invests in a riskless asset and a risky asset. The risky asset is subject to proportional transaction costs and its price follows a jump diffusion. In this situation, the optimal trading strategy is to maintain the fraction of the dollar amount invested in the riskless asset to the dollar amount invested in the risky asset in between two bounds. In contrast to the pure diffusion setting where the investor faces no jump risk, this fraction can jump discontinuously outside the bounds which is optimally followed by a transaction to the boundary. We character- ize the value function and provide bounds on the trading boundaries. Our numerical results show the introduction of jumps ("event risk") dramatically affect the optimal transaction strategy. In particular jumps tend to reduce the amount of stock the investor holds and increase the width of the no transaction region. We also show that the boundaries are affected not only by the size of the jump but can be very sensitive to the uncertainty in the jump size. We also examine how the optimal transaction boundaries vary through time for investors with deterministic horizons by looking at the optimal policies for investors with Erlang distributed horizons, which has been shown to provide good approximations to the deterministic horizon optimal policies.
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