"The perpetual American put option for jumpdiffusions with applications"
Research, Article, Preprint
OPEN
Aase, Knut K
(2005)
 Publisher: eScholarship, University of California

Subject:
optimal stopping  Optimal exercise policy; American put option; perpetual option; optimal stopping; incomplete markets; equity premiums; CCAPM.  Optimal exercise policy  equity premiums  perpetual option  American put option  incomplete markets  Optimal exercise policy, American put option, perpetual option, optimal stopping, incomplete markets, equity premiums, CCAPM  CCAPM
In this paper, we solve an optimal stopping problem with an infinite time horizon, when the state variable follows a jumpdiffusion. Under certain conditions our solution can be interpreted as the price of an American perpetual put option, when the underlying asset follows this type of process.
We present several examples demonstrating when the solution can be interpreted as a perpetual put price. This takes us into a study of how to risk adjust jumpdiffusions. One key observation is that the probabililty distribution under the risk adjusted measure depends on the equity premium, which is not the case for the standard, continuous version. This difference may be utilized to find intertemporal, equilibrium equity premiums, for example
Our basic solution is exact only when jump sizes can not be negative. We investigate when our solution is an approximation also for negative jumps.
Various market models are studied at an increasing level of complexity, ending with the incomplete model in the last part of the paper.