Scandinavian Working Papers in Economics

Discussion Papers,
Norwegian School of Economics, Department of Business and Management Science

No 2005/12: The perpetual American put option for jump-diffusions with applications

Knut K. Aase ()
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Knut K. Aase: Dept. of Finance and Management Science, Norwegian School of Economics and Business Administration, Postal: NHH , Department of Finance and Management Science, Helleveien 30, N-5045 Bergen, Norway

Abstract: In this paper we solve an optimal stopping problem with an infinite time horizon, when the state variable follows a jump-diffusion. 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 jump-diffusions. One key observation is that the probability 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.

Keywords: Optimal exercise policy; American put option; perpetual option; optimal stopping; incomplete markets; equity premiums; CCAPM.

JEL-codes: G00

34 pages, November 30, 2005

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