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Risks 2018, 6(4), 127; https://doi.org/10.3390/risks6040127

Perpetual American Defaultable Options in Models with Random Dividends and Partial Information

1
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK
2
Department of Mathematics, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
*
Author to whom correspondence should be addressed.
Received: 30 December 2017 / Revised: 17 October 2018 / Accepted: 25 October 2018 / Published: 6 November 2018
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Abstract

We present closed-form solutions to the perpetual American dividend-paying put and call option pricing problems in two extensions of the Black–Merton–Scholes model with random dividends under full and partial information. We assume that the dividend rate of the underlying asset price changes its value at a certain random time which has an exponential distribution and is independent of the standard Brownian motion driving the price of the underlying risky asset. In the full information version of the model, it is assumed that this time is observable to the option holder, while in the partial information version of the model, it is assumed that this time is unobservable to the option holder. The optimal exercise times are shown to be the first times at which the underlying risky asset price process hits certain constant levels. The proof is based on the solutions of the associated free-boundary problems and the applications of the change-of-variable formula. View Full-Text
Keywords: perpetual American options; random dividends; optimal stopping problem; Brownian motion; hidden Markov chain; filtering estimate; innovation process; free-boundary problem; a change-of-variable formula with local time on surfaces perpetual American options; random dividends; optimal stopping problem; Brownian motion; hidden Markov chain; filtering estimate; innovation process; free-boundary problem; a change-of-variable formula with local time on surfaces
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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V. Gapeev, P.; Al Motairi, H. Perpetual American Defaultable Options in Models with Random Dividends and Partial Information. Risks 2018, 6, 127.

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