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

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.

Perpetual American Standard and Lookback Options with Event Risk and Asymmetric Information

“…From the point of view of financial mathematics and credit risk theory, the models in which the event or default times happen at the last passage times do not fall into the classical reduced form framework. More precisely, unlike in the existing models studied in Szimayer [50], Gapeev and Al Motairi [21], Glover and Hulley [25], Dumitrescu, Quenez, and Sulem [16], and Grigorova, Quenez, and Sulem [28], neither the immersion hypothesis nor the density hypothesis is satisfied (see Aksamit and Jeanblanc [1;Remark 5.31]), so that the default intensity process simply does not exist in our setting (see, e.g. Bielecki and Rutkowski [12; Chapter VIII] and Jeanblanc and Li [31] for the description of these concepts).…”

Perpetual American Cancellable Standard Options in Models with Last Passage Times

“…The model studied here differs from models studied in existing works such as Szimayer [6], Gapeev and Al Motairi [7], Glover and Hulley [8], Dumitrescu et al [9], and Grigorova et al [10], as neither the immersion hypothesis nor the density hypothesis is satisfied by the random times (or default times) θ and η, and the default intensity process simply does not exists in our setting (see, e.g., Bielecki and Rutkowski [11]). We see clearly in ( 6) and ( 7) that, in the case of zero recovery, this leads to a modified discounting factors, which are no longer functions of the sum of the interest rate and the default intensity rate.…”

Discounted optimal stopping problems in first-passage time models with random thresholds

“…Note that other applications of the concept described above include the consideration of perpetual American dividend-paying options with credit risk which are defaulted at the times when the underlying risky asset price processes reach such random thresholds. Other perpetual American defaultable and withdrawable dividend-paying options were recently considered in [14] and [15] in some other diffusion-type models of financial markets with full and partial information.…”