Perpetual American Cancellable Standard Options in Models with Last Passage Times

Abstract

We derive explicit solutions to the perpetual American cancellable standard put and call options in an extension of the Black–Merton–Scholes model. It is assumed that the contracts are cancelled at the last hitting times for the underlying asset price process of some constant upper or lower levels which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price reaches some lower or upper constant levels. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit conditions.

Last-Passage American Cancelable Option in Lévy Models

“…Our article continues the research conducted by Gapeev et al (2020), where the authors also evaluate the American-style option (1), but they carry this out by solving an appropriate HJB system of equations. In Gapeev et al (2020), the underlying asset price was described by geometric Brownian motion, for which the above approach is very natural due to the locality of the diffusive generator of the asset price process S t . Still, in the context of nonlocal generators, a 'guess-and-verify' method used in this paper seems to be more efficient.…”

“…We set the intensity λ of the N t process from Equation (4) equal to zero, and thus X t becomes the arithmetic Brownian motion with drift parameter µ = r − σ 2 2 . This example corresponds to the option evaluated in Gapeev et al (2020). The scope of the numerical analysis here is to find the optimal exercise level a * and the fair price V(s) of the option.…”

“…Most of them reduce the original optimal stopping problems to the associated free boundary problems. Wu and Li (2022) valuate the American put option in a very similar model as Gapeev et al (2020) did, that is, any stopping rules must be before the last exit time of the price of the underlying assets at any fixed level. This paper focuses only on the Black-Scholes market as well.…”

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

“…In this work, the upper bounds for the maximum process as well as the lower bounds for the minimum process are given exogenously, by virtue of the presence of the linear recovery amounts in the appropriate reward functionals. The case of perpetual American defaultable standard options in models with last passage times of constant levels for the underlying asset prices and zero recoveries was recently considered in Gapeev, Li, and Wu [20].…”

Optimal stopping problems for maxima and minima in models with asymmetric information

“…Moreover, we show that the appearance of the withdrawal opportunities for the writers of the contracts at the times θ and η may essentially change the behaviour of the optimal exercise boundaries for the holders of the options. The other problems of perpetual American cancellable or defaultable standard and lookback options in models with last passage times of constant and random levels for the underlying asset prices and zero or linear recoveries were recently considered in Gapeev, Li and Wu [21] and Gapeev and Li [16], respectively.…”