Executive Stock Option Exercise with Full and Partial Information on a Drift Change Point

Abstract

We analyse the optimal exercise of an American call executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point, and seek to optimally exercise the option by maximising its discounted payoff under the physical measure. The first agent has full information, and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration F in which the stock drift becomes a diffusion driven by the innovations process, an F-Brownian motion also driving the stock under F, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterise the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.

On the monotonicity of the stopping boundary for time-inhomogeneous optimal stopping problems

“…The common feature of these stopping problems is a random variable whose outcome is unknown to the optimiser and which affects the drift of the underlying process and/or the payoff function. The literature is vast and diverse in this field and we cite, among others, [30], [8], [10], [11], [12], [13], [16], [17] [18]. We focus, in particular, on models as in [12] and [17] where a random variable affects the drift of the underlying process and, in a Bayesian formulation of the problem, only the prior distribution of the random variable is known to the optimiser.…”

Stopping problems with an unknown state

“…Then V = v/(1 + ϕ), where ϕ = π/(1 − π ). Moreover, if τ ∈ T X is an optimal stopping in (13), then it is also optimal in the original problem (4).…”

“…For example, American options with incomplete information about the drift of the underlying process have been studied in [4] and [7], and a liquidation problem has been studied in [5]. The related literature includes optimal stopping for regime-switching models (see [12] and [23]), studies of models containing change points [9,13], a study allowing for an arbitrary distribution of the unknown state [6], problems of stochastic control [16] and singular control [2], and stochastic games [3] under incomplete information. Stopping problems with a random time horizon are studied in, for example, [1] and [17], where the authors consider models with a random finite time horizon but with state-independent distributions; for a study with a state-dependent random horizon, see [8].…”

A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems

“…However, when solving HJB equation, the monotonicity also plays a key role to ensure the convergence of the numerical scheme toward the viscosity solution. Indeed in high dimensional Merton's control problem, the matrix in the diffusion part is lower rank near the origin and it has been found in Be ´ne ´zet et al (2019) and Henderson et al (2020) that the standard finite difference schemes become non monotone and may not converge to the viscosity solution of the HJB. To solve the degeneracy issue, a fitted finite volume have been proposed in Dleuna Nyoumbi and Tambue (2021) for one and two dimensional optimal control problems.…”