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

Abstract

We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American withdrawable standard and lookback put and call options in an extension of the Black-Merton-Scholes model with asymmetric information. It is assumed that the contracts are withdrawn by their writers at the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval which are not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 process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original necessarily two-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and normalreflection conditions. We prove that the optimal exercise boundaries are the maximal and minimal solutions of some first-order nonlinear ordinary differential equations.

Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions

“…Discounted optimal stopping problems for certain reward functionals depending on the running maxima and minima of continuous Markov (diffusion-type) processes were initiated by Shepp and Shiryaev (1993) and further developed by Pedersen (2000); Guo and Shepp (2001); Gapeev (2007); Guo and Zervos (2010); Peskir (2012Peskir ( , 2014; Glover et al (2013); Rodosthenous and Zervos (2017); Gapeev (2019Gapeev ( , 2020; ; Gapeev and Li (2021); Gapeev and Al Motairi (2021); Gapeev (2022) among others. The main feature in the analysis of such optimal stopping problems was that the normal-reflection conditions hold for the value functions at the diagonals of the state spaces of the multi-dimensional continuous Markov processes having the initial processes and the running extrema as their components.…”

“…Furthermore, by virtue of the asymptotic distributional properties of the running maximum S and minimum Q from (5) of the geometric Brownian motion X from ( 3)-(4) on the infinitesimally small time intervals (see, e.g. [Dubins et al (1993); Subsection 3.3] for similar arguments applied to the running maxima of the Bessel processes, [Peskir (1998); Proposition 2.1] for similar properties of the running maxima of general diffusion processes, and [Gapeev and Li (2021); Theorem 2.1, Part (i)] for similar properties of the running maxima and minima of geometric Brownian motions), it follows that the reward functionals in ( 21) and ( 22) infinitesimally increase when X t = Q t or X t = S t , for each t ≥ 0 . This fact also shows that all points (x, s, q) from the diagonals 23), respectively.…”

Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

“…Discounted optimal stopping problems for certain reward functionals depending on the running maxima and minima of continuous Markov (diffusion-type) processes were initiated by Shepp and Shiryaev (1993) and further developed by Pedersen Pedersen (2000), Guo and Shepp (2001), Gapeev (2007), Guo and Zervos (2010), Peskir (2012), Peskir (2014), Glover et al (2013), Rodosthenous and Zervos (2017), Gapeev (2019Gapeev ( , 2020, , Gapeev and Al Motairi (2021), Gapeev and Li (2021), and Gapeev et al (2022) among others. The main feature in the analysis of such optimal stopping problems was that the normal-reflection conditions hold for the value functions at the diagonals of the state spaces of the multi-dimensional continuous Markov processes having the initial processes and the running extrema as their components.…”

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.…”