scholarly journals On optimal stopping problems with positive discounting rates and related Laplace transforms of first hitting times in models with geometric Brownian motions

2021 ◽  
Keyword(s):  
Optimal Stopping ◽  
Laplace Transforms ◽  
Hitting Times ◽  
Brownian Motions ◽  
Optimal Stopping Problems ◽  
Geometric Brownian Motions ◽  
First Hitting Times

scholarly journals Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

2021 ◽  
Vol 53 (1) ◽  
pp. 189-219
Author(s):  
Pavel V. Gapeev ◽  
Peter M. Kort ◽  
Maria N. Lavrutich
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Sunk Costs ◽  
Boundary Problems ◽  
Closed Form Solutions ◽  
Lookback Options ◽  
Normal Reflection ◽  
Running Maximum ◽  
Optimal Stopping Problems ◽  
Geometric Brownian Motions

AbstractWe present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

First Passage Times of Constant-Elasticity-of-Variance Processes with Two-Sided Reflecting Barriers

2012 ◽  
Vol 49 (04) ◽  
pp. 1119-1133
Author(s):  
Lijun Bo ◽  
Chen Hao
Keyword(s):  
Laplace Transforms ◽  
First Passage Times ◽  
Square Root ◽  
First Passage ◽  
Brownian Motions ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Geometric Brownian Motions ◽  
Reflecting Barriers ◽  
Passage Times

In this paper we explore the first passage times of constant-elasticity-of-variance (CEV) processes with two-sided reflecting barriers. The explicit Laplace transforms of the first passage times are derived. Our results can include analytic formulae concerning Laplace transforms of first passage times of reflected Ornstein–Uhlenbeck processes, reflected geometric Brownian motions, and reflected square-root processes.

scholarly journals Perpetual American Cancellable Standard Options in Models with Last Passage Times

Algorithms ◽  
2020 ◽  
Vol 14 (1) ◽  
pp. 3
Author(s):  
Pavel V. Gapeev ◽  
Libo Li ◽  
Zhuoshu Wu
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Asset Price ◽  
Stopping Times ◽  
Hitting Times ◽  
Explicit Solutions ◽  
Boundary Problems ◽  
Price Process ◽  
Optimal Stopping Problems ◽  
Passage Times

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.

scholarly journals First Passage Times of Constant-Elasticity-of-Variance Processes with Two-Sided Reflecting Barriers

2012 ◽  
Vol 49 (4) ◽  
pp. 1119-1133 ◽  
Author(s):  
Lijun Bo ◽  
Chen Hao
Keyword(s):  
Laplace Transforms ◽  
First Passage Times ◽  
Square Root ◽  
First Passage ◽  
Brownian Motions ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Geometric Brownian Motions ◽  
Reflecting Barriers ◽  
Passage Times

In this paper we explore the first passage times of constant-elasticity-of-variance (CEV) processes with two-sided reflecting barriers. The explicit Laplace transforms of the first passage times are derived. Our results can include analytic formulae concerning Laplace transforms of first passage times of reflected Ornstein–Uhlenbeck processes, reflected geometric Brownian motions, and reflected square-root processes.

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (1) ◽  
pp. 66-73 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Maximal Inequality ◽  
Stopping Times ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Real Analysis ◽  
Optimal Stopping Problems ◽  
Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

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