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.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (03) ◽  
pp. 799-817 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (3) ◽  
pp. 799-817 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

scholarly journals Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes

2007 ◽  
Vol 44 (03) ◽  
pp. 713-731 ◽  
Author(s):  
Pavel V. Gapeev
Keyword(s):  
Optimal Stopping ◽  
Diffusion Processes ◽  
Free Boundary Problems ◽  
Compound Poisson Process ◽  
Jump Diffusion ◽  
Dimensional System ◽  
Closed Form Solutions ◽  
Jump Diffusion Processes ◽  
Lookback Options ◽  
Jump Diffusion Model

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

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 Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes

2007 ◽  
Vol 44 (3) ◽  
pp. 713-731 ◽  
Author(s):  
Pavel V. Gapeev
Keyword(s):  
Optimal Stopping ◽  
Diffusion Processes ◽  
Free Boundary Problems ◽  
Compound Poisson Process ◽  
Jump Diffusion ◽  
Dimensional System ◽  
Closed Form Solutions ◽  
Jump Diffusion Processes ◽  
Lookback Options ◽  
Jump Diffusion Model

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (03) ◽  
pp. 799-817
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (03) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (03) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

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