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 (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.

Discounted optimal stopping problems for the maximum process

2000 ◽  
Vol 37 (4) ◽  
pp. 972-983 ◽  
Author(s):  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Stopping Times ◽  
Value Functions ◽  
Black Scholes Model ◽  
Lookback Options ◽  
Black Scholes ◽  
Maximum Process ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

Discounted optimal stopping problems for the maximum process

2000 ◽  
Vol 37 (04) ◽  
pp. 972-983 ◽  
Author(s):  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Stopping Times ◽  
Value Functions ◽  
Black Scholes Model ◽  
Lookback Options ◽  
Black Scholes ◽  
Maximum Process ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

scholarly journals Discrete-type approximations for non-Markovian optimal stopping problems: Part I

2019 ◽  
Vol 56 (4) ◽  
pp. 981-1005 ◽  
Author(s):  
Dorival Leão ◽  
Alberto Ohashi ◽  
Francesco Russo
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Rates Of Convergence ◽  
Stopping Times ◽  
Full Generality ◽  
Optimal Values ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Discrete Type

AbstractWe present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

scholarly journals On Asymptotics of Optimal Stopping Times

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 194
Author(s):  
Hugh N. Entwistle ◽  
Christopher J. Lustri ◽  
Georgy Yu. Sofronov
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Upper Bound ◽  
Stopping Time ◽  
Stopping Times ◽  
Independent Random Variables ◽  
Algebraic Form ◽  
Asymptotic Expressions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

Optimal stopping problems with generalized objective functions

1990 ◽  
Vol 27 (04) ◽  
pp. 828-838
Author(s):  
T. P. Hill ◽  
D. P. Kennedy
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Objective Functions ◽  
Monotone Functions ◽  
Optimal Stopping Problems ◽  
Universal Bounds ◽  
Optimal Stopping Times ◽  
Entire Vector

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions ofEXt, wheretis a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··,E[XnI{t=n}]),such as the minimax objective to maximize minj{E[XjI{t=j}]}.Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.

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.

On Optimal Stopping of Inhomogeneous Standard Markov Processes

1995 ◽  
Vol 2 (4) ◽  
pp. 335-346
Author(s):  
B. Dochviri
Keyword(s):  
Markov Process ◽  
Optimal Stopping ◽  
Value Function ◽  
Stopping Times ◽  
Homogeneous Markov Process ◽  
Limit Procedure ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
The Value Function

Abstract The connection between the optimal stopping problems for inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. An excessive characterization of the value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous standard Markov process is given. The form of ε-optimal (optimal) stopping times is also found.

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.

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