scholarly journals Callable Russian Options and Their Optimal Boundaries

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Atsuo Suzuki ◽  
Katsushige Sawaki
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimal Stopping Problem ◽  
Dynkin Game ◽  
Russian Option ◽  
The Value Function

We deal with the pricing of callable Russian options. A callable Russian option is a contract in which both of the seller and the buyer have the rights to cancel and to exercise at any time, respectively. The pricing of such an option can be formulated as an optimal stopping problem between the seller and the buyer, and is analyzed as Dynkin game. We derive the value function of callable Russian options and their optimal boundaries.

scholarly journals American options in an imperfect complete market with default

2018 ◽  
Vol 64 ◽  
pp. 93-110 ◽  
Author(s):  
Roxana Dumitrescu ◽  
Marie-Claire Quenez ◽  
Agnès Sulem
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
American Options ◽  
American Option ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Nonlinear Expectation ◽  
Portfolio Strategy ◽  
Reflected Bsde ◽  
The Value Function

We study pricing and hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process (ξt). We define the seller's price of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with a nonlinear expectation 𝓔g (induced by a BSDE), which corresponds to the solution of a nonlinear reflected BSDE with obstacle (ξt). Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer's price of the American option, which is defined as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy φ so that he/she is superhedged. We show that the buyer's price is equal to the value function of an optimal stopping problem with a nonlinear expectation, and that it can be characterized via the solution of a reflected BSDE with obstacle (ξt). Under the additional assumption of left upper semicontinuity along stopping times of (ξt), we show the existence of a super-hedge (τ, φ) for the buyer.

THE SWING OPTION ON THE STOCK MARKET

2005 ◽  
Vol 08 (01) ◽  
pp. 123-139 ◽  
Author(s):  
MARTIN DAHLGREN ◽  
RALF KORN
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Additional Constraint ◽  
Numerical Implementation ◽  
Optimal Stopping Problem ◽  
Existence Of A Solution ◽  
Swing Option ◽  
Time Distance ◽  
The Value Function

The valuation of a Swing option for stocks under the additional constraint of a minimum time distance between two different exercise times is considered. We give an explicit characterization of its pricing function as the value function of a multiple optimal stopping problem. The solution of this problem is related to a system of variational inequalities. We prove existence of a solution to this system and discuss the numerical implementation of a valuation algorithm.

scholarly journals Dynkin game under g-expectation in continuous time

2020 ◽  
Vol 9 (2) ◽  
pp. 459-470
Author(s):  
Helin Wu ◽  
Yong Ren ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Saddle Points ◽  
Value Functions ◽  
Dynkin Game ◽  
The Value Function

Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (1) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

Reflected BSDEs when the obstacle is predictable and nonlinear optimal stopping problem

2021 ◽  
pp. 2150049
Author(s):  
Siham Bouhadou ◽  
Youssef Ouknine
Keyword(s):  
Optimal Stopping ◽  
Existence And Uniqueness ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Predictable Process ◽  
Optimal Stopping Theory ◽  
Reflected Bsdes

In the first part of this paper, we study RBSDEs in the case where the filtration is non-quasi-left-continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part, we introduce an optimal stopping problem indexed by predictable stopping times with the nonlinear predictable [Formula: see text] expectation induced by an appropriate backward stochastic differential equation (BSDE). We establish some useful properties of [Formula: see text]-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.

scholarly journals An Optimal Stopping Problem for Jump Diffusion Logistic Population Model

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Yang Sun ◽  
Xiaohui Ai
Keyword(s):  
Optimal Stopping ◽  
Critical State ◽  
Explicit Solution ◽  
Population Model ◽  
Value Function ◽  
Jump Diffusion ◽  
Optimal Value Function ◽  
Optimal Stopping Problem ◽  
Optimal Value ◽  
Poisson Jump

This paper examines an optimal stopping problem for the stochastic (Wiener-Poisson) jump diffusion logistic population model. We present an explicit solution to an optimal stopping problem of the stochastic (Wiener-Poisson) jump diffusion logistic population model by applying the smooth pasting technique (Dayanik and Karatzas, 2003; Dixit, 1993). We formulate this as an optimal stopping problem of maximizing the expected reward. We express the critical state of the optimal stopping region and the optimal value function explicitly.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (01) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

Optimal stopping with random intervention times

2002 ◽  
Vol 34 (01) ◽  
pp. 141-157 ◽  
Author(s):  
Paul Dupuis ◽  
Hui Wang
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Value Function ◽  
Value Functions ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary ◽  
Time Optimal ◽  
Optimal Stopping Problems ◽  
Necessary And Sufficient ◽  
The Value Function

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞 0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞 2 otherwise, both contradicting the usual 𝒞 1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

An Optimal Stopping Problem Arising from a Decision Model with Many Agents

1998 ◽  
Vol 12 (3) ◽  
pp. 393-408 ◽  
Author(s):  
Bruno Bassan ◽  
Claudia Ceci
Keyword(s):  
Markov Process ◽  
Stefan Problem ◽  
Optimal Stopping ◽  
Economic Model ◽  
Decision Model ◽  
Value Function ◽  
Lower Semicontinuous ◽  
Social Planner ◽  
Optimal Stopping Problem ◽  
Reward Function

We study an optimal stopping problem for a nonhomogeneous Markov process, with a reward function that is lower semicontinuous everywhere and smooth in certain regions. We prove that the payoff (value function) is lower semicontinuous as well and solves a so-called generalized Stefan problem in each of these regions. We provide some results for the geometry of the “stopping observations” set. Our results generalize those in Bassan, Brezzi, and Scarsini (1996). The problem we consider stems from an economic model in which several self-interested agents desire information, whereas a social planner, although benevolent toward the agents, might decide to withhold information in order to induce diversification in their behavior.

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