On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes (original) (raw)

Formulas for the Laplace Transform of Stopping Times based on Drawdowns and Drawups

2009

In this work we study drawdowns and drawups of general diffusion processes. The drawdown process is defined as the current drop of the process from its running maximum, while the drawup process is defined as the current increase over its running minimum. The drawdown and the drawup are the first hitting times of the drawdown and the drawup processes respectively. In particular, we derive a closed-form formula for the Laplace transform of the probability density of the drawdown of a units when it precedes the drawup of b units. We then separately consider the special case of drifted Brownian motion, for which we derive a closed form formula for the above-mentioned density by inverting the Laplace transform. Finally, we apply the results to a problem of interest in financial risk-management and to the problem of transient signal detection and identification of two-sided changes in the drift of general diffusion processes.

Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups

Stochastic Processes and Their Applications, 2009

This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size a. As a consequence, we obtain the probabilities that a drawdown of size a precedes a drawup of size b and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein-Uhlenbeck process, and Cox-Ingersoll-Ross process.

Diffusion transformations, Black–Scholes equation and optimal stopping

The Annals of Applied Probability, 2018

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf [Probab. Theory Related Fields 164 (2016) 1027-1069] as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation, we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an alternative Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing in the presence of bubbles. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.

On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process

Methodology and Computing in Applied Probability, 2010

It is studied the first-passage time (FPT) of a time homogeneous onedimensional diffusion, driven by the stochastic differential equation dX(t) = μ(X(t))dt + σ (X(t))dB t , X(0) = x 0 , through b + Y(t), where b > x 0 and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B t. In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B t , for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately the FPT density; some examples and numerical results are also reported.

3 Threshold Strategies in Optimal Stopping Problem for Diffusion Processes and

2016

We study a problem when a solution to optimal stopping problem for one-dimensional diffusion will generate by threshold strategy. Namely, we give necessary and sufficient conditions under which an optimal stopping time can be specified as the first time when the process exceeds some level (threshold), and a continuation set is a semi-interval. We give also second-order conditions, which allow to discard such solutions to free-boundary problem that are not the solutions to optimal stopping problem.

First hitting time of integral diffusions and applications

Stochastic Models, 2017

We study the first hitting time of integral functionals of time-homogeneous diffusions, and characterize their Laplace transforms through a stochastic time change. We obtain explicit expressions of the Laplace transforms for the geometric Brownian motion (GBM) and the mean-reverting GBM process. We also introduce a novel probability identity based on an independent exponential randomization and obtain explicit Laplace transforms of the price of arithmetic Asian options and other derivative prices that non-linearly depend on the integral diffusions. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.

Optimal Multiple Stopping of Linear Diffusions

Mathematics of Operations Research, 2008

Motivated by the analysis of financial instruments with multiple exercise rights of American type and mean reverting underlyers, we formulate and solve the optimal multiple-stopping problem for a general linear regular diffusion process and a general reward function. Instead of relying on specific properties of geometric Brownian motion and call and put option payoffs as in most of the existing literature, we use general theory of optimal stopping for diffusions, and we illustrate the resulting optimal exercise policies by concrete examples and constructive recipes.

Threshold Strategies in Optimal Stopping Problem for Diffusion Processes and Free-Boundary Problem

arXiv: Probability, 2013

We study a problem when a solution to optimal stopping problem for one-dimensional diffusion will generate by threshold strategy. Namely, we give necessary and sufficient conditions under which an optimal stopping time can be specified as the first time when the process exceeds some level (threshold), and a continuation set is a semi-interval. We give also second-order conditions, which allow to discard such solutions to free-boundary problem that are not the solutions to optimal stopping problem.

Variational View to Optimal Stopping Problems for Diffusion Processes and Threshold Strategies

arXiv: Probability, 2015

We describe a variational approach to solving optimal stopping problems for diffusion processes, as an alternative to the traditional approach based on the solution of the free-boundary problem. We study smooth pasting conditions from a variational point of view, and give some examples when the solution to free-boundary problem is not the solution to optimal stopping problem. A special attention is paid to threshold strategies which allow reduce optimal stopping problem to more simple one-parametric optimization. Necessary and sufficient conditions for threshold structure of optimal stopping time are derived. We apply these results to both investment timing and optimal abandon models.

Construction of the value function and optimal rules in optimal stopping of one-dimensional diffusions

Advances in Applied Probability, 2010

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 opti...