Integral Options in Models with Jumps

We present an explicit solution to the formulated in [17] optimal stopping problem for a geometric compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the smooth fit may break down and then be replaced by the continuous fit. The result can be interpreted as pricing perpetual integral options in a model with jumps.Jump process, stochastic differential equation, optimal stopping problem, integral American option, compound Poisson process, Shiryaev´s process, Girsanov´s theorem, Ito´s formula, integrodifferential free-boundary problem, smooth and continuous fit, hypergeometric functions

Multiple Disorder Problems for Wiener and Compound Poisson Processes With Exponential Jumps

The multiple disorder problem consists of finding a sequence of stopping times which are as close as possible to the (unknown) times of "disorder" when the distribution of an observed process changes its probability characteristics. We present a formulation and solution of the multiple disorder problem for a Wiener and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial optimal switching problems to the corresponding coupled optimal stopping problems and solving the equivalent coupled free-boundary problems by means of the smooth- and continuous-fit conditions.Multiple disorder problem, Wiener process, compound Poisson process, optimal switching, coupled optimal stopping problem, (integro-differential) coupled free-boundary problem, smooth and continuous fit, Ito-Tanaka-Meyer formula.

Discounted Optimal Stopping for Maxima of some Jump-Diffusion Processes

We present solutions to 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 may break down and the latter then be replaced by the continuous fit. The results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.Discounted optimal stopping problem, Brownian motion, compound Poisson process, maximum process, integro-differential free-boundary problem, continuous and smooth fit, normal reflection, a change-of-variable formula with local time on surfaces, perpetual lookback American options

On Maximal Inequalities for some Jump Processes

We present a solution to the considered in [5] and [22] optimal stopping problem for some jump processes. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The derived result is applied for determining the best constants in maximal inequalities for a compound Poisson process with linear drift and exponential jumps.Jump process, stochastic differential equation, maximum process, optimal stopping problem, compound Poisson process, Ito’s formula, integro-differential free-boundary problem, normal reflection, continuous and smooth fit, maximality principle, maximal inequalities

Optimal stopping problems in diffusion-type models with running maxima and drawdowns

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 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

Perpetual Barrier Options in Jump-Diffusion Models

We present a closed form solution to the perpetual American double barrier call option problem 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 irregular optimal stopping problem to an integro-differential free-boundary problem and solving the latter by using continuous and smooth fit. The obtained solution of the nontrivial free-boundary problem gives the possibility to observe some special analytic properties of the value function at the optimal stopping boundaries.American double barrier options, optimal stopping problem, jump-diffusion model, integro-differential free-boundary problem, continuous and smooth fit, Ito-Tanaka-Meyer formula

Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon

We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon.Discounted optimal stopping problem, finite horizon, geometric Brownian motion, maximum process, parabolic free-boundary problem, smooth fit, normal reflection, a nonlinear Volterra integral equation of the second kind, boundary surface, a change-of-variable formula with local time on surfaces, American lookback option problem

Nonadditive disorder problems for some diffusion processes

We study nonadditive Bayesian problems of detecting a change in drift of an observed diffusion process where the cost function of the detection delay has the same structure as in [27] and construct a finite-dimensional Markovian sufficient statistic for that case. We show that when the cost function is linear the optimal stopping time is found as the first time when the a posteriori probability process hits a stochastic boundary depending on the observation process. It is shown that under some nontrivial relationships on the coefficients of the observed diffusion the problem admits a closed form solution. The method of proof is based on embedding the initial problem into a two-dimensional optimal stopping problem and solving the equivalent free-boundary problem by means of the smooth-fit conditions

Discounted optimal stopping for maxima of some jump-diffusion processes

We present solutions to 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 may break down and the latter then be replaced by the continuous fit. The results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model