Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes

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.

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Optimal insurance control for insurers with jump-diffusion risk processes

Annals of Actuarial Science ◽

10.1017/s1748499518000192 ◽

2018 ◽

Vol 13 (1) ◽

pp. 198-213

Author(s):

Linlin Tian ◽

Lihua Bai

Keyword(s):

Diffusion Processes ◽

Compound Poisson Process ◽

Linear Quadratic Regulator ◽

Jump Diffusion ◽

Analytic Solutions ◽

Linear Quadratic ◽

Discounted Cost ◽

Optimal Insurance ◽

Jump Diffusion Processes ◽

Risk Processes

AbstractIn this paper, we model the surplus process as a compound Poisson process perturbed by diffusion and allow the insurer to ask its customers for input to minimize the distance from some prescribed target path and the total discounted cost on a fixed interval. The problem is reduced to a version of a linear quadratic regulator under jump-diffusion processes. It is treated using three methods: dynamic programming, completion of square and the stochastic maximum principle. The analytic solutions to the optimal control and the corresponding optimal value function are obtained.

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PRICING OF FIRST TOUCH DIGITALS UNDER NORMAL INVERSE GAUSSIAN PROCESSES

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024906003834 ◽

2006 ◽

Vol 09 (06) ◽

pp. 915-949 ◽

Cited By ~ 13

Author(s):

OLEG KUDRYAVTSEV ◽

SERGEI LEVENDORSKIǏ

Keyword(s):

Diffusion Model ◽

Diffusion Processes ◽

Jump Diffusion ◽

Spot Price ◽

Inverse Gaussian ◽

Jump Diffusion Processes ◽

Double Exponential ◽

Motion Approximation ◽

Jump Diffusion Model ◽

Normal Inverse Gaussian

We calculate prices of first touch digitals under normal inverse Gaussian (NIG) processes, and compare them to prices in the Brownian model and double exponential jump-diffusion model. Numerical results are produced to show that for typical parameters values, the relative error of the Brownian motion approximation to NIG price can be 2–3 dozen percent if the spot price is at the distance 0.05–0.2 from the barrier (normalized to one). A similar effect is observed for approximations by the double exponential jump-diffusion model, if the jump component of the approximation is significant. We show that two jump-diffusion processes can give approximately the same results for European options but essentially different results for first touch digitals and barrier options. A fast approximate pricing formula under NIG is derived.

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On Optimal Stopping Problems for Matrix-Exponential Jump-Diffusion Processes

Journal of Applied Probability ◽

10.1239/jap/1339878803 ◽

2012 ◽

Vol 49 (2) ◽

pp. 531-548 ◽

Cited By ~ 4

Author(s):

Yuan-Chung Sheu ◽

Ming-Yao Tsai

Keyword(s):

Explicit Formula ◽

Optimal Stopping ◽

Diffusion Processes ◽

Jump Diffusion ◽

Optimal Level ◽

Matrix Exponential ◽

Call Type ◽

Jump Diffusion Processes ◽

Averaging Problem ◽

Optimal Stopping Problems

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

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First passage times of a jump diffusion process

Advances in Applied Probability ◽

10.1239/aap/1051201658 ◽

2003 ◽

Vol 35 (2) ◽

pp. 504-531 ◽

Cited By ~ 204

Author(s):

S. G. Kou ◽

Hui Wang

Keyword(s):

Diffusion Process ◽

Diffusion Processes ◽

Jump Diffusion ◽

First Passage Times ◽

First Passage ◽

Jump Diffusion Processes ◽

Lookback Options ◽

Double Exponential ◽

Jump Diffusion Process ◽

Passage Times

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

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Power Exchange Option with a Hybrid Credit Risk under Jump-Diffusion Model

Mathematics ◽

10.3390/math10010053 ◽

2021 ◽

Vol 10 (1) ◽

pp. 53

Author(s):

Junkee Jeon ◽

Geonwoo Kim

Keyword(s):

Credit Risk ◽

Structural Model ◽

Risk Model ◽

Diffusion Processes ◽

Jump Diffusion ◽

Reduced Form ◽

Jump Diffusion Processes ◽

Power Exchange ◽

Jump Diffusion Model ◽

Exchange Option

In this paper, we study the valuation of power exchange options with a correlated hybrid credit risk when the underlying assets follow the jump-diffusion processes. The hybrid credit risk model is constructed using two credit risk models (the reduced-form model and the structural model), and the jump-diffusion processes are proposed based on the assumptions of Merton. We assume that the dynamics of underlying assets have correlated continuous terms as well as idiosyncratic and common jump terms. Under the proposed model, we derive the explicit pricing formula of the power exchange option using the measure change technique with multidimensional Girsanov’s theorem. Finally, the formula is presented as the normal cumulative functions and the infinite sums.

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On Optimal Stopping Problems for Matrix-Exponential Jump-Diffusion Processes

Journal of Applied Probability ◽

10.1017/s0021900200009256 ◽

2012 ◽

Vol 49 (02) ◽

pp. 531-548

Author(s):

Yuan-Chung Sheu ◽

Ming-Yao Tsai

Keyword(s):

Explicit Formula ◽

Optimal Stopping ◽

Diffusion Processes ◽

Jump Diffusion ◽

Optimal Level ◽

Matrix Exponential ◽

Call Type ◽

Jump Diffusion Processes ◽

Averaging Problem ◽

Optimal Stopping Problems

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

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Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

Advances in Applied Probability ◽

10.1017/apr.2020.57 ◽

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.

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