Optimal selling rules in a regime-switching exponential Gaussian diffusion model

This paper develops optimal selling rules in asset trading using a regime-switching exponential Gaussian diffusion model. The optimization problem is solved by a combined approach of boundary value problems and probabilistic analysis. A system of linear differential equations with variable coefficients and two-point boundary conditions, satisfied by the objective function of the problem, is derived. The existence and uniqueness of the solution are proved. A closed-form solution in terms of Weber functions is obtained for one-dimensional cases. For m-dimensional cases, a stochastic recursive algorithm for numerically searching the optimal value is developed. Numerical results are reported

Optimal stopping problems with regime switching: a viscosity solution method

We employ the viscosity solution technique to analyze optimal stopping problems with regime switching. Specifically, we obtain the viscosity property of value functions, the uniqueness of viscosity solutions, the regularity of value functions and the form of optimal stopping intervals. Finally, we provide an application of the results.Comment: 29 pages, 1 figur

Optimal Strategies for Round-Trip Pairs Trading Under Geometric Brownian Motions

This paper is concerned with an optimal strategy for simultaneously trading a pair of stocks. The idea of pairs trading is to monitor their price movements and compare their relative strength over time. A pairs trade is triggered by the divergence of their prices and consists of a pair of positions to short the strong stock and to long the weak one. Such a strategy bets on the reversal of their price strengths. A round-trip trading strategy refers to opening and closing such a pair of security positions. Typical pairs-trading models usually assume a difference of the stock prices satisfies a mean-reversion equation. However, we consider the optimal pairs-trading problem by allowing the stock prices to follow general geometric Brownian motions. The objective is to trade the pairs over time to maximize an overall return with a fixed commission cost for each transaction. Initially, we allow the initial pairs position to be either long or flat. We then consider the problem when the initial pairs position may be long, flat, or short. In each case, the optimal policy is characterized by threshold curves obtained by solving the associated HJB equations.Comment: 47 pages, 5 figure

Upper and lower solutions for regime-switching diffusions with applications in financial mathematics

This paper develops a method of upper and lower solutions for a general system of second-order ordinary differential equations with two-point boundary conditions. Our motivation of study stems from a class of financial mathematics problems under regime-switching diffusion models. Two examples are double barrier option valuation and optimal selling rules in asset trading. We establish the existence of a unique C2 solution of the two-point boundary value problem. We construct monotone sequences of upper and lower solutions that are shown to converge to the unique solution of the boundary value problem. This construction provides a feasible numerical method to compute approximate solutions. An important feature of the proposed numerical method is that the unique solution is bracketed by the upper and lower approximate solutions, which provide an interval estimate of the unique solution function. We apply the general results to a regime-switching mean-reverting model and improve related results already reported in the literature. For the mean-reverting model, explicit upper and lower solutions are obtained and numerical integration methods are employed. In another case (Example 3 in section 5) a different regime-switching model is considered, where the general results apply, but only the upper solution is explicitly obtained. In that example, only the sequence of upper solutions is numerically constructed using finite difference methods. Numerical results are reported

Discrete Ornstein-Uhlenbeck process in a stationary dynamic enviroment

The thesis is devoted to the study of solutions to the following linear recursion: \beq X_{n+1}=\gamma X_n+ \xi_n, \feq where $\gamma \in (0,1)$ is a constant and (\xi_n)_{n\in\zz} is a stationary and ergodic sequence of normal variables with \emph{random} means and variances. More precisely, we assume that \beq \xi_n=\mu_n+\sigma_n\veps_n, \feq where (\veps)_{n\in\zz} is an i.i.d. sequence of standard normal variables and (\mu_n,\sigma_n)_{n\in\zz} is a stationary and ergodic process independent of (\veps_n)_{n\in\zz}, which serves as an exogenous dynamic environment for the model. This is an example of a so called SV (stands for stochastic variance or stochastic volatility) time-series model. We refer to the stationary solution of this recursion as a discrete Ornstein-Uhlenbeck process in a stationary dynamic environment. \par The solution to the above recursion is well understood in the classical case, when $\xi_n$ form an i.i.d. sequence. When the pairs mean and variance form a two-component finite-state Markov process, the recursion can be thought as a discrete-time analogue of the Langevin equation with regime switches, a continuous-time model of a type which is widely used in econometrics to analyze financial time series. \par In this thesis we mostly focus on the study of general features, common for all solutions to the recursion with the innovation/error term $\xi_n$ modulated as above by a random environment $(\mu_n,\sigma_n),$ regardless the distribution of the environment. In particular, we study asymptotic behavior of the solution when $\gamma$ approaches $1.$ In addition, we investigate the asymptotic behavior of the extreme values $M_n=\max_{1\leq k\leq n} X_k$ and the partial sums $S_n=\sum_{k=1}^n X_k.$ The case of Markov-dependent environments will be studied in more detail elsewhere. \par The existence of general patterns in the long-term behavior of $X_n,$ independent of a particular choice of the environment, is a manifestation of the universality of the underlying mathematical framework. It turns out that the setup allows for a great flexibility in modeling yet maintaining tractability, even when is considered in its full generality. We thus believe that the model is of interest from both theoretical as well as practical points of views; in particular, for modeling financial time series

Localized kernel-based approximation for pricing financial options under regime switching jump diffusion model

In this paper, we consider European and American option pricing problems under regime switching jump diffusion models which are formulated as a system of partial integro-differential equations (PIDEs) with fixed and free boundaries. For free boundary problem arising in pricing American option, we use operator splitting method to deal with early exercise feature of American option. For developing a numerical technique we employ localized radial basis function generated finite difference (RBF-FD) approximation to overcome the ill-conditioning and high density issues of discretized matrices. The proposed method leads to linear systems with tridiagonal and diagonal dominant matrices. Also, in this paper the convergence and consistency of the proposed method are discussed. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options. Published by Elsevier B.V. on behalf of IMACS