We study the regularity of the stochastic representation of the solution of a class of initial-boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.

We consider a pair (X, Y ) of stochastic processes satisfying the equation dX = a(X)Y dB driven by a Brownian motion and study the monotonicity and continuity in y of the value function v(x, y) = sup τ Ex,y[e −qτ g(Xτ )], where the supremum is taken over stopping times with respect to the filtration generated by (X, Y ). Our results can successfully be applied to pricing American options where X is the discounted price of an asset while Y is given by a stochastic volatility model such as those proposed by Heston or Hull and White. The main method of proof is based on time-change and coupling.

We characterize the price of an Asian option, a financial contract, as a fixedpoint of a non-linear operator. In recent years, there has been interest in incorporating changes of regime into the parameters describing the evolution of the underlying asset price, namely the interest rate and the volatility, to model sudden exogenous events in the economy. Asian options are particularly interesting because the payoff depends on the integrated asset price. We study the case of both floating-and fixed-strike Asian call options with arithmetic averaging when the asset follows a regime-switching geometric Brownian motion with coefficients that depend on a Markov chain. The typical approach to finding the value of a financial option is to solve an associated system of coupled partial differential equations. Alternatively, we propose an iterative procedure that converges to the value of this contract with geometric rate using a classical fixed-point theorem.Suppose that under P , the underlying asset price follows the regime-switching geometric Brownian motionwhere r(i) > 0 and σ(i) > 0 denote the risk-free interest rate and the volatility at regime i, respectively, and δ ≥ 0 is the dividend rate. Denote by F t the sigma-algebra generated by {(X u , Y u ) : 0 ≤ u ≤ t}. Throughout this paper we fix a time t 0 ∈ [0, T ), and define the integrated process

We study the problem of utility maximization from terminal wealth in which an agent optimally builds her portfolio by investing in a bond and a risky asset. The asset price dynamics follow a diffusion process with regime-switching coefficients modeled by a continuous-time finite-state Markov chain. We consider an investor with a Constant Relative Risk Aversion (CRRA) utility function. We deduce the associated Hamilton-Jacobi-Bellman equation to construct the solution and the optimal trading strategy and verify optimality by showing that the value function is the unique constrained viscosity solution of the HJB equation. By means of a Laplace transform method, we show how to explicitly compute the value function and illustrate the method with the two-and three-states cases. This method is interesting in its own right and can be adapted in other applications involving hybrid systems and using other types of transforms with basic properties similar to the Laplace transform.

A variable annuity is an equity-linked financial product typically offered by insurance companies. The policyholder makes an upfront payment to the insurance company and, in return, the insurer is required to make a series of payments starting at an agreed upon date. For a higher premium, many insurance companies offer additional guarantees or options which protect policyholders from various market risks. This research is centered around two of these options: the guaranteed minimum income benefit (GMIB) and the reset option. The sensitivity of various parameters on the value of the GMIB is explored, particularly the guaranteed payment rate set by the insurer. Additionally, a critical value for future interest rates is calculated to determine the rationality of exercising the reset option. This will be able to provide insight to both the policyholder and policy writer on how their future projections on the performance of the stock market and interest rates should guide their respective actions of exercising and pricing variable annuity options. This can help provide details into the value of adding options to a variable annuity for companies that are looking to make variable annuity policies more attractive in a competitive market.