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 $v(x,y)=\sup_{\tau}E_{x,y}[e^{-q\tau}g(X_{\tau})]$, 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.

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.