  This paper extends the valuation and optimal surrender framework for variable
annuities with guaranteed minimum benefits in a L\'evy equity market
environment by incorporating a stochastic interest rate described by the
Hull-White model. This approach frames a more dynamic and realistic financial
setting compared to previous literature. We exploit a robust valuation
mechanism employing a hybrid numerical method that merges tree methods for
interest rate modeling with finite difference techniques for the underlying
asset price. This method is particularly effective for addressing the
complexities of variable annuities, where periodic fees and mortality risks are
significant factors. Our findings reveal the influence of stochastic interest
rates on the strategic decision-making process concerning the surrender of
these financial instruments. Through comprehensive numerical experiments, and
by comparing our results with those obtained through the Longstaff-Schwartz
Monte Carlo method, we illustrate how our refined model can guide insurers in
designing contracts that equitably balance the interests of both parties. This
is particularly relevant in discouraging premature surrenders while adapting to
the realistic fluctuations of financial markets. Lastly, a comparative statics
analysis with varying interest rate parameters underscores the impact of
interest rates on the cost of the optimal surrender strategy, emphasizing the
importance of accurately modeling stochastic interest rates.
