Multi-step double barrier options under time-varying interest rates

Double barrier options are popular in the over-the-counter market due to their flexible investment strategies, opportunities to capitalize on volatility, and potential for increased leverage and more significant price movements, enhancing possible payoffs by incorporating two barrier levels. Multi-step double barrier options are particularly useful since they allow investors to set the barrier levels in a flexible manner while they are computationally efficient due to the explicit pricing formulas. In our study, we propose a method for pricing multi-step double barrier options under time-varying interest rates, acknowledging the potential unrealistic nature of employing a constant interest rate in economic scenarios marked by frequent adjustments in central bank monetary policies, such as during the COVID-19 pandemic. The method we employ to introduce a time-varying feature to the interest rate entails incorporating random jumps at various time points as needed. Our setup allows us to incorporate jumps not only in the interest rate dynamics but also in the asset price, so we can utilize jumps to more comprehensively depict the random nature of underlying price movement.

This paper derives the explicit pricing formula for the multi-step double barrier options with an arbitrary European-style payoff and obtains the non-crossing probability for the multi-step double boundaries of a Brownian motion with piecewise constant drift. We include multi-step double barrier put/call option prices when both the interest rate and the underlying asset jump. Also, our results are illustrated by some numerical examples showing the effect of different jump sizes of interest rates and the underlying asset price.