Barrier option pricing and hedging model under stochastic conditions

The application of barrier options is wide and diverse. This article formulates a pricing and hedging framework for barrier options within a stochastic environment, where the dynamics of interest rates are captured through the application of the Vasicek model. We further utilize the Dubins-Schwarz theorem and Girsanov theorem to transform the measure. Using the joint distribution of Brownian motion, the analytical solutions of the model are obtained. Finally, this article provides explicit solutions for barrier option price, delta and gamma under random interest rate conditions. A comparative analysis between the PDE method and our method validates our solution. In the numerical analysis, we take the up-and-out call option as an example to analyse the sensitivities of barrier level and volatility of the barrier asset, then discuss the impact of the relativity of the barrier asset and no-default zero-coupon on the price. This conclusion gives different inspirations for hedging.