On the pricing of vulnerable Parisian options

Parisian options, serving as substitutes for barrier options, are frequently embedded in complex financial derivatives. Despite the considerable mathematical challenges in pricing, their significant applications in finance and insurance have generated extensive studies in the literature. However, the impact of counterparty risk has not been taken into account so far. To address this gap, we develop a closed-form pricing framework for vulnerable Parisian options. Utilizing the Laplace transform and measure-change technique, we derive closed-form pricing formulas of Parisian options incorporating counterparty credit risk. Finally, we conduct numerical analyses to verify our pricing formulas’ accuracy and efficiency.