Pricing multiple barrier derivatives under stochastic volatility

Abstract

This work generalizes existing one- and two-dimensional pricing formulas with an equal number of barriers to a setting of n dimensions and up to two barriers in the presence of stochastic volatility. This allows for the consideration of multidimensional single-barrier derivatives with, for example, a collateral triggered by a barrier default of the issuing company. We introduce stochastic volatility to a multidimensional Black–Scholes framework via the common Cox–Ingersoll–Ross process and present semianalytical solutions for collateralized structured products with two barriers, one representing default and the other a market-related option. Our model accommodates implied volatility skew along the lines of displaced diffusions. We show that our semianalytical formulas are more efficient in terms of computational speed than Monte Carlo simulations, particularly for tail scenarios. Moreover, our proposed analytical simplifications permit a twenty-fivefold gain in time savings compared with the results given by the main theorem. These multidimensional structured products gained increasing popularity after the subprime and financial crises. Therefore, we perform comprehensive sensitivity analyses with respect to stochastic volatility parameters and contribute to a better understanding of multidimensional barrier derivatives in a stochastic volatility framework.