On the pricing of double barrier options under stochastic volatility models: A probabilistic approach

We study the pricing of double barrier knock-out options under stochastic volatility using a conditional Monte Carlo method based on the local time-space formula of Peskir. A valuation formula including an early knock-out discount is provided, where the discount depends on the local time of the underlying stochastic processes and the deltas of the option at the barriers. The latter solve a system of coupled Volterra integral equations of the first kind. This characterization leads to an efficient numerical method for general volatility diffusion models. An algorithm for numerical implementation, based on a conditional quasi-Monte Carlo simulation method, is presented and shown to converge numerically to the true value of the claim. A numerical study is performed to illustrate properties of double barrier knock-out calls in the Heston stochastic volatility model. In our calibration, we find that short-dated (long-dated) at-the-money (ATM) knock-out call prices increase (decrease) when the speed of mean reversion increases, long run mean volatility increases, and vol-of-vol decreases. We also find that the deltas and vegas of short-dated ATM double barrier calls can be very sensitive to volatility in contrast to long-dated ones.