Pricing Bounds for Volatility Derivatives via Duality and Least Squares Monte Carlo

Abstract

Derivatives on the Chicago Board Options Exchange volatility index have gained significant popularity over the last decade. The pricing of volatility derivatives involves evaluating the square root of a conditional expectation which cannot be computed by direct Monte Carlo methods. Least squares Monte Carlo methods can be used, but the sign of the error is difficult to determine. In this paper, we propose a new model-independent technique for computing upper and lower pricing bounds for volatility derivatives. In particular, we first present a general stochastic duality result on payoffs involving convex (or concave) functions. This result also allows us to interpret these contingent claims as a type of chooser options. It is then applied to volatility derivatives along with minor adjustments to handle issues caused by the square root function. The upper bound involves the evaluation of a variance swap, while the lower bound involves estimating a martingale increment corresponding to its hedging portfolio. Both can be achieved simultaneously using a single linear least square regression. Numerical results show that the method works very well for futures, calls and puts under a wide range of parameter choices.