Optimal robust bounds for variance options

Robust, or model-independent properties of the variance swap are wellknown, and date back to Dupire [19] and Neuberger [37], who showed that, given the price of co-terminal call options, the price of a variance swap was exactly specified under the assumption that the price process is continuous. In Cox and Wang [11] we showed that a lower bound on the price of a variance call could be established using a solution to the Skorokhod embedding problem due to Root [45]. In this paper, we provide a construction, and a proof of optimality of the upper bound, using results of Rost [46] and Chacon [9], and show how this proof can be used to determine a super-hedging strategy which is model-independent. In addition, we outline how the hedging strategy may be computed numerically. Using these methods, we also show that the Heston-Nandi model is ‘asymptotically extreme’ in the sense that, for large maturities, the Heston-Nandi model gives prices for variance call options which are approximately the lowest values consistent with the same call price data.