Realized Volatility Options

We will in some places restrict attention to puts, by put-call parity: for realized variance options, a long-call short-put combination pays [X]T −Q, equal to a Q-strike variance swap; and for realized volatility options, a long-call short-put combination pays [X] T − Q1/2, equal to a Q1/2-strike volatility swap. Unlike variance swaps [EQF07/024, EQF07/045], which admit exact model-free (assuming only continuity of Y ) hedging and pricing in terms of Europeans, variance and volatility options have a range of values consistent with given prices of Europeans. With no further assumptions, there exist sub/super-replication strategies and lower/upper pricing bounds (section 4). Under an independence condition, there exist exact pricing formulas in terms of Europeans (section 2). Under specific models, there exist exact pricing formulas in terms of model parameters (section 1). Unless otherwise noted, all prices are denominated in units of a T -maturity discount bond. The results apply to dollar-denominated prices, provided that interest rates vary deterministically, because if Y ′ is a dollar-denominated share price and Y is that share’s bond-denominated price, then log Y − log Y ′ has finite variation, so [log Y ] = [log Y ′]. Expectations E will be with respect to martingale measure P.