Realized Volatility and Variance: Options via Swaps

Let St denote the value of a stock or stock index at time t. Given a variance/volatility option to be priced and hedged, let us designate as time 0 the start of its averaging period, and time T the end. For t ∈ [0, T ], and τ ≤ t, let R2 τ,t denote the realized variance of returns over the time interval [τ, t]. The mathematical results about the synthesis of volatility and variance swaps will hold exactly if R2 refers to the continuously-monitored variance. This means the quadratic variation of logS, ∗Bloomberg L.P. and NYU Courant Institute pcarr@nyc.rr.com. †University of Chicago. RL@math.uchicago.edu. We thank the International Securities Exchange, Kris Monaco,