Arithmetic variance swaps

Abstract

Biases in standard variance swap rates (VSRs) can induce substantial deviations below market rates. Defining realized variance as the sum of squared price (not log-price) changes yields an ‘arithmetic’ variance swap with no such biases. Its fair value has advantages over the standard VSR: no discrete monitoring or jump biases; and the same value applies for any monitoring frequency, even irregular monitoring and to any underlying, including those taking zero or negative values. We derive the fair value for the arithmetic variance swap and compare it with the standard VSR by: analysing errors introduced by interpolation and integration techniques; numerical experiments for approximation accuracy; and using 23 years of FTSE 100 options data to explore the empirical properties of arithmetic variance (and higher moment) swaps. The FTSE 100 variance risk has a strong negative correlation with the implied third moment, which can be captured using a higher moment arithmetic swap.