Arithmetic variance swaps

Abstract

Biases in standard variance swap rates can induce substantial deviations below market rates. Defining realised 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 variance swap rate: 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 with the standard variance swap rate 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.

Risk‐neutral skewness and commodity futures pricing

“…Since BKM estimators are weighted by the underlying's squared or cubed strike price, it might cause estimation bias, especially during the illiquid period in which the call options part will deteriorate, and the put options part will be overstated. Put options price increases rapidly when market exception falls in downside way, resulting in more negative value in estimation (Kozhan et al, 2013; Leontsinis & Alexander, 2017). To account for the jump, discrete, and downside risks (errors) under the BKM method, Kozhan et al (2013) present a new estimation measure.…”

“…Suppose there is a contract with exact time‐to‐maturity equal to 30, the corresponding RNSK value is used directly; otherwise, Hermite cubic spline is used to interpolate constant maturity RNSK. The Hermite cubic spline method is argued to account for calendar arbitrage issue and nonlinear trend for long‐maturity data fitting and provides excellent shape‐preserving merit (Leontsinis & Alexander, 2017).…”

A general property for time aggregation

“…the discretisation error (the formula assumes the RV is monitored continuously) and the jump error (the integral is derived by assuming F t follows a diffusion -not necessarily geometric Brownian motion). However, the fair value of an arithmetic variance swap, for which the realised characteristic is based on (1), has neither jump nor discretisation bias [Leontsinis and Alexander, 2017]. Indeed, any swap having a realised characteristic which satisfies the AP has neither discretisation error nor jump error.…”