A Correction to the CBOE VIX Calculation Formula

Abstract

The volatility index (VIX) documented in the Cboe white paper (The Cboe Volatility Index, 2003-2020) involves the use of quoted option prices to approximate implied variances for the S&P 500 index. Cboe uses the average price of both call and put options at the strike that is just below or at the forward price of the underlying. This procedure overestimates implied variances that are used to compute the VIX index. The error is less noticeable when quoted strikes around forward index value are significantly close to each other such as in the case of the Cboe 30-day VIX index, but becomes larger as strikes are further apart from each other as in the Cboe VVIX index (the VIX of VIX). We show that better accuracy can be achieved by using only the put price at the strike , and replacing the strike price used in the adjustment term with the forward price or the average of the two adjacent strikes that are above and below the forward price. Whatever error remaining is purely due to numerical integration. We identified two types of numerical integration errors, one from the curvature of options prices and the other from the kink that joins the put and call segments of out-of-money options at the forward price. We proposed methods to reduce both type of errors.