Appendix - Variance: The Standard Measure of Deviation
Derivatives-based Portfolio Solutions
VARIANCE, NOT VOLATILITY, IS THE APPROPRIATE MEASURE FOR DEVIATIONThere are three reasons why variance, not volatility, should be used as the correct measure for volatility. However, despite these reasons, even variance swaps are normally quoted as the square root of variance for an easier comparison with the implied volatility of options (but we note that skew and convexity mean the fair price of variance should always trade above ATM options). - Variance takes into account implied volatility at all stock prices. Variance takes into account the implied volatility of all strikes with the same expiry (while ATM implied will change with spot, even if volatility surface does not change).
- Deviations need to be squared to avoid cancelling. Mathematically, if deviations were simply summed then positive and negative deviations would cancel. This is why the sum of squared deviations is taken (variance) to prevent the deviations from cancelling. Taking the square root of this sum (volatility) should be considered a derivative of this pure measure of deviation (variance).
- Profit from a delta-hedged option depends on the square of the return. Due to the convexity of an option, if the volatility is doubled, the profits from delta hedging are multiplied by a factor of four. For this reason, variance (which looks at squared returns) is a better measure of deviation than volatility
VARIANCE TAKES INTO ACCOUNT VOLATILITY AT ALL STOCK PRICESWhen looking at how rich or cheap options with the same maturity are, rather than looking at the implied volatility for a certain strike (ie, ATM or another suitable strike) it is better to look at the implied variance as it takes into account the implied volatility of all strikes. For example, if an option with a fixed strike that is initially ATM is bought, then as soon as spot moves it is no longer ATM. However, if a variance swap (or log contract of options in the absence of a variance swap market) is bought, then its traded level is applicable no matter what the level of spot. The fact a variance swap (or log contract) payout depends only on the realised variance and is not path dependent makes it the ideal measure for deviation. PROFIT FROM DELTA HEDGING PROPORTIONAL TO RETURN SQUAREDAssuming constant volatility, zero interest rates and dividend, the daily profit and loss (P&L) from delta hedging an option is given below: Delta-hedged P&L from option = S ^{2}γ /2 + cost term where: γ = gamma VOLATILITY SHOULD BE CONSIDERED A DERIVATIVE OF VARIANCEThe three examples above show why variance is the natural measure for deviation. Volatility, the square root of variance, should be considered a derivative of variance rather than a pure measure of deviation. It is variance, not volatility, that is the second moment of a distribution (the first moment is the forward or expected price). For more details on moments, read the section How to Measure Skew and Smile. VIX AND VDAX MOVED FROM OLD ATM CALCULATION TO VARIANCEDue to the realisation that variance, not volatility, was the correct measure of deviation, on Monday, September 22, 2003, the VIX index moved away from using ATM implied towards a variance-based calculation. Variance-based calculations have also been used for by other volatility index providers. The old VIX, renamed VXO, took the implied volatility for strikes above and below spot for both calls and puts. As the first two-month expiries were used, the old index was an average of eight implied volatility measures as 8 = 2 (strikes) × 2 (put/call) × 2 (expiry). We note that the use of the first two expiries (excluding the front month if it was less than eight calendar days) meant the maturity was on average 1.5 months, not one month as for the new VIX. Similarly, the VDAX index, which was based on 45-day ATM-implied volatility, has been superseded by the V1X index, which, like the new VIX, uses a variance swap calculation. All recent volatility indices, such as the vStoxx (V2X), VSMI (V3X), VFTSE, VNKY and VHSI, use a variance swap calculation, although we note the recent VIMEX index uses a similar methodology to the old VIX (potentially due to illiquidity of OTM options on the Mexbol index). VARIANCE TERM STRUCTURE IS NOT ALWAYS FLATWhile average variance term structure should be flat in theory, in practice supply and demand imbalances can impact variance term structure. The buying of protection at the long end should mean that variance term structure is on average upward sloping, but in turbulent markets it is usually inverted. |

### Variance as the Standard

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