Strike (%) of zero delta straddle = e^{(r + (σ.σ)/2).T}

As the delta of a portfolio is equal to the sum of the deltas of the securities in the portfolio, a position can be delta hedged by purchasing, or going short, a number of shares (or futures in the case of an index) equal to the delta. For example, if ten call options have been bought with a delta of 40%, then four shares (10 × 40% = 4) have to be shorted to create a portfolio of zero delta. The shares have to be shorted as a call option has positive delta; hence, the delta hedge has to be negative for the sum of the two positions to have zero delta. If we were long a put (which has negative delta), then we would have to buy stock to ensure the overall delta was zero.

As an investor who is long gamma can delta hedge by sitting on the bid and offer, this trade can pin an underlying to the strike. This is a side effect of selling if the stock rises above the strike, and buying if the stock falls below the strike. The amount of buying and selling has to be significant compared with the traded volume of the underlying, which is why pinning normally occurs for relatively illiquid stocks or where the position is particularly sizable  Given the high trading volume of indices, it is difficult to pin a major index. Pinning is more likely to occur in relatively calm markets, where there is no strong trend to drive the stock away from its pin.

To calculate option premiums and volatility exactly is typically too difficult to do without the aid of a tool. However, there are some useful rules of thumb that can be used to give an estimate. These are a useful sanity check in case an input to a pricing model has been entered incorrectly.

While a simple view on both volatility and equity market direction can be implemented via a long or short position in a call or put, a far wider set of payoffs is possible if two or three different options are used. We investigate strategies using option structures (or option combos) that can be used to meet different investor needs.

ATM option premium in percent is roughly (2 × π)^-1/2  × σ × t^1/2

Call price = S.N(d1) – K.N(d2) e-rT

ATM call price = S N(σ × √T / 2) – S N(-σ × √T / 2) as K=S (as ATM)

ATM call price = S × σ × √T / √(2π)

ATM call price = σ × √T / √(2π) in percent

ATM call price ≈ 0.4 × σ × √T in percent

^where:

We assume zero interest rates and dividends (r = 0)

^{Definition of d1 and d2 is the standard Black-Scholes formula.}

^{σ = implied volatility}

^{S = spot}

^{K = strike}

^{R = interest rate}

^{T = time to expiry}

^{N(z) = cumulative normal distribution}

<sup>OTM options can be calculated by assuming 50% delta
If an index is 3000 pts and has a 20% implied then the price of a 1y ATM option is approximately 240pts (3000×8% as per the formula above). A 3200 call is therefore approximately 240 - 50% (3200-3000) = 140 pts assuming a 50% delta. Similarly, a 3200 put is approximately 340 pts.

Profit from delta hedging is proportional to percentage move squared
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. Assuming constant volatility, zero interest rates and dividend, the daily profit and loss (P&L) from delta hedging an option is given below.</sup>

Daily P&L from option = Delta P&L + Gamma P&L + Theta P&L

Daily P&L from option = S.δ + S.2γ/2 + tθ where S is change in Stock and t is time

Daily P&L from option - S.δ = S.2γ/2 + tθ = Delta hedged P&L from option

Delta hedged P&L from option = S.2γ/2 + cost term (tθ does not depend on stock price)

where:

δ = delta

γ = gamma

t = time

θ = theta

If the effect of theta is ignored (as it is a cost that does not depend on the size of the stock price movement), the profit of a delta hedged option position is equal to a scaling factor (γ/2) multiplied by the square of the return. This means that the profit from a 2% move in a stock price is four times the profit from a 1% move in stock price. 

Partly due to its use in Black-Scholes, historically, volatility has been used as the measure of deviation for financial assets. However, the correct measure of deviation is variance (or volatility squared). Volatility should be considered to be a derivative of variance. The realization that variance should be used instead of volatility led volatility indices, such as the VIX, to move away from ATM volatility (VXO index) towards a variance-based calculation.

There 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).

1. 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 volatility will change with spot, even if volatility surface does not change).When 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 is bought, then its traded level is applicable no matter what the level of spot. The fact that a variance swap (or log contract) payout depends only on the realized variance and is not path dependent makes it the ideal measure for deviation.
2. 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).
3. 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.

The three points 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. 

VIX and VDAX moved from old ATM calculation to variance
Due to the realization 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).

The variance term structure is not always flat
While 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 the variance term structure is on average upward sloping, but in turbulent markets it is usually inverted.