### Building risk-neutral position with index puts

From the point of view of my option pricer far-out-of-money index puts are almost always overbought, sometimes in magnitude of tens of times. Here, for example, is the picture from the last trading day for July contract of SPXpm:

We would like to sell those far puts, but common sense tells us, that naked sell can be extremely dangerous. Some flash-crash, alien invasion or zombie-apocalypse with market down at 10% may kill our deposit. We really want to do it risk-neutral.

From the current option price chain we can get an estimate of observing market delta, i.e. derived function of option prices by strike price, you can see it in pricer column “Market delta”. The problem is, when market moves, far-out-of-money put price will be mostly depend on implied volatility change. The IV surge will sometimes add to put price more, than price movement itself.

Here we can use the “leverage effect”, which is very strong on indexes. The effect is that volatility correlated with price change. Price drops – volatility rises, price rises – volatility drops.

As a result, additional delta should be added to market delta to get real delta to take into account price change together with volatility change. Let’s find out this additional delta.

Let’s assume, that volatility index VIX change describes well the impact of volatility change on put price.

Let’s x is a logarithm of S&P 500 index daily change, normalized on current VIX.
Let’s y is a logarithm of VIX daily change.

After gathering data and solving linear regression y on x we have:
y = -93 * x, correlation between x and y is -0.74

The coefficient k = - 93 links relative change of base asset price P (normalized on volatility) and relative change of option price Z.

dZ/Z = k * ((dP/P) / VIX)

To get the delta:

D = dZ / dP = k * Z / (P * VIX)

For example, put 1830 priced 2.8 will have additional delta:

D = -93 * 2.8 / (1950 * 12) =  -0.011

Together with market delta of -0.034 it gives us full delta -0.045. So, 11 sold 1830 puts can be hedged with one ES future contract (ES futures have multiplier of 50, while SPX(pm) options have multiplier of 100).

This research has entirely investigative nature; the author doesn’t bear any responsibility for results of its use.