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).

Cognitum Option Pricer is used.

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

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