Following on from the last post we will now look at how the concept of adversity can lead to the variance premium.

Let’s think about a $100 stock and the $100 strike call
and put. Assuming no interest rates or dividends, and a volatility of 30%, both
the one-year call and put will each be worth $9.92. One trader sells the put,
and another buys the call. Now let’s look at two price paths for the stock.

Case One:

On the first day, the stock jumps to $119.84. It stays there
until expiration. Each trader makes $9.92 at expiry, but the PL evolves
slightly differently over time.

Figure One: PL for short put.

Figure Two: PL for long call.

Now consider the case when the jump happens right at
expiration.

Figure Three: PL for short put.

Figure Four: PL for long call

My contention is that the shape of Figure Four is the one
most preferred by investors. That the long option maintains the ability to
“snatch victory from the jaws of defeat” is important. People prefer this, which
points to options being over-priced. Hence the variance premium.

Clearly these price paths are extreme and highly artificial
but a similar effect occurs if the price is a GBM. I simulated ten thousand
paths where the stock price was a GBM with a return of 20% and a volatility of
30%. Again the long call initially fell behind before catching up (the same
behavior as with the toy example). The median put advantage is shown in Figure
Five.

Figure Five: The median advantage of a one year short put,
over a one year long call.

This psychological effect gives another plausible reason for
the existence of the variance premium. However, viewed like this the “variance”
premium is really a gamma effect. The late jump provides the redemption. This
explains why the variance premium is greatest for short dated at-the-money options, and also why the premium tends to be higher
when volatility is low (Sinclair, 2013). For indices these are both known
phenomena, but if my theory is correct it should also show up in the
cross-section of stock options. Lower volatility stocks should have higher
variance premia than high volatility stocks.

There is another direct trading prediction. The
psychological component of the variance premium should be greatest for low
volatility underlyings. That is the type of premium that can be harvested by
selling options. It is a strategy that bets against the irrational avoidance of
regret. But if we find a high volatility underlying with an apparently high
variance premium, it is probably a sign of real risk being present. I would
expect selling options on these stocks to be losers.

Here are my predictions:

·
The variance premium is greater for low
volatility stocks.

·
Selling options is more profitable on low
volatility underlyings.

·
Selling options when a high volatility
underlying has a large implied/historical spread will not be profitable.

take a look at the goyal sorting by vol into deciles paper on ssrn - TonyC

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