# Adversity and the Variance Premium Part 2

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.

Obviously this is very, very early in the investigation but I think it is promising.