The skew

In yesterday's post, I took the example of setting up an option position and letting it run until maturity. The choice was to either delta hedge it dynamically (to profit from a difference between the realized and implied volatilities) or not (to take a gamble with low risk on possible stock value at expiration).

But one can also buy an option and sell it later. Most options have a negative theta. This means that their value decreases as time passes if nothing else changes. So to sell an option at a higher price than I bought it, I need the underlying, or the implied volatility, to have moved significantly to offset such loss.
Trading options like that is mostly a leveraged way of trading the underlying as it is the factor the most influential in the option price.

But there is an alternative: if the option implied volatility matches the realized volatility (in a BS world), the loss/gain of value due to time and underlying moves will be compensated by the profit/loss from dynamic delta hedging. So if I further sell that option at an implied volatility higher, I know I made a profit.

So that's another way to make money with options: if you think the implied volatility is low and will rise: buy the option, delta hedge it to offset the loss of value due to time and sell it with a profit when implied volatility rises.

This is a source of profit for the market makers: they quote prices on the screen for options, once they trade an option, they delta hedge it and try to flatten the position, locking a profit in volatility terms.
If they delta hedged frequently such profit in terms of volatility translate into a profit in money.

Obviously there is a trade off as we do not trade in BS world: if you delta hedge too often, you pay a lot of transaction costs, and you may pay a lot the bid-ask spread of the stock. But if you do not delta hedge frequently enough, the profit you make is mostly due to random fluctuations of the stock rather than careful trading of options. Not to mention that the stock does not follow a lognormal process.

As we are not in BS world, we should not expect that all strikes have the same volatility. A friend of mine was kind enough to provide some fresh option data.
On the following picture you see the implied volatility corresponding to the bid and ask prices of calls and puts for the AEX index options. Data was taken the 1st of march 2013 around 9:05 and the expiration was 14 days from then. There was no dividends and I estimated the risk free rate to be around 0.2%. Index was trading at 339.855.

The blue dots are the calls implied volatilities and the greens come from the puts. As one can see, volatility can be very high for extreme strikes. For the tightest strike (340), I get 14.5% and 14.7% for the call bid and ask. On the following picture is a zoom:

There is many explanations for the presence of the skew in the market. None of them really says what exact shape the skew should be.

I mean, we can price options with other models than BS, then calculate implied volatility and that will give a skew. Different choice of models will give different skew shape depending if the model includes jumps, random volatility and things like that. Most of them are model for the stock price.

We could try to use the model that seem to describe the stock path the best, decide that's the one we should use and if the skew it gives is different than the one from the market, put a trade on.

1/ It is quite difficult to assess what will be the frequency of jumps of a stock until expiration, the volatility of volatility and so on.
2/ Remember about the drift, we discussed largely that the "real world" drift should not be used in pricing because there would be an arbitrage (call-put parity). What about the real world drift of the random volatility and other quantities like that?

Another approach could be to take the skew independently of the stock and say that the skew as a "standard" shape and when it deviates from it, it will come back to the standard shape latter on. In order to characterize the standard shape, we could use the same models I was talking about but tweak the parameters until the skew they give matches the one we have, without really looking at the stock price. Or we could even directly parametrize the volatility curve to get a standard shape.

Let's see the main difference: If we pick a model, say with jumps and random volatility. For given parameters for jump probability, average jump size, volatility of volatility... it will give a unique skew. Then we play on these until the skew it gives is close to what we see.
If we work from the skew, we pick a function we think will work and adjust its parameters until if fits what we see.
The problem with the second approach is that we have to be careful some constraints to prevent arbitrage while the first approach prevents them from the start. However the flexibility of the second method can be needed when the precision has to be of 0.2% (we saw in the example that's the difference between ask price and bid price, surely our theoretical price should be in between).

In a following post we will discuss how we can use splines for the second approach.