After going through the BS model and deriving an equation for vanilla options, it is tempting to believe all of the assumptions that have gone into it. This post will be the first in a series examining some of those assumptions, extending them where possible, and looking at how they fail in some other cases.

Today I’m going to write about the vol smile. If you go to the market and examine quoted vanilla option prices at a given expiry, and put these into an implied vol solver (for example the one on my page!!), if the market believed the BS model was correct, we’d expect to get the same value for each option, no matter what the strike. Alas, not so! In fact, what we will see is that options that have strikes further away from the forward price will usually have higher implied vols than those near the forward price (ie. ‘at-the-money’). This is called the ‘vol smile’ because as it increases away from the money in either direction, it looks something like a smile! In the picture below I’ve shown some toy vol smiles and the sort of evolution that is typically seen with time-to-expiry.

What does this mean? A higher implied vol means a higher price for the option, so we’re saying options far in-the-money or out-of-the-money cost more in real life than expected by the BS model. In the BS model, the log of the stock price was normally distributed at expiry, but more expensive options at distant strikes means that the real distribution has a higher-than-expected probability of ending up at extreme strikes, so the real probability has ‘fat tails’ relative to a normal distribution.

As described in the post on Risk-Free Valuation, we can go a step further and back out the market-implied distribution from the observed call prices from the equation

Taking the smile shown above at , and calculating prices at each strike (using the standard BS vanilla equation, but with the implied vol given by the smile as the vol input), and taking the second derivative with respect to strike, gives the following risk-neutral distribution

Note that the real distribution does indeed have fat tails at distant strikes. Since they have the same expectation and variance, this means that it also has a central peak, and intermediate values suppressed relative to the normal. The graph below shows the tails in more focus

How can we alter the BS model to accommodate a vol smile? One possibility is to allow the vol parameter to vary deterministically with spot and time. This approach can indeed match observed vol smiles, also has several weaknesses, I’ll explore it in depth in a later post (it’s called the Local Volatility model, by the way).

A more interesting idea is to allow the volatility itself to be a random variable. This seems intuitive – the volatility, as well as the stock price, responds to information arriving randomly and unpredictably and thus probably should be stochastic. Why would this give us a vol smile? Well – option prices can be seen as the average payoff over all of the different paths that the spot might take. For paths in which the vol stays low, the price won’t go very far. On the other hand, if the vol increases lots there’s a much higher chance that we will end up far away from the money. Looking at this in reverse, if at the expiry date we’re far from the money, it’s much more likely that we followed a higher volatility path to get here, so the implied vol away from the money will be higher.

Stochastic vol models are widely used by practitioners, and there are many different types and models used with many different strengths and weaknesses. I will return to this topic again, most likely repeatedly! The take-home lesson for today though is that vol smiles are important: they imply fat-tailed distributions relative to a lognormal, and they are significant, real features of markets – we need our models to match them or we will lose lots of money to other people who are!

[An interesting historical point is that before the market crash in 1987, there was no vol smile – options indeed tended to have the same vol regardless of strike – people believed the BS model more than they do now. Could the crash have made people realise that large moves were much more likely in real life than BS suggested, and adjusted accordingly? Or do higher prices at extreme strikes represent traders insuring themselves against the possibilities of more market crashes? There are parallels with the present – another assumption of BS is that it is possible to borrow unlimited amounts at the risk-free borrowing rate. This was almost true for big banks before the 2007 crash, but not so any more, and once again a lot of what we do now is trying to understand how to price options correctly in a price where there isn’t really such a thing at the risk-free rate. Each crash seems to lead to belated better understanding of the BS model weaknesses, and because markets often follow the models that participants are using to model them, this improved understanding itself has an effect on the market!]