This post discusses what is the volatility smile and why should you care as a trader about it. I’ll keep it as simple as possible in order not to get into technicalities which won’t add anything to the understanding of the subject anyway.
Black and Schole’s World
If we consider the Black and Scholes’ world, we have to assume volatility is constant through the life of an option. The experienced trader knows this is in fact not the case. Not that Black and Scholes wouldn’t want to assume that volatility also changes as time goes on, but it makes the math easier and we get in the end a nice formula. As an aside note, there are models that assume volatility changes but that is beyond the scope of this post.
Volatility Does Not fit
One of the (many?) inconsistencies of the Black and Scholes’ world can be seen as what is called the volatility smile. The volatility is the only parameter we need to plug in the formula that is not easily observed from the market. Is it historical volatility we put in the formula? or is it some assumption we have towards the market. We can also tackle the problem differently by assuming the price of an option currently traded at the market is the fair price. Then from that price, we can see what volatility we need to plug in the formula that brings us to the traded price. This is called the implied volatility.
The Smile\Skew
So far so good. We can do that for different maturities and get a graph with the x-axis being the maturity, and the y-axis the volatility. This is where it gets tricky. For some reason, the volatility is not constant as assumed by the Black and Scholes’ world.
We get a smile or sometimes a skew. This skew seems consistent in most of the world markets. Here is an example of the smile in Israel:
The above options are on the index which was trading at about 500 at the time (year 2000). As the option gets into the money, the volatility gets smaller. Does it mean that out-of-the-money options are over-priced? or is it that the model is incorrect all together?
Stochastic Volatility
If the assumption that volatility is constant over the life time of the option, it is reasonable to expect that scholars have tried to change that assumption to what is known as stochastic volatility. This fancy word just means that the volatility is no longer constant, but random. The problems with the different models out there is that they no longer give us a nice formula to just plug in numbers like the Black and Scholes formula. But the good thing is that they are closer to reality(sometimes).
Volatility Arbitrage
If we have a more accurate model that supposedly describes the distribution of the option better, then an arbitrage opportunity must arise. Let me now introduce what is known as skewness. Taken from wikipedia:
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.
I am not sure if it helped are got you more confused. Here is a picture also taken from wikipedia which helps:
To capture an undervalued skew, we can buy OTM calls and sell OTM puts!
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