Volatility Smiles using R. An example with UBS equity options.

Volatility is one of the key points of finance and just for this reason academics and finance professionals have come up with lots of different approaches to tackle it. Among all of them, for me, the implicit volatility is the most elegant way to figure it out. I should admit that its mysterious smile always has seduced me. So, now that the SNB has settled that UBS is a healthy bank again, I have decided to have a quick look and check out if UBS is smiling again.

UBSimatge

First of all, for all those who are not familiar with this concept, the implicit volatility is the volatility that one can estimate assuming the Black-Scholes model (BS), taking the option’s market price as an input. In other words, BS priced one option assuming constant volatility. So, if we get a traded price for one option, then, we can use this information and the BSM to find out its volatility. Theoretically, the volatility of one option should be constant for all strikes at same expiration date – That is, options must meet the put-call parity -, but we will see that this rule is far from true.

Thanks to yahoo finance we have a lot of financial data that one can easily download and handle. The CBOE also provide data about options, but for me it’s much more user-friendly the yahoo website.

For this exercise, I get option data expired on 18th January 2014 published the 18th November 2013 (9:00 CET) on the Yahoo webpage. The strike prices goes from 5$ to 30$, but some of them are out of the market. The UBS stock’s price in that moment was 18.53 $. The free interest rate was 0.8% (friday 15th nov). – I get the 3 months T-bill and it turns out extremely low -.I used the Bisection method because it is very easy to implement and understand. Of course, there are many methods we can use here, but I believe that introduce complexity when is not strictly required is a bad practice.

Vol

Here you are the R Code and Data.

As you can see, UBS is smiling again. The graph draws two happy smiles, one for put options and another one for call options. The first thing we must notice is that both smiles agree only in one point, being S = K (at the money). So, we can say that market only meet the BS assumption in one point. But then, what happens with the other strikes? Why do investors only trade these options at such high prices? – Remember that relation between volatility and price (vega) is always positive for plain vanilla options – If there are not any other hidden reasons like illiquidity, it means that investors are expected huge movements on the stock market within next two month. Concretely, investors expected a volatility over 50%, the same up or down.

Definitely, it seems too much. It is supposed that good news will bring stability for UBS, not more wild fluctuations. But to be completely sure, we only must wait until the 18th of January. Then, we’ll see.