# Volatility arbitrage

If you really get into the Heston stochastic volatility model you will find yourself having to do some numerical integration in the complex plane (thanks to the transform methods used to solve the governing equation). This can be quite tricky to do in practice. Is all that effort worth it? Well, in part this depends on how good the model is. So you might think people would test the accuracy of the model against the data. Do they do this? Rarely. It is deemed sufficient to calibrate to a static dataset of option values regardless of the accuracy of the dynamics of that dataset. Yes, I know you then hedge with vanillas to reduce model risk, but this is a fudge that is completely inconsistent with the initial modelling. The cynic in me says that the benefit of modelling in such oblivion is truly tested by the state of your bank balance at the end of the year. If you get a bonus, does it matter? I don't have too much of a problem with that, depending on where you are in the management structure. However, I suspect that this is not most people's justification for their inaccurate modelling. I suspect that people really do believe that they are doing good work, and the more complicated the mathematics the better.

So, many know all the ins and outs of the most advanced volatility models based in the classical no-arbitrage world. Well, what if your job is to find volatility arbitrage opportunities? "There's no such thing as a free lunch" is drummed into most quants, thanks to academics and authors who take an almost axiomatic approach to our subject (see Dermanās recent blog). Those who know the details of volatility arbitrage are few and far between. Take the example of how to hedge when you think that options are mispriced.

You forecast volatility to be much higher or lower than current implied volatility. Clearly this is an arbitrage opportunity. But to get at that profit you must hedge stock risk. Now, working within an otherwise very simple Black-Scholes world but with two volatilities, implied and forecast, how should you hedge and how much profit will you make?

I have attached an audio recording (MP3) of the lecture I gave on this topic at a recent conference in Amsterdam.

In my next blog I will give some of the details of this problem.

P