With the stock market freaking out and all, I figured I should take a look at how volatility was being priced in the option market. The CBOE generously provides snapshots of market data for anyone interested to download. By using this data, we can calculate the markets ‘implied volatility’, or level of ‘freaking out’. For those not familiar with the concept of implied volatility, essentially we can take the prices of options in the market and back out the volatility implied by those prices using the Black-Scholes formula. Its been shown over and over again that the assumptions of the Black-Scholes model don’t hold up to empirical data; but its an easy calculation to perform, and so implied volatility is a widely used metric. Anyway, below is my Black-Scholes option pricing function and the function used to back out implied volatility (written in R of course). Since implied volatility can only be found numerically, I used the Bisection Method to calculate it since it was easy to implement, but there are faster methods out there.
If you go to Google and search for “Black Scholes” you are bound to come across a long list of articles that derive the Black-Scholes PDE and Call Price formula. Before I learned about the more technical issues of stochastic calculus and martingale measures, I would read these derivations and assume the authors were the experts. There always seemed to be some hand waving going on, but I figured it was just a complex subject and I just didn’t fully understand all the details. Now, some years later and after having formally learned the material, I find myself in disbelief at how sloppy and often wrong some of these derivations are. As an example, take a look at Wikipedia’s article on the Black Scholes model. Now forget everything you just read. This article is my attempt to straighten things out. I will try to be more rigorous than most, but I may skip over some of the regularity conditions which concern the pure math types.
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