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.
I find options fascinating because they deal with the abstract ideas of volatility and correlation, both of which are unobservable and can often seem like wild animal spirits (take the current stock market as an example). Understanding these subtle concepts is never easy, but it is essential in pricing some of the more exotic options which involve multiple underlying stocks. To set the scene, let’s pretend that your neighbor wants to make a bet with you where he will pay you $100 if Google (GOOG) and Apple (APPL) are above 500 and 240 respectively after 1 year, but you have to pay him $25 today. How would we determine if $25 is a good deal or not?