July 28th, 2001, 5:59 am
Yes, just an inverse problem, but one that is very sensitive to the initial data, the market prices. The market price does indeed come from the real market. It's common practice to quote option prices in terms of implied volatility, rather than a $ amount. So you hear people talking about an option having a price of 22% vol. That just means take the Black-Scholes formula and plug in the known stock price, interest rate, strike...and a volatility of 0.22 and whatever answer you get is the real $ amount. So you can think of the inverse problem as inferring local sigma(S,t) from implied volatility!Why is this important? Because you have to know how to interpret and use the information 'contained in' the smile. Does the smile tell you what volatility is going to do in the future, for instance. It's common practice to calibrate your pricing model so that it matches market prices. But there are many ways of doing that depending on your vol model: deterministic vol, stochastic vol, uncertain vol, utility theory, 'equilibrium' pricing...Some make more sense than others, and some are easier to implement than others. Unfortunately, the easy-to-implement models don't make much sense and the good models are hard to implement! (Reminds me of "Your work is both original and true. Unfortunately, the parts that are true are not original, and the parts that are original are not true." Edgar Allan Poe???)