I. Introduction(Heading 1)
Volatility is a central theme of financial studies. Ever since Black-Scholes (BS) came up with their framework for pricing and replication, traders focused on volatility as a prime factor for uncertainty. They converted market price information to implied volatility. This was particularly convenient for the purpose of price interpolation and extrapolation to different maturities and strikes. Famously, since the crash of 1987, market players realized that the inversion of the BS formula from observed market prices led to a smile shaped form for the volatility -volatility smile-reflecting that the implied volatility changed with the exercise price.