Successful trading, hedging and risk managing of option portfolios crucially depends on the accuracy of the underlying pricing models. Consequently, new valuation approaches are continuously developed in departing from the foundations of option theory laid by Black and Scholes (1973), Merton (1973) and Harrison and Kreps (1979), and existing models are refined. However, despite these pervasive developments, the model of Black and Scholes (1973) remains a pivot in modern financial theory and an important benchmark for more sophisticated models, be it from a theoretical or practical point of view.
The popularity of the Black and Scholes (BS) model is likely due to its clear and easy-to communicate set of assumptions. Based on the geometric Brownian motion for the underlying asset price dynamics, and continuous trading in a complete and frictionless market, simple closed form solutions for plain vanilla calls and puts are derived: given the current underlying price at time, the option’s strike price, its expiry date the prevailing riskless interest rate, and an estimate of the (expected) market volatility, option prices are straightforward to compute.
The crucial parameter in option valuation by BS is the market volatility. Since it is unknown, one studies implied volatility, which is derived by inverting the BS formula for a cross section of options with different strikes and maturities traded at the same point in time. As is visible in the left panel of Figure 1 for May 2, 2000 (i.e. 20000502, a notation we will use from now on), implied volatilities display a remarkable curvature across the strike dimension, and – albeit to a lesser degree – a term structure across time to maturity. For a given time to maturity the phenomenon is called smile or smirk.
Download
A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics
