Benchmark Models

In this chapter we give a very succint primer of basic models for reference derivative markets: equity derivatives (Black–Scholes model and stochastic volatility or/and jump extensions), interest rate derivatives (Libor market model) and credit derivatives (one-factor Gaussian copula model). Can we really call these models? They are more quotation devices, used by traders for conveying the information regarding value of derivatives in units of measurements with a more physical significance than simply a number of dollars or euros. We want to speak of the implied volatility, which gives the “temperature” of the markets at a given level of maturity and strike for an option, and of implied correlation, which is the measure of (credit) “contagion”. More serious (calibratable, see Chap. 9 ) models are affine models (affine drift, covariance and jump intensity coefficients). In these models, semi-explicit Fourier formulas are available for the prices and Greeks of European vanilla options. Yes, there is a lot of jargon in this theory, such as “Greeks” in reference to the letters (e.g. Δ above) which are used for various risk sensitivities, and European (or American above, but also Russian, Parisian, and somewhere in the book I contrived “Hawaiian”), or vanilla (as opposed to exotic), in reference to different kinds of options (remember, this is a global world). In Chap. 9 we come to understand why we need such fast pricing formulas at the stage of model calibration, which involves intensive pricing of vanilla options. However, as far as pricing exotics or dealing with less standard models is concerned, the pricing equations have to be solved numerically, which is the object of the next few chapters.