Section: Scientific Foundations

Model calibration

One of the most important research directions in mathematical finance after Merton, Black and Scholes is the modeling of the so called implied volatility smile , that is, the fact that different traded options on the same underlying have different Black-Scholes implied volatilities. The smile phenomenon clearly indicates that the Black-Scholes model with constant volatility does not provide a satisfactory explanation of the prices observed in the market and has led to the appearance of a large variety of extensions of this model aiming to overcome the above difficulty. Some popular model classes are: the local volatility models (where the stock price volatility is a deterministic function of price level and time), diffusions with stochastic volatility, jump-diffusions, and so on. An essential step in using any such approach is the model calibration , that is, the reconstruction of model parameters from the prices of traded options. The main difficulty of the calibration problem comes from the fact that it is an inverse problem to that of option pricing and as such, typically ill-posed.

The calibration problem is yet more complex in the interest rate markets since in this case the empirical data that can be used includes a wider variety of financial products from standard obligations to swaptions (options on swaps). The underlying model may belong to the class of short rate models like Hull-White [70] , [54] , CIR [59] , Vasicek [84] ,etc. or to the popular class of LIBOR (London Interbank Offered Rates) market models like BGM [55] .

The choice of a particular model depends on the financial products available for calibration as well as on the problems in which the result of the calibration will be used.

The calibration problem is of particular interest for MathFi project because due to its high numerical complexity, it is one of the domains of mathematical finance where efficient computational algorithms are most needed.