Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems

Abstract

We study two calibration problems for the lognormal SABR model using the moment method and some new formulae for the moments of the logarithm of the forward prices/rates variable. The lognormal SABR model is a special case of the SABR model [1]. The acronym “SABR” means “Stochastic-αβρ” and comes from the original names of the model parameters (i.e., α,β,ρ) [1]. The SABR model is a system of two stochastic differential equations widely used in mathematical finance whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The lognormal SABR model corresponds to the choice β = 1 and depends on three quantities: the parameters α,ρ and the initial stochastic volatility. In fact the initial stochastic volatility cannot be observed and can be regarded as a parameter. A calibration problem is an inverse problem that consists in determineing the values of these three parameters starting from a set of data. We consider two different sets of data, that is: i) the set of the forward prices/rates observed at a given time on multiple independent trajectories of the lognormal SABR model, ii) the set of the forward prices/rates observed on a discrete set of known time values along a single trajectory of the lognormal SABR model. The calibration problems corresponding to these two sets of data are formulated as constrained nonlinear least-squares problems and are solved numerically. The formulation of these nonlinear least-squares problems is based on some new formulae for the moments of the logarithm of the forward prices/rates. Note that in the financial markets the first set of data considered is hardly available while the second set of data is of common use and corresponds simply to the time series of the observed forward prices/rates. As a consequence the first calibration problem although realistic in several contexts of science and engineering is of limited interest in finance while the second calibration problem is of practical use in finance (and elsewhere). The formulation of these calibration problems and the methods used to solve them are tested on synthetic and on real data. The real data studied are the data belonging to a time series of exchange rates between currencies (euro/U.S. dollar exchange rates).