Multivariate Fractional Brownian Motion and Generalizations of SABR Model

The SABR model is a generalization of the Constant Elasticity of Variance (CEV) model. It was introduced and analyzed by Hagan et al. [1]. Rapidly it has become the market standard for quoting cap and swaption volatilities thanks to the approximate formula for implied volatility which allowed real time risk management of large books of caps and swaptions. Later on it was also used in FX and equity markets.The generalization introduces stochastic volatility to the CEV model. The volatility process is assumed to follow a geometric Brownian motion with zero drift, i.e., a martingale and no mean reversion. This assumption differs significantly from other models of stochastic volatility, where mean reversion is introduced.In this chapter, another generalization of CEV and also of SABR is proposed. Namely, the Brownian motion defining the volatility process is replaced with a fractional Brownian motion. Such modification leads to the question of how one should define the dependence structure between the Brownian motion of the CEV model and the fractional Brownian motion of the new volatility process.We link the choice to multivariate self-similarity property of stochastic drivers in the model.