Marek Musiela, Deputy Director, Oxford-Man Institute of Quantitative Finance, University of Oxford

Commonwealth Bank, Level 19, Darling Park Tower 1, 201 Sussex Street, Sydney NSW 2000

ABSTRACT

Mathematical models are developed to capture market behaviour at a point in time and are used to gain competitive advantage over time. In the option business, for example, they are calibrated to liquid information and used to price and trade more exotic and hence less liquid products. However market liquidity changes over time, it can increase or evaporate depending on the economic conditions. This is one of the factors that drive evolution of models which need to be adapted to the changing market conditions.

Moreover, models that are successful in pricing and managing risk in one area of the market are often adapted to the markets for which they were not designed. For example, Black and Scholes model was originally developed to price equity option but soon after it was adapted to price FX and interest rate options.

Sigma alpha beta rho model, known also as SABR, is perhaps the only option pricing model, after the Black and Scholes model, that was universally accepted as the market standard for quoting cap and swaption volatilities. At the time it fitted the data relatively well, however, its success was driven primarily by the approximate formula for implied volatility developed by Pat Hagan.

More recently SABR was applied to price equity and FX options. This generated new challenges for the SABR framework and suggested a new class of models inspired by SABR, where the process defining the noise is a bivariate fractional Brownian motion with parameter (1/2, H).

In the classical SABR model correlation parameter between the two Brownian motions determines many of its mathematical properties. In the modified framework Brownian motion defining the dynamics of stochastic volatility is replaced by a fractional Brownian motion. This leads to the first question: how one should define the dependence structure between the Brownian motion driving the asset dynamics and the fractional Brownian motion used to define the volatility. Once a choice is made the question is: how does it affect the model behaviour.

In this talk I will analyse mathematical properties of the classical SABR model and will propose a dependence structure between the two processes that is consistent with the definition of a multivariate fractional Brownian motion with multivariate self-similarity parameter H. I will also analyse some properties of the new fractional SABR model.

**Sponsored by Commonwealth Bank**

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