  In this paper, we introduce and analyze the fractional Barndorff-Nielsen and
Shephard (BN-S) stochastic volatility model. The proposed model is based upon
two desirable properties of the long-term variance process suggested by the
empirical data: long-term memory and jumps. The proposed model incorporates the
long-term memory and positive autocorrelation properties of fractional Brownian
motion with $H>1/2$, and the jump properties of the BN-S model. We find
arbitrage-free prices for variance and volatility swaps for this new model.
Because fractional Brownian motion is still a Gaussian process, we derive some
new expressions for the distributions of integrals of continuous Gaussian
processes as we work towards an analytic expression for the prices of these
swaps. The model is analyzed in connection to the quadratic hedging problem and
some related analytical results are developed. The amount of derivatives
required to minimize a quadratic hedging error is obtained. Finally, we provide
some numerical analysis based on the VIX data. Numerical results show the
efficiency of the proposed model compared to the Heston model and the classical
BN-S model.
