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In this paper, we introduce and analyze the fractional Barndorff-Nielsen andShephard (BN-S) stochastic volatility model. The proposed model is based upontwo desirable properties of the long-term variance process suggested by theempirical data: long-term memory and jumps. The proposed model incorporates thelong-term memory and positive autocorrelation properties of fractional Brownianmotion with $H>1/2$, and the jump properties of the BN-S model. We findarbitrage-free prices for variance and volatility swaps for this new model.Because fractional Brownian motion is still a Gaussian process, we derive somenew expressions for the distributions of integrals of continuous Gaussianprocesses as we work towards an analytic expression for the prices of theseswaps. The model is analyzed in connection to the quadratic hedging problem andsome related analytical results are developed. The amount of derivativesrequired to minimize a quadratic hedging error is obtained. Finally, we providesome numerical analysis based on the VIX data. Numerical results show theefficiency of the proposed model compared to the Heston model and the classicalBN-S model.

Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging