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

TL;DR

This paper introduces a fractional BN-S stochastic volatility model capturing long-term memory and jumps, providing arbitrage-free prices for variance and volatility swaps, with analytical expressions and numerical validation against real data.

Contribution

The paper develops a novel fractional BN-S model combining long-term memory and jumps, deriving new distribution formulas, and analyzing hedging and pricing in a unified framework.

Findings

Efficient pricing formulas for variance and volatility swaps.

The fractional BN-S model outperforms classical models in numerical tests.

Analytical results for Gaussian integrals aid in derivative pricing.

Abstract

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.