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

Nicholas Salmon; Indranil SenGupta

doi:10.1007/s10436-021-00394-4

Home/Archive/arXiv

arXiv · 2105.02325

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

Nicholas Salmon·Indranil SenGupta

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

q-fin.MF q-fin.PR

Keep this discovery

BibTeX RIS

Nicholas Salmon, Indranil SenGupta. 2021-05-05. Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging. https://doi.org/10.1007/s10436-021-00394-4

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Variance-Optimal Hedging in the Rough Hawkes--Heston Model

We study variance-optimal stock hedging and the convergence of approximate strategies in the rough Hawkes--Heston model. Starting from the model's affine conditional transform and the affine Volterra jump framework, we obtain semi-explicit hedges for European calls and a representation of the minimum quadratic error through the Galtchouk--Kunita--Watanabe projection. Our main approximation result keeps the original stock, variance driver, and information flow fixed while regularizing the kernel used to evaluate the hedge. To handle singular memory and common marked jumps, we construct the approximate holdings from histories available before trading and preserve the conditional transform's random modulus envelope. Riccati--Volterra stability and weighted truncation then yield convergence in the original stock's trading norm on compact Fourier intervals. For calls, a joint choice of kernel regularization and Fourier cutoff gives convergence of the initial capitals and strategies, uniform-in-time square-mean convergence of continuous-time gains, and convergence of the terminal mean-square error to the variance-optimal value. A numerical experiment with shifted fractional kernels illustrates the construction on common original-market paths.

q-fin.MF↗

arXiv2026

Numeraire Invariance of Entropy-Projected Martingale Measures

Let $P$ be a fixed physical law and let $Q$ be an equivalent martingale measure selected from the martingale-measure set associated with a chosen numeraire. A change of numeraire maps $Q$ to $T_LQ$, where $d(T_LQ)=L\,dQ$ and $L$ is the terminal likelihood ratio. The forward relative-entropy projection minimizing $D_{\mathrm{KL}}(P\Vert Q)$ commutes with this transform because its objective changes only by the constant $-E_P\log L$. The minimal entropy martingale measure (MEMM) orientation $D_{\mathrm{KL}}(Q\Vert P)$ does not have this property, and a trinomial counterexample shows that independently recomputed MEMMs need not be likelihood compatible. We make two economic consequences explicit. First, the two entropy orientations are precisely the $Q$-dependent terms in the classical convex-dual objectives for logarithmic and exponential utility, respectively. Second, likelihood compatibility is equivalent to equality of the pricing functionals obtained in the two numeraires. Hence the forward selectors value every integrable claim consistently across numeraires, whereas the two MEMMs in the counterexample assign different prices to a nonreplicable digital claim. We also prove a finite-state class-level characterization: uniform invariance over the elementary one-period likelihood-ratio families forces a smooth convex $f$-divergence to be logarithmic, up to scaling and affine equivalence. Finally, in finite-state markets, the forward projection exists under the usual strictly positive feasible-point condition; its density $dP/dQ^*$ is attainable log-optimal terminal wealth, and the minimum forward entropy equals maximal expected log growth.

q-fin.MF↗

arXiv2026

The Delta of a Variance Swap

We define the variance swap delta as the sensitivity of the price of variance to a change in underlying price. We use Carr-Madan spanning formulas to analyze this sensitivity when the implied volatility smile curve may depend on the underlying price. We show that the variance swap total delta is zero for the class of smile curves that are pure functions of (log) moneyness, which goes against the empirical observation that variance is up when the market is down. We propose a simple modification of the smile to correct this issue.

q-fin.MF↗