Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models

Abstract

We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP does not require strict self-similarity assumptions on the Volterra process. For this reason, we are able to apply such an LDP to two notable examples of non-self-similar rough volatility models: models where the volatility is given as a function of a log-modulated fractional Brownian motion (Bayer, C., F. Harang, and P. Pigato. 2021. “Log-Modulated Rough Stochastic Volatility Models.” SIAM Journal on Financial Mathematics 12 (3): 1257–1284), and models where it is given as a function of a fractional Ornstein–Uhlenbeck (fOU) process (Gatheral, J., T. Jaisson, and M. Rosenbaum. 2018. “Volatility is Rough.” Quantitative Finance 18 (6): 933–949). In both cases, we derive consequences for short-maturity European option prices implied volatility surfaces and implied volatility skew. In the fOU case, we also discuss moderate deviations pricing and simulation results.

KEYWORDS:

• Rough volatility
• implied volatility
• European option pricing
• short-time asymptotics
• fractional Ornstein-Uhlenbeck
• modulated models

Acknowledgments

The authors are grateful to Christian Bayer and Lucia Caramellino for discussion and support and to two referees for helpful remarks and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Note that both fBM and RLp (as in Rough Bergomi) are non-stationary and give rise to non-stationary volatility processes

2 RLp is the stochastic process driving the volatility in the rough Bergomi model (Bayer, Friz, and Gatheral 2016)

3 Let us mention that, for other purposes, one could consider the stationary solution to the fractional SDE above (see for example Gatheral, Jaisson, and Rosenbaum 2018), explicitly given by ${\int }_{-\mathrm{\infty }}^t{e}^{-a(t-u)}\mathrm{d}{B}_u^H.$ However, we are interested in this paper in option valuation, so we take as volatility driver the process $V_t$ above, with $V_0=0$, so that ${\sigma }_0=\sigma \left(V_0\right)=\sigma \left(0\right)$ is spot volatility in (20(20) $\{\begin{array}{l}\mathrm{d}{S}_t=S_t\sigma \left(V_t\right)d(\overline{\rho }{\overline{B}}_t+\rho B_t),\\ \mathrm{d}{V}_t=-a{V}_t\mathrm{d}t+\mathrm{d}{B}_t^H,\end{array}$(20) ).

4 This expansion is given in Friz, Gassiat, and Pigato (2022, Lemma 6.1), where the kernel $C(t-s{)}^{H-1/2}$ is used. However, in the proof of this result, the specific shape of the kernel is not used, but only self-similarity, and therefore it holds for $K(t,s)$ in (13(13) $\begin{array}{rl}{K}_H(t,s)& =c_H\left[{\left(\frac{t}{s}\right)}^{H-1/2}\right(t-s{)}^{H-1/2}-(H-\frac{1}{2})s^{1/2-H}{\int }_s^t{u}^{H-3/2}(u-s{)}^{H-1/2}\mathrm{d}u],\end{array}$(13) ) as well.

Additional information

Funding

BP acknowledges the support of Indam-GNAMPA, the support of MUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C23000330006) and the support of University of Rome Tor Vergata (project “Asymptotic Properties in Probability” (CUP E83C22001780005)).