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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.Rough VolatilityNew insights about the regularity of the instantaneous variance obtainedfrom realized variance data (see Gatheral, Jaisson, and Rosenbaum (2018), Bennedsen, Lunde, and Pakkanen (2021, to appear), Fukasawa, Takabatake, and Westphal (2019)), have inspiredthe development of so-called rough stochastic volatility modelsin the financial literature. In simple terms, such a model can bedescribed by the following SDE [\begin{equation} \label{eq:stoch-vol} dS_t = S_t \sqrt{v_t}dB_t,\end{equation}] where the logarithm of the instantaneous variance process(v) behaves similarly to a fractional Brownian motion (fBm) with Hurstindex (0 < H < 1/2). One of the attractive features of rough volatilitymodels is that they can explain the long-established power-law explosionof the at-the-money (ATM) skew of options as time-to-maturity (T \to 0)and, thus, provide excellent fits to the implied volatility surface, aswas observed in Bayer, Friz, and Gatheral (2016), but already anticipated much earlier in Alòs, León, and Vives (2007),Fukasawa (2011). In mathematical terms, let (\sigma_{BS}(T,k)) denote thethe implied volatility of an option with time to maturity (T) andlog-moneyness (k). We define the ATM skew as [\begin{equation} \text{ATM-skew}(T) = \partial_k \sigma_{BS}(T,k)|_{k=0}, \end{equation}] and it is this quantity which can be observed to have apower law explosion when (T\rightarrow 0).
Rough volatility models provide a framework which allows to getexcellent fits to market data simultaneously w.r.t. to time series ofprices of the underlying and to option prices, with few parameters.
Super rough volatility?Empirical studies of realized variance data as well as studies of theATM skew in the implied volatility surfaces tend to conclude that(H \ll 1/2), often even (H < 0.1). As both these estimates involve acertain kind of smoothing – realized variance being an estimate of(\int_t^{t+h} v_sds) rather than (v_t) itself, option prices and theirimplied skews being in general not available or reliable very close tomaturity – this begs the question, if (H) actually might even be equalto (0).
From the realized variance viewpoint, Fukasawa, Takabatake, and Westphal (2019) indeed seems to suggestthat (H) could be (0). However, for a fractional Brownian motion, (H=0)is not allowed since in this case the the integral kernel of thefractional Brownian motion is no longer square integrable, and thus newprocesses or techniques must be employed in order to study the behaviorof stochastic volatility models in this regime. Some interesting effortshas been made to understand this problem. In particular, the theory ofGaussian multiplicative chaos has been used to this end, see forinstance the review paper Rhodes and Vargas (2014). Indeed, a proper scaling limit of fBm(W^H) as (H \to 0) produces a log-correlated Gaussian field (see, forinstance, Neuman and Rosenbaum (2018), Hager and Neuman (2020)). However, the resultingrandom element is no longer a (continuous) stochastic process and onlymakes sense in terms of a generalized function (distribution), and it istherefore more challenging to use in practice.
One way to overcome this problem is to use techniques related to thetheory of Gaussian multiplicative chaos, see for instance thereview paper Rhodes and Vargas (2014). With this advanced framework, a proper scaling limitof fBm (W^H) as (H \to 0) produces a log-correlated Gaussian field (see,for instance, Neuman and Rosenbaum (2018), Hager and Neuman (2020)).
Logarithmic modulation of the fBmIn our recent work Bayer, Harang, and Pigato (2021) we investigate the (H=0) problemfrom a different point of view. We consider an actual continuousstochastic process with (H=0) by introducing a logarithmic term in thedefinition of the kernel (K:\mathbb{R}_+\rightarrow \mathbb{R}), whichfor small (r>0) behaves similarly to [\begin{align} r^{H-\frac{1}{2}}\log(1/r)^{-p}\label{eq1}\tag{1}\end{align}] for some parameter (p>1). This modification ensures that (K)remains square integrable for all (H \in [0,1/2)). Hence, the resultingfamily of Gaussian Volterra processes (\hat{W}) will be continuous andwith finite variance even for (H=0), and the ambiguities of theasymptotic analysis for (H\to0) cease to matter, as we can simply do theasymptotic for (H=0). We stress again that (\hat{W}) is a proper,continuous Gaussian process even for (H=0).
At the same time, as we apply our logarithmic modification only close tothe singularity of the power-law kernel, we may expect that theresulting rough volatility models are close to the correspondingstandard rough volatility models for (H \gg 0), see Figure 1 where wecompare the classical rough Bergomi model with the so called super roughBergomi model (created with the logarithmic modification of the fBm).
Above: Comparisons between the ATM skews of a rough Bergomi model and acorresponding super-rough Bergomi model for(H \in {0.01, 0.05, 0.09}). Skews are computed by Monte Carlosimulation.
Above: ATM implied volatility skews (absolute values) in the (super-)rough Bergomi model plotted against expiry (t) and Hurst index (H).Skews are computed by Monte Carlo simulation based on exact simulationof the underlying (log-modulated) fBm. Note that (H = 0) is included inthe plot in the super-rough case. Note how this seems in keeping withthe findings in Forde et al. (2020), of a vanishing skewness as(H\downarrow 0) in rough Bergomi.
The power law explosion of the ATM skewThe process we propose here can be seen as an extension of the logBrownian motion studied in Mocioalca and Viens (2005), to include a fractionalpower. This allows for a better comparison with classical fractionalprocesses, such as the Riemann-Liouville fractional Brownian motion,typically used in rough volatility models.
In this way, we are able to obtain rough volatility models which allowcontinuous interpolation for (H \in [0,1/2)), in the sense that all suchchoices of (H) are valid within the same model, with no apparent breaksbetween them. To illustrate this observation, we consider asuper-rough Bergomi model, which is simply obtained by replacing theRiemann-Liouville fBm by the process (\hat{W}) in the rough Bergomimodel of Bayer, Friz, and Gatheral (2016). Figure 2 shows the ATM-skew for various expires andvalues of (H) between – and including – (0) and (0.1). Indeed, thesurface “looks” smooth in (H), visually indicating a smooth transitionfrom the power law explosion (T^{H-1/2}) for (H>0) to the skew behaviourat (H = 0).
In contrast, the skew-behaviour changes remarkably for the standardrough Bergomi model for small (H), see Figure 1. In particular, the skewflattens significantly for very small (H). The log-modulated version inFigure 2 shows no signs of this flattening. To the contrary, a morerefined analysis, which is the main goal of our article, shows that theskew behaves like (T^{H-1/2}) – up to logarithmic terms – and, hence,steepens as (H \to 0).
Note in particular that the log-fBm does not have a scale invarianceproperty due to the logarithmic term in the kernel, thereby making anyshort time asymptotics very difficult. However, by employing thevol-of-vol expansion in Fukasawa (2011) we obtain an asymptotic formulafor the ATM skew when the volatility-of-volatility (\epsilon) is small.Indeed, we obtain a skew formula of the form [\begin{equation} \label{eq:skew-asymptotic} \text{ATM-skew} \approx a_{H,\zeta,p}\, \rho \,\log(1/T)^{-p} \, T^{H-1/2} \epsilon, \text{ as } T \to 0,\end{equation}] with Hurst parameter (H\in[0,1/2)), see Theorem 5.4 inBayer, Harang, and Pigato (2021). Here, (p>1) is a parameter of the kernel definedin (\eqref{eq1}), and (a_{H,\zeta,p}) is a constant depending on (H) –and other parameters – which is smooth in (H) with(a_{0,\zeta,p} \neq 0). This shows us that the explosion of the ATM skewin a stochastic volatility model generated from the log modulatedversion of a fBm behaves similarly to (T^{H-\frac{1}{2}}), the power lawexplosion of the rough Bergomi model when (H>0), while still preservingthis property when (H=0), providing a simple extension of the roughBergomi model to capture an important feature of the volatility surface.
Disclaimer: The following material is based on the published articleBayer, Harang, and Pigato (2021), and the reuse of figures and material for the blogpost has been approved by the journal SIAM Journal of FinancialMathematics.
BibliographyAlòs, Elisa, Jorge A León, and Josep Vives. 2007. “On the Short-Time Behavior of the Implied Volatility for Jump-Diffusion Models with Stochastic Volatility.” Finance and Stochastics 11 (4): 571–89.Bayer, Christian, Peter K. Friz, and Jim Gatheral. 2016. “Pricing Under Rough Volatility.” Quantitative Finance 16 (6): 887–904.Bayer, Christian, Fabian A. Harang, and Paolo Pigato. 2021. “Log-Modulated Rough Stochastic Volatility Models.” SIAM J. Financial Math. 12 (3): 1257–84. https://doi.org/10.1137/20M135902X.Bennedsen, Mikkel, Asger Lunde, and Mikko S Pakkanen. 2021, to appear. “Decoupling the Short-and Long-Term Behavior of Stochastic Volatility.” Journal of Financial Econometrics, 2021, to appear.Forde, Martin, Masaaki Fukasawa, Stefan Gerhold, and Benjamin Smith. 2020. “The Rough Bergomi Model as (H\to 0) – Skew Flattening/Blow up and Non-Gaussian Rough Volatility.” Available at Https://Nms.kcl.ac.uk/Martin.forde/.Fukasawa, Masaaki. 2011. “Asymptotic Analysis for Stochastic Volatility: Martingale Expansion.” Finance and Stochastics 15: 635–54. https://doi.org/https://doi.org/10.1007/s00780-010-0136-6.Fukasawa, Masaaki, Tetsuya Takabatake, and Rebecca Westphal. 2019. “Is Volatility Rough?” arXiv Preprint arXiv:1905.04852.Gatheral, Jim, Thibault Jaisson, and Mathieu Rosenbaum. 2018. “Volatility Is Rough.” Quantitative Finance 18 (6): 933–49.Hager, Paul, and Eyal Neuman. 2020. “The Multiplicative Chaos of (H=0) Fractional Brownian Fields.” Preprint arXiv:2008.01385.Mocioalca, Oana, and Frederi Viens. 2005. “Skorohod Integration and Stochastic Calculus Beyond the Fractional Brownian Scale.” J. Funct. Anal. 222 (2): 385–434. https://doi.org/10.1016/j.jfa.2004.07.013.Neuman, Eyal, and Mathieu Rosenbaum. 2018. “Fractional Brownian Motion with Zero Hurst Parameter: A Rough Volatility Viewpoint.” Electronic Communications in Probability 23.Rhodes, Rémi, and Vincent Vargas. 2014. “Gaussian Multiplicative Chaos and Applications: A Review.” Probability Surveys 11: 315–92.To leave a comment for the author, please follow the link and comment on their blog: YoungStatS.
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