Log-modulated rough stochastic volatility models
LLOG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS
C. BAYER, F. HARANG, AND P. PIGATO
Abstract.
We propose a new class of rough stochastic volatility models obtained by modu-lating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmicterm, such that the kernel retains square integrability even in the limit case of vanishingHurst index H . The so-obtained log-modulated fractional Brownian motion (log-fBm) is acontinuous Gaussian process even for H = 0 . As a consequence, the resulting super-roughstochastic volatility models can be analysed over the whole range ≤ H < / without theneed of further normalization. We obtain skew asymptotics of the form log(1 /T ) − p T H − / as T → , H ≥ , so no flattening of the skew occurs as H → . Introduction
Prompted by new insights about the regularity of instantaneous variance obtained fromrealized variance data (see [16, 5, 15]), rough stochastic volatility models have become moreand more popular in the financial literature. Loosely speaking, these are stochastic volatilitymodels d S t = S t √ v t d Z t , (1.1)where the logarithm of the instantaneous variance process v roughly behaves like a fractionalBrownian motion (fBm) with Hurst index < H < / . One of the attractive features ofrough volatility models is that they can explain the long-established power-law explosion ofthe ATM skew of options as time-to-maturity T → and, thus, provide excellent fits tothe implied volatility surface, as was observed in [2], but already anticipated much earlier in[1, 12]. Hence, rough volatility models provide a framework which allows to get excellent fitsto market data simultaneously w.r.t. to time series of prices of the underlying and to optionprices, with few parameters.Popular rough volatility models are either explicitly defined in terms of fBm, or rather interms of a Volterra equation. Examples of the former case include the rough Bergomi modelof [2], where the variance process is given of the form v t = ξ ( t ) exp (cid:18) η (cid:102) W t − t H η (cid:19) . (1.2) Mathematics Subject Classification.
Primary 91G30; Secondary 60G22.
Key words and phrases. rough volatility models, stochastic volatility, rough Bergomi model, implied skew,fractional Brownian motion, log Brownian motion.
Acknowledgments.
We are grateful to M. Fukasawa, J. Gatheral and A. Gulisashvili for valuable commentsand support. F. Harang gratefully acknowledges financial support from the STORM project 274410, fundedby the Research Council of Norway. C. Bayer gratefully acknowledges support by the German research councilDFG via the cluster of excellence MATH+, project AA4-2. a r X i v : . [ q -f i n . M F ] A ug OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 2
Here ξ ( t ) denotes the forward variance curve, (cid:102) W t denotes the Riemann-Liouville fBm, i.e.,the Volterra process defined by (cid:102) W t := (cid:90) t K ( t − s )d W s , K ( r ) := √ Hr H − / , r > , (1.3)where W denotes a standard Bm correlated with the Bm Z with correlation coefficient ρ . Asan example for the second type of model, in [7] the authors consider a rough Heston model,where v t = v + 1Γ( H + 1 / (cid:90) t ( t − s ) H − / λ ( θ − v s )d s + 1Γ( H + 1 / (cid:90) t ( t − s ) H − / λν √ v s d W s . (1.4)We note that the roughness of the fBm (or the singularity of the Volterra kernel in (1.3)and (1.4)) causes considerable analytical and numerical difficulties, owing to the fact thatthe variance process v fails to be a semimartingale or a Markov process in rough volatilitymodels. Due to these technical difficulties, results holding for both the aforementioned classesof models are difficult to achieve. We refer to [3, 22] for attempts at unifying the treatmentof rough volatility models.Empirical studies of realized variance data as well as studies of the ATM skew in impliedvolatility surfaces tend to conclude that H (cid:28) / , often even H < . . As both attemptsinvolve a certain kind of smoothing – realized variance being an estimate of (cid:82) t + ht v s d s ratherthan v t itself, option prices and their implied skews being in general not available or reliablevery close to maturity – this begs the question, if H actually might even be equal to zero.From the realized variance viewpoint, [15] indeed seems to suggest that H could be . Ofcourse, H = 0 is not allowed in the rough volatility models suggested above, but the casehas been studied before in the literature on Gaussian multiplicative chaos , see for instancethe review paper [27]. Indeed, a proper scaling limit of fBm W H as H → produces alog-correlated Gaussian field (see, for instance, [25, 18]).Despite the well-established literature, some important financial questions regarding the H → limit are not very well understood yet. In particular, what happens with the ATMskew of implied volatility as H → . On the one hand, given that the skew behaves like T H − / as time-to-maturity T → in rough volatility models with H > , one might expecta power law explosion as T − / in the limiting case H = 0 . However, a closer look atthe asymptotic results for H > , casts some doubt on this conjecture. Indeed, taking themoderate deviation asymptotics of [4] as one example of such an expansion, we have theasymptotic formula skew ∼ const ρη √ H ( H + 1 / H + 3 / T H − / (1.5)as T → . Of course, the factor √ H ( H +1 / H +3 / → as H → , so that (1.5) entails twolimits ( H → , T → ), which cannot necessarily be interchanged. Note that √ H appearsin (1.5) by requiring the underlying fBm to have variance equal to one at time t = 1 . Indeed,some standardization of this type is needed in order to make models for different values of H comparable – even though the choice of standardization may be quite important. OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 3
Remark . As described above, in this paper we vary the Hurst index H while keepingthe other model parameters – in particular, the vol-of-vol η – fixed. An alternative point ofview motivated from the shape of the skew itself is to keep √ Hη rather then η fixed, whichleads to more stable behavior of the skew. The second alternative, however, has undesirableeffects on other properties of the model. Fixing √ Hη implies exploding variance of log v t inthe model (1.2) as H → . Consequently, we expect an explosion of the kurtosis of the assetprice as well as of the volatility of VIX options.The multiplicative-chaos approach in [25] is used in [8] to establish a H → limit for roughBergomi, for which the limit skewness vanishes or blows-up depending on the renormalization.Using continuity of Volterra integral equations, a H → limit for driftless rough Heston isconsidered in [9], in this case with a non-symmetric limit behavior. However, in [8, 9] noexplicit formula for the skew of implied volatility is given. Moreover, in both cases the limitvolatility is not a process, but is defined as a distribution. Hyper-rough volatility, in a senseanalogous to a H < model, has also been considered [21, 20], but also in this case spotvolatility is not defined. The extreme T − / speed of explosion for the skew expected in the H → limit has been shown to be a model-free bound [23, 11], and is reached under localvolatility through a volatility function with a singularity ATM [26], but this poses the problemof time-consistency (see also [10]). To the best of our knowledge, this extreme behavior ofthe skew has not been shown for any (time-consistent) stochastic volatility model, where thevolatility is a proper process.1.1. Our contribution.
In this paper, we consider an actual process with H = 0 by in-troducing a logarithmic term in the definition of the kernel K that ensures that K remainssquare integrable for all H ∈ [0 , / , see (2.2) for the precise definition. Note that we ignore H ≥ / in this paper, as what we are interested in is the H → limit, but there would beno real difficulties in considering H ∈ [0 , . Hence, the resulting family of Gaussian Volterraprocesses (cid:99) W will be continuous and with finite variance even for H = 0 , and the ambiguitiesof the asymptotic analysis for H → and T → cease to matter, as we can simply do theasymptotic for H = 0 . We stress again that (cid:99) W is a proper, continuous Gaussian process evenfor H = 0 . At the same time, as we apply our logarithmic modification only close to thesingularity of the power-law kernel, we may expect that the resulting rough volatility modelsare close to the corresponding standard rough volatility models for H (cid:29) , see Figure 7.4.The process we propose here can be seen as an extension of the log Brownian motion studiedin [24], to include a fractional power. This allows for a better comparison with classical frac-tional processes, such as the Riemann-Liouville fractional Brownian motion, typically used inrough volatility models. We also mention that the standard log-Brownian motion (withoutthe fractional power) has recently been analysed in the context of rough volatility models in[17], as well as in the context of regularization by noise for ill-posed ODEs in [19].In this way, we are able to obtain rough volatility models which allow continuous inter-polation for H ∈ [0 , / , in the sense that all such choices of H are valid within the samemodel, with no apparent breaks between them. To illustrate this observation, we consider OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 4 T H A b s o l u t e v a l ue o f sk e w (a) Rough Bergomi model T H A b s o l u t e v a l ue o f sk e w (b) Super-rough Bergomi model
Figure 1.1.
ATM implied volatility skews (absolute values) in the (super-)rough Bergomi model plotted against expiry t and Hurst index H . Skewsare computed by Monte Carlo simulation based on exact simulation of theunderlying (log-moderated) fBm. Note that H = 0 is included in the plotin the super-rough case. Parameter values of the rough Bergomi model are η = 2 . , ρ = − . , ξ ( t ) ≡ . . Additional parameters for the log-fBm (seeSection 2 for details) are ζ = 0 . , p = 2 . . Note how this seems in keepingwith the findings in [8], of a vanishing skewness as H ↓ in rough Bergomi.a super-rough Bergomi model , which is simply obtained by replacing the Riemann-LiouvillefBm by the log-fBm defined in (2.1) below in the rough Bergomi model of [2]. Figure 1.1bshows the ATM-skew for various expiries and values of H between – and including – and . . Indeed, the surface “looks” smooth in H , visually indicating a smooth transition fromthe power law explosion T H − / for H > to the skew behaviour at H = 0 . In contrast,the skew-behaviour changes remarkably for the standard rough Bergomi model for small H ,see Figure 1.1a. In particular, the skew flattens significantly for very small H . On the otherhand, the log-moderated version in Figure 1.1b shows no signs of flattening. To the contrary,a more refined analysis, which is the main purpose of this paper, shows that the skew behaveslike T H − / – up to logarithmic terms – and, hence, steepens as H → .Note that the log-fBm does not have a scale invariance property, thereby making any shorttime asymptotics very difficult. Hence, in this paper we use the vol-of-vol expansion in [12]to obtain an asymptotic formula for the ATM skew when the volatility-of-volatility is small.Indeed, we obtain a skew formula of the formskew ∼ a H,ζ,p ρ log(1 /T ) − p T H − / (cid:15), as T → , (1.6) OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 5 for small vol-of-vol (cid:15) , Hurst parameter H ∈ [0 , / , see Theorem 5.4. Here, p > is aparameter of the kernel defined in (2.2), and a H,ζ,p is a constant depending on H – andother parameters – which is smooth in H with a ,ζ,p (cid:54) = 0 . Then, we prove that the short-time asymptotics corresponding to (1.6) at the Edgeworth CLT regime holds even withoutconsidering the small vol-of-vol regime, for log-modulated models (in the sense of regularvariation) with H > .1.2. Outline of the paper.
In Section 2 we introduce a class of Gaussian processes, ex-tending the notion of fractional Brownian motion through a modulation with a log term.In Section 3 we compute some essential probabilistic features of the log fractional Brownianmotion (log-fBm) such as variance and covariance, which will be crucial for applications toasymptotic expansions of the implied volatility corresponding to certain rough volatility mod-els as well as for simulation of the processes. In Section 4 we provide a short overview of theMartingale expansion developed by Fukasawa in [12], and its application towards analysis ofthe implied volatility surface. Furthermore, we provide explicit computations of the covari-ance terms appearing in the asymptotic expansion in the case when the volatility is drivenby a log-fBm. Section 5 deals with a particular skew expansion, asymptotic in vol-of-vol,using Fukasawa’s Martingale approach. Here, we also include an asymptotic expansion forthe rough Bergomi model, when driven by a log-fBm. In Section 6 we consider a slightly moregeneral kernel and the asymptotics for the skew at the Edgeworth CLT regime, that holdsfor any vol-of-vol parameter, generalising a result in [14]. At last, in Section 7 we providesome details on numerical simulations and computations of the skew.2.
Rough and super rough volatility modelling
The fractional Brownian motion (fBm) is a well studied Gaussian processes. A simplifiedversion of this process, called the Riemann-Liouville fBm is given as a Volterra type stochasticintegral with respect to a Brownian motion, i.e. B Ht := (cid:90) t ( t − s ) H − d W s , where { W t } t ∈ ,T ] denotes a standard Brownian motion. However this process is only definedfor H ∈ (0 , , and thus excludes the case when H = 0 . To overcome this challenge, wepropose to modulate the Riemann-Liouville fBm with a log term to control the singularityin the kernel ( t − s ) H − . In this section we therefore will construct a particular fractionalprocesses which allows to generalize the Riemann-Liouville fBm to H ∈ [0 , . We considerthe Gaussian Volterra process (cid:99) W t := (cid:90) t K ( t − s )d W s , (2.1) OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 6 t K ( t ) log−fBm kernel, z = 0.1log−fBm kernel, z = 0.25Riemann−Liouville kernel Figure 2.1.
The logarithmic kernel (2.2) with H = 0 . , p = 2 , and ζ ∈{ . , . } compared against the Riemann-Liouville kernel K ( t ) (cid:39) t H − / .All kernels are normalized to have Var (cid:99) W = 1 .and the kernel K satisfies K ( r ) := Cr H − / max ( ζ log (1 /r ) , − p = Cr H − / ζ − p log(1 /r ) − p , ≤ r ≤ e − /ζ ,Cr H − / , r > e − /ζ . (2.2)We assume that ≤ H < / , ζ > , and p > . C is a constant which will be chosen tonormalize the process, i.e., to guarantee Var( (cid:99) W ) = 1 . Therefore, C will depend on all the other parameters. We call the process { (cid:99) W t } t ∈ ,T ] a log-fractional Brownian motion (log-fBm) . We note that by the choice of these parameters, { (cid:99) W t } t ∈ ,T ] is a continuous Gaussian process with vanishing expectation. Indeed, it is readilychecked that K ∈ L ([0 , T ]) (see in particular Lemma 3.2 for explicit computations), andthus { (cid:99) W } t ∈ ,T ] is well defined as a Wiener integral. Moreover, due to the assumption that p > , the continuity can be verified by Fernique’s continuity condition (see e.g. [17, Lem.2.3] for a recent overview). For ease of notation we introduce χ := e − /ζ . Remark . As χ depends exponentially on /ζ and the log-fBm-kernel introduced in (2.2)only differs from the standard Riemann-Liouville kernel on (0 , χ ) , one may be tempted to OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 7 expect that the corresponding process (cid:99) W behaves very similar to the Riemann-Liouville fBmas often used in rough volatility models. This is undoubtedly true for H (cid:29) , and motivatesthe whole paper, but note that there are profound differences as H → , as witnessed byFigure 3.1. In particular, despite the very localized changes, the log-fBm has finite varianceeven for H = 0 .We see the super-rough Bergomi model mostly as a perturbation of the rough Bergomimodel, which stays close to the rough Bergomi model when H (cid:29) , but still has nice prop-erties for H → , see Figure 7.4 for a comparison of skews in the (super-) rough Bergomimodel. This implies that the super-rough Bergomi model differes substantially from therough Bergomi model as H → , as already seen in Figure 1.1. Remark . The super-rough Bergomi model adds two more parameters ( ζ > and p > ) tothe rough Bergomi model. If we want to keep close to the rough Bergomi model for not-too-small H , then we need both ζ and p to be chosen small within their admissible ranges. This,however, may very well introduce numerical difficulties for H ≈ , as the kernel approachesa kernel which fails to be square integrable as ζ → or p → .3. Moments of the log-fractional Brownian motion
The skew formulas to be derived in later section will depend on formulas for some momentsof the log-fractional Brownian motion and the underlying Brownian motion. Computingthese moments will also give us an explicit formula for the constant C in the kernel (2.2).Throughout this section, we shall often use the following elementary lemma. Lemma 3.1.
Consider < u < , a ≤ , and b > . Then we have (cid:90) u r − a log (cid:18) r (cid:19) − b d r = log (cid:18) u (cid:19) − b E b (cid:18) (1 − a ) log (cid:18) u (cid:19)(cid:19) , where E b denotes the exponential integral, given by E b ( x ) := (cid:90) ∞ e − xt t − b d t, x ≥ . Note that the exponential integral E b ( x ) is infinite for negative x , which is excluded byour assumptions. Using the above lemma, second moments of { (cid:99) W t } t ∈ ,T ] can then computedexplicitly. Lemma 3.2.
Let { (cid:99) W t } t ∈ ,T ] be the log-fBm given in (2.1) with kernel given in (2.2) . Thenthe variance of (cid:99) W satisfies Var( (cid:99) W t ) = C (cid:20) ζ − p log (cid:16) t ∧ χ (cid:17) − p E p (cid:16) H log (cid:16) t ∧ χ (cid:17)(cid:17) + t H − ( t ∧ χ ) H H (cid:21) , H > ,C (cid:20) ζ − p p − log (cid:16) t ∧ χ (cid:17) − p + log (cid:16) tt ∧ χ (cid:17)(cid:21) , H = 0 . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 8
Assuming χ < , the scaling constant C = C H,ζ,p required to ensure
Var (cid:99) W = 1 satisfies C H,ζ,p := (cid:104) ζ E p (2 H/ζ ) + − χ H H (cid:105) − / , H > , (cid:104) p (2 p − ζ (cid:105) − / , H = 0 . Proof.
By definition, we have
Var( (cid:99) W t ) = (cid:90) t K ( t − r ) d r = (cid:90) t K ( r ) d r = C (cid:34) ζ − p (cid:90) t ∧ χ r H − log (cid:18) r (cid:19) − p d r + (cid:90) tt ∧ χ r H − d r (cid:35) . Applying Lemma 3.1, the integral gives (cid:90) t ∧ χ r H − log (cid:18) r (cid:19) − p d r = log (cid:18) t ∧ χ (cid:19) − p E p (cid:18) H log (cid:18) t ∧ χ (cid:19)(cid:19) , which simplifies in the case H = 0 to the expression (cid:90) t ∧ χ r − log (cid:18) r (cid:19) − p d r = 12 p − (cid:18) t ∧ χ (cid:19) − p . For the second integral, we have by standard computations (cid:90) tt ∧ χ r H − d r = t H − ( t ∧ χ ) H H , H > , log( t ) − log ( t ∧ χ ) , H = 0 . (cid:3) Note that the scaling factors are continuous in H on [0 , / , see also Figure 3.1.Unfortunately, we have not been able to find closed form expressions for the covariances cov (cid:16)(cid:99) W t , (cid:99) W s (cid:17) of the log-fractional Bm. Nonetheless, numerical integration is relatively easyusing the double exponential method to take care of the singularity at the boundary of theintegral. Explicit formulas do, however, exist for the covariances with (correlated) Brownianmotions. Lemma 3.3.
Let { Z t } t ∈ ,T ] be a standard Brownian motion correlated with the Brownianmotion { W t } t ∈ ,T ] driving the log-fBm { (cid:99) W t } t ∈ ,T ] in (2.1) , and let ρ ∈ [ − , denote thecorrelation parameter. Denote u := t − t ∧ s and v := ( u ∨ χ ) ∧ t . Then for s, t ∈ [0 , T ] , thecovariance between (cid:99) W t and Z s is given by cov (cid:16)(cid:99) W t , Z s (cid:17) = Cρ (cid:26) ζ − p (cid:20) log (cid:18) v (cid:19) − p E p (cid:18) ( H + 1 /
2) log (cid:18) v (cid:19)(cid:19) − u> log (cid:18) u (cid:19) − p E p (cid:18) ( H + 1 /
2) log (cid:18) u (cid:19)(cid:19)(cid:21) + t H +1 / − v H +1 / H + 1 / (cid:27) . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 9 . . . . . . H C H log−fBmRiemann−Liouville−fBm Figure 3.1.
The scaling factor C H,ζ,p needed to achieve
Var (cid:99) W = 1 as shownin Lemma 3.2 compared to the corresponding scaling factor √ H for theRiemann-Liouville kernel K ( r ) (cid:39) r H − / . Parameters for the log-fBm are ζ = 0 . , p = 2 , giving χ ≈ × − . Proof.
Direct computations reveal that cov (cid:16)(cid:99) W t , Z s (cid:17) = E (cid:104)(cid:99) W t Z s (cid:105) = ρ (cid:90) t ∧ s K ( t − r )d r = ρ (cid:90) tu K ( r )d r = ρ (cid:90) vu K ( r )d r + ρ (cid:90) tv K ( r )d r = Cζ − p ρ (cid:90) vu r H − / log(1 /r ) − p d r + Cρ (cid:90) tv r H − / d r. For the first integral, we use Lemma 3.1 with a = 1 / − H and b = p , to obtain (cid:90) vu r H − / log(1 /r ) − p d r = (cid:20) log (cid:18) v (cid:19) − p E p (cid:18) ( H + 1 /
2) log (cid:18) v (cid:19)(cid:19) − log (cid:18) u (cid:19) − p E p (cid:18) ( H + 1 /
2) log (cid:18) u (cid:19)(cid:19) (cid:21) , (3.1)the second integral is trivial. (cid:3) Fukasawa’s method
We give a short introduction to the asymptotic expansion for stochastic volatility modelsoutlined in [12], adapted to the case of Gaussian noise driving the asset price and the volatility.
OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 10
This simplifies certain computations and conditions, and thus the results have been slightlychanged accordingly.Let (Ω , F n , {F nt } t ∈ [0 ,T ] , P ) be a filtered probability space for each n ∈ N , where a continuousmartingale X n lives. Consider an asset price process S n : [0 , T ] × Ω → R d given by S nt = exp( Z nt ) Z nt = Z + R ( t ) + A nt + X nt + (cid:90) t g ns d W s . (4.1)Here { g nt } t ∈ [0 ,T ] be an adapted process to the filtration {F nt } . The Brownian motion { W t } t ∈ [0 ,T ] is independent of the martingale X n , and is correlated with the stochastic process t (cid:55)→ g nt , inorder to capture the leverage effect. The function R is supposed to reflect the interest rateand is often assumed to be constant, and in applications typically chosen to be zero. A n isa drift term, such that S n is a Martingale. Denote by M nt the Martingale part of Z n , i.e. M nt = X nt + (cid:82) t g ns d W s . It is readily seen that the quadratic variation of M n is given by (cid:104) M n (cid:105) = (cid:104) X n (cid:105) + (cid:90) · | g ns | d s. Throughout the text we will refer to ( R, A n , X n , g n ) as a stochastic volatility model. Inapplications, we will assume that the following hypothesis holds for the model ( R, A n , X n , g n ) : Hypothesis 4.1.
For a given null sequence { (cid:15) n } , there exists a sequence Σ n with Σ =lim n →∞ Σ n > such that for all n ∈ N D n := (cid:15) − n (Σ − n (cid:104) M n (cid:105) T −
1) and 1 (cid:82) t | g ns | d s are bounded in L p (Ω) for any p > Moreover, D n and Σ − n M nT converges weakly to a randomvariable, say ( N , N ) . Fukasawa derives expansions of claims constructed from the model ( R, A n , X n , g n ) , whichcan be written as E [ F ( Z T )] , where F ( z ) = e − R ( T ) f ( S exp( z )) . A particular case of interestfor the current article is when the martingale part M n satisfies Hypothesis 4.1 and theRandom variables N and N are normally distributed. In this case, the Martingale expansioncan be used to give an asymptotic expansion of the implied volatility in terms of the vol-of-volparameter. The following theorem is a combination of Theorem 2.4 and Corollary 2.6 foundin [12]. Theorem 4.2.
Suppose F is a Borel measurable function of polynomial growth, and thatHypothesis 4.1 holds with N and N being normally distributed. Denote by σ n T = Σ n . Thenthe Black-Scholes implied volatility can be expanded as σ BS = σ n (cid:16) (cid:15) n δ − ρd ] (cid:17) + o ( (cid:15) n ) , (4.2) where δ := E [ N ] , ρ := E [ N N ] , OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 11 and d := log( S/K ) + r + Σ n / √ Σ n , d := d − (cid:112) Σ n . The following example is an application of the above theorem to the case when the volatilityis assumed to be driven by a Gaussian Volterra process. This particular example will motivatethe subsequent discussions on volatility models driven by super-rough processes. As thisexample is essentially [12, Sec. 3.3] adapted to general Volterra processes, we will sometimesrefer to this particular case as
Fukasawa’s example . Example 4.3 (Fukasawa’s example with volatility driven by Gaussian Volterra processes.) . Consider the asset price dynamics given by S t = S exp( Z t ) Z t = R ( t ) − (cid:90) t g ( Y ns ) d s + (cid:90) t g ( Y ns )[ ρ d W (cid:48) s + (cid:112) − ρ d W s ] , where for a null sequence { (cid:15) n } we specify Y ns = y + (cid:15) n (cid:99) W s , (cid:99) W t := (cid:90) t K ( t − s )d W (cid:48) s , and K is a possibly singular Volterra kernel, and the two processes { W t } t ∈ [0 ,T ] and { W (cid:48) t } t ∈ [0 ,T ] are independent Brownian motions, and ρ ∈ (0 , is the coefficient determining the correla-tion between Y n and Z . Observe also that ( (cid:82) T (cid:99) W s d s, W (cid:48) T ) is normally distributed with E [ (cid:90) T (cid:99) W s d s W (cid:48) T ] = (cid:90) T (cid:90) t K ( t − s )d s d r. Referring to (4.1) , we now have X n = (cid:82) · g ( Y ns ) (cid:112) − ρ d W s and A n = R ( · ) − (cid:82) · g ( Y ns ) d s .Furthermore, M nt := (cid:82) t g ( Y ns )[ ρ d W (cid:48) s + (cid:112) − ρ d W s ] , and we see that (cid:104) M (cid:105) T = (cid:90) T E [ g ( Y ns )]d s Assuming that g is twice differentiable and bounded away from , we set Σ n = Σ = g ( y ) T we see that for n ∈ N (cid:15) − n ( (cid:104) M (cid:105) T g ( y ) T −
1) and 1 (cid:82) T g ( Y ns )d s are both bounded in L p (Ω) for any p > . Furthermore Σ n M nT and (cid:15) − n ( (cid:104) M (cid:105) T g ( y ) T − are bothseen to be normally distributed, and thus the conditions in Hypothesis 4.1 are satisfied. Itfollows from Theorem 4.2 that the Black-Scholes implied Volatility is given by σ BS = g ( y ) (cid:16) − (cid:15) n ρ d (cid:17) , (4.3) where we have used that E [ N ] = 0 . A first order Taylor expansion of g reveals that ρ ( T ) := g (cid:48) ( y ) ρg ( y ) T / (cid:90) T E [ (cid:99) W s W (cid:48) T ]d s = g (cid:48) ( y ) ρg ( y ) T / (cid:90) T (cid:90) t K ( t − s )d s d t. (4.4) In subsequent sections, we will investigate this term in in more detail for a particular choiceof the Volterra kernel K . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 12 Skew expansions with log-fractional Brownian motion
We now apply the small vol-of-vol expansion to log-moderated rough volatility models. Inthe first step, we compute the term ρ for such models.5.1. ρ in the case of log-fractional Brownian motion. We will compute the term ρ given in Example 4.3 when the Volterra process is given as a log-fractional Brownian motion.Recall from (4.4) that ρ is given by ρ = g (cid:48) ( y ) ρg ( y ) T / (cid:90) T E (cid:104)(cid:99) W s W T (cid:105) d s, (5.1)where ρ is the correlation coefficient between the Brownian noises. We compute the integralon the r.h.s. under the assumption that T is small, more precisely, T ≤ χ . (Keep in mindthat we are eventually going to look for asymptotics for T → .) Lemma 5.1.
Let
T > and { W t } t ∈ [0 ,T ] be a Brownian motion, and define the log-fractionalBrownian motion (cid:99) W t = (cid:82) t K ( t − s )d W s , where the kernel is given as in (2.2) . Then we have (cid:90) T E (cid:104) W T (cid:99) W s (cid:105) d s = (cid:90) T (cid:90) s K ( s − r )d r d s = I ( T ∧ χ ) + T >χ ( I ( T, χ ) + I ( T, χ )) , where I ( T ) := Cζ − p log (cid:18) T (cid:19) − p (cid:20) T E p (cid:18) ( H + 1 /
2) log (cid:18) T (cid:19)(cid:19) − E p (cid:18) ( H + 3 /
2) log (cid:18) T (cid:19)(cid:19)(cid:21) ,I ( T, χ ) := CH + 1 / (cid:32) T H +3 / − χ H +3 / H + 3 / − ( T − χ ) χ H +1 / (cid:33) ,I ( T, χ ) := Cζ − p ( T − χ ) log (cid:18) χ (cid:19) − p E p (cid:18) ( H + 1 /
2) log (cid:18) χ (cid:19)(cid:19) . Proof.
It is tempting to integrate the formula in Lemma 3.3, but we were not able to find aclosed form expression this way. Rather, let us start from scratch. Clearly, we have that (cid:90) T E [ W T (cid:99) W s ]d s = (cid:90) T (cid:90) s K ( s − r ) d r d s =: I ( T ) . We first assume that T ≤ χ . Using the representation of the kernel given in (2.2) we seethat (cid:90) T (cid:90) s K ( s − r )d r d s = (cid:90) T Cζ − p (cid:90) s ( s − r ) H − / log (cid:18) s − r (cid:19) − p d r d s = Cζ − p (cid:90) T (cid:90) s r H − / log (cid:18) r (cid:19) − p d r d s = Cζ − p (cid:90) T (cid:90) Tr d s r H − / log (cid:18) r (cid:19) − p d r = Cζ − p (cid:34) T (cid:90) T r H − / log (cid:18) r (cid:19) − p d r − (cid:90) T r H +1 / log (cid:18) r (cid:19) − p d r (cid:35) . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 13
By Lemma 3.1 it then follows that (cid:90) T r H − / log (cid:18) r (cid:19) − p d r = log (cid:18) T (cid:19) − p E p (cid:18) ( H + 1 /
2) log (cid:18) T (cid:19)(cid:19) , (cid:90) T r H +1 / log (cid:18) r (cid:19) − p d r = log (cid:18) T (cid:19) − p E p (cid:18) ( H + 3 /
2) log (cid:18) T (cid:19)(cid:19) . Putting the terms together, we obtain I ( T ) = I ( T ) = Cζ − p log (cid:18) T (cid:19) − p (cid:20) T E p (cid:18) ( H + 1 /
2) log (cid:18) T (cid:19)(cid:19) − E p (cid:18) ( H + 3 /
2) log (cid:18) T (cid:19)(cid:19)(cid:21) . Let us now consider the case
T > χ . The integral can then naturally be split as I ( T ) = (cid:90) χ (cid:90) s K ( s − r )d r d s + (cid:90) Tχ (cid:90) s − χ K ( s − r )d r d s + (cid:90) Tχ (cid:90) ss − χ K ( s − r )d r d s = I ( χ ) + (cid:90) Tχ (cid:90) sχ K ( u )d u d s + (cid:90) Tχ (cid:90) χ K ( u )d u d s, noting that I ( χ ) is already known. An elementary calculation gives us the second term, (cid:90) Tχ (cid:90) sχ K ( u ) d u d s = C (cid:90) Tχ (cid:90) sχ u H − / d u d s = CH + 1 / (cid:90) Tχ (cid:16) s H +1 / − χ H +1 / (cid:17) d s = CH + 1 / (cid:32) T H +3 / − χ H +3 / H + 3 / − ( T − χ ) χ H +1 / (cid:33) . Finally, regarding the third term we do a substitution of variables and apply Lemma 3.1 toobtain (cid:90) Tχ (cid:90) χ K ( u )d u d s = Cζ − p ( T − χ ) (cid:90) χ log (cid:18) u (cid:19) − p u H − / d u = Cζ − p ( T − χ ) log (cid:18) χ (cid:19) − p E p (cid:18) ( H + 1 /
2) log (cid:18) χ (cid:19)(cid:19) . (cid:3) Asymptotic expansion for Example 4.3.
We continue with a discussion of Exam-ple 4.3, when the volatility depends on a log-fBm. As we have already computed ρ , wehave all ingredients for the asymptotic expansion in terms of small vol-of-vol. We are alsointerested in the short time behaviour of this term, which relies on the following well knownasymptotic expansion of the exponential integral E p : E p ( x ) ∼ e − x x (cid:20) − px + p ( p + 1) x ± · · · (cid:21) as x → ∞ . (5.2) Lemma 5.2.
The term ρ in (5.1) satisfies the asymptotic expansion ρ = g (cid:48) ( y ) g ( y ) Cζ − p ρ ( H + 1 / H + 3 /
2) log (cid:18) T (cid:19) − p T H (1 + o (1)) as T → . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 14
Proof.
Using the asymptotic expansion in (5.2) we have E p (cid:18) ( H + 1 /
2) log (cid:18) T (cid:19)(cid:19) = T H +1 / ( H + 1 /
2) log (cid:0) T (cid:1) (1 + o (1)) , (5.3) E p (cid:18) ( H + 3 /
2) log (cid:18) T (cid:19)(cid:19) = T H +3 / ( H + 3 /
2) log (cid:0) T (cid:1) (1 + o (1)) , (5.4)as T → . Hence (as T < χ eventually), we obtain (cid:90) T E (cid:104) W T (cid:99) W s (cid:105) d s = Cζ − p ( H + 1 / H + 3 /
2) log (cid:18) T (cid:19) − p T H +3 / (1 + o (1)) . Recalling (5.1), we have ρ = g (cid:48) ( y ) ρg ( y ) T / (cid:90) T E (cid:104) W T (cid:99) W s (cid:105) d s = g (cid:48) ( y ) g ( y ) Cζ − p ρ ( H + 1 / H + 3 /
2) log (cid:18) T (cid:19) − p T H (1 + o (1)) . (cid:3) Let us now look again at the implied volatility found in Theorem 4.2 and in Example 4.3.By the formula (4.2) in we have σ BS = σ (cid:18) − ρ d (cid:15) n (cid:19) (1 + o ( (cid:15) n )) , Following Example 4.3, setting R ( t ) := rt for a constant r > , σ := g ( y ) , Σ := g ( y ) T , and d = log (cid:0) S K (cid:1) + r − Σ2 √ Σ . Hence, the part of the leading order term depending on log-moneyness log (cid:0) S K (cid:1) is − σ ρ log (cid:0) S K (cid:1) g ( y ) √ T (cid:15) n = − aρ log (cid:18) T (cid:19) − p T H − / log (cid:18) S K (cid:19) (cid:15) n (1 + o T (1)) , with a := 12 g (cid:48) ( y ) g ( y ) Cζ − p ( H + 1 / H + 3 / g (cid:48) ( y ) g ( y ) (cid:20)(cid:113) E(2
H/ζ ) /ζ + − exp( − H/ζ )2 H ζ p ( H + 1 / H + 3 / (cid:21) − , H > , g (cid:48) ( y ) g ( y ) ζ / − p (cid:113) p − p , H = 0 . (5.5) Remark . As the skew asymptotic is linear in a = a H,ζ,p , we may think of these modelparameters to contribute to vol-of-vol . It turns out that a varies considerably as a functionof ζ and p for fixed roughness H . The actual asymptotic skew formula is, fortunately, muchmore stable, see Figure 7.3.These considerations leads to the following theorem regarding the ATM volatility skew forsmall vol-of-vol and short maturity T : Theorem 5.4.
The implied volatility in Example 4.3 with log-moderated fBm satisfies σ BS = g ( y ) (cid:18) − ρ d (cid:15) n (cid:19) (1 + o ( (cid:15) n )) OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 15 with ρ given by (5.1) together with Lemma 5.1 and d = log (cid:0) S K (cid:1) + r − g ( y ) T g ( y ) √ T .
For log-moneyness k, k (cid:48) ∈ R , short maturity T , and any ≤ H ≤ / , the skew thereforebehaves like σ BS ( T, k ) − σ BS ( T, k (cid:48) ) k − k (cid:48) ≈ − aρ log (cid:18) T (cid:19) − p T H − / (cid:15) n , with a defined in (5.5) . The rough Bergomi model.
As a practical example, we consider here the roughBergomi model, when the driving noise of the instantaneous variance is given as a log-fBm.To this end, denote by ξ ( u ) = E Q [ v u |F ] , where v denotes the instantaneous variance. Therough Bergomi model is given by S nt = S E (cid:18)(cid:90) t (cid:112) v ns dB s (cid:19) v nt = ξ ( t ) E (cid:18) (cid:15) n (cid:90) t K ( t − s )d W (cid:48) s (cid:19) (5.6)where B t = ρW (cid:48) t + (cid:112) − ρ W t . Theorem 5.5.
Let for n ∈ N , let ( S n , v n ) be a stochastic volatility model given with roughBergomi dynamics as in (5.6) , where { (cid:15) n } n ∈ N is a null sequence, representing vol-of-vol η ,and where the Volterra kernel K is given as in (2.2) . Then the following expansion holds forthe implied volatility surface σ BS ( T, k ) = k (cid:15) n T − (cid:18)(cid:90) T ξ ( s )d s (cid:19) − (cid:90) T ξ ( s ) (cid:90) s K ( s − r ) (cid:112) ξ ( r )d r d s + o ( (cid:15) n ) , where k = log( S /K ) denotes log-moneyness and T is maturity time. Furthermore, the ATMvolatility skew behaves like σ BS ( T, k ) − σ BS ( T, k (cid:48) ) k − k (cid:48) ≈ (cid:15) n T − (cid:18)(cid:90) T ξ ( s )d s (cid:19) − (cid:90) T ξ ( s ) (cid:90) s K ( s − r ) (cid:112) ξ ( r )d r d s. Proof.
For the proof of this theorem we will apply the martingale expansion of Theorem 4.2to obtain the implied volatility expansion. To this end, we need to verify that Hypothesis 4.1holds for this particular model. Since g ( y ) ∼ e y is unbounded, we cannot apply Lemma 5.2directly, and we need to verify that the conditions in Hypothesis 4.1 indeed holds. We beginto specify the terms of Fukasawa’s expansion.In this case we have A nt = − (cid:90) t v ns d sM nt = (cid:112) − ρ (cid:90) t (cid:112) v ns d W (cid:48) s + ρ (cid:90) t (cid:112) v ns d W s . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 16
It is readily seen that (cid:104) M n (cid:105) = − A n . We set Σ n = Σ = (cid:82) T ξ ( s )d s and we see that for each n ∈ N D n = (cid:15) − n (Σ − (cid:90) T ξ ( s ) E (cid:18) (cid:15) n (cid:90) s K ( s − r )d W r (cid:19) d s − is bounded in D n ∈ L p (Ω) for any p > . Furthermore, by Jensen’s inequality, it follows that (cid:16)(cid:82) T v ns d s (cid:17) − is bounded in L p (Ω) . Indeed, we see that E (cid:34)(cid:18)(cid:90) T v ns d s (cid:19) − p (cid:35) ≤ (cid:90) T E (cid:2) ( v ns ) − p (cid:3) d s < ∞ , where we have used that ( v ns ) − = v nt = ξ ( t ) E (cid:16) − (cid:15) n (cid:82) t K ( t − s )d W s (cid:17) which is contained in L p (Ω) . Moreover, we see that D n and Σ − / M nT converge weakly to the normal randomvariables N and N . In particular, we have that N = (cid:82) T ξ ( s ) (cid:82) t K ( s − r )d W r d s Σ and N = (cid:82) T (cid:112) ξ ( s )d B s √ Σ . We can therefore apply Theorem 4.2 to the rough Bergomi model. To this end, we need tocompute ρ = E [ N N ] , and we observe that ρ = ρ Σ − (cid:90) T ξ ( s ) (cid:90) s K ( s − r ) (cid:112) ξ ( r )d r d s. Explicit computations of this term is more difficult, due to the integration over the variancecurve. Of course, if ξ ( s ) = ξ is constant, then ρ is computed identically as in Lemma 5.2.It follows from Theorem 4.2, using that σ n = (cid:113) Σ T , that the implied Volatility is given by σ BS = (cid:114) Σ T (cid:16) − (cid:15) n ρ d (cid:17) + o ( (cid:15) n ) Inserting the values for d and ρ , considering the leading order term involving the log-moneyness k = log (cid:0) SK (cid:1) , we find that σ BS ( T, k ) = k (cid:15) n T − − (cid:90) T ξ ( s ) (cid:90) s K ( s − r ) (cid:112) ξ ( r )d r d s + o ( (cid:15) n ) , where k = log (cid:0) SK (cid:1) . Furthermore, from the above formula, it is straight forward to see thatthe ATM volatility skew behaves like (cid:15) n T − − (cid:90) T ξ ( s ) (cid:90) s K ( s − r ) (cid:112) ξ ( r )d r d s. Substituting
Σ = (cid:82) T ξ ( s )d s , concludes the proof. (cid:3) Corollary 5.6.
In the super-rough Bergomi model with constant forward variance curve ξ ,the ATM skew behaves like σ BS ( T, k ) − σ BS ( T, k (cid:48) ) k − k (cid:48) ≈ − aρ log (cid:18) T (cid:19) − p T H − / (cid:15) n , as T → , with a given in (5.5) , substituting g ( x ) ≡ ξ (0) exp( x ) . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 17 Asymptotic skew under log-fractional volatility
In Section 5 we show an expansion for the implied skew in small time and small vol-of-volusing the martingale expansion outlined in Section 4. We attempt here to understand theshort time behavior of a log-modulated rough stochastic volatility model without consideringthe small vol-of-vol regime, but just the short time asymptotics. For this, we adapt Fukasawa’sframework of [13, 6, 14] to log-fractional volatility, using the “regular variation” language. Letus consider, similarly to (5.6), a stochastic volatility model of the form S t = S E (cid:18)(cid:90) t √ v s d B s (cid:19) ,v t = ξ ( t ) E (cid:18) η (cid:90) t K ( t, s )d W (cid:48) s (cid:19) where B s = ρW (cid:48) s + (cid:112) − ρ W s , η > . For now, we do not assume the specific form (2.1)for K , but only square integrability. Let s ( t ) := (cid:16)(cid:82) t K ( t, s ) ds (cid:17) / . To allow logarithmiccorrections to the fractional power-law type kernels we assume s ( t ) → as t → and s to beregularly varying at : for some L slowly varying, s ( t ) = t H L ( t ) (so L ( t ) → if H = 0 ). We also assume H ∈ [0 , / (rough but also super-rough volatility).Let ¯ ξ ( t ) := 1 t (cid:90) t ξ ( u )d u, K := lim t → (cid:82) t K ( t, s )d s (cid:113) t (cid:82) t K ( t, s )d sα ( z ) := z ρη K√ v (2 H + 3) where v is spot volatility and ξ ( · ) is continuous at . The following theorem and corollaryare inspired by [14, Theorem 2.1 and Corollary 2.1], modified in order to be applicable tolog-fractional volatility. Theorem 6.1.
Denoting σ BS ( k, T ) the Black-Scholes implied volatility at time with expiry T and log-moneyness k . For z ∈ R and T → , σ BS (cid:16) z √ T , T (cid:17) = (cid:113) ¯ ξ ( T ) (cid:0) α ( z ) s ( T ) (cid:1) + o ( s ( T )) Corollary 6.2.
The implied skew behaves as follows: for z (cid:48) (cid:54) = z , if K (cid:54) = 0 , σ BS (cid:16) z √ T , T (cid:17) − σ BS (cid:16) z (cid:48) √ T , T (cid:17) z √ T − z (cid:48) √ T ∼ ρη K H + 3 s ( T ) √ T , where ∼ denotes asymptotic equivalence as T → . If K = 0 , √ Ts ( T ) σ BS (cid:16) z √ T , T (cid:17) − σ BS (cid:16) z (cid:48) √ T , T (cid:17) z √ T − z (cid:48) √ T → . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 18
Remark . It always holds
K ≤ . This corollary gives the exact scaling of the impliedskew if K (cid:54) = 0 , otherwise just gives an upper bound.
Proof.
This proof is based on [14, Appendix A], [13, Theorem 1]. Note that (cid:18) √ t (cid:18) S t S − (cid:19) , ηs ( t ) (cid:18) v t ξ ( t ) − (cid:19)(cid:19) → ( γ, δ ) in law as t → , where ( γ, δ ) is a centred 2-dim Gaussian with covariance Σ = (cid:32) v ρ √ v K ρ √ v K (cid:33) . (6.1)For t > , u ∈ [0 , let us write X tu = 1 √ t (cid:18) S ut S − (cid:19) Note that X t is a martingale in u for fixed t , with quadratic variation d (cid:104) X t (cid:105) u = ( S tu /S ) v ut d u = (1 + √ tX tu ) v ut d u. We write ∆ = ( e z √ t − / √ t . We use the Bachelier pricing equation as in [13, 14], ∂p∂u ( x, u ) + 12 ξ ( ut ) ∂ p∂x ( x, u ) = 0 , p ( x,
1) = (∆ − x ) + whose explicit solution is given by p ( x, u ) = (∆ − x )Φ (cid:32) t (∆ − x ) (cid:82) tut ξ ( s )d s (cid:33) + φ (cid:32) t (∆ − x ) (cid:82) tut ξ ( s )d s (cid:33) t (cid:90) tut ξ ( s )d s (6.2)with Φ , φ standard normal distribution function and density. By Itô’s formula we rewrite thefollowing rescaled put option price in terms of X t E [( S e z √ t − S t ) + ] S √ t = E [(∆ − X t ) + ] = E [ p ( X t , p (0 ,
0) + 12 E (cid:20)(cid:90) ∂ p∂x ( X tu , u )( v ut − ξ ( ut ))d u (cid:21) + 12 E (cid:20)(cid:90) ∂ p∂x ( X tu , u )(2 √ tX tu + t ( X tu ) ) v ut d u (cid:21) (6.3)Since (as in [14]) (cid:90) E (cid:20) ∂ p∂x ( X tu , u ) X tu v ut (cid:21) d u → v z √ v φ (cid:18) z √ v (cid:19) we have E (cid:20)(cid:90) ∂ p∂x ( X tu , u )(2 √ tX tu + t ( X tu ) ) v ut d u (cid:21) = O ( √ t ) so this term is negligible in (6.3). Now we use our different (possibly logarithmic) scalingassumption for the volatility and get (cid:18) X tu , v tu − ξ ( tu ) ηs ( tu ) (cid:19) t → −−→ ( √ uγ, v δ ) . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 19
Again as in [14], as t → we have ∂ p∂x ( X tu , u ) t → −−→ (cid:112) v (1 − u ) φ (cid:32) z − √ uγ (cid:112) v (1 − u ) (cid:33) in law for each u ∈ [0 , . We have s ( t ) (cid:90) E (cid:20) ∂ p∂x ( X tu , u )( v ut − ξ ( ut )) (cid:21) d u = (cid:90) s ( ut ) s ( t ) u H u H s ( ut ) E (cid:20) ∂ p∂x ( X tu , u )( v ut − ξ ( ut )) (cid:21) d u. Regular variation of s ( · ) implies s ( ut ) ∼ u H s ( t ) as t → . So, lim t s ( t ) (cid:90) E (cid:20) ∂ p∂x ( X tu , u )( v ut − ξ ( ut )) (cid:21) d u = lim t (cid:90) u H s ( ut ) E (cid:20) ∂ p∂x ( X tu , u )( v ut − ξ ( ut )) (cid:21) d u = (cid:90) E (cid:34) u H ηv δ (cid:112) v (1 − u ) φ (cid:32) z − √ uγ (cid:112) v (1 − u ) (cid:33)(cid:35) d u. The joint (Gaussian) density of γ and δ is given in (6.1). Explicit computations give lim t s ( t ) (cid:90) E (cid:20) ∂ p∂x ( X tu , u )( v ut − ξ ( ut )) (cid:21) d u = zρη K H + 3 / φ (cid:18) z √ v (cid:19) . Now, from the definition of α and (6.2), we write the rescaled put option with expiry t as E [( S e z √ t − S t ) + ] S √ t = p (0 ,
0) + α ( z ) √ v φ (cid:18) z √ v (cid:19) s ( t ) + o ( s ( t ))= ∆Φ (cid:32) ∆ (cid:112) ¯ ξ ( t ) (cid:33) + (cid:113) ¯ ξ ( t ) φ (cid:32) ∆ (cid:112) ¯ ξ ( t ) (cid:33) (1 + α ( z ) s ( t )) + o ( s ( t )) . (6.4)Let p BS ( K, t, σ ) denote the price under the Black-Scholes model of a put option with strike K , expiry t and volatility σ . We have the following Taylor expansion, holding for fixed a ,analogous to [14, Equation (6)] p BS ( S e z √ t , t, σ + as ( t )) S √ t = ∆Φ (cid:18) ∆ σ (cid:19) + σφ (cid:18) ∆ σ (cid:19) (cid:16) aσ s ( t ) (cid:17) + o ( s ( t )) . We have equality with (6.4) with σ = (cid:113) ¯ ξ ( t ) , a = α ( z ) (cid:113) ¯ ξ ( t ) , and the implied volatility expansion follows taking T = t (at least formally). (cid:3) Asymptotic skew of the (super) rough Bergomi model.
We consider now themodel with K given in (2.1) and (2.2). We have, using Lemma 3.1, Lemma 3.2 and (5.3) (cid:90) t K ( t, s ) ds ∼ C ζ − p p − log(1 /t ) − p for H = 0 C ζ − p H log(1 /t ) − p t H for H > (cid:90) t K ( t, s ) ds ∼ Cζ − p H + 1 / /t ) − p t H +1 / for H ≥ OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 20
We get K = √ H/ ( H + 1 / for H ∈ [0 , / . So, writing “skew” in the sense of Corollary6.2, skew ∼ ρηCζ − p (2 H + 3)( H + 1 / T H − / log(1 /T ) − p , for H > , and we recover the analogous result to (1.6) and Theorem 5.5. For H = 0 , we cansay T / log(1 /T ) p − / skew → , which gives an upper bound, but we do not get the precise time-scaling of the skew. However,this upper bound is consistent with the small vol-of-vol result (1.6), even for H = 0 . Moreover,from Figure 1.1b, it seems reasonable to expect that the same asymptotics should hold forthe skew at H = 0 . The question remains open, whether it is possible to obtain a preciseshort-time asymptotic result without using a small vol-of-vol expansion.As a sanity check, note that when K is the classical Riemann-Liouville kernel we recoverthe well known constant in the explosion of the skew, see e.g. [4, 13].7. Numerical analysis . . . . T sk e w ( ab s o l u t e v a l ue ) modelasymptotic (a) η = 0 . . . . . . T sk e w ( ab s o l u t e v a l ue ) modelasymptotic (b) η = 2 Figure 7.1.
Asymptotic formula for the ATM-skew for small vol-of-vol inthe super-rough Bergomi model with H = 0 , ρ = − . , ζ = 0 . , p = 2 , andsmall vol-of-vol η = 0 . vs “normal” vol-of-vol η = 2 . The asymptotic formulais compared against skews computed by Monte Carlo simulation.We supplement the theoretical results by some numerical experiments. In all these ex-amples, we use the super-rough Bergomi model (5.6). Skews are computed based on MonteCarlo simulation with exact simulation of the log-fractional Brownian motion (2.1) togetherwith (2.2). More precisely, we compute the covariance function of ( W, (cid:99) W ) using the formulasin Section 3 as well as numerical integration for the auto-covariance of (cid:99) W . Exact simulationfrom ( W, (cid:99) W ) is then done by the Cholesky method. Given samples from the stochastic vari-ance, the asset price processes is computed by Euler discretization. We start by comparingthe small vol-of-vol expansion with the skews obtained in the model, see Theorem 5.5. OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 21
In Figure 7.1, we compare the asymptotic formula with the actual skew for two differentvalues of the vol-of-vol parameter η . Clearly, for small η (left), the accuracy is extremelygood, and the fit deteriorates noticeably when η is increased. Note that we concentrate onthe case H = 0 , as here the behaviour obviously differs most from the rough Bergomi case. T sk e w ( ab s o l u t e v a l ue ) small vol−of−volshort time of small vol−of−vol Figure 7.2.
Short time expansion of the asymptotic formula for the ATM-skew for small vol-of-vol, see Theorem 5.4 together with Theorem 5.5. Theparameters correspond to Figure 7.1aNext we consider the short-time asymptotic of the asymptotic skew formula obtained inTheorem 5.5 together with Theorem 5.4, see Figure 7.2. We should note that Theorem 5.4only provides the short time asymptotic for < T < χ = e − /ζ ≈ × − in our example.Hence, we need to zoom in very closely for the asymptotic formula to hold. The Figureindicates that the convergence of the short-time asymptotics is very slow. OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 22 . . . . T sk e w ( ab s o l u t e v a l ue ) z = 0.1, p = 1.01 z = 0.1, p = 2 z = 0.2, p = 1.01 z = 0.2, p = 2 Figure 7.3.
Asymptotic skew formulas for small vol-of-vol in the super-roughBergomi model for different values of ζ and p . The remaining parameters are H = 0 , ρ = − . , η = 2 , ξ ≡ . .Coming back to the discussion of the additional parameters ζ and p in Remarks 2.2 and 5.3,we compare the small vol-of-vol skew formulas of Theorem 5.5 for different values of ζ and p , see Figure 7.3. Clearly, the absolute value of the ATM skew is increasing in both ζ and p , which indicates that one of these parameters could be easily removed – by fixing it to acanonical value. In this case, we suggest to fix p to a value close to , such as p = 1 . asused in the plot.Finally, we compare the super-rough Bergomi model with the standard rough Bergomimodel. Figure 7.4 compares ATM-skews – as computed by Monte Carlo simulation – forboth models and different values of H . As expected, the curves differ substantially for verysmall H , but move closely together for H large. In this sense, the super-rough Bergomimodel can be seen as a perturbation of the rough Bergomi model for H (cid:29) , which is stillwell-defined in the limit H = 0 – naturally departing from the rough Bergomi model in theprocess, i.e., as H → . OG-MODULATED ROUGH STOCHASTIC VOLATILITY MODELS 23 . . . . . . T sk e w ( ab s o l u t e v a l ue ) sBergomi, H = 0.01sBergomi, H = 0.05sBergomi, H = 0.09rBergomi, H = 0.01rBergomi, H = 0.05rBergomi, H = 0.09 Figure 7.4.
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Christian Bayer: Weierstrass Institute of Applied Analysis and Stochastics, Mohrenstr.39, 10117 Berlin, Germany
E-mail address : [email protected] Fabian A. Harang: Department of Mathematics, University of Oslo, P.O. box 1053, Blin-dern, 0316, OSLO, Norway
E-mail address : [email protected] Paolo Pigato: Department of Economics and Finance, University of Rome Tor Vergata,Via Columbia 2, 00133 Roma, Italy
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