Hybrid scheme for Brownian semistationary processes

Let \((\varOmega,\mathcal{F},(\mathcal{F}_{t})_{t \in \mathbb {R}}, \mathbb{P})\) be a filtered probability space, satisfying the usual conditions, supporting a (two-sided) standard Brownian motion \(W=(W_{t})_{t \in \mathbb {R}}\). We consider a Brownian semistationary process where \(\sigma= (\sigma_{t})_{t\in \mathbb {R}}\) is an \((\mathcal{F}_{t})_{t \in \mathbb {R}}\)-predictable process with locally bounded trajectories, which captures the stochastic volatility (intermittency) of \(X\), and where \(g: (0,\infty) \to[0,\infty)\) is a Borel-measurable kernel function.

To ensure that the integral (2.1) is well defined, we assume that the kernel function \(g\) is square-integrable, that is, \(\int _{0}^{\infty}g(x)^{2} \mathrm {d}x < \infty\). In fact, we shortly introduce some more specific assumptions on \(g\) that imply its square-integrability. Throughout the paper, we also assume that the process \(\sigma\) has finite second moments, \(\mathbb {E}[\sigma_{t}^{2}] < \infty\) for all \(t \in \mathbb {R}\), and that the process is covariance-stationary, namely, These assumptions imply that also \(X\) is covariance-stationary, that is, However, the process \(X\) need not be strictly stationary as the dependence between the volatility process \(\sigma\) and the driving Brownian motion \(W\) may be time-varying.

As mentioned above, we consider a kernel function that satisfies \(g(x) \propto x^{\alpha}\) for some \(\alpha\in(-\frac{1}{2},\frac{1}{2})\setminus\{0 \}\) when \(x>0\) is near zero. To make this idea rigorous and to allow additional flexibility, we formulate our assumptions on \(g\) using the theory of regular variation [15] and, more specifically, slowly varying functions.

To this end, recall that a measurable function \(L : (0,1] \rightarrow [0,\infty)\) is slowly varying at 0 if for any \(t>0\), Moreover, a function \(f(x) = x^{\beta}L(x)\), \(x \in(0,1]\), where \(\beta \in \mathbb {R}\) and \(L\) is slowly varying at 0, is said to be regularly varying at 0, with \(\beta\) being the index of regular variation.

A key feature of slowly varying functions, which will be very important in the sequel, is that they can be sandwiched between polynomial functions as follows. If \(\delta>0\) and \(L\) is slowly varying at 0 and bounded away from 0 and \(\infty\) on every interval \((u,1]\), \(u \in(0,1)\), then there exist constants \(\overline{C}_{\delta} \geq \underline{C}_{\delta} >0\) such that The inequalities above are an immediate consequence of the so-called Potter bounds for slowly varying functions; see [15, Theorem 1.5.6(ii)] and (4.1) below. Making \(\delta\) very small, we see that slowly varying functions are asymptotically negligible in comparison with polynomially growing/decaying functions. Thus, by multiplying power functions and slowly varying functions, regular variation provides a flexible framework to construct functions that behave asymptotically like power functions.

Our assumptions concerning the kernel function \(g\) are as follows: (Here, and in the sequel, we use the notation \(f(x) = \mathcal {O}(h(x))\), \(x \rightarrow a\), to indicate that \(\limsup_{x \rightarrow a} |\frac{f(x)}{h(x)}| < \infty\). Additionally, analogous notation is later used for sequences and computational complexity.) In view of the bound (2.2), these assumptions ensure that \(g\) is square-integrable. It is worth pointing out that (A1) accommodates functions \(L_{g}\) with \(\lim_{x \rightarrow 0} L_{g}(x) = \infty\), e.g. \(L_{g}(x) = 1 - \log x\).

The assumption (A1) influences the short-term behavior and roughness of the process \(X\). A simple way to assess the roughness of \(X\) is to study the behavior of its variogram (also called second-order structure function in the turbulence literature) as \(h \rightarrow0\). Note that by covariance-stationarity, Under our assumptions, we have the following characterization of the behavior of \(V_{X}\) near zero, which generalizes a result of Barndorff-Nielsen [3, p. 9] and implies that \(X\) has a locally Hölder-continuous modification. Therein, and in what follows, we write \(a(x) \sim b(x)\), \(x \rightarrow y\), to indicate that \(\lim_{x \rightarrow y} \frac{a(x)}{b(x)} = 1\). The proof of this result is carried out in Sect. 4.1.

Motivated by Proposition 2.2, we call \(\alpha\) the roughness index of the process \(X\). Ignoring the slowly varying factor \(L_{g}(h)^{2}\) in (2.2), we see that the variogram \(V(h)\) behaves like \(h^{2\alpha+1}\) for small values of \(h\), which is reminiscent of the scaling property of the increments of a fractional Brownian motion (fBm) with Hurst index \(H = \alpha+ \frac{1}{2}\). Thus, the process \(X\) behaves locally like such an fBm, at least when it comes to second order structure and roughness. (Moreover, the factor \(\frac{1}{2\alpha + 1} + \int_{0}^{\infty}( (y+ 1)^{\alpha}- y^{\alpha})^{2} \mathrm {d}y\) coincides with the normalization coefficient that appears in the Mandelbrot–Van Ness representation [23, Theorem 1.3.1] of an fBm with \(H = \alpha+ \frac{1}{2}\).)

Let us now look at two examples of a kernel function \(g\) that satisfies our assumptions.

Let \(t \in \mathbb {R}\) and consider discretizing \(X_{t}\) based on its integral representation (2.1) on the grid \(\mathcal{G}^{n}_{t} := \{t,t-\frac {1}{n}, t-\frac{2}{n},\ldots\}\) for \(n \in \mathbb {N}\). To derive our discretization scheme, let us first note that if the volatility process \(\sigma\) does not vary too much, then it is reasonable to use the approximation that is, we keep \(\sigma\) constant in each discretization cell. (Here, and in the sequel, “≈” stands for an informal approximation used for purely heuristic purposes.) If \(k\) is “small”, then due to (A1), we may approximate as the slowly varying function \(L_{g}\) varies “less” than the power function \(y \mapsto y^{\alpha}\) near zero; cf. (2.2). If \(k\) is “large”, or at least \(k \geq2\), then choosing \(b_{k} \in [k-1,k]\) provides an adequate approximation by (A2). Applying (2.4) to the first \(\kappa\) terms, where \(\kappa= 1,2,\ldots{}\), and (2.5) to the remaining terms in the approximating series in (2.3) yields For completeness, we also allow \(\kappa= 0\), in which case we require that \(b_{1} \in(0,1]\) and interpret the first sum on the right-hand side of (2.6) as zero. To make numerical implementation feasible, we truncate the second sum on the right-hand side of (2.6) so that both sums have \(N_{n} \geq\kappa+1\) terms in total. Thus, we arrive at a discretization scheme for \(X_{t}\), which we call a hybrid scheme, given by where and \(\mathbf{b}:=(b_{k})_{k=\kappa+1}^{\infty}\) is a sequence of real numbers, evaluation points, that must satisfy \(b_{k} \in[k-1,k]\setminus \{0\}\) for each \(k\geq\kappa+1\), but otherwise can be chosen freely.

As it stands, the discretization grid \(\mathcal{G}^{n}_{t}\) depends on the time \(t\), which may seem cumbersome with regard to sampling \(X^{n}_{t}\) simultaneously for different times \(t\). However, note that whenever times \(t\) and \(t'\) are separated by a multiple of \(\frac{1}{n}\), the corresponding grids \(\mathcal{G}^{n}_{t}\) and \(\mathcal{G}^{n}_{t'}\) will intersect. In fact, the hybrid scheme defined by (2.7) and (2.8) can be implemented efficiently, as we shall see in Sect. 3.1, below. Since the degenerate case \(\kappa=0\) with \(b_{k} = k\) for all \(k \geq1\) corresponds to the usual Riemann-sum discretization scheme of \(X_{t}\) with (Itô type) forward sums from (2.8). Henceforth, we denote the associated sequence \(( k)_{k=\kappa+1}^{\infty}\) by \(\mathbf {b}_{\mathrm{FWD}}\), where the subscript “\(\mathrm{FWD}\)” alludes to forward sums. However, including terms involving Wiener integrals of a power function given by (2.7), that is, having \(\kappa\geq1\), improves the accuracy of the discretization considerably, as we shall see. Having the leeway to select \(b_{k}\) within the interval \([k-1,k]\setminus\{ 0 \}\), so that the function \(g(t-\cdot)\) is evaluated at a point that does not necessarily belong to \(\mathcal {G}^{n}_{t}\), leads additionally to a moderate improvement.

The truncation in the sum (2.8) entails that the stochastic integral (2.1) defining \(X_{t}\) is truncated at \(t - \frac{N_{n}}{n}\). In practice, the value of the parameter \(N_{n}\) should be large enough to mitigate the effect of truncation. To ensure that the truncation point \(t - \frac{N_{n}}{n}\) tends to \(-\infty\) as \(n \rightarrow\infty\) in our asymptotic results, we introduce the following assumption:

We are now ready to state our main theoretical result, which gives a sharp description of the asymptotic behavior of the mean square error (MSE) of the hybrid scheme as \(n \rightarrow\infty\). We defer the proof of this result to Sect. 4.2.

When the hybrid scheme is used to simulate the \(\mathcal{BSS}\) process \(X\) on an equidistant grid \(\{0,\frac{1}{n},\frac{2}{n},\ldots,\frac {\lfloor nT \rfloor}{n} \}\) for some \(T>0\) (see Sect. 3.1 on the details of the implementation), the following consequence of Theorem 2.5 ensures that the covariance structure of the simulated process approximates that of the actual process \(X\).

In Theorem 2.5, the asymptotics of the MSE (2.11) are determined by the behavior of the kernel function \(g\) near zero, as specified in (A1). The condition (2.9) ensures that the error from approximating \(g\) near zero is asymptotically larger than the error induced by the truncation of the stochastic integral (2.1) at \(t - \frac{N_{n}}{n}\). In fact, a different kind of asymptotics of the MSE, where truncation error becomes dominant, could be derived when (2.9) does not hold, under some additional assumptions, but we do not pursue this direction in the present paper.

While the rate of convergence in (2.11) is fully determined by the roughness index \(\alpha\), which may seem discouraging at first, it turns out that the quantity \(J(\alpha,\kappa,\mathbf{b})\), which we call the asymptotic MSE, can vary a lot, depending on how we choose \(\kappa\) and \(\mathbf{b}\), and can have a substantial impact on the precision of the approximation of \(X\). It is immediate from (2.12) that increasing \(\kappa\) will decrease \(J(\alpha,\kappa ,\mathbf{b})\). Moreover, for given \(\alpha\) and \(\kappa\), it is straightforward to choose \(\mathbf{b}\) so that \(J(\alpha,\kappa,\mathbf {b})\) is minimized, as shown in the following result.

To understand how much increasing \(\kappa\) and using the optimal sequence \(\mathbf{b}^{*}\) from Proposition 2.8 improves the approximation, we study numerically the asymptotic root mean square error (RMSE) \(\sqrt{J(\alpha,\kappa,\mathbf{b})}\). In particular, we assess how much the asymptotic RMSE decreases relative to the RMSE of the forward Riemann-sum scheme (\(\kappa=0\) and \(\mathbf {b} = \mathbf{b}_{\mathrm{FWD}}\)) by using the quantity The results are presented in Fig. 1. We find that employing the hybrid scheme with \(\kappa\geq1\) leads to a substantial reduction in the asymptotic RMSE relative to the forward Riemann-sum scheme when \(\alpha\in(-\frac{1}{2},0)\). Indeed, when \(\kappa\geq 1\), the asymptotic RMSE, as a function of \(\alpha\), does not blow up as \(\alpha\rightarrow-\frac{1}{2}\), while with \(\kappa= 0\) it does. This explains why the reduction in the asymptotic RMSE approaches \(100\% \) as \(\alpha\rightarrow-\frac{1}{2}\). When \(\alpha\in(0,\frac {1}{2})\), the improvement achieved using the hybrid scheme is more modest, but still considerable. Figure 1 also highlights the importance of using the optimal sequence \(\mathbf{b}^{*}\), instead of \(\mathbf{b}_{\mathrm{FWD}}\), as evaluation points in the scheme, in particular when \(\alpha\in(0,\frac{1}{2})\). Finally, we observe that increasing \(\kappa\) beyond 2 does not appear to lead to a significant further reduction. Indeed, in our numerical experiments, reported in Sects. 3.2 and 3.3 below, we observe that using \(\kappa= 1,2\) already leads to good results.

It is useful to extend the hybrid scheme to a class of non-stationary processes that are closely related to \(\mathcal{BSS}\) processes. This extension is important in connection with an application to the so-called rough Bergomi model which we discuss in Sect. 3.3 below. More precisely, we consider processes of the form where the kernel function \(g\), volatility process \(\sigma\) and driving Brownian motion \(W\) are as before. We call \(Y\) a truncated Brownian semistationary (\(\mathcal{TBSS}\)) process, as \(Y\) is obtained from the \(\mathcal{BSS}\) process \(X\) by truncating the stochastic integral in (2.1) at 0. Of the preceding assumptions, only (A1) and (A2) are needed to ensure that the stochastic integral in (2.14) exists—in fact, of (A2), only the requirement that \(g\) is differentiable on \((0,\infty)\) comes into play.

The \(\mathcal{TBSS}\) process \(Y\) does not have covariance-stationary increments, so we define its (time-dependent) variogram as Extending Proposition 2.2, we can describe the behavior of \(h \mapsto V_{Y}(h,t)\) near zero as follows. The existence of a locally Hölder-continuous modification is then a straightforward consequence. We omit the proof of this result, as it would be a straightforward adaptation of the proof of Proposition 2.2.

Note that while the increments of \(Y\) are not covariance-stationary, the asymptotic behavior of \(V_{Y}(h,t)\) is the same as that of \(V_{X}(h)\) as \(h \rightarrow0\) (cf. Proposition 2.2) for any \(t>0\). Thus, the increments of \(Y\) (apart from increments starting at time 0) are locally like the increments of \(X\).

We define the hybrid scheme to discretize \(Y_{t}\), for any \(t \geq0\), as where In effect, we simply drop the summands in (2.7) and (2.8) that correspond to integrals and increments on the negative real line. We make remarks on the implementation of this scheme in Sect. 3.1 below.

The MSE of the hybrid scheme for the \(\mathcal{TBSS}\) process \(Y\) has the following asymptotic behavior as \(n \rightarrow\infty\), which is in fact identical to the asymptotic behavior of the MSE of the hybrid scheme for \(\mathcal{BSS}\) processes. We omit the proof of this result, which would be a simple modification of the proof of Theorem 2.5.

Simulating the \(\mathcal{BSS}\) process \(X\) on the equidistant grid \(\{0,\frac{1}{n},\frac{2}{n},\ldots,\frac{\lfloor nT \rfloor}{n} \}\) for some \(T>0\) using the hybrid scheme entails generating Provided that we can simulate the random variables we can compute (3.1) via the formula In order to simulate (3.2) and (3.3), it is instrumental to note that the \(\kappa +1\)-dimensional random vectors are i.i.d. according to a multivariate Gaussian distribution with mean zero and covariance matrix \(\varSigma\) given by for \(j = 2,\ldots,\kappa+1\), and for \(j\), \(k = 2,\ldots,\kappa+1\) such that \(j < k\), where \({}_{2} F_{1}\) stands for the Gauss hypergeometric function; see e.g. [17, Sect. 2.1.1] for the definition. (When \(k < j\), set \(\varSigma _{j,k} = \varSigma_{k,j}\).) For the convenience of the reader, we provide a proof of (3.5) in Sect. 4.3.

Thus, \((\mathbf{W}^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\) can be generated by taking independent draws from the multivariate Gaussian distribution \(N_{\kappa+1}(\mathbf{0},\varSigma)\). If the volatility process \(\sigma\) is independent of \(W\), then \((\sigma ^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\) can be generated separately, possibly using exact methods. (Exact methods are available e.g. for Gaussian processes, as mentioned in the introduction, and diffusions; see [14].) In the case where \(\sigma\) depends on \(W\), simulating \((\mathbf{W}^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\) and \((\sigma^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\) is less straightforward. That said, if \(\sigma\) is driven by a standard Brownian motion \(Z\), correlated with \(W\), say, one could rely on a factor decomposition where \(\rho\in[-1,1]\) is the correlation parameter and \((W^{\perp}_{t})_{t \in[0,T]}\) is a standard Brownian motion independent of \(W\). Then one would first generate \((\mathbf{W}^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\), use (3.6) to generate \(( Z_{\frac {i+1}{n}}-Z_{\frac{i}{n}} )_{i=-N_{n}}^{\lfloor nT \rfloor-1}\), and employ some appropriate approximate method to produce \((\sigma ^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\) thereafter. This approach has, however, the caveat that it induces an additional approximation error, not quantified in Theorem 2.5.

In the hybrid scheme, it typically suffices to take \(\kappa\) to be at most 3. Thus, in (3.4), the first sum \(\check{X}^{n}_{\frac{i}{n}}\) requires only a negligible computational effort. By contrast, the number of terms in the second sum \(\hat{X}^{n}_{\frac {i}{n}}\) increases as \(n \rightarrow\infty\). It is then useful to note that where and \(\varGamma\star\varXi\) stands for the discrete convolution of the sequences \(\varGamma\) and \(\varXi\). It is well known that the discrete convolution can be evaluated efficiently using a fast Fourier transform (FFT). The computational complexity of simultaneously evaluating \((\varGamma\star\varXi)_{i}\) for all \(i = 0,1,\ldots,\lfloor nT \rfloor\) using an FFT is \(\mathcal{O}(N_{n} \log N_{n})\), see [22, Sect. 3.3.4], which under (A4) translates to \(\mathcal{O}(n^{\gamma+1} \log n)\). The computational complexity of the entire hybrid scheme is then \(\mathcal {O}(n^{\gamma+1} \log n)\), provided that \((\sigma ^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\) is generated using a scheme with complexity not exceeding \(\mathcal{O}(n^{\gamma+1} \log n)\). As a comparison, we mention that the complexity of an exact simulation of a stationary Gaussian process using circulant embeddings is \(\mathcal {O}(n \log n)\) [2, Chapter XI, Sect. 3], whereas the complexity of the Cholesky factorization is \(\mathcal{O}(n^{3})\) [2, Chapter XI, Sect. 2].

Figure 2 presents examples of trajectories of the \(\mathcal{BSS}\) process \(X\) using the hybrid scheme with \(\kappa= 1, 2\) and \(\mathbf{b} = \mathbf{b}^{*}\). We choose the kernel function \(g\) to be the gamma kernel (Example 2.3) with \(\lambda= 1\). We also discretize \(X\) using the Riemann-sum scheme, \(\kappa= 0\) with \(\mathbf{b} \in\{ \mathbf{b}_{\mathrm{FWD}},\mathbf{b}^{*}\}\) (that is, the forward Riemann-sum scheme and its counterpart with optimally chosen evaluation points). We can make two observations. Firstly, we see how the roughness parameter \(\alpha\) controls the regularity properties of the trajectories of \(X\)—as we decrease \(\alpha\), the trajectories of \(X\) become increasingly rough. Secondly, and more importantly, we see how the simulated trajectories coming from the Riemann-sum and hybrid schemes can be rather different, even though we use the same innovations for the driving Brownian motion. In fact, the two variants of the hybrid scheme (\(\kappa= 1, 2\)) yield almost identical trajectories, while the Riemann-sum scheme (\(\kappa= 0\)) produces trajectories that are comparatively smoother, this difference becoming more apparent as \(\alpha\) approaches \(-\frac{1}{2}\). Indeed, in the extreme case with \(\alpha= -0.499\), both variants of the Riemann-sum scheme break down and yield anomalous trajectories with very little variation, while the hybrid scheme continues to produce accurate results. The fact that the hybrid scheme is able to reproduce the fine properties of rough \(\mathcal {BSS}\) processes, even for values of \(\alpha\) very close to \(-\frac{1}{2}\), is backed up by a further experiment reported in the following section.

Suppose that we have observations \(X_{\frac{i}{m}}\), \(i = 0,1,\ldots ,m\), of the \(\mathcal {BSS}\) process \(X\) given by (2.1), for some \(m \in \mathbb {N}\). Barndorff-Nielsen et al. [6] and Corcuera et al. [16] discuss how the roughness index \(\alpha\) can be estimated consistently as \(m \rightarrow\infty\). The method is based on the change-of-frequency (COF) statistics which compare the realized quadratic variations of \(X\), using second-order increments, with two different lag lengths. Corcuera et al. [16] have shown that under some assumptions on the process \(X\), which are similar to (A1)–(A3) albeit slightly more restrictive, it holds that An in-depth study of the finite sample performance of this COF estimator can be found in [12].

To examine how well the hybrid scheme reproduces the fine properties of the \(\mathcal {BSS}\) process in terms of regularity/roughness, we apply the COF estimator to discretized trajectories of \(X\), where the kernel function \(g\) is again the gamma kernel (Example 2.3) with \(\lambda= 1\), generated using the hybrid scheme with \(\kappa= 1,2,3\) and \(\mathbf{b} = \mathbf{b}^{*}\). We consider the case where the volatility process satisfies \(\sigma_{t} = 1\), that is, the process \(X\) is Gaussian. This allows us to quantify and control the intrinsic bias and noisiness, measured in terms of standard deviation, of the estimation method itself, by initially applying the estimator to trajectories that have been simulated using an exact method based on the Cholesky factorization. We then study the behavior of the estimator when applied to a discretized trajectory, while decreasing the step size of the discretization scheme. More precisely, we simulate \(\hat{\alpha}(X^{n},m)\), where \(m=500\) and \(X_{n}\) is the hybrid scheme for \(X\) with \(n = ms\) and \(s \in\{1,2,5\}\). This means that we compute \(\hat{\alpha}(X^{n},m)\) using \(m\) observations obtained by subsampling every \(s\)th observation in the sequence \(X^{n}_{\frac{i}{n}}\), \(i = 0,1,\ldots,n\). As a comparison, we repeat these simulations substituting the hybrid scheme with the Riemann-sum scheme, using \(\kappa= 0\) with \(\mathbf {b}\in\{\mathbf{b}_{\mathrm{FWD}},\mathbf{b}^{*}\}\).

The results are presented in Fig. 3. We observe that the intrinsic bias of the estimator with \(m = 500\) observations is negligible, and hence the bias of the estimates computed from discretized trajectories is then attributable to approximation error arising from the respective discretization scheme, where positive (resp. negative) bias indicates that the simulated trajectories are smoother (resp. rougher) than those of the process \(X\). Concentrating first on the baseline case \(s = 1\), we note that the hybrid scheme produces essentially unbiased results when \(\alpha\in(-\frac {1}{2},0)\), while there is moderate bias when \(\alpha\in(0,\frac {1}{2})\), which disappears when passing from \(\kappa=1\) to \(\kappa=3\), even for values of \(\alpha\) very close to \(\frac{1}{2}\). (The largest value of \(\alpha\) considered in our simulations is \(\alpha= 0.49\); one would expect the performance to weaken as \(\alpha\) approaches \(\frac {1}{2}\), cf. Fig. 1, but this range of parameter values seems to be of limited practical interest.) The standard deviations exhibit a similar pattern. The corresponding results for the Riemann-sum scheme are clearly inferior, exhibiting significant bias, while using optimal evaluation points (\(\mathbf{b} = \mathbf{b}^{*}\)) improves the situation slightly. In particular, the bias in the case \(\alpha\in(-\frac{1}{2},0)\) is positive, indicating too smooth discretized trajectories, which is connected with the failure of the Riemann-sum scheme with \(\alpha\) near \(-\frac{1}{2}\), illustrated in Fig. 2. With \(s=2\) and \(s=5\), the results improve with both schemes. Notably, in the case \(s=5\), the performance of the hybrid scheme even with \(\kappa= 1\) is on a par with the exact method. However, the improvements with the Riemann-sum scheme are more meager, as considerable bias persists when \(\alpha\) is near \(-\frac{1}{2}\).

As another experiment, we study Monte Carlo option pricing in the rough Bergomi (rBergomi) model of Bayer et al. [10]. In the rBergomi model, the logarithmic spot variance of the price of the underlying is modeled by a rough Gaussian process, which is a special case of (2.14). By virtue of the rough volatility process, the model fits well to observed implied volatility smiles [10, Sect. 5].

More precisely, the price of the underlying in the rBergomi model with time horizon \(T>0\) is defined, under an equivalent martingale measure identified with ℙ, as using the spot variance process Above, \(S_{0}>0\), \(\eta>0\) and \(\alpha\in(-\frac{1}{2},0)\) are deterministic parameters, and \(Z\) is a standard Brownian motion given by where \(\rho\in(-1,1)\) is the correlation parameter and \((W^{\perp}_{t})_{t \in[0,T]}\) is a standard Brownian motion independent of \(W\). The process \((\xi^{0}_{t})_{t \in[0,T]}\) is the so-called forward variance curve [10, Sect. 3], which we assume here to be flat, \(\xi^{0}_{t} = \xi>0\) for all \(t \in[0,T]\).

We aim to compute by using Monte Carlo simulation the price of a European call option struck at \(K>0\) with maturity \(T\), which is given by The approach suggested by Bayer et al. [10] involves sampling the Gaussian processes \(Z\) and \(Y\) on a discrete time-grid using exact simulation and then approximating \(S\) and \(v\) using Euler discretization. We modify this approach by using the hybrid scheme to simulate \(Y\), instead of the computationally more costly exact simulation. As the hybrid scheme involves simulating increments of the Brownian motion \(W\) driving \(Y\), we can conveniently simulate the increments of \(Z\), needed for the Euler discretization of \(S\), by using the representation (3.9).

We map the option price \(\mathrm{C}(S_{0},K,T)\) to the corresponding Black–Scholes implied volatility \(\mathrm{IV}(S_{0},K,T)\). Reparametrizing the implied volatility using the log-strike \(k := \log (K/S_{0})\) allows us to drop the dependence on the initial price, so we abuse notation slightly and write \(\mathrm{IV}(k,T)\) for the corresponding implied volatility. Figure 4 displays implied volatility smiles obtained from the rBergomi model using the hybrid and Riemann-sum schemes to simulate \(Y\), as discussed above, and compares these to the smiles obtained using an exact simulation of \(Y\) via Cholesky factorization. The parameter values are given in Table 1. They have been adopted from Bayer et al. [10], who demonstrate that they result in realistic volatility smiles. We consider two different maturities: “short”, \(T = 0.041\), and “long”, \(T = 1\).

We observe that the Riemann-sum scheme (\(\kappa=0\), \(\mathbf{b} \in\{\mathbf{b}_{\mathrm{FWD}},\mathbf{b}^{*}\}\)) is able to capture the shape of the implied volatility smile, but not its level. But the method even breaks down with more extreme log-strikes (the prices are so low that the root-finding algorithm used to compute the implied volatility would return zero). In contrast, the hybrid scheme with \(\kappa= 1,2\) and \(\mathbf{b}=\mathbf{b}^{*}\) yields implied volatility smiles that are indistinguishable from the benchmark smiles obtained using exact simulation. Further, there is no discernible difference between the smiles obtained using \(\kappa=1\) and \(\kappa=2\). As in the previous section, we observe that the hybrid scheme is indeed capable of producing very accurate trajectories of \(\mathcal{TBSS}\) processes, in particular in the case \(\alpha\in(-\frac {1}{2},0)\), even when \(\kappa= 1\).

(i) By the covariance-stationarity of the volatility process \(\sigma\), we may express the variogram \(V(h)\) for any \(h \geq0\) as Invoking (A1) and Lemma 4.1, we find that We may clearly assume that \(h<1\), which allows us to work with the decomposition where According to (A2), there exists \(M>1\) such that \(x \mapsto |g'(x)|\) is nonincreasing on \([M,\infty)\). Thus, using the mean value theorem, we deduce that where \(\xi= \xi(x,h) \in[x,x+h]\). It follows then that which in turn implies that Making the substitution \(y=\frac{x}{h}\), we next obtain where By the definition of slow variation at 0, We show below that the functions \(G_{h}\), \(h \in(0,1)\), have an integrable majorant. Thus, by the dominated convergence theorem, Since \(\alpha< \frac{1}{2}\), we have \(\lim_{h \rightarrow0} \frac {A'_{h}}{h^{2\alpha+ 1 } L_{g}(h)^{2}}=0\) by (2.2) and (4.5), so we get from (4.4) and (4.6) that which together with (4.3) implies the assertion.

It remains to justify the use of the dominated convergence theorem to deduce (4.6). For any \(y \in(0,1]\), we have by the Potter bound (4.1) and the elementary inequality \((u+v)^{2} \leq 2u^{2}+2v^{2}\) that where we choose \(\delta_{1} \in(0,\alpha+\frac{1}{2})\) to ensure that \(2(\alpha-\delta_{1}) > -1\). Consider then \(y \in[1,\infty)\). By adding and subtracting the term \((y+1)^{\alpha}\frac{L_{g}(hy)}{L_{g}(h)}\) and using again the inequality \((u+v)^{2} \leq2u^{2}+2v^{2}\), we get We recall that \(\underline{L}_{g} := \inf_{x \in(0,1]} L_{g}(x) >0\) by (A1), so Using the mean value theorem and the bound for the derivative of \(L_{g}\) from (A1), we observe that where \(\xi= \xi(y,h) \in[hy,h(y+1)]\). Noting that the constraint \(y < \frac{1}{h}-1\) is equivalent to \(h < \frac{1}{y+1}\), we obtain further as \(y\geq1\), which we then use to deduce that Additionally, we observe that by (4.1) and (4.2), where we choose \(\delta_{2} \in(0,\frac{1}{2}-\alpha)\), ensuring that \(2(\alpha-1+\delta_{2}) < -1\). We may finally define the function which satisfies \(0 \leq G_{h}(y) \leq G(y)\) for any \(y \in(0,\infty)\) and \(h \in(0,1)\), and is integrable on \((0,\infty)\) with the aforementioned choices of \(\delta_{1}\) and \(\delta_{2}\).

(ii) To show existence of the modification, we need a localization procedure that involves the ancillary process We first check that \(F\) is locally bounded under (A1) and (A2), which is essential for localization. To this end, let \(T \in(0,\infty)\) and write, for any \(t \in[-T,T]\), where and \(M>1\) is such that \(x \mapsto|g'(x)|\) is nonincreasing on \([M,\infty)\), as in the proof of (i). Since \(g'\) is continuous on \((0,\infty)\) and \(\sigma\) is locally bounded, we have for any \(t \in[-T,T]\) that Further, when \(t \in[-T,T]\), where \(g'(t-u)^{2} \leq g'(-T-u)^{2}\) since the arguments satisfy Thus, for any \(t \in[-T,T]\) almost surely, as we have where we change the order of expectation and integration relying on Tonelli’s theorem and where the final equality follows from the covariance-stationarity of \(\sigma\). So we can conclude that \(F\) is indeed locally bounded.

Let now \(m \in \mathbb {N}\) and, for localization, define the sequence of stopping times that satisfies \(\tau_{m,n} \uparrow\infty\) almost surely as \(n \rightarrow\infty\) since both \(F\) and \(\sigma\) are locally bounded. (We follow the usual convention that \(\inf\varnothing= \infty\).) Consider now the modified \(\mathcal {BSS}\) process that coincides with \(X\) on the stochastic interval \([\![-m,\tau _{m,n} ]\!]\). The process \(X^{m,n}\) satisfies the assumptions of [5, Lemma 1]; so for any \(p>0\), there exists a constant \(\hat{C}_{p}>0\) such that Using the upper bound in (2.2), we can deduce from (i) that for any \(\delta> 0\), there are constants \(\tilde {C}_{\delta}>0\) and \(\underline{h}_{\delta}> 0\) such that Applying (4.8) to (4.7), we get for \(s,t \in[-m,\infty)\), \(|s-t| < \underline{h}_{\delta}\) that We may note that \(p(\alpha+ \frac{1}{2}-\frac{\delta}{2}-\frac {1}{p})>0\) for small enough \(\delta\) and large enough \(p\), and in particular, as \(\delta\downarrow0\) and \(p \uparrow\infty\). Thus it follows from the Kolmogorov–Chentsov theorem [21, Theorem 3.22] that \(X^{m,n}\) has a modification with locally \(\phi\)-Hölder-continuous trajectories for any \(\phi\in(0,\alpha+ \frac{1}{2})\). Moreover, a modification of \(X\) on ℝ, having locally \(\phi\)-Hölder-continuous trajectories for any \(\phi\in(0,\alpha+ \frac{1}{2})\), can then be constructed from these modifications of \(X^{m,n}\), \(m \in \mathbb {N}\), \(n \in \mathbb {N}\), by letting first \(n \rightarrow\infty\) and then \(m \rightarrow \infty\).  □

As a preparation, we first establish an auxiliary result that deals with the asymptotic behavior of certain integrals of regularly varying functions.

In order to prove (3.5), we rely on the following integral identity for the Gauss hypergeometric function \({}_{2} F_{1}\).