The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model. Motivated by the statistical analysis of realized volatility by Gatheral, Jaisson and Rosenbaum [20] and the theoretical results on implied volatility [2,18], rough stochastic volatility has emerged as a new paradigm in quantitative finance, overcoming the observed limitations of diffusive stochastic volatility models. In these models, the trajectories of the volatility have lower Hölder regularity than the trajectories of standard Brownian motion [4,20]. In fact, they are based on fractional Brownian motion (fBm), which is a centered Gaussian process, whose covariance structure depends on the so-called Hurst parameter, H (we refer to [26,16,11] for more details regarding the fBm processes). In the rough volatility case, where 0 < H < 1/2, the fBm has negatively correlated increments and rough sample paths. Gatheral, Jaisson, and Rosenbaum [20] empirically demonstrated the advantages of such models. For instance, they showed that the log-volatility in practice has a similar behavior to fBm with the Hurst exponent H ≈ 0.1 at any reasonable time scale (see also [19]). These results were confirmed by Bennedsen, Lunde and Pakkanen [8], who studied over a thousand individual US equities and showed that H lies in (0, 1/2) for each equity. Other works [8,4,20] showed further benefits of such rough volatility models over standard stochastic volatility models, in terms of explaining crucial phenomena observed in financial markets.The rough Bergomi (rBergomi) model, proposed by Bayer, Friz and Gatheral [4], was one of the first developed rough volatility models. This model, depending on only three parameters, shows remarkable fit to empirical implied volatility surfaces. The construction of the rB...

Abstract. In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics is dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (explicit-TL ) method, can be very slow. Implicit approximations have been developed to improve numerical stability and provide efficient simulation algorithms for those systems. Here, we propose an efficient Multilevel Monte Carlo (MLMC) method in the spirit of the work by Anderson and Higham (2012). This method uses split-step implicit tau-leap (SSI-TL ) at levels where the explicit-TL method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.

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When approximating the expectation of a functional of a certain stochastic process, the efficiency and performance of deterministic quadrature methods such as sparse grids, and hierarchical variance reduction methods such as multilevel Monte Carlo (MLMC), may be highly deteriorated in different ways by the low regularity of the integrand with respect to the input parameters. To overcome this issue, a smoothing procedure is needed to uncover the available regularity and improve the performance of the aforementioned numerical methods. In this work, we consider cases where we cannot perform an analytic smoothing. Thus, we introduce a novel numerical smoothing technique based on root-finding combined with a one dimensional integration with respect to a single well-chosen variable. We prove that under appropriate conditions, the resulting function of the remaining variables is a highly smooth function, potentially allowing a higher efficiency of adaptive sparse grids quadrature (ASGQ), in particular when combined with hierarchical transformations (Brownian bridge and Richardson extrapolation on the weak error) to treat high dimensionality effectively. Our study is motivated by option pricing problems and our main focus is on dynamics where a discretization of the asset price is needed. Through our analysis and numerical experiments, we illustrate the advantage of combining numerical smoothing with ASGQ compared to the Monte Carlo (MC) approach. Furthermore, we demonstrate how numerical smoothing significantly reduces the kurtosis at the deep levels of MLMC, and also improves the strong convergence rate, when using Euler scheme. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, TOL, this results in an improvement of the complexity from O TOL −2.5 in the standard case to O TOL −2 log(TOL) 2 . Finally, we show how our numerical smoothing combined with MLMC enables us also to estimate density functions, which standard MLMC (without smoothing) fails to achieve.

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