Rough Bergomi Model Research Articles

Two-factor Rough Bergomi Model: American Call Option Pricing and Calibration by Interior Point Optimization Algorithm

Two-factor Rough Bergomi Model: American Call Option Pricing and Calibration by Interior Point Optimization Algorithm

Functional quantization of rough volatility and applications to volatility derivatives

We develop a product functional quantization of rough volatility. Since the optimal quantizers can be computed offline, this new technique, built on the insightful works by [Luschgy, H. and Pagès, G., Functional quantization of Gaussian processes. J. Funct. Anal., 2002, 196(2), 486–531; Luschgy, H. and Pagès, G., High-resolution product quantization for Gaussian processes under sup-norm distortion. Bernoulli, 2007, 13(3), 653–671; Pagès, G., Quadratic optimal functional quantization of stochastic processes and numerical applications. In Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 101–142, 2007 (Springer: Berlin Heidelberg)], becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis on the pricing of options on the VIX and realized variance in the rough Bergomi model [Bayer, C., Friz, P.K. and Gatheral, J., Pricing under rough volatility. Quant. Finance, 2016, 16(6), 887–904] and compare our results to other benchmarks recently suggested.

Approximation of Stochastic Volterra Equations with kernels of completely monotone type

In this work, we develop a multifactor approximation for d d -dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L 2 L^2 -estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in ( 0 , 1 / 2 ) (0,1/2) , we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

Local volatility under rough volatility

AbstractSeveral asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small‐maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, supporting their calibration power to SP500 option data. Rough volatility models also generate a local volatility surface, via the so‐called Markovian projection of the stochastic volatility. We complement the existing results on implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated “1/2 skew rule” linking the short‐term at‐the‐money skew of the implied volatility to the short‐term at‐the‐money skew of the local volatility, a consequence of the celebrated “harmonic mean formula” of [Berestycki et al. (2002). Quantitative Finance, 2, 61–69], is replaced by a new rule: the ratio of the at‐the‐money implied and local volatility skews tends to the constant (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.

VIX pricing in the rBergomi model under a regime switching change of measure

A generalised rough Bergomi model with regime switching stochastic change of measure yields upward sloping VIX smiles

From Stochastic to Rough Volatility: A New Deep Learning Perspective on Hedging

The Black–Scholes model assumes that volatility is constant, and the Heston model assumes that volatility is stochastic, while the rough Bergomi (rBergomi) model, which allows rough volatility, can perform better with high-frequency data. However, classical calibration and hedging techniques are difficult to apply under the rBergomi model due to the high cost caused by its non-Markovianity. This paper proposes a gated recurrent unit neural network (GRU-NN) architecture for hedging with different-regularity volatility. One advantage is that the gating network signals embedded in our architecture can control how the present input and previous memory update the current activation. These gates are updated adaptively in the learning process and thus outperform conventional deep learning techniques in a non-Markovian environment. Our numerical results also prove that the rBergomi model outperforms the other two models in hedging.

WEAK ERROR RATES FOR OPTION PRICING UNDER LINEAR ROUGH VOLATILITY

In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [C. Bayer, P. K. Friz & J. Gatheral (2016) Pricing under rough volatility, Quantitative Finance 16 (6), 887–904, doi:10.1080/14697688.2015.1099717], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, [Formula: see text], over reasonable timescales. Both time series of asset prices and option-derived price data indicate that [Formula: see text] often takes values close to [Formula: see text] or less, i.e. rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only [Formula: see text]. We prove rate [Formula: see text] for the weak convergence of the Euler method for the rough Stein–Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses Talay–Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.

Multiscaling and rough volatility: An empirical investigation

Pricing derivatives goes back to the acclaimed Black and Scholes model. However, such a modelling approach is known not to be able to reproduce some of the financial stylised facts, including the dynamics of volatility. In the mathematical finance community, it has therefore emerged a new paradigm, named rough volatility modelling, that represents the volatility dynamics of financial assets as a fractional Brownian motion with Hurst exponent very small, which indeed produces rough paths. At the same time, prices’ time series have been shown to be multiscaling, characterised by different Hurst scaling exponents. This paper assesses the interplay, if present, between price multiscaling and volatility roughness, defined as the (low) Hurst exponent of the volatility process. In particular, we perform extensive simulation experiments by using one of the leading rough volatility models present in the literature, the rough Bergomi model. A real data analysis is also conducted to test if the rough volatility model reproduces the same relationship. We find that the model can reproduce multiscaling features of the prices’ time series when a low value of the Hurst exponent is used, but it fails to reproduce what the real data says. Indeed, we find that the dependency between prices’ multiscaling and the Hurst exponent of the volatility process is diametrically opposite to what we find in real data, namely a negative interplay between the two.

Moving average options: Machine learning and Gauss-Hermite quadrature for a double non-Markovian problem

Evaluating moving average options is a tough computational challenge for the energy and commodity market as the payoff of the option depends on the prices of a certain underlying observed on a moving window so, when a long window is considered, the pricing problem becomes high dimensional. We present an efficient method for pricing Bermudan style moving average options, based on Gaussian Process Regression and Gauss-Hermite quadrature, thus named GPR-GHQ. Specifically, the proposed algorithm proceeds backward in time and, at each time-step, the continuation value is computed only in a few points by using Gauss-Hermite quadrature, and then it is learned through Gaussian Process Regression. We test the proposed approach in the Black-Scholes model, where the GPR-GHQ method is made even more efficient by exploiting the positive homogeneity of the continuation value, which allows one to reduce the problem size. Positive homogeneity is also exploited to develop a binomial Markov chain, which is able to deal efficiently with medium-long windows. Secondly, we test GPR-GHQ in the Clewlow-Strickland model, the reference framework for modeling prices of energy commodities. Then, we investigate the performance of the proposed method in the Heston model, which is a very popular model among the non-Gaussian ones. Finally, we consider a challenging problem which involves double non-Markovian feature, that is the rough-Bergomi model. In this case, the pricing problem is even harder since the whole history of the volatility process impacts the future distribution of the process. The manuscript includes a numerical investigation, which displays that GPR-GHQ is very accurate in pricing and computing the Greeks with respect to the Longstaff-Schwartz method and it is able to handle options with a very long window, thus overcoming the problem of high dimensionality.

Multilevel Monte Carlo simulation for VIX options in the rough Bergomi model

We consider the pricing of VIX options in the rough Bergomi model [Bayer, Friz, and Gatheral, Pricing under rough volatility, Quantitative Finance 16(6), 887-904, 2016]. In this setting, the VIX random variable is defined by the one-dimensional integral of the exponential of a Gaussian process with correlated increments, hence approximate samples of the VIX can be constructed via discretization of the integral and simulation of a correlated Gaussian vector. A Monte-Carlo estimator of VIX options based on a rectangle discretization scheme and exact Gaussian sampling via the Cholesky method has a computational complexity of order $\mathcal O(\varepsilon^{-4})$ when the mean-squared error is set to $\varepsilon^2$. We demonstrate that this cost can be reduced to $\mathcal O(\varepsilon^{-2} \log^2(\varepsilon))$ combining the scheme above with the multilevel method [Giles, Multilevel Monte Carlo path simulation, Oper. Res. 56(3), 607-617, 2008], and further reduced to the asymptotically optimal cost $\mathcal O(\varepsilon^{-2})$ when using a trapezoidal discretization. We provide numerical experiments highlighting the efficiency of the multilevel approach in the pricing of VIX options in such a rough forward variance setting.

A rough SABR formula

<p style='text-indent:20px;'>Following an approach originally suggested by Balland in the context of the SABR model, we derive an ODE that is satisfied by normalized volatility smiles for short maturities under a rough volatility extension of the SABR model that extends also the rough Bergomi model. We solve this ODE numerically and further present a very accurate approximation to the numerical solution that we dub the <i>rough SABR formula</i>.</p>

The Riemann–Liouville field and its GMC as [formula omitted], and skew flattening for the rough Bergomi model

We consider a re-scaled Riemann–Liouville (RL) process ZtH=∫0t(t−s)H−12dWs, and using Lévy’s continuity theorem for random fields we show that ZH tends weakly to an almost log-correlated Gaussian field Z as H→0. Away from zero, this field differs from a standard Bacry–Muzy field by an a.s.Hölder continuous Gaussian process, and we show that ξγH(dt)=eγZtH−12γ2V ar(ZtH)dt tends to a Gaussian multiplicative chaos (GMC) random measure ξγ for γ∈(0,1) as H→0. We also show convergence in law for ξγH as H→0 for γ∈[0,2) using tightness arguments, and ξγ is non-atomic and locally multifractal away from zero. In the final section, we discuss applications to the Rough Bergomi model; specifically, using Jacod’s stable convergence theorem, we prove the surprising result that (with an appropriate re-scaling) the martingale component Xt of the log stock price tends weakly to Bξγ([0,t]) as H→0, where B is a Brownian motion independent of everything else. This implies that the implied volatility smile for the full rough Bergomi model with ρ≤0 is symmetric in the H→0 limit, and without re-scaling the model tends weakly to the Black–Scholes model as H→0. We also derive a closed-form expression for the conditional third moment E((Xt+h−Xt)3|Ft) (for H>0) given a finite history, and E(XT3) tends to zero (or blows up) exponentially fast as H→0 depending on whether γ is less than or greater than a critical γ≈1.61711 which is the root of 14+12logγ−316γ2. We also briefly discuss the pros and cons of a H=0 model with non-zero skew for which Xt/t tends weakly to a non-Gaussian random variable X1 with non-zero skewness as t→0.

Pathwise large deviations for the rough Bergomi model: Corrigendum

AbstractThis note corrects an error in the definition of the rate function in Jacquier, Pakkanen, and Stone (2018) and slightly simplifies some proofs.

Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing

The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate a more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To overcome these difficulties, we show that the rBergomi model can be well-approximated by the forward-variance Bergomi model with wisely chosen weights and mean-reversion speed parameters (aBergomi), which has the Markovian property. We establish an explicit bound on the L2-error between the respective kernels of these two models, which is explicitly controlled by the number of terms in the aBergomi model. We establish and describe the affine structure of the rBergomi model, and show the convergence of the affine structure of the aBergomi model to the one of the rBergomi model. We demonstrate the efficiency and accuracy of our method by implementing a classical Markovian Monte Carlo simulation scheme for the aBergomi model, which we compare to the hybrid scheme of the rBergomi model.

Refinement by reducing and reusing random numbers of the Hybrid scheme for Brownian semistationary processes

We revisit the Hybrid scheme proposed by Bennedsen et al. (2017) for numerical simulations of Brownian semistationary processes, and propose a Refinement by Reducing and Reusing random numbers of the Hybrid scheme (3R Hybrid scheme). The key idea is to reuse random variables through orthogonal projections. An application to the analysis of the rough Bergomi model is also given.

A rough SABR formula

Following an approach originally suggested by Balland in the context of the SABR model, we derive an ODE that is satisfied by normalized volatility smiles for short maturities under a rough volatility extension of the SABR model that extends also the rough Bergomi model. We solve this ODE numerically and further present a very accurate approximation to the numerical solution that we dub the rough SABR formula.

Functional quantization of rough volatility and applications to the VIX

We develop a product functional quantization of rough volatility. Since the quantizers can be computed offline, this new technique, built on the insightful works by Luschgy and Pages, becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis to pricing VIX Futures in the rough Bergomi model and compare our results to other recently suggested benchmarks.

Log-Modulated Rough Stochastic Volatility Models

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 10 August 2020Accepted: 19 May 2021Published online: 28 September 2021Keywordsrough volatility models, stochastic volatility, rough Bergomi model, implied skew, fractional Brownian motion, log Brownian motionAMS Subject Headings91G30, 60G22Publication DataISSN (online): 1945-497XPublisher: Society for Industrial and Applied MathematicsCODEN: sjfmbj

GMM DCKE - Semi-Analytic Conditional Expectations

We introduce a data driven and model free approach for computing conditional expectations. The new method combines Gaussian Mean Mixture models with classic analytic techniques based on the properties of the Gaussian distribution. We also incorporate a proxy hedge that leads to analytic expressions for the hedge with respect to the chosen proxy. This essentially makes use of the representation of the hedge sensitivity measuring the part of the variance that is attributed to the proxy. If we take the underlying, this corresponds to a time discrete minimal variance delta hedge. We apply our method to the calibration of pricing and hedging of (multi-dimensional) exotic Bermudan options, the calibration of stochastic local volatility models and applications to xVA/exposure calculation. For illustration we have chosen the rough Bergomi model and high-dimensional Heston models. Finally, we discuss issues when increasing the dimensionality and propose solutions using established statistical learning methods.

Dynamically Controlled Kernel Estimation

We introduce a data driven and model free approach for computing conditional expectations. The new method is based on classical techniques combined with machine learning methods. In particular, we consider kernel density estimation based on simulated risk factors combined with a control variate. This is used in a Gaussian process regression for finally approximating the conditional expectation. In this way we increase not only the stability of the estimator but we also need a significantly lower amount of simulations due to the variance reduction. Since we apply Gaussian process regression, we do not only get a point estimate, but also the full distribution. It turns out that the optimal coefficient for the control variate is the minimal variance delta. Thus, in this way we obtain model free and purely data driven hedges. Finally, we apply our method to several examples from option pricing including exotic option payoffs and payoffs with multiple underlyings for different models including the rough Bergomi model. A discussion on the challenges to extend the method to a large dimensional settings is provided and is partially solved by using Quasi random number sequences.