The characteristic function of rough Heston models

In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H > 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H < 1 / 2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.

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.

High-order methods for the option pricing under multivariate rough volatility models

We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small‐time, large‐time, and (i.e., ) limits. We show that the short‐maturity smile scales in qualitatively the same way as a general rough stochastic volatility model , and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of and so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log‐moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher‐order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher‐order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large‐maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price tends weakly to a nonsymmetric random variable as (i.e., ) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with , and we show that tends weakly to a nonsymmetric random variable as , which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log‐moneyness as , and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the process converges on pathspace to a random tempered distribution which has the same law as the hyper‐rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).

Fractional Brownian Motions, Fractional Noises and Applications

We show that when the price process [Formula: see text] represents a fully incomplete market, the optimal super-replication of any Markovian claim [Formula: see text] with [Formula: see text] being nonnegative and lower semicontinuous is of buy-and-hold type. Since both (unbounded) stochastic volatility models and rough volatility models are examples of fully incomplete markets, one can interpret the buy-and-hold property when super-replicating Markovian claims as a natural phenomenon in incomplete markets.

In recent years, researches on the fractional Riccati equation have attracted extensive attention, and many achievements have emerged. However, most of the current works are carried out on the basis of constant fractional order. Here we study the Cauchy problem of one simple type Riccati equation with variable fractional order. By using Gronwall-Bellman inequality and other mathematical tools, we will prove the uniqueness of solutions of this problem.

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.

We propose a dynamic fractional volatility model that incorporates a time-varying Hurst exponent estimated via Daubechies-4 wavelet analysis on 252-day rolling windows to capture evolving market memory effects in equity markets. This approach overcomes the limitations of traditional GARCH-type and static fractional volatility models, which assume a constant memory parameter and struggle during regime shifts and market stress. The model is applied to daily closing prices of the S&P 500 Index over 1,258 trading days from January 1, 2015 to December 31, 2019, yielding statistically significant improvements in forecasting performance. Empirical results indicate a 12.3 % reduction in RMSE and a 9.8 % improvement in MAPE, with an out-of-sample R-squared exceeding 0.72 compared to benchmark models. Maximum likelihood estimation with Fisher scoring is used for daily parameter updates, ensuring the model remains responsive to rapidly changing market conditions. Additionally, the model achieves an average absolute option pricing error of 1.8 %, markedly lower than that of traditional specifications. These enhancements are further corroborated by pairwise Diebold–Mariano tests, which confirm the statistical significance of the forecast improvements. Overall, this framework offers a rigorous and computationally efficient method for real-time volatility forecasting that delivers substantial benefits for risk management, derivative pricing, and automated trading strategies, grounded in robust statistical methodology.

Linear Stochastic Differential Games with State Dependent Fractional Brownian Motion

Inspired by the work of Al`os, Le ́on and Vives [ALV07] and Fukasawa [Fuk17], who showed that a volatility process driven by a fractional Brownian motion generates the power law at-the-money volatility skew observed in financial market data, Gatheral, Jaisson and Rosenbaum [GJR18a] spawned a class of models now known as rough volatility models. We study the asymptotic behaviour of such models, and investigate how convolutional neural networks can be used for their calibration. Chapter 1 serves as an introduction. We begin with implied volatility, and then intro- duce a number of model classes, starting with local volatility models and ending with rough volatility models, and discuss their associated asymptotic behaviour. We also introduce the theoretical tools used to prove the main results. In Chapter 2 we study the small-time behaviour of the rough Bergomi model, introduced by Bayer, Friz, and Gatheral [BFG16]. We prove a pathwise large deviations principle for a small-noise version of the model, and use this result to establish the small-time behaviour of the rescaled log stock price process. This, in turn, allows us to characterise the small-time implied volatility behaviour of the model. Using the same theoretical framework, we are also able to establish the small-time implied volatility behaviour of the lognormal fSABR model of Akahori, Song, and Wang [ASW17]. In Chapter 3 we present small-time implied volatility asymptotics for realised variance (RV) options for a number of (rough) stochastic volatility models via a large deviations principle. We interestingly discover that these (rough) volatility models, together with others proposed in the literature, generate linear smiles around the money. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we develop an approximation scheme for the density of the realised variance, which in turn allows the volatility swap density to be expressed in closed form. Lastly, we investigate different constructions of multi-factor models and how their construction affects the convexity of 4 the implied volatility smile. Remarkably, we identify a class of models that can generate non-linear smiles around-the-money. Additionally, we establish small-noise asymptotic behaviour of a general class of VIX options in the large strike regime. In Chapter 4, which is self-contained, we give an introduction to machine learning and neural networks. We investigate the use of convolutional neural networks to find the H ̈older exponent of simulated sample paths of the rough Bergomi model, a method which performs extremely well and is found to be robust when applied to trajectories of a fractional Brownian motion and an Ornstein-Uhlenbeck process. We then propose a novel calibration scheme for the rough Bergomi model based on our results.

<p style='text-indent:20px;'>In this paper, we study the existence and uniqueness of mild solutions for neutral delay Hilfer fractional integrodifferential equations with fractional Brownian motion. Sufficient conditions for controllability of neutral delay Hilfer fractional differential equations with fractional Brownian motion are established. The required results are obtained based on the fixed point theorem combined with the semigroup theory, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.</p>

<p>Since the fractional $ G $-Brownian motion (fGBm) generalizes the concepts of the standard Brownian motion, fractional Brownian motion, and $ G $-Brownian motion, while it can exhibit long-range dependence or antipersistence and feature the volatility uncertainty simultaneously, it can be a better alternative stochastic process in the financial applications. Thus, in this paper, some empirical studies for the financial applications of the fGBm were carried out, where the recent high-frequency data for some selected assets in the financial market are from the Oxford-Man Institute of Quantitative Finance Realized Library. There are two main empirical findings. One was that the H-$ G $-normal distributions associated with the fGBm are more suitable in describing the dynamics of daily returns and increments of log-volatility for these assets than the usual distributions, since they not only characterize the properties of skewness, excess kurtosis, and long-range dependence $ \left(\frac{1}{2} < H < 1\right) $ or antipersistence $ \left(0 < H < \frac{1}{2}\right) $, but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different $ \underline{\sigma}^2 $ and $ \overline{\sigma}^2 $, but Hurst parameters $ H < \frac{1}{2} $, at any reasonable time scale. Then a generalized stochastic model for the dynamics of the assets called rough fractional stochastic volatility model driven by fGBm (RFSV-fGBm) was developed. Finally, some parameter estimates and numerical experiments for the RFSV-fGBm model were investigated and carried out.</p>

Affine Forward Variance Models

This article introduces a discrete‐time fractional stochastic volatility model (FSV) based on fractional Gaussian noise. The new model includes the standard stochastic volatility model as a special case and has the same limit as the fractional integrated stochastic volatility (FISV) model, which is the continuous‐time fractional Ornstein–Uhlenbeck process. A simulated maximum likelihood method, which maximizes the time‐domain log‐likelihood function calculated by the importance sampling technique, and a frequency‐domain quasi maximum likelihood method (or quasi Whittle) are employed to estimate the model parameters. Simulation studies suggest that, while both estimation methods can accurately estimate the model, the simulated maximum likelihood method outperforms the quasi Whittle method. As an illustration, we fit the FSV and FISV models with the proposed estimation techniques to the S&P 500 composite index over a sample period spanning 45 years.