A new model reconciles rough volatility and jumps.
problem Combining rough volatility and jump processes.
method Developed a reversionary Heston model with fast mean reversions and large vol-of-vols.
result The reversionary Heston model converges to Lévy jump processes for certain values of the parameter.
Efficient simulation scheme for rough Heston model reduces computational cost.
problem Accurate and efficient simulation of the rough Heston model for option pricing.
method Weak simulation scheme based on Markovian approximations of the rough Heston process.
result The new scheme exhibits second order weak convergence with linear computational cost.
Extends rough Heston model solution to general λ.
problem Improving the rough Heston model for various λ values.
method Generalized rational approximation for Mittag-Leffler kernel.
result Convergence of the solution for general λ.
We present a number of related comparison results, which allow to compare moment explosion times, moment generating functions and critical moments between rough and non-rough Heston models of stochastic volatility. All results are based on a comparison principle for certain non-linear Volterra integral equations. Our u…
Deep learning calibrates a rough Heston model to match implied volatilities.
problem Calibrating the quadratic rough Heston model to match market implied volatilities.
method Multi-factor approximation and deep learning for efficient calibration.
result The model accurately reproduces SPX and VIX implied volatilities.
Market maker optimizes SPX and VIX spread using quadratic rough Heston model.
problem Maximizing profit from SPX and VIX spread while managing inventory risk.
method Uses quadratic rough Heston model to optimize multi-asset market making problem, approximating high-dimensional optimization.
result Asymptotic closed-form solution for optimization problem.
Expanding the rough Heston model in H
problem Analyzing the dependence of the fractional Riccati equation on the Hurst parameter H method Deriving a Taylor expansion of the Riccati solution in H result Local uniform convergence and analyticity of the fractional Riccati solution
New model captures asymmetric rough volatility with Zumbach effect.
problem Capturing asymmetric rough volatility and Zumbach effect.
method Proposes a bivariate QHawkes process to model asymmetric buying and selling actions.
result Derives a super-rough-Heston model preserving the Zumbach effect.
It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Br…
Extends Heston model with local volatility for better fit to market volatilities.
problem Fitting stochastic volatility models to market volatilities.
method Adds local volatility term to rough-Heston model, preserving stylized results.
result Provides a proper extrapolation scheme for calibration.
How to reconcile the classical Heston model with its rough counterpart? We introduce a lifted version of the Heston model with n multi-factors, sharing the same Brownian motion but mean reverting at different speeds. Our model nests as extreme cases the classical Heston model (when n = 1), and the rough Heston model (w…
Using microscopic price models based on Hawkes processes, it has been shown that under some no-arbitrage condition, the high degree of endogeneity of markets together with the phenomenon of metaorders splitting generate rough Heston-type volatility at the macroscopic scale. One additional important feature of financial…
Improved approximations for rough Heston model reduce errors.
problem Lack of Markov and semimartingale properties in rough Heston model.
method Markovian approximations with weak error analysis.
result Super-polynomial convergence of new approximations.
Rough volatility models are known to reproduce the behavior of historical volatility data while at the same time fitting the volatility surface remarkably well, with very few parameters. However, managing the risks of derivatives under rough volatility can be intricate since the dynamics involve fractional Brownian mot…
Fitting simultaneously SPX and VIX smiles is known to be one of the most challenging problems in volatility modeling. A long-standing conjecture due to Julien Guyon is that it may not be possible to calibrate jointly these two quantities with a model with continuous sample-paths. We present the quadratic rough Heston m…
We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arxiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions…
Previous literature has identified an effect, dubbed the Zumbach effect, that is nonzero empirically but conjectured to be zero in any conventional stochastic volatility model. Essentially this effect corresponds to the property that past squared returns forecast future volatilities better than past volatilities foreca…
Study approximates rough stochastic volatility models using diffusion processes.
problem High computational cost in simulating rough stochastic volatility models.
method Approximates stochastic Volterra equations with an N-dimensional diffusion process.
result Approximations converge strongly with superpolynomial rate in N.
The ADO-Heston model approximates market implied skew in vanilla options.
problem Reproduce market implied skew in vanilla options using a Markovian approximation.
method Derived characteristic function under risk-neutral and real measures, chose market price of risk, found closed form for log-price CF and implied skew.
result The ADO-Heston model can approximate the vanilla implied skew at small T but not exactly as rough volatility models. New methods for volatility modeling using rough paths and signatures.
problem Calibrating implied volatility surfaces in various stochastic models.
method Analytical approximations and signature-based models based on rough path theory.
result Signature-based models achieve comparable accuracy to analytical expansions and can capture more complex dynamics.
The rough Heston model emerges from scaling bivariate INAR processes, linking microstructure to option pricing.
problem Modeling and pricing financial options with heavy-tailed and cumulative processes.
method Scaling limit of bivariate INAR processes converging to rough Heston model, explicit formulas linking asymmetry parameters to volatility.
result Weak-error estimates and FFT-accelerated simulation for European and path-dependent options.
This paper investigates Merton's portfolio problem in a rough stochastic environment described by Volterra Heston model. The model has a non-Markovian and non-semimartingale structure. By considering an auxiliary random process, we solve the portfolio optimization problem with the martingale optimality principle. Optim…
Model captures rough volatility and jump clustering in stock vol dynamics.
problem Capturing the joint evolution of S&P 500 and VIX implied vol smiles.
method Rough Hawkes Heston model with affine Volterra dynamics, power kernel, and exponential jump law.
result Model accurately captures S&P 500 and VIX implied vol smiles with low power kernel.
A universal LSTM model outperforms asset-specific models in forecasting stock volatilities.
problem Forecasting stock volatilities across different assets.
method Trained an LSTM network on a pooled dataset of liquid stocks to forecast daily realized volatilities.
result The LSTM model consistently outperforms other asset-specific parametric models in volatility forecasting.
We show that typical behaviors of market participants at the high frequency scale generate leverage effect and rough volatility. To do so, we build a simple microscopic model for the price of an asset based on Hawkes processes. We encode in this model some of the main features of market microstructure in the context of…
The paper derives formulas for pricing geometric Asian options in the Volterra-Heston model.
problem Pricing geometric Asian options in the Volterra-Heston model.
method Derives semi-closed formulas using Fourier transforms and Riccati-Volterra equations.
result Derives formulas for pricing geometric Asian options with fixed and floating strikes.
The paper speeds up and improves pricing and calibration for the rough Heston model.
problem Improving the accuracy and speed of pricing vanilla options under the rough Heston model.
method Combining modified Adams method with SINH-acceleration method for Fourier inversion.
result The model implied vol surface is much flatter and fits market data poorly, indicating ghost calibration.
Motivated by empirical evidence for rough volatility models, this paper investigates continuous-time mean-variance (MV) portfolio selection under the Volterra Heston model. Due to the non-Markovian and non-semimartingale nature of the model, classic stochastic optimal control frameworks are not directly applicable to t…
New methods price American options in rough volatility models.
problem Pricing American options under rough volatility.
method Integrating deep-signature and signature-kernel learning into optimal stopping problem solutions.
result Performance comparison in rough Heston and rough Bergomi models.
We develop high-order approximations for the Heston model.
problem Modeling the Heston model with high accuracy and efficiency.
method Combining approximation schemes on different random grids to achieve any order of convergence.
result Achieve any order of convergence for the Heston model.
We consider a fractional version of the Heston volatility model which is inspired by [16]. Within this model we treat portfolio optimization problems for power utility functions. Using a suitable representation of the fractional part, followed by a reasonable approximation we show that it is possible to cast the proble…
Efficiently simulates the Heston model with large time steps using a novel method.
problem Challenges in simulating the Heston model with large time steps.
method Implicit integrated variance scheme exploiting the near-linear nature between stochastic driver and conditional integrated variance process.
result Achieves near-exact accuracy with coarse discretizations, efficient for large time steps.
Study models market volatility with persistent and temporary impacts.
problem Microstructure of rough volatility models driven by Poisson measures.
method Existence and uniqueness of solutions for stochastic path-dependent Volterra equations.
result Volatility process converges to fractional Heston model with spikes.
Rough volatility models are very appealing because of their remarkable fit of both historical and implied volatilities. However, due to the non-Markovian and non-semimartingale nature of the volatility process, there is no simple way to simulate efficiently such models, which makes risk management of derivatives an int…
This thesis investigates Merton's portfolio problem under two different rough Heston models, which have a non-Markovian structure. The motivation behind this choice of problem is due to the recent discovery and success of rough volatility processes. The optimisation problem is solved from two different approaches: firs…
We characterize the behaviour of the Rough Heston model introduced by Jaisson\&Rosenbaum \cite{JR16} in the small-time, large-time and α→1/2 (i.e. H→0) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model (cf.\ \cite{FZ17}, \cite{FGP…
Paper solves Merton's portfolio problem in a non-Markovian, non-semimartingale model.
problem Merton's portfolio optimization in a fake stationary Volterra-Heston model.
method Stochastic factor solution to a Riccati BSDE, combined with martingale optimality principle.
result Derives semi-closed form optimal strategies and value function.
Develops multifactor approximations for SVEs with completely monotone kernels.
problem Approximating SVEs with kernels of completely monotone type.
method Multifactor approximation, Euler discretization, L2-estimation, convergence analysis. result New multifactor Euler scheme reduces computational cost and outperforms SVEs for option pricing.
Enhanced volatility forecasting using options data and rough volatility model.
problem Improving realized volatility forecasting accuracy.
method Infer spot volatility from options data using rough stochastic volatility model, accelerate estimation with deep learning, benchmark against traditional models.
result Augmented HAR-RV-RHeston model outperforms traditional models in daily and long-term forecasting.
A new deep learning method for option pricing in rough volatility models.
problem Efficient pricing of European options in high-dimensional rough volatility models.
method Time-stepping deep gradient flow method reformulating the option pricing PDE as an energy minimization problem.
result The method respects asymptotic behavior and known bounds for option prices.
A new simulation method for Volterra processes improves convergence for rough kernels.
problem Simulating Volterra processes with singular kernels.
method iVi (integrated Volterra implicit) scheme based on Inverse Gaussian distribution.
result The iVi scheme achieves weak convergence with few time steps, especially for rough kernels.
This paper optimizes portfolio selection for multivariate affine and quadratic Volterra models with rough volatilities.
problem Optimizing portfolio selection for multivariate models with rough volatilities and stochastic correlations.
method Investigates continuous-time Markowitz mean-variance problem for multivariate affine and quadratic Volterra models using Riccati backward stochastic differential equations (BSDEs).
result Derives explicit solutions for BSDEs in affine Volterra models and new analytic formulae for quadratic models.
The paper values variable annuities using complex stochastic models and deep learning.
problem Valuation of variable annuities with early surrender options under non-Markovian models.
method Developed a deep signature Least Squares Monte Carlo approach to handle path-dependent continuation values.
result Fair fees increase with Hurst parameters of stock volatility and mortality force.
A new SINC method for fast and accurate option pricing.
problem Computing option prices efficiently and accurately.
method SINC approach based on Shannon Sampling Theorem.
result SINC provides the most accurate and fast pricing computation.
The paper analyzes robustness and sensitivity of rough Volterra stochastic volatility models.
problem Analyzing the robustness and sensitivity of stochastic volatility models.
method Statistical tests and empirical analysis on Apple Inc. equity options.
result Comparison of different models' robustness and sensitivity to option data structure.
We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterized by the affine form of their cumulant generating function, which can be obtained as solution of a convolution Riccati equat…
Study shows how heavy-tailed Hawkes processes can model rough volatility in financial markets.
problem Modeling rough volatility in financial markets with heavy-tailed Hawkes processes.
method Established weak convergence of Hawkes process with power-law kernel, derived scaling limit for financial market model.
result Price-volatility process converges weakly to a rough Heston model after rescaling.
We consider the fractional Heston model originally proposed by Comte, Coutin and Renault. Inspired by recent ground-breaking work on rough volatility, which showed that models with volatility driven by fractional Brownian motion with short memory allows for better calibration of the volatility surface and more robust e…