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.
Researchers compute Greeks for rough Volterra SV models using Malliavin calculus.
problem Computing Greeks under rough Volterra stochastic volatility models.
method Malliavin calculus techniques, extending integration by parts to non-square integrable functionals.
result Formulas for computing Greeks (Delta, Gamma, Rho, Vega) under various rough Volterra SV models.
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.
This paper improves simulation methods for rough Volterra stochastic volatility models.
problem Inefficient techniques in Monte-Carlo simulations for rough Volterra volatility models.
method Comparison and modification of three simulation methods: Cholesky, Hybrid, and rDonsker schemes.
result Suggests modifications to improve simulation accuracy and efficiency.
Study small-time CLTs for stochastic Volterra equations with various kernels.
problem Understanding the behavior of stochastic Volterra equations with different kernels.
method Proved convergence of finite-dimensional distributions, functional CLT, and limit theorems for smooth transformations.
result Derived asymptotic pricing formulae for digital calls in rough volatility models.
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.
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 consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the con…
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…
Volterra square-root process boundary behavior and martingale measures
problem Boundary behavior of the Volterra square-root process
method Comparison principles for Volterra integral equations and generalized Riemann-Liouville fractional equations
result Finiteness of negative p-moments and atom at the boundary for rough kernels In this paper, we consider equilibrium strategies under Volterra processes and time-inconsistent preferences embracing mean-variance portfolio selection (MVP). Using a functional Itô calculus approach, we overcome the non-Markovian and non-semimartingale difficulty in Volterra processes. The equilibrium strategy is the…
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…
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.
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.
Investigates mean-variance portfolio selection in non-Markovian markets.
problem Continuous-time Markowitz mean-variance portfolio selection in fake stationary affine Volterra models.
method Stochastic factor solution to a Riccati BSDE, deriving explicit solutions as multi-dimensional Riccati-Volterra equations.
result Analytical closed-form expressions for optimal portfolio policies and mean-variance efficient frontier.
Study provides LDP for non self-similar stochastic volatility models.
problem Analyzing non self-similar stochastic volatility models.
method Short-time large deviation principle (LDP) for models with Volterra process.
result Derives consequences for option prices, implied volatility surfaces, and skew.
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.
We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where log-volatility follows a Gaussian Volterra process. While providing a good fit for European…
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.
Unified approach to stochastic Volterra systems' deviations.
problem Large and moderate deviations for stochastic Volterra systems.
method Weak convergence approach by Budhijara, Dupuis and Ellis.
result Unified treatment of deviations for a broad class of stochastic Volterra equations.
New IBP formulae for rough stochastic Volterra processes.
problem Deriving IBP formulae for path-dependent stochastic Volterra processes.
method Developed a new fractional IBP formula that interpolates between standard and Bismut-Elworthy-Li formulae.
result For rough noise, the expectation is differentiable along constant directions under certain Hölder continuity conditions.
Investigates optimal investment strategies in financial markets with jumps.
problem Optimal portfolio selection for investors in multi-asset financial markets with jumps.
method Uses martingale optimality principle and Riccati backward stochastic differential equations with jumps.
result Derives semi-closed form optimal strategies and value function for Merton's problem.
We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L1-kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is b…
Neural SVEs model complex systems with memory, outperforming traditional methods.
problem Modeling systems with memory effects and irregular behavior.
method Introducing neural stochastic Volterra equations as a physics-inspired architecture.
result Neural SVEs outperform neural SDEs and DeepONets in various applications.
Study of coupled Hawkes processes with rough-volatility limits.
problem Understanding coupled Hawkes processes with rough-volatility limits.
method Proving weak convergence of rescaled intensity vector to stochastic Volterra equations.
result Limiting components exhibit different degrees of roughness and cross-decorrelation law.
The research presented in this article provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenge…
Study proves optimal controls for stochastic Volterra equations with singular kernels.
problem Existence of optimal controls for stochastic Volterra equations with singular kernels.
method Sufficient conditions based on integrability and growth hypotheses.
result Existence of optimal relaxed and strict controls under classical convexity assumptions.
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…
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.
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.
We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a nonnegative function of a Volterra type continuous Gaussian process that may have very rough sample paths. The main results obtained in the paper are sample path and small-noise large deviation principles for the log…
We prove strong existence and uniqueness, and Hölder regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised ke…
The paper analyzes the stationarity of stochastic Volterra integral equations and introduces fake stationary regimes.
problem Analyzing the stationarity of non-Markovian dynamical systems described by SVIEs.
method Investigates the properties of SVIE solutions, focusing on stationarity over finite and long time horizons, and introduces a deterministic stabilizer to induce a fake stationary regime.
result SVIEs do not exhibit a strong stationary regime unless the kernel is constant or degenerate, but a fake stationary regime can be achieved with a deterministic stabilizer.
Model for high-frequency trading with rough volatility.
problem High-frequency trading dynamics and rough volatility modeling.
method Stochastic partial differential equation (SPDE) with rough volatility driven by a Hawkes process.
result The volatility path of the SPDE is rougher than that driven by a standard Brownian motion.
Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that under no-arbitrage assumption, the market impact function can only be of power-law type. Furthermore, we prove that this implies that the macroscopic price is diffusive with roug…
Developed scalable Monte Carlo method for VIX option pricing.
problem VIX option pricing in stochastic Volterra rough volatility models with non-Markovian vol-of-vol.
method Infinite dimensional Markovian representation to devise scalable least squares Monte Carlo.
result Efficient VIX option pricing method for generalized models.
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.
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.
The non-Markovian nature of rough volatility processes makes Monte Carlo methods challenging and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker's approximation of Brownian motion to the f…
This work studies nonnegativity-preserving kernels for stochastic equations and their applications.
problem Nonnegativity preservation in stochastic Volterra equations and related processes.
method Characterization and application of completely monotone kernels; approximation schemes for weak error.
result Positive linear combinations of decaying exponentials can be used for second-order approximation schemes.
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.
Study rough volatility models using path-dependent PDEs and fractional Brownian motions.
problem Modeling and analyzing rough volatility in financial markets.
method Showed conditional expectations are unique classical solutions to path-dependent PDEs derived from functional Itô formula. Leverage these to study weak rates of convergence for discretized stochastic integrals.
result Obtained optimal weak error rates for approximating log-stock prices in rough volatility models.
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
The study uses response theory to understand RNNs processing input signals.
problem Understanding how RNNs process sequential data.
method Deriving a Volterra series representation for SRNNs output using response theory from nonequilibrium statistical mechanics.
result SRNNs can be viewed as kernel machines operating on a reproducing kernel Hilbert space associated with the response feature.
The square root of Fredholm determinants causes numerical instabilities in option pricing models.
problem Numerical instabilities in Fourier-based option pricing for the Volterra Stein-Stein model.
method Characterization of determinant crossing behavior, derivation of transform to handle crossings, efficient algorithms.
result Significant improvement in accuracy and reduction in computational cost for Fourier-based pricing.
We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and…
Path-dependent PDEs model VIX and Realised Variance options.
problem Modeling volatility derivatives with path-dependence.
method Continuous stochastic volatility model with Gaussian Volterra process, proving well-posedness of PDEs.
result Formulae for greeks and implied volatility provided, finite-dimensional pricing PDEs obtained in Markovian models.
A new model fits SPX and VIX volatility surfaces and term structures efficiently.
problem Calibrating SPX and VIX volatility models to market data.
method Gaussian polynomial volatility models, joint calibration, functional quantization, Neural Networks.
result A conventional one-factor Markovian model outperforms rough and non-rough models.