Derives new equations for stochastic volatility models.
problem Modeling local-stochastic-volatility models and their derivatives.
method Conditional forward equation, Dupire stochastic PDE, rolling expiry vanilla option SPDE.
result New equations for LSV models and their derivatives.
Derives new equations for volatility models and option pricing.
problem Modeling and pricing options in local-stochastic-volatility models.
method Develops conditional forward equations and Dupire stochastic PDEs.
result Derives new SPDE for vanilla options.
Extends unbiased simulation method to Asian options.
problem Simulating path-dependent dynamics for Asian options.
method Extension of unbiased simulation method for SDEs to path-dependent dynamics.
result Extension applies to numerical resolution of path-dependent PDEs.
Survey revisits Bachelier and Dupire, highlighting optimal transport's role.
problem Finding arbitrage-free models calibrated to volatility surfaces.
method Revisits mathematical finance principles, uses optimal transport results.
result Optimal transport provides rigorous foundations for Dupire's model.
There are several (mathematical) reasons why Dupire's formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire's local …
Develops a new calculus for stochastic processes with occupation flows.
problem Analyzing the behavior of stochastic processes with occupation flows.
method Itô calculus for occupied processes, Feynman-Kac approach.
result Unified Markovian lifts for pricing financial derivatives.
We extend Dupire's formula for stochastic interest rates and local volatility.
problem Deriving formulas for stochastic interest rates and local volatility.
method Generalizations of Dupire's formula for stochastic drift and local volatility.
result Validated the limits of the generalized Dupire formulae for specific cases.
We derive a forward equation for arbitrage-free barrier option prices, in terms of Markovian projections of the stochastic volatility process, in continuous semi-martingale models. This provides a Dupire-type formula for the coefficient derived by Brunick and Shreve for their mimicking diffusion and can be interpreted …
Neural networks improve financial derivative pricing accuracy.
problem Improving accuracy in financial derivative pricing.
method Use neural networks to model drift and volatility in SDE models, optimize using SGD for European options and PDE for American options.
result Neural network models outperform traditional models in pricing derivatives.
Paper provides an explicit formula for local volatility in Cheyette models.
problem Approximating local volatility in Cheyette interest rate models.
method Extended Dupire framework, perturbation methods, probabilistic techniques.
result Explicit analytical formula for local volatility in Cheyette models.
Paper improves stochastic collocation for local volatility models.
problem Improving local volatility models for assets with boundaries.
method Applied stochastic collocation to lognormal distributions, derived analytical local volatility.
result Simple analytical Dupire local volatility derived from option prices.
We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in …
A robust implementation of a Dupire type local volatility model is an important issue for every option trading floor. Typically, this (inverse) problem is solved in a two step procedure : (i) a smooth parametrization of the implied volatility surface; (ii) computation of the local volatility based on the resulting call…
New algorithm calibrates local volatility from option prices using deep neural networks.
problem Calibrating local volatility from market option prices with reduced interpolation and reprice errors.
method Deep self-consistent learning using neural networks to approximate both option prices and local volatility.
result Improved performance in terms of reduced interpolation and reprice errors compared to existing methods.
Develops a deep learning method for enforcing no-arbitrage in local volatility surfaces.
problem No-arbitrage conditions not enforced in deep learning approaches for local volatility.
method Jointly interpolates European vanilla option prices, enforcing no-arbitrage through modified loss functions or network architectures.
result Demonstrates the effectiveness of enforcing no-arbitrage in local volatility surfaces using deep learning.
These notes are the first half of the contents of the course given by the second author at the Bachelier Seminar (February 8-15-22 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis.
Derives functional Itô formula for non-anticipative maps of rough paths.
problem Functional Itô formula for non-anticipative maps of càdlàg rough paths.
method Approximation properties of the signature and Marcus transformation.
result Functional Taylor expansion for sufficiently regular non-anticipative maps.
We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. A uniqueness theorem is given for the solutions of this equation. This result generalizes Dupire…
In this paper we provide evidence that financial option markets for equity indices give rise to non-trivial dependency structures between its constituents. Thus, if the individual constituent distributions of an equity index are inferred from the single-stock option markets and combined via a Gaussian copula, for examp…
Develops ML method for solving financial equations.
problem Solving financial equations efficiently and accurately.
method Combines semi-analytical and numerical techniques.
result Significantly faster and more accurate solutions.
We propose two main applications of Gyöngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous Itô processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility mod…
We create consistent option surfaces without arbitrage.
problem Constructing consistent option surfaces free of arbitrage across different maturities.
method Combining PCA-Smolyak approximation with chain-consistent diffusion and c-EMOT bridge.
result Computable certificates for strong convexity, solver correctness, and Dupire/Greeks stability.
Extends Itô's formula for path-dependent functions in finance.
problem Modeling and hedging of path-dependent financial options.
method Functional extension of Itô's formula for C^{0,1}-functions of continuous weak Dirichlet processes.
result Validates the hedging or superhedging problems for path-dependent options.
We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general…
A new method infers parameters from PDEs using Gaussian processes.
problem Estimating unknown parameters in PDEs from noisy data.
method PDE-Informed Gaussian Process (PIGP) method.
result The method bypasses numerical solvers for PDEs and provides uncertainty quantification.
The Bass model is calibrated to vanilla options using a fixed-point equation.
problem Calibration of the Bass local volatility model to vanilla options.
method Solving a fixed-point equation to achieve calibration.
result Existence and uniqueness of the solution to the fixed-point equation, and linear convergence of the fixed-point iteration scheme.
Neural-PDE learns PDEs from data using LSTM, outperforming traditional methods.
problem Solving time-dependent PDEs numerically is challenging.
method Bidirectional LSTM encoder to learn governing rules from data.
result Neural-PDE efficiently predicts PDE dynamics with minimal parameters.
A new framework for SPX and VIX hedging that combines AI and market dynamics.
problem Jointly hedging SPX and VIX exposures under transaction costs and regime shifts.
method Integrates an SSVI-based implied-volatility surface and a Cboe-compliant VIX computation with a control layer that enforces safety as constraints.
result Reduces expected shortfall while suppressing nuisance turnover in a reproducible synthetic environment.
LOV model calibrates European and American options with path-dependent volatility.
problem Calibrating European and American options with path-dependent volatility.
method Designing a local volatility model that incorporates path-dependent shocks through an occupation sensitivity function.
result LOV model successfully calibrates options chains with automatic European vanilla option calibration and path-dependent flexibility.
Neural Q-learning tackles high-dimensional PDEs.
problem Solving high-dimensional PDEs is computationally challenging.
method Adapting Q-learning from reinforcement learning to solve PDEs.
result The neural network approximator converges to the PDE solution as the network width increases.
Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net …
PDE-DKL combines NNs and GPs for high-dimensional PDE problems.
problem High-dimensional PDE problems with scarce data.
method PDE-constrained Deep Kernel Learning (PDE-DKL) framework.
result High accuracy with reduced data requirements.
We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market port…
By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic v…
Solves second-order PDEs using quotients and differential invariants.
problem Solving second-order PDEs with first-order quotients.
method Solve the quotient PDE using differential invariants, then add new constraints to solve the original PDE.
result New method for solving second-order scalar PDEs with infinite-dimensional symmetry algebras.
PRISMA uses PDE residuals for fast, robust, and accurate inference.
problem Slow gradient-based optimization and instability in PDE residual-based methods.
method Integrates PDE residuals directly into the model's architecture via attention mechanisms in the spectral domain.
result Competitive accuracy with significantly lower inference costs and faster speeds.
We present a detailed analysis and implementation of a splitting strategy to identify simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying model consists of a jump-diffusion driven asset with time and price dependent volatility. Our approach uses a forwa…
Meta-learning base distributions for efficient PDE solutions.
problem Efficiently solving parametric parabolic PDEs across different scenarios.
method Meta-learning base distributions to compute PDE solutions.
result Improves generalization to new parameter regimes.
Kernel method learns PDEs from noisy data.
problem Discovering and solving PDEs from noisy data.
method Kernel smoothing, regression, and operator learning.
result Competitive performance compared to state-of-the-art algorithms.
Solves a PDE for Landsberg surfaces using new Finsler surface insights.
problem Solving the Landsberg's PDE for Finsler surfaces.
method Reduces the system of non-linear PDEs to a single PDE, the Landsberg's PDE, and solves it.
result Obtains a class of solutions for the Landsberg's PDE.
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
Using the theory of the symmetry group for PDEs [15, 17], we derive the symmetry group G associated to surfaces PDE. Several group invariant solutions of the surfaces PDE are given by solving a reduced system of partial differential equations.
Develops arithmetic PDE geometry concepts like curvature and cohomology.
problem Creating a geometry framework for arithmetic PDEs.
method Introducing arithmetic analogues of Levi-Civita and Chern connections, then developing curvature and characteristic classes.
result Arithmetic analogues of curvature and characteristic classes have been developed.
Unified framework solves nonlinear PDEs and IPs using Gaussian processes.
problem Solving and identifying parameters in nonlinear PDEs and inverse problems.
method Gaussian process framework approximating solutions as MAP estimators, reducing to finite-dimensional optimization problem.
result Unified method converges in a small number of iterations for various PDEs.
The paper presents a PDE method for xVA incorporation in financial derivatives.
problem Incorporating value adjustments (xVA) in financial derivative pricing.
method Analytical solution of PDEs in the Black-Scholes framework.
result New semi-closed formulas for xVA are derived and compared to Monte-Carlo and numerical methods.
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
Recent work of Dupire and Carr and Lee has highlighted the importance of understanding the Skorokhod embedding originally proposed by Root for the model-independent hedging of variance options. Root's work shows that there exists a barrier from which one may define a stopping time which solves the Skorokhod embedding p…
We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these net…