Extend classical theory of affine processes to path-dependent setting
problem Path-dependent affine processes
method Introduce path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform
result Define path-dependent affine processes through their exponential-affine Fourier--Laplace transform and establish a characterization theorem
PDGM uses neural nets to solve complex financial equations.
problem Solving path-dependent partial differential equations (PPDEs)
method Generalized Deep Galerkin Method (PDGM) combining feed-forward and LSTM architectures
result PDGM successfully models solutions to various PPDEs, including financial derivatives.
We investigate the computational aspects of the basket CDS pricing with counterparty risk under a credit contagion model of multinames. This model enables us to capture the systematic volatility increases in the market triggered by a particular bankruptcy. The drawback of this problem is its analytical complication due…
In this paper, we introduce and develop the theory of semimartingale optimal transport in a path dependent setting. Instead of the classical constraints on marginal distributions, we consider a general framework of path dependent constraints. Duality results are established, representing the solution in terms of path d…
Novel signature approach for pricing and hedging path-dependent options with market frictions.
problem Pricing and hedging path-dependent options with market frictions.
method Signature approach, mean-quadratic variation criterion, non-standard infinite-dimensional Riccati equations, time-augmented signature, non-Markovian stochastic control problem.
result Effective hedging strategies in frictional markets with low-truncated signature approximations.
New control theory for self-path-dependent problems solves unique constraints.
problem Optimal control with self-path-dependent constraints in stochastic systems.
method Introduces new HJB equations for variational inequalities with historical maximum controls.
result Value functions are viscosity solutions to HJB equations under Lipschitz conditions.
Solves infinite horizon portfolio problem with path-dependent labor income.
problem Infinite horizon portfolio choice with path-dependent labor income.
method Solves an infinite dimensional stochastic optimal control problem using explicit solutions to the HJB equation.
result Explicit solutions to the optimal controls in feedback form are found.
Proves global well-posedness for superquadratic BSDEs without Markovian assumption.
problem Global well-posedness of multidimensional superquadratic BSDEs without Markovian assumption.
method Interplay between local well-posedness of FBSDEs and backward iterations of superquadratic BSDEs.
result Global well-posedness of superquadratic BSDEs proved.
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.
The study examines insurance demand under rough volatility and path-dependent shocks.
problem Optimal insurance and investment strategies under rough volatility and path-dependent shocks.
method Rough volatility model and Hawkes process with power kernel, Functional Ito formula extension.
result Individuals demand more catastrophe insurance when path-dependent effects are considered.
The paper solves optimal control problems for stochastic delay equations.
problem Optimal control of stochastic delay differential equations.
method Rewriting the problem in an infinite-dimensional Hilbert space, using dynamic programming and viscosity solutions.
result Characterizes the value function as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.
Deep signature/log-signature FBSDE algorithm improves accuracy and training time.
problem Solving FBSDEs with state and path dependent features.
method Incorporates deep signature/log-signature transformation into RNN model.
result Improves accuracy and training time compared to existing methods.
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.
New method for hedging path-dependent options with price impact using probabilistic arguments.
problem Hedging of path-dependent options with price impact.
method Dual formulation using probabilistic arguments, proving existence of perfect hedging portfolios.
result Existence of a perfect hedging portfolio for path-dependent options with price impact.
New simulation approaches to evaluating path-dependent options without matrix inversion issues nor Euler bias are evaluated. They employ three main contributions: Stochastic approximation replaces regression in the LSM algorithm; Explicit weak solutions to stochastic differential equations are developed and applied to …
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.
New SDEs from affine and polynomial perspectives for path-dependent processes.
problem Characterizing path-dependent stochastic processes.
method Affine and polynomial processes, signature SDEs, Fourier-Laplace transform, Riccati and linear ODEs.
result Explicit formulas for the Fourier-Laplace transform and expected values of entire functions of signature processes.
Deep learning improves probabilistic PPDE solution accuracy.
problem Approximating solutions to path-dependent PDEs with limited basis selection.
method Deep learning for conditional expectation estimation with error bounds.
result Deep learning yields more accurate PPDE solutions, especially in high dimensions.
Deep learning predicts path-dependent processes from historical data.
problem Predicting path-dependent processes using historical data.
method Nonparametric regression with deep neural networks.
result Deep learning method converges to theoretical predictions as observation frequency increases.
Derives FPDE for equity-linked insurance pricing.
problem Calculating prices for insurance policies with complex payment histories.
method Variational techniques in functional Itô calculus.
result Derives a functional partial differential equation.
A new method for portfolio allocation in continuous-time markets.
problem Path-dependent portfolio allocation in continuous-time markets.
method Path-by-path framework, self-financing concept, partial differential equation, continuous-time algorithms.
result General explicit solution for wealth evolution in generic markets.
New method uses LSTM and signature theory to solve complex financial PDEs.
problem Solving path-dependent PDEs for financial derivatives pricing.
method Combining LSTM networks and rough paths theory.
result Efficient algorithms for pricing and hedging path-dependent derivatives.
Deep learning models price convertible bonds with complex reset and call features.
problem Pricing convertible bonds with path-dependent reset and call provisions.
method Formulated as a PPDE, deep learning approximates conditional expectations.
result Deep learning produces stable and accurate prices across various model specifications.
Study path-dependent affine models under uncertain parameters for financial applications.
problem Valuation of path-dependent financial derivatives under parameter uncertainty.
method Developed path-dependent setting for value function, established dynamic programming principle, approximated functional derivatives with neural networks.
result Efficient numerical methods for valuation of complex financial derivatives under parameter uncertainty.
The paper develops methods to price and hedge options in path-dependent stock models.
problem Pricing and hedging options under complex stock models.
method Develops a path-dependent PDE for option pricing and differentiability of path-dependent SDE solutions.
result Provides formulas for option Greeks and differentiability of path-dependent SDE solutions.
Guyon-Lekeufack model accurately predicts market volatility.
problem Modeling and predicting market volatility accurately.
method Path-dependent volatility model with weighted past price returns and squared volatility.
result Wellposedness of the coupled system of stochastic differential equations for all parameter values.
The paper analyzes optimal consumption with past spending maximum as a reference.
problem Optimal consumption with past spending maximum as a reference.
method Path-dependent exponential utility, Hamilton-Jacobi-Bellman (HJB) equation, dual transform, smooth-fit principle.
result Closed-form solutions for optimal investment and consumption strategies in each region.
In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Leão, Ohashi and Russo and, in contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartinga…
Develops a numerical scheme for solving path-dependent FBSDEs and PDEs.
problem Solving path-dependent FBSDEs and PDEs numerically.
method Picard iteration method for FBSDEs, concentration inequality for estimator, supervised learning with neural networks for PDEs.
result Proves convergence and rate of convergence for the Picard iteration method.
Dupire's functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Name…
Extends PD-NJ-ODE to noisy observations and dependent observation times.
problem Predicting continuous-time stochastic processes with irregular and noisy observations.
method Extends PD-NJ-ODE to handle conditional independence and noisy observations.
result Theoretical guarantees and empirical examples for handling noisy observations and dependent observation times.
Robust PDE method for path-dependent Asian-style options using MPDATA.
problem Valuation of path-dependent Asian-style options.
method Non-oscillatory forward-in-time second-order MPDATA finite-difference scheme for solving 2D PDEs.
result MPDATA scheme improves solution over first-order upwind step, highlighting its importance.
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.
Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a derivative-asset. The payoff of the derivative-asset may be path-dependent. Additionally, the process underlying the derivative may exhibit killing (i.e. jum…
This paper optimizes dividend payout rates with a drawdown constraint in a stochastic model.
problem Optimizing dividend payout rates while avoiding drawdowns in a stochastic model.
method Solving a path-dependent stochastic control problem using Hamilton-Jacobi-Bellman equations and PDE methods.
result Explicit characterization of an optimal feedback control strategy, including two free boundaries and the running maximum surplus process.
We study stochastic differential equations (SDEs) whose drift and diffusion coefficients are path-dependent and controlled. We construct a value process on the canonical path space, considered simultaneously under a family of singular measures, rather than the usual family of processes indexed by the controls. This val…
Study proves existence, uniqueness, and positivity of solutions to a complex volatility model.
problem Modeling equity index and spot volatility with path-dependent features and general kernels.
method Proved existence and uniqueness of a continuous solution to a Stochastic Volterra Equation (SVE) with non-convolutional, non-bounded kernels and non-Lipschitz coefficients.
result Positivity of the volatility process under certain conditions on the kernels.
A new method in finance without probabilities or integrals.
problem Creating a model-free approach to continuous-time finance.
method Pathwise approach using causal functional calculus and transition principle of Isaacs.
result A fully non-linear path-dependent equation characterizes optimal solutions.
Global approximation for piecewise linear paths via signatures.
problem Global approximation theorems for piecewise linear paths.
method Using signatures of piecewise linear paths and their density in Lp-norms. result Linear functionals of signatures are dense in Lp-norms under an integrability condition. Deep signature algorithm for pricing path-dependent options.
problem Pricing path-dependent options with complex payoff functions.
method Extended backward scheme for state-dependent FBSDEs with reflections, incorporating signature layer for path-dependent FBSDEs.
result Convergence analysis of the algorithm with explicit dependence on truncation order and neural network approximation errors.
Deep learning solves complex volatility equations.
problem Solving path-dependent PDEs in rough volatility.
method Interpreting PDE as BSDE, using neural network reservoir approach.
result Proved theoretical convergence for least-square regression.
In this paper, we extend the first-order asymptotics analysis of Fouque et al. to general path-dependent financial derivatives using Dupire's functional Ito calculus. The main conclusion is that the market group parameters calibrated to vanilla options can be used to price to the same order exotic, path-dependent deriv…
The paper provides an efficient method to price path-dependent derivatives using multiscale stochastic volatility models.
problem Pricing path-dependent derivatives under multiscale stochastic volatility models.
method Derives a Malliavin representation for the first-order approximation of the price of path-dependent derivatives.
result An efficient Monte Carlo approximation for pricing path-dependent derivatives is derived.
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.
In this paper, we give a numerical method for pricing long maturity, path dependent options by using the Markov property for each underlying asset. This enables us to approximate a path dependent option by using some kinds of plain vanillas. We give some examples whose underlying assets behave as some popular Levy proc…
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.
The paper introduces a new volatility model using Fourier techniques for pricing and hedging.
problem Pricing and hedging of financial derivatives with stochastic volatility.
method A Fourier-based approach to price and hedge European and path-dependent options in a stochastic volatility model.
result The model includes and extends popular volatility models like Stein-Stein, Bergomi, and Heston.
Survey and new results link hydrodynamics, molecular physics, and financial engineering.
problem Understanding financial engineering topics like Asian options and volatility swaps.
method Linking Kevin waves, Klein-Kramers, and Kolmogorov equations to financial models.
result Corrected the original solution of the Kolmogorov equation.