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We apply a physics-informed deep-learning approach the PINN approach to the Black-Scholes equation for pricing American and European options. We test our approach on both simulated as well as real market data, compare it to…
In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free…
Deep hedging is a framework for hedging derivatives in the presence of market frictions. In this study, we focus on the problem of hedging a given target option by using multiple options. To extend the deep hedging framework to this…
Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well…
The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This…
In the paper, we propose a new calculation scheme for American options in the framework of a forward backward stochastic differential equation (FBSDE). The well-known decomposition of an American option price with that of a European option…
We develop a kernel-based solver for path-dependent PDEs (PPDEs) along with a convergence theory. Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal…
Option pricing, a fundamental problem in finance, often requires solving non-linear partial differential equations (PDEs). When dealing with multi-asset options, such as rainbow options, these PDEs become high-dimensional, leading to…
This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T.…
We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal…
This work studies the deep learning-based numerical algorithms for optimal hedging problems in markets with general convex transaction costs on the trading rates, focusing on their scalability of trading time horizon. Based on the…
We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price…
We introduce the feedforward neural network to attack the sign problem via the path optimization method. The variables of integration is complexified and the integration path is optimized in the complexified space by minimizing the cost…
In this work, we propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained not only…
In this paper, we investigate reflected backward stochastic differential equations driven by rough paths (rough RBSDEs), which can be viewed as probabilistic representations of nonlinear rough partial differential equations (rough PDEs) or…
Building on the functional-analytic framework of operator-valued kernels and un-truncated signature kernels, we propose a scalable, provably convergent signature-based algorithm for a broad class of high-dimensional, path-dependent hedging…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
High-dimensional parabolic partial integro-differential equations (PIDEs) appear in many applications in insurance and finance. Existing numerical methods suffer from the curse of dimensionality or provide solutions only for a given…
Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the form $$\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v-…
We propose a new multistep deep learning-based algorithm for the resolution of moderate to high dimensional nonlinear backward stochastic differential equations (BSDEs) and their corresponding parabolic partial differential equations (PDE).…