Time Deep Gradient Flow Method for Option Pricing
J.G. Rou (TU Delft - Applied Probability)
A. Papapantoleon – Promotor (TU Delft - Applied Probability)
F.H.J. Redig – Promotor (TU Delft - Applied Probability)
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Abstract
Pricing financial derivatives such as options is a key challenge in financial mathematics. A common approach is to express the option price as the solution to a partial differential equation (PDE). As option pricing models become increasingly complicated, the resulting PDEs become high-dimensional, rendering classical numerical methods computationally infeasible. Recent advances suggest that deep learning may offer efficient alternatives for solving such problems.
This dissertation develops a novel deep learning approach for pricing options, which can efficiently handle high-dimensional problems. We call our method the Time Deep Gradient Flow (TDGF). The TDGF reformulates the PDE as a time-discretized variational problem, which is then approximated using deep neural networks. We focus on three option pricing models: the Black--Scholes model with constant volatility; the Heston model with stochastic volatility and the lifted Heston model, a Markovian approximation of rough volatility. We compare our method in these models with classical numerical methods such as Monte Carlo and Fourier methods and another deep learning method called the Deep Galerkin Method (DGM).