Beyond Monte Carlo: Harnessing Diffusion Models to Simulate Financial Market Dynamics

We propose a highly efficient and accurate methodology for generating synthetic financial market data using a diffusion model approach. The synthetic data produced by our methodology align closely with observed market data in several key aspects: (i) they pass the two-sample Cramer - von Mises test for portfolios of assets, and (ii) Q - Q plots demonstrate consistency across quantiles, including in the tails, between observed and generated market data. Moreover, the covariance matrices derived from a large set of synthetic market data exhibit significantly lower condition numbers compared to the estimated covariance matrices of the observed data. This property makes them suitable for use as regularized versions of the latter. For model training, we develop an efficient and fast algorithm based on numerical integration rather than Monte Carlo simulations. The methodology is tested on a large set of equity data.

A generative flow for conditional sampling via optimal transport

Sampling conditional distributions is a fundamental task for Bayesian inference and density estimation. Generative models, such as normalizing flows and generative adversarial networks, characterize conditional distributions by learning a transport map that pushes forward a simple reference (e.g., a standard Gaussian) to a target distribution. While these approaches successfully describe many non-Gaussian problems, their performance is often limited by parametric bias and the reliability of gradient-based (adversarial) optimizers to learn these transformations. This work proposes a non-parametric generative model that iteratively maps reference samples to the target. The model uses block-triangular transport maps, whose components are shown to characterize conditionals of the target distribution. These maps arise from solving an optimal transport problem with a weighted $L^2$ cost function, thereby extending the data-driven approach in [Trigila and Tabak, 2016] for conditional sampling. The proposed approach is demonstrated on a two dimensional example and on a parameter inference problem involving nonlinear ODEs.