Generating drawdown-realistic financial price paths using path
  signatures
Emiel Lemahieu Kris Boudt Maarten Wyns
A novel generative machine learning approach for the simulation of sequences
of financial price data with drawdowns quantifiably close to empirical data is
introduced. Applications such as pricing drawdown insurance options or
developing portfolio drawdown control strategies call for a host of
drawdown-realistic paths. Historical scenarios may be insufficient to
effectively train and backtest the strategy, while standard parametric Monte
Carlo does not adequately preserve drawdowns. We advocate a non-parametric
Monte Carlo approach combining a variational autoencoder generative model with
a drawdown reconstruction loss function. To overcome issues of numerical
complexity and non-differentiability, we approximate drawdown as a linear
function of the moments of the path, known in the literature as path
signatures. We prove the required regularity of drawdown function and
consistency of the approximation. Furthermore, we obtain close numerical
approximations using linear regression for fractional Brownian and empirical
data. We argue that linear combinations of the moments of a path yield a
mathematically non-trivial smoothing of the drawdown function, which gives one
leeway to simulate drawdown-realistic price paths by including drawdown
evaluation metrics in the learning objective. We conclude with numerical
experiments on mixed equity, bond, real estate and commodity portfolios and
obtain a host of drawdown-realistic paths.