A Path Shadowing Monte-Carlo method provides prediction of future paths given any generative model.

At a given date, it averages future quantities over generated price paths whose past history matches, or “shadows”, the actual (observed) history.

We test our approach using paths generated from a maximum entropy model of financial prices,

based on the recently introduced “Scattering Spectra” which are multi-scale analogues of the standard skewness and kurtosis.

This model promotes diversity of generated paths while reproducing the main statistical properties of financial prices, including stylized facts on volatility roughness.

Our method yields state-of-the-art predictions for future realized volatility. It also allows one to determine conditional option smiles for the S&P500.

These smiles depend only on the distribution of the price process, and are shown to outperform both the current version of the Path Dependent Volatility model and the option market itself.

Random walks on random graphs are associated with diffusion in disordered media. In this talk, the graphs of interest are uniform spanning tree (UST) and loop-erased random walk (LERW). First I will demonstrate some asymptotic behavior of the simple random walk on the three-dimensional UST. Next I will discuss annealed transition probability of the simple random walk on high-dimensional LERWs.