HPC

9. From HMC to NUTS: Automating the Hamiltonian In the previous post we visited the (simplified) version of a Bayesian estimator of the $GARCH(1,1)$ volatility model parameters distributions using the Hamiltonian Monte Carlo technique. We suggested then, that implementing the No U-Turn Sampler (NUTS) would provide an added bonus of better convergence and faster run times. This article serves as the second part of that series - and will number the sections accordingly. ...

1. The Model: GARCH(1,1) We assume a time series of returns $y_t$. We model them as: $$y_t = \sigma_t \epsilon_t \quad \text{where} \quad \epsilon_t \sim \mathcal{N}(0, 1)$$ The variance $\sigma_t^2$ evolves according to the GARCH(1,1) recursion: $$\sigma_t^2 = \omega + \alpha y_{t-1}^2 + \beta \sigma_{t-1}^2$$ Constraints for Stability: $\omega \gt 0, \alpha \ge 0, \beta \ge 0$ (Positivity) $\alpha + \beta \lt 1$ (Stationarity/Mean Reversion) 2. The Bayesian Objective (Log-Posterior) In HMC, we don’t just want the “best” parameters; we want to sample from the posterior distribution $P(\theta | y)$. By Bayes’ Theorem: $$\log P(\theta | y) = \log P(y | \theta) + \log P(\theta)$$ ...

Mechanical Sympathy in Quant Finance: Optimizing Monte Carlo Post-Processing The Hook: The Latency of Decision In quantitative finance, the difference between a simulation that takes 10 minutes and one that takes 30 seconds isn’t just “convenience”—it’s a competitive edge. When processing millions of Monte Carlo paths to calculate Sharpe Ratios and volatility, you quickly hit the “Memory Wall.” The CPU is so fast that it spends most of its time idling, waiting for data to arrive from RAM. ...