Gaussian Processes for VaR

A bank can tell you there is a 1% chance a trading desk loses more than $10 million tomorrow and still have no answer to the obvious follow-up: if that 1% happens, is the loss $11 million or $110 million? That gap is why researchers are pushing Gaussian Process Regression into market risk. (bis.org) Value-at-Risk is the cutoff line. If a portfolio has a one-day 99% Value-at-Risk of $10 million, the model is saying 99 days out of 100 the loss should stay below $10 million. (bis.org) Expected Shortfall is the average of the bad days beyond that cutoff. If Value-at-Risk is the edge of the cliff, Expected Shortfall is the average depth after you fall over it. (personal.ntu.edu.sg) Banks care about both numbers because trading books contain options, rates products, currencies, and other positions whose losses do not move in straight lines. A small move in markets can produce a very different loss than a large move in markets. (mdpi.com) The hard part is not writing down the definition of risk. The hard part is revaluing a large portfolio across thousands or millions of market scenarios, because each scenario can require another full pass through a pricing engine. (mdpi.com) Gaussian Process Regression is one way to avoid rerunning that full engine every time. It learns a smooth map from market inputs, like prices or volatility levels, to portfolio values, then predicts values for new scenarios from that learned map. (ingentaconnect.com) The useful twist is that a Gaussian process does not just give one estimate. It also gives a probability distribution around that estimate, which means the model can say both “here is the likely portfolio value” and “here is how uncertain I am about it.” (ingentaconnect.com) That matters in risk because Value-at-Risk and Expected Shortfall live in the tail, where data are sparse and mistakes are expensive. A model that exposes its own uncertainty fits more naturally with stress testing and model validation than a black box that only emits one number. (arxiv.org) This is not a brand-new idea. A 2011 paper by I. Garcia and J. Jimenez used piecewise Gaussian processes to estimate Value-at-Risk and Expected Shortfall on daily Standard & Poor’s 500 data and reported satisfactory Basel-style backtesting results. (arxiv.org) A 2019 paper by Sascha Wilkens applied Gaussian Process Regression directly to portfolio revaluation for Value-at-Risk and Expected Shortfall and reported risk figures identical to full revaluation on test portfolios with vanilla and barrier options, while outperforming Taylor expansion with limited training data. (ingentaconnect.com) The new push is more practical and more institutional. A June 3, 2025 paper by N. Lehdili, P. Oswald, and H. D. Nguyen studied Gaussian Process Regression and multi-fidelity modeling for bank market risk and found that standard and multi-fidelity Gaussian process models beat both traditional bank approaches and a neural network benchmark on pricing accuracy and risk-calculation efficiency in their tests. (mdpi.com) The regulatory backdrop helps explain the timing. The Basel Committee’s market risk framework uses Expected Shortfall in the internal models approach, so any machine learning method that can estimate portfolio values quickly while still producing defensible tail-risk numbers gets immediate attention inside banks. (bis.org) What researchers are really selling is a bridge between two worlds that usually sit apart. Probabilistic machine learning is good at saying “here is the whole distribution,” while regulation asks for specific tail numbers like Value-at-Risk and Expected Shortfall, and Gaussian processes connect those two demands in one workflow. (mdpi.com) The catch is scale. Gaussian processes are elegant on small and medium training sets, but large trading books can create high-dimensional inputs and heavy computational loads, which is why recent work leans on approximations and multi-fidelity tricks rather than a naïve one-model-fits-all setup. (mdpi.com) So the story is not that Gaussian processes suddenly replaced standard risk systems in 2026. The story is that a line of research running from 2011 through 2025 is turning a statistically neat idea into something banks can actually test against pricing engines, backtesting rules, and capital models. (arxiv.org, ingentaconnect.com, mdpi.com)