Kalman filter
Quantitative Finance glossary · Quantitative Finance
Definition
Recursive Bayesian estimator for linear-Gaussian state-space models boldsymbolx_t = boldsymbolFboldsymbolx_t-1 + boldsymbolw_t, boldsymboly_t = boldsymbolHboldsymbolx_t + boldsymbolv_t. Two-step recursion: predict (hatboldsymbolx_t|t-1 = boldsymbolFhatboldsymbolx_t-1|t-1, propagate covariance) then update with Kalman gain boldsymbolK_t = boldsymbolP_t|t-1boldsymbolH^T(boldsymbolHboldsymbolP_t|t-1boldsymbolH^T + boldsymbolR)^-1. Used in finance for dynamic beta estimation, factor-model state, and yield-curve latent-factor models (Nelson-Siegel state-space). Extended/Unscented variants relax linearity.
Kalman filter in plain English
The Kalman filter is a clever way to track something you can't see directly using only noisy, imperfect measurements. Picture tracking a plane on a fuzzy radar: each new blip is a bit off, so instead of trusting it fully, you blend it with where you predicted the plane would be — leaning more on whichever you trust more right now. It repeats two steps forever: predict where things are headed, then nudge that guess toward the fresh measurement by an amount (the 'Kalman gain') that automatically balances how noisy the data is against how confident your prediction is. In finance it's used to track hidden, drifting quantities like how sensitive a stock is to the market, or the invisible forces shaping interest rates, updating smoothly as new data arrives.
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