# Kalman Filter in Interest Rate Models

A couple questions regarding the use of Kalman filtering in estimating parameters of short rate models:

1) In Duan & Simonato (1995), which seems to be one of the earliest applications of the Kalman filter to short rate models, they state on page 5,

"To deal with the estimation problem, it is reasonable to assume that the yields for different maturities are observed with errors of unknown magnitudes"

Note yields are the output of the observation equation. Mathematically, they mean the yield $R$ at time $t$ for a $T$-bond may be written as $$R_t(r_t, T) = -\frac{1}{T-t} \ln P_t(r_t,T) + \epsilon_t, \qquad (1)$$ where $r_t$ is the short rate value (the output of the state equation) at time $t$ and $\epsilon_t$ is the "error of unknown magnitude." Why is there an error term? The relationship (1) without the $\epsilon_t$ is simply the definition of the yield. Are we supposed to assume the "error" represents things like illiquidity or quotes with finite decimals?

2) With the exception of the Vasicek model, most short rate models are non-Gaussian. In filtering parlance, the conditional distribution of the current state given the past states is non-Gaussian. How is Kalman filter justified in this case, given it assumes both the state and the observation equations are Gaussian?

## 1 Answer

There are two cases depending on what P is in your formula.

1. P is the market price of the bond: There exists an idealized, true but unobserved yield curve, but we observe bonds that have some measurement error. This can be because of microstructure imperfections (bid/ask spreads, liquidity and other premia, non synchroneity of quotes since bond markets are OTC). This is purely measurement noise.

2. P is the theoretical price based on a model and a parameter set: Then the noise stems from the fact that the model is only an approximation of the true DGP. Since here you claim that P is a function of the short rate, I suspect that this is the correct interpretation of the error.

Re the second question: You are right, more sophisticated non-affine models are non linear in the parameters. There are various options. Perhaps use the Extended Kalman Filter where you linearize locally. In non-affine cases you might not even have bonds in closed form, therefore you might employ appriximations for pricing that extend to filtering. Or use non-parametric methods like the paper by Takamizawa (link below). You can also discretize the state space and apply Hamilton's Markov Chain filter (google Hamilton and flexible nonlinear inference, link below for an example). The Unscented Kalman Filter/ Particle Filters which have been used for Stoch Vol filters should also be useful.

Hamilton's nonlinear inference: http://www.socialsciences.manchester.ac.uk/medialibrary/economics/discussionpapers/EDP-0401.pdf