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Could anyone explain why the Vasicek model isn't an arbitrage-free model?

Additionally, which interest rate model is arbitrage-free and why?

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Short rate models are broadly divided into equilibrium models and no-arbitrage models. The models from Vasicek, Dothan and Cox, Ingersoll and Ross are examples of equilibrium short rate models. The models from Ho-Lee, Hull-White and Black-Karasinski are no-arbitrage models.

Take Vasicek and Hull-White as an example. The short rate processes are $\mathrm{d}r_t=\kappa(\theta-r_t)\mathrm{d}t+\sigma \mathrm{d}W_t$ for Vasicek and $\mathrm{d}r_t=\kappa(\theta_t-r_t)\mathrm{d}t+\sigma \mathrm{d}W_t$ for Hull and White. Both model imply a normally distributed, mean-reverting short rate process but as you see, Hull and White consider a time-inhomogeneous SDE where the coefficients are allowed to be time-dependent (but still deterministic). The extra degrees of freedom are used for calibration: with Hull and White you can fit the observed term structure of interest rate (zero-coupon bond prices) perfectly.

Put differently, for no-arbitrage models, the discount curve is an input factor which you use to find your model specification whereas for equilibrium models, the discount curve is an output which is determined by the small set of model parameters.

It is also possible to make the speed of mean-reversion $\kappa$ and the volatility $\sigma$ time-dependent in order to have more model parameters and improve the fit with observed cap or swaption volatilites. But you always run into the risk of overfitting... Hull and White describe this trade-off as the difference between an academic's view and a trader's view.

The extension from Brigo and Mercurio combines both approaches and takes a classical equilibrium model ($x_t$) and adds a time-dependent shift ($\varphi_t$) to it ($r_t=x_t+\varphi_t$) which allows for a perfect fit with respect to the observed zero-coupon bond prices but keeps many desirable properties from the equilibrium model. You may want to have a look at their excellent book on interest rate modelling.

Other classifications of short rate models include whether they allow for closed-form solutions of zero-coupon bond (option) prices and whether such closed-forms are affine linear in the short rate.

Finally, note that the term ``no-arbitrage model'' has nothing to with no-arbitrage pricing. Of course, you can still use the no-arbitrage principle to value cash flows et cetera regardless which short rate model you choose. The problem is that if you use equilibrium models and have mismatches between the model implied term structure and the observed term structure, arbitrage possibilities arise.

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