How Nelson–Siegel Curves Differ from Short-Rate Models

# Link between short rate model and Nelson-Siegel


# Link between short rate model and Nelson-Siegel












I have just read the paper about the Nelson-Siegel model and I am a bit confused about its relation to short rate models, such as the Vasicek model.

Nelson-Siegel model

The Nelson-Siegel model is a parsimonious model of the term structure, which is flexible enough to generate monotonic, humped or S shaped relationships between bond yield and maturities. It consists of an equation for the instantaneous forward rate $r$ for maturity $m$: $$f(m) = \beta_0 + \beta_1 exp(-m/\tau) + \beta_2 [(m/\tau) exp(-m/\tau)]$$ which in turn implies that the yield $R$ for maturity $m$ is: $$R(m) = \frac{1}{m} \int_0^m f(x) dx = \beta_0 + (\beta_1 + \beta_2) [1 - exp(-m/\tau)] (m/\tau) - \beta_2 exp(-m/\tau)$$

Short rate models

On the other hand, we have short rate models, which assume that the short rate follows a stochastic process. For example, for the Vasicek model we have that, under the risk neutral measure: $$dr(t) = \kappa [\theta - r(t)] dt + \sigma dW(t) \quad r(0) = r_0$$ which in turn implies that the price $P$ of a bond with maturity $m$ is: $$P(T) = E \left\{ exp \left( \int_0^m r(s) ds \right) \right\}$$ From this, we can calculate the yield $R(m)$.

Question

Could you help me understand if Nelson-Siegel is a complement or a substitute to short rate models? The source of my confusion is the following. On one hand, I see that short rate models are about the evolution of the short rate in time, while the Nelson-Siegel is about the cross section of maturities. But short rate models are able to generate a term structure, and can even fit the current one if adequately extended. So why do we need Nelson-Siegel? Just because it fits the observed datapoints more flexibly?

## Answer by NC520 (score 0, accepted)

https://quant.stackexchange.com/a/80669

Nelson-Siegel, like other models that aim to fit the term structure of interest rates (e.g. Svensson), are a complement to, not a substitute of, short rate models. The former fits a curve to bond prices observed in the market and generates a best-fitting yield curve $m \rightarrow R^M(m)$, or equivalently, a best-fitting instantaneous forward-rate curve $m \rightarrow f^M(m)$. The superscript $M$ indicates that these are market curves. The latter instead are pure theoretical models, where the yield curve $m \rightarrow R(m)$ and the forward-rate curve $m \rightarrow f(m)$ are endogenously generated, i.e. are an output of the model.

It is true that short rate models can be extended to fit the current (market) term structures. This is exactly what the difference between the two is: calibration of the short-rate models occurs (for example, and roughly speaking) by matching the observed market forward-rate curve $f^M(m)$ and the model implied one $f(m)$.

For a more detailed explanation see `Anderson Sleath 1999`.