# Why isn't the Nelson-Siegel model arbitrage-free?

Assume $X_t$ is a multivariate Ornstein-Uhlenbeck process, i.e. $$dX_t=\sigma dB_t-AX_tdt$$ and the spot interest rate evolves by the following equation: $$r_t=a+b\cdot X_t.$$ After solving for $X_t$ using $e^{tA}X_t$ and Ito and looking at $\int_0^T{r_s\;ds}$, it turns out that $$\int_0^T{r_s\;ds} \sim \mathcal{N}(aT+b^{T}(I-e^{-TA})A^{-1}X_0,b^{T}V_Tb)$$ where $V_t$ is the covariance matrix of $\int_0^T(I-e^{-(T-u)A})A^{-1}\sigma dB_u$.

This gives us the yield curve $$y(t)=a+\frac{b^{T}(I-e^{-tA})A^{-1}X_0}{t}+\frac{b^{T}V_tb}{2t}$$ and by plugging in $A= \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \\ \end{pmatrix}$ we finally arrive at $$y(t)=a+\frac{1-e^{-\lambda t}}{\lambda t}C_0+e^{-\lambda t}C_1+\frac{b^{T}V_tb}{2t}.$$ The formula above without $\frac{b^{T}V_tb}{2t}$ is known as the Nelson-Siegel yield curve model. Could somebody clarify why neglecting $\frac{b^{T}V_tb}{2t}$ leads to arbitrage opportunities?

So I am essentially asking the following question:

Why is the above model (with $\frac{b^{T}V_tb}{2t}$) arbitrage free?

## 2 Answers

The original Nelson Siegel paper describes a parsimonious model of the term structure using only four or three (if $\lambda_t$ is fixed). Filipovic (1999) proves that this model can never be used in a arbitrage free context, paraphrasing the abstract:

We introduce the class of consistent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson–Siegel (NS) family. (We show that) there exists no nontrivial interest rate model driven by a consistent state space Itō process.

This problem is solved by Christensen et al. (2009). They provide some ODE's which must hold for an AFNS and write that the "key difference between Dynamic NS and AFNS is the maturity dependent yield-adjustment term" and show how to solve for this term.

They show that the yield adjustment term is empirically small and that their model

fares well in out-of-sample prediction, consistently outperforming, for example, the canonical $A_0(3)$ model (of Duffee 2002).

Let $P(t,T)$ be the time-$t$ price of the zero-coupon bond expiring at $T$.

The no-arbitrage condition forces: $$e^{-\int_0^tr_sds}P(t,T)=\mathbb{E}[e^{-\int_0^Tr_sds}|\mathcal{F_t}],$$ where $\mathcal{F_t}$ is the filtration of the Brownian motion up to time $t$. Note that the expression on the right is a martingale by the tower property of expectations, so by the First Theorem of Asset pricing, there is no arbitrage. It immediately follows that $$P(t,T)=\mathbb{E}[e^{-\int_t^Tr_sds}],$$ which will further result in the yield curve specified as above (with $V_t$ term). Therefore neglecting the covariance term could result in arbitrage.

In fact, I was told that there is a proof showing it is not indeed arbitrage-free, but I am not going to go into that.