Nelson–Siegel Yield Curves and No-Arbitrage Adjustments
Summary
The document explains how a Gaussian short-rate model with a multivariate Ornstein–Uhlenbeck state process produces a yield curve. Its derivation includes a maturity-dependent covariance adjustment; omitting that term gives the familiar Nelson–Siegel curve form. The discussion connects this difference to the question of whether a dynamic Nelson–Siegel specification can represent bond prices without arbitrage.
The cited explanations distinguish the original parsimonious curve from arbitrage-free Nelson–Siegel models, describing later work that imposes consistency conditions and adds a yield adjustment. They also derive bond prices from conditional expectations of discounted future cash flows, with the discounted bond price forming a martingale under the stated assumptions. The document gives no detailed proof that every omitted adjustment creates a specific trade or arbitrage, and it notes that the adjustment may be empirically small. Its main lesson is about model consistency, not a practical trading strategy.
Key ideas
- A Gaussian short-rate model implies a yield adjustment involving the covariance of integrated rates.
- The standard Nelson–Siegel curve omits that maturity-dependent adjustment.
- Arbitrage-free Nelson–Siegel models require consistency restrictions on the state dynamics and yields.
- Bond prices can be expressed as conditional expectations of discounted future values under the model assumptions.
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# Why isn't the Nelson-Siegel model arbitrage-free?
# 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?
## Answer by Bob Jansen (score 8)
https://quant.stackexchange.com/a/3496
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).
## Answer by Tom Artiom Fiodorov (score 4)
https://quant.stackexchange.com/a/3494
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.