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{{expert-subject|1=finance and investment|date=December 2012|reason=Confirmation, details on the Affine Term Structure Model
An '''affine term structure model''' is a [[financial model]] that relates [[zero-coupon bond]] prices (i.e. the discount curve) to a [[spot rate]] model. It is particularly useful for
== Background ==
Start with a stochastic [[Short-rate model|short rate]] model <math>r(t)</math> with dynamics:▼
▲Start with a stochastic short rate model <math>r(t)</math> with dynamics
▲: <math>
dr(t)=\mu(t,r(t)) \, dt + \sigma(t,r(t)) \, dW(t)
</math>
and a risk-free zero-coupon bond maturing at time <math>T</math> with price <math>
where <math>A</math> and <math>B</math> are deterministic functions, then the short rate model is said to have an '''affine term structure'''. The yield of a bond with maturity <math>\tau</math>, denoted by <math>y(t,\tau)</math>, is given by:<math display="block">y(t,\tau) = -{1\over{\tau}}\log P(t,\tau)</math>▼
=== Feynman-Kac formula ===
▲where <math>A</math> and <math>B</math> are deterministic functions, then the short rate model is said to have an '''affine term structure'''.
For the moment, we have not yet figured out how to explicitly compute the bond's price; however, the bond price's definition implies a link to the [[Feynman–Kac formula|Feynman-Kac formula]], which suggests that the bond's price may be explicitly modeled by a [[partial differential equation]]. Assuming that the bond price is a function of <math>x\in\mathbb{R}^{n}</math> [[Latent variable|latent factors]] leads to the PDE:<math display="block">-{\partial P\over{\partial \tau}} + \sum_{i=1}^{n}\mu_{i}{\partial P\over{\partial x_{i}}} + {1\over{2}}\sum_{i,j=1}^{n} \Omega_{ij}{\partial^{2} P\over{\partial x_{i}\partial x_{j}}} - rP = 0, \quad P(0,x) = 1</math>where <math>\Omega</math> is the [[covariance matrix]] of the latent factors where the latent factors are driven by an Ito [[stochastic differential equation]] in the risk-neutral measure:<math display="block">dx = \mu^{\mathbb{Q}}dt + \Sigma dW^{\mathbb{Q}}, \quad \Omega = \Sigma\Sigma^{T}</math>Assume a solution for the bond price of the form:<math display="block">P(\tau,x) = \exp\left[A(\tau) + x^{T}B(\tau) \right], \quad A(0) = B_{i}(0) = 0</math>The derivatives of the bond price with respect to maturity and each latent factor are:<math display="block">\begin{aligned}
{\partial P\over{\partial \tau}} &= \left[ A'(\tau) + x^{T}B'(\tau)\right]P \\
{\partial P\over{\partial x_{i}}} &= B_{i}(\tau)P \\
{\partial^{2} P\over{\partial x_{i}\partial x_{j}}} &= B_{i}(\tau)B_{j}(\tau)P\\
\end{aligned}</math>With these derivatives, the PDE may be reduced to a series of ordinary differential equations:<math display="block">-\left[A'(\tau) + x^{T}B'(\tau) \right] + \sum_{i=1}^{n}\mu_{i}B_{i}(\tau) + {1\over{2}}\sum_{i,j=1}^{n} \Omega_{ij}B_{i}(\tau)B_{j}(\tau) - r = 0, \quad A(0) = B_{i}(0) = 0</math>To compute a closed-form solution requires additional specifications.
== Existence ==
Using [[Itô's lemma|Ito's formula]] we can determine the constraints on <math>\mu</math> and <math>\sigma</math> which will result in an affine term structure. Assuming the bond has an affine term structure and <math>
: <math>A_t(t,T)-(1+B_t(t,T))r-\mu(t,r)B(t,T)+\frac{1}{2}\sigma^2(t,r)B^2(t,T)=0</math>
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The boundary value
:
implies
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\end{align}
</math>
== Arbitrage-Free Nelson-Siegel ==
One approach to affine term structure modeling is to enforce an [[arbitrage-free]] condition on the proposed model. In a series of papers,<ref>{{Cite journal|last1=Christensen|first1=Jens H. E.|last2=Diebold|first2=Francis X.|last3=Rudebusch|first3=Glenn D.|date=2011-09-01|title=The affine arbitrage-free class of Nelson–Siegel term structure models|url=http://www.sciencedirect.com/science/article/pii/S0304407611000388|journal=Journal of Econometrics|series=Annals Issue on Forecasting|language=en|volume=164|issue=1|pages=4–20|doi=10.1016/j.jeconom.2011.02.011|issn=0304-4076|url-access=subscription}}</ref><ref>{{Cite journal|last1=Christensen|first1=Jens H. E.|last2=Rudebusch|first2=Glenn D.|date=2012-11-01|title=The Response of Interest Rates to US and UK Quantitative Easing|url=https://academic.oup.com/ej/article/122/564/F385/5079473|journal=The Economic Journal|language=en|volume=122|issue=564|pages=F385–F414|doi=10.1111/j.1468-0297.2012.02554.x|s2cid=153927550 |issn=0013-0133|url-access=subscription}}</ref><ref>{{Cite journal|last1=Christensen|first1=Jens H. E.|last2=Krogstrup|first2=Signe|date=2019-01-01|title=Transmission of Quantitative Easing: The Role of Central Bank Reserves|url=http://www.frbsf.org/economic-research/files/wp2014-18.pdf|journal=The Economic Journal|language=en|volume=129|issue=617|pages=249–272|doi=10.1111/ecoj.12600|s2cid=167553886 |issn=0013-0133}}</ref> a proposed dynamic yield curve model was developed using an arbitrage-free version of the famous Nelson-Siegel model,<ref>{{Cite journal|last1=Nelson|first1=Charles R.|last2=Siegel|first2=Andrew F.|date=1987|title=Parsimonious Modeling of Yield Curves|journal=The Journal of Business|volume=60|issue=4|pages=473–489|doi=10.1086/296409|jstor=2352957|issn=0021-9398}}</ref> which the authors label AFNS. To derive the AFNS model, the authors make several assumptions:
# There are three latent factors corresponding to the ''level'', ''slope'', and ''curvature'' of the [[yield curve]]
# The latent factors evolve according to multivariate [[Ornstein–Uhlenbeck process|Ornstein-Uhlenbeck processes]]. The particular specifications differ based on the measure being used:
##<math>dx = K^{\mathbb{P}}(\theta-x)dt + \Sigma dW^{\mathbb{P}}</math> (Real-world measure <math>\mathbb{P}</math>)
##<math>dx = -K^{\mathbb{Q}}xdt + \Sigma dW^{\mathbb{Q}}</math> (Risk-neutral measure <math>\mathbb{Q}</math>)
# The volatility matrix <math>\Sigma</math> is diagonal
# The short rate is a function of the level and slope (<math>r = x_{1} + x_{2}</math>)
From the assumed model of the zero-coupon bond price:<math display="block">P(\tau,x) = \exp\left[A(\tau) + x^{T}B(\tau) \right]</math>The yield at maturity <math>\tau</math> is given by:<math display="block">y(\tau) = -{A(\tau)\over{\tau}} - {x^{T}B(\tau)\over{\tau}}</math>And based on the listed assumptions, the set of ODEs that must be solved for a closed-form solution is given by:<math display="block">-\left[A'(\tau) + B'(\tau)^{T}x \right] - B(\tau)^{T}K^{\mathbb{Q}}x + {1\over{2}}B(\tau)^{T}\Omega B(\tau) - \rho^{T}x = 0, \quad A(0) = B_{i}(0) = 0</math>where <math>\rho = \begin{pmatrix} 1 & 1 & 0 \end{pmatrix}^{T}</math> and <math>\Omega</math> is a diagonal matrix with entries <math>\Omega_{ii} = \sigma_{i}^{2}</math>. Matching coefficients, we have the set of equations:<math display="block">\begin{aligned}
-B'(\tau) &= \left(K^{\mathbb{Q}}\right)^{T}B(\tau) + \rho, \quad B_{i}(0) = 0 \\
A'(\tau) &= {1\over{2}}B(\tau)^{T}\Omega B(\tau), \quad A(0) = 0
\end{aligned}</math>To find a tractable solution, the authors propose that <math>K^{\mathbb{Q}}</math> take the form:<math display="block">K^{\mathbb{Q}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & \lambda & -\lambda \\ 0 & 0 & \lambda \end{pmatrix}</math>Solving the set of coupled ODEs for the vector <math>B(\tau)</math>, and letting <math>\mathcal{B}(\tau) = -{1\over{\tau}}B(\tau)</math>, we find that:<math display="block">\mathcal{B}(\tau) = \begin{pmatrix} 1 & {1-e^{-\lambda\tau}\over{\lambda \tau}} & {1-e^{-\lambda\tau}\over{\lambda \tau}} - e^{-\lambda\tau} \end{pmatrix}^{T}</math>Then <math>x^{T}\mathcal{B}(\tau)</math> reproduces the standard Nelson-Siegel yield curve model. The solution for the yield adjustment factor <math>\mathcal{A}(\tau) = -{1\over{\tau}}A(\tau)</math> is more complicated, found in Appendix B of the 2007 paper, but is necessary to enforce the arbitrage-free condition.
=== Average expected short rate ===
One quantity of interest that may be derived from the AFNS model is the average expected short rate (AESR), which is defined as:<math display="block">\text{AESR} \equiv {1\over{\tau}}\int_{t}^{t+\tau}\mathbb{E}_{t}(r_{s})ds = y(\tau) - \text{TP}(\tau)</math>where <math>\mathbb{E}_{t}(r_{s})</math> is the [[conditional expectation]] of the short rate and <math>\text{TP}(\tau)</math> is the term premium associated with a bond of maturity <math>\tau</math>. To find the AESR, recall that the dynamics of the latent factors under the real-world measure <math>\mathbb{P}</math> are:<math display="block">dx = K^{\mathbb{P}}(\theta-x)dt + \Sigma dW^{\mathbb{P}}</math>The general solution of the multivariate Ornstein-Uhlenbeck process is:<math display="block">x_{t} = \theta + e^{-K^{\mathbb{P}}t}(x_{0}-\theta) + \int_{0}^{t} e^{-K^{\mathbb{P}}(t-t')}\Sigma dW^{\mathbb{P}}</math>Note that <math>e^{-K^{\mathbb{P}}t}</math> is the [[matrix exponential]]. From this solution, it is possible to explicitly compute the conditional expectation of the factors at time <math>t+\tau</math> as:<math display="block">\mathbb{E}_{t}(x_{t+\tau}) = \theta + e^{-K^{\mathbb{P}}\tau}(x_{t}-\theta)</math>Noting that <math>r_{t} = \rho^{T}x_{t}</math>, the general solution for the AESR may be found analytically:<math display="block">{1\over{\tau}}\int_{t}^{t+\tau}\mathbb{E}_{t}(r_{s})ds = \rho^{T}\left[ \theta + {1\over{\tau}}\left( K^{\mathbb{P}} \right)^{-1}\left(I - e^{-K^{\mathbb{P}}\tau}\right)(x_{t}-\theta) \right]</math>
== References ==
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