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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 deriving the [[yield curve]] – the process of determining spot rate model inputs from observable [[bond market]] data. The affine class of term structure models implies the convenient form that log bond prices are linear functions of the spot rate<ref>{{Cite journal|last1=Duffie|first1=Darrell|last2=Kan|first2=Rui|date=1996|title=A Yield-Factor Model of Interest Rates|journal=Mathematical Finance|language=en|volume=6|issue=4|pages=379–406|doi=10.1111/j.1467-9965.1996.tb00123.x|issn=1467-9965}}</ref> (and potentially additional state variables).
== Background ==▼
▲== Background ==
Start with a stochastic [[Short-rate model|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>
:<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 ===
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 \\
▲where <math>A</math> and <math>B</math> are deterministic functions, then the short rate model is said to have an '''affine term structure'''.
{\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>
The boundary value
:<math>
implies
: <math>
\begin{align}
A(T,T)&=0\\
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Next, assume that <math>\mu</math> and <math>\sigma^2</math> are affine in <math>r</math>:
: <math>
\begin{align}
\mu(t,r)&=\alpha(t)r+\beta(t)\\
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The differential equation then becomes
: <math>
A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)-\left[1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)\right]r=0
</math>
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Because this formula must hold for all <math>r</math>, <math>t</math>, <math>T</math>, the coefficient of <math>r</math> must equal zero.
: <math>
1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)=0
</math>
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Then the other term must vanish as well.
: <math>
A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)=0
</math>
Then, assuming <math>\mu</math> and <math>\sigma^2</math> are affine in <math>r</math>, the model has an affine term structure where <math>A</math> and <math>B</math> satisfy the system of equations:
: <math>\begin{align}
1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)&=0\\
B(T,T)&=0\\
Line 77 ⟶ 82:
=== Vasicek ===
The [[Vasicek model]] <math>dr=(b-ar)\,dt+\sigma \,dW</math> has an affine term structure where
: <math>
\begin{align}
p(t,T)&=e^{A(t,T)-B(t,T)r(
B(t,T)&=\frac{1}{a}\left(1-e^{-a(T-t)}\right)\\
A(t,T)&=\frac{(B(t,T)-T+t)(ab-\frac{1}{2}\sigma^2)}{a^2}-\frac{\sigma^2B^2(t,T)}{4a}
\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 ==
<references />
== Further reading ==
*{{cite book | author=Bjork, Tomas | title=Arbitrage Theory in Continuous Time, third edition| year=2009 | publisher = New York, NY: [[Oxford University Press]] | isbn = 978-0-19-957474-2}}
[[Category:Interest rates]]
[[Category:Financial models]]
[[Category:Fixed income analysis]]
[[Category:Stochastic models]]
[[Category:Short-rate models]]
[[Category:Mathematical and quantitative methods (economics)]]
[[Category:Mathematical finance]]
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