Affine term structure model: Difference between revisions

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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|lastlast1=Duffie|firstfirst1=Darrell|last2=Kan|first2=Rui|date=1996|title=A Yield-Factor Model of Interest Rates|url=https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9965.1996.tb00123.x|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 ==

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== 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|lastlast1=Christensen|firstfirst1=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|lastlast1=Christensen|firstfirst1=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|lastlast1=Christensen|firstfirst1=Jens H. E.|last2=Krogstrup|first2=Signe|date=2019-01-01|title=Transmission of Quantitative Easing: The Role of Central Bank Reserves|url=httpshttp://academicwww.oupfrbsf.comorg/ejeconomic-research/articlefiles/129/617/249/5250963wp2014-18.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|lastlast1=Nelson|firstfirst1=Charles R.|last2=Siegel|first2=Andrew F.|date=1987|title=Parsimonious Modeling of Yield Curves|url=http://www.jstor.org/stable/2352957|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>)

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=== 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 conditionconditional expecationexpectation 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 ==