Affine term structure model: Difference between revisions

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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 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 ==