Affine term structure model

Start with a stochastic short rate model r(t) with dynamics: : dr(t)=\mu(t,r(t)) \, dt + \sigma(t,r(t)) \, dW(t) and a risk-free zero-coupon bond maturing at time T with price P(t,T) at time t. The price of a zero-coupon bond is given by:P(t,T) = \mathbb{E}^{\mathbb{Q}}\left\{ \exp\left[ -\int_{t}^{T}r(t')dt' \right] \right\}where T=t+\tau, with \tau being is the bond's maturity. The expectation is taken with respect to the risk-neutral probability measure \mathbb{Q}. If the bond's price has the form: :P(t,T)=e^{A(t,T)-rB(t,T)} where A and B are deterministic functions, then the short rate model is said to have an affine term structure. The yield of a bond with maturity \tau, denoted by y(t,\tau), is given by:y(t,\tau) = -{1\over{\tau}}\log P(t,\tau) 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, 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 x\in\mathbb{R}^{n} latent factors leads to the PDE:-{\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) = 1where \Omega 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:dx = \mu^{\mathbb{Q}}dt + \Sigma dW^{\mathbb{Q}}, \quad \Omega = \Sigma\Sigma^{T}Assume a solution for the bond price of the form:P(\tau,x) = \exp\left[A(\tau) + x^{T}B(\tau) \right], \quad A(0) = B_{i}(0) = 0The derivatives of the bond price with respect to maturity and each latent factor are:\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}With these derivatives, the PDE may be reduced to a series of ordinary differential equations:-\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) = 0To compute a closed-form solution requires additional specifications. == Existence ==

Using Ito's formula we can determine the constraints on \mu and \sigma which will result in an affine term structure. Assuming the bond has an affine term structure and P satisfies the term structure equation, we get: : 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 The boundary value :P(T,T)=1 implies : \begin{align} A(T,T)&=0\\ B(T,T)&=0 \end{align} Next, assume that \mu and \sigma^2 are affine in r: : \begin{align} \mu(t,r)&=\alpha(t)r+\beta(t)\\ \sigma(t,r)&=\sqrt{\gamma(t)r+\delta(t)} \end{align} The differential equation then becomes : 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 Because this formula must hold for all r, t, T, the coefficient of r must equal zero. : 1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)=0 Then the other term must vanish as well. : A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)=0 Then, assuming \mu and \sigma^2 are affine in r, the model has an affine term structure where A and B satisfy the system of equations: : \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\\ A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)&=0\\ A(T,T)&=0 \end{align} == Models with ATS ==

Vasicek The Vasicek model dr=(b-ar)\,dt+\sigma \,dW has an affine term structure where : \begin{align} p(t,T)&=e^{A(t,T)-B(t,T)r(t)}\\ 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} == 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, a proposed dynamic yield curve model was developed using an arbitrage-free version of the famous Nelson-Siegel model, 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 processes. The particular specifications differ based on the measure being used: • dx = K^{\mathbb{P}}(\theta-x)dt + \Sigma dW^{\mathbb{P}} (Real-world measure \mathbb{P}) • dx = -K^{\mathbb{Q}}xdt + \Sigma dW^{\mathbb{Q}} (Risk-neutral measure \mathbb{Q}) • The volatility matrix \Sigma is diagonal • The short rate is a function of the level and slope (r = x_{1} + x_{2}) From the assumed model of the zero-coupon bond price:P(\tau,x) = \exp\left[A(\tau) + x^{T}B(\tau) \right]The yield at maturity \tau is given by:y(\tau) = -{A(\tau)\over{\tau}} - {x^{T}B(\tau)\over{\tau}}And based on the listed assumptions, the set of ODEs that must be solved for a closed-form solution is given by:-\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) = 0where \rho = \begin{pmatrix} 1 & 1 & 0 \end{pmatrix}^{T} and \Omega is a diagonal matrix with entries \Omega_{ii} = \sigma_{i}^{2}. Matching coefficients, we have the set of equations:\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}To find a tractable solution, the authors propose that K^{\mathbb{Q}} take the form:K^{\mathbb{Q}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & \lambda & -\lambda \\ 0 & 0 & \lambda \end{pmatrix}Solving the set of coupled ODEs for the vector B(\tau), and letting \mathcal{B}(\tau) = -{1\over{\tau}}B(\tau), we find that:\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}Then x^{T}\mathcal{B}(\tau) reproduces the standard Nelson-Siegel yield curve model. The solution for the yield adjustment factor \mathcal{A}(\tau) = -{1\over{\tau}}A(\tau) 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:\text{AESR} \equiv {1\over{\tau}}\int_{t}^{t+\tau}\mathbb{E}_{t}(r_{s})ds = y(\tau) - \text{TP}(\tau)where \mathbb{E}_{t}(r_{s}) is the conditional expectation of the short rate and \text{TP}(\tau) is the term premium associated with a bond of maturity \tau. To find the AESR, recall that the dynamics of the latent factors under the real-world measure \mathbb{P} are:dx = K^{\mathbb{P}}(\theta-x)dt + \Sigma dW^{\mathbb{P}}The general solution of the multivariate Ornstein-Uhlenbeck process is:x_{t} = \theta + e^{-K^{\mathbb{P}}t}(x_{0}-\theta) + \int_{0}^{t} e^{-K^{\mathbb{P}}(t-t')}\Sigma dW^{\mathbb{P}}Note that e^{-K^{\mathbb{P}}t} is the matrix exponential. From this solution, it is possible to explicitly compute the conditional expectation of the factors at time t+\tau as:\mathbb{E}_{t}(x_{t+\tau}) = \theta + e^{-K^{\mathbb{P}}\tau}(x_{t}-\theta)Noting that r_{t} = \rho^{T}x_{t}, the general solution for the AESR may be found analytically:{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] == References ==