Affine model

A Bank of Japan working paper writes the short rate as an affine function of two factors $x_t$, $r_t = \rho + x_{1t} + x_{2t}$. It takes as the basic setup of the affine model that the factors follow Gaussian processes and that the market price of risk is also an affine function of the factors. In this setup, under the no-arbitrage condition, the $T$-month zero-coupon rate is an affine function of the factors, $y_{t,T}(x_t) = a_T + b_T' x_t$, where $a_T$ and $b_T$ are functions of the parameters and the remaining maturity¹.

From the yield, the zero-coupon bond price can be written as $P = e^{-T y}$ (for continuous compounding), so it takes the form of an exponential of a linear expression in the factors. The Hull-White model is one affine model, and the price of the zero-coupon bond maturing at $S$ can be written with the short rate $r$ at time $t$ as $A(t,S)\,e^{B(t,S)\,r}$².

Consider a hypothetical example in which there is no mean reversion, an extreme case ($\kappa = 0$ in Itkin et al.'s equations), with $t = 0$ and $S = 5$. Then $B(t,S) = t - S$, so $B = -5$. If the short rate rises by 1 percentage point (0.01), the price becomes $e^{-5 \times 0.01} = 0.951$ times its former value (a fall of about 4.9%), and if it falls by 1 percentage point, the price becomes $e^{5 \times 0.01} = 1.051$ times its former value (a rise of about 5.1%). If the speed of mean reversion $\kappa$ is a positive constant, then $B(t,S) = -(1 - e^{-\kappa(S-t)})/\kappa$, and the size of $B$ is smaller than $S - t$.

The same 2013 working paper reports that, using data for Japan, the United States and the United Kingdom up to March 2013, affine models that ignore the zero lower bound showed a strong tendency to overestimate the long-run level of the short rate³.

Notes

1. Hibiki Ichiue and Yoichi Ueno, "Estimating Term Premia at the Zero Bound: An Analysis of Japanese, US, and UK Yields" (Bank of Japan Working Paper 13-E-8, May 2013; pages and equations are from the Japanese version, Working Paper 13-J-6), section 2.1 (PDF pages 9 to 10), equations (1) to (4) and (7) to (9). The text simplifies some of the source's symbols. The source cites the analytical solution of the affine model as due to Duffie and Kan (1996). ↩

2. Andrey Itkin and Dmitry Muravey, "Semi-closed form prices of barrier options in the Hull-White model" (arXiv, version of 17 September 2020), page 3, equations (6) and (7). The source's symbol $\kappa$ corresponds to $a$ in this site's Hull-White model. ↩

3. The abstract of the same working paper (PDF page 2). ↩