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Affine Short-Rate Models with a Mean-Reverting State Variable

Article Quant Q&A · Author: Carl
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Summary

The document formulates a two-state stochastic differential equation model in which the short rate mean-reverts toward a second state variable, while that state variable mean-reverts toward a long-run level. It asks whether the resulting affine system admits closed-form solutions and whether its zero-coupon bond price has an exponential-affine representation under the risk-neutral measure.

The question also rewrites the drift in matrix form and asks whether the parameterization creates identification problems during calibration. It notes a comparison with the two-factor Vasicek model and specifies independent Brownian shocks, but supplies no derivation, pricing solution, calibration analysis, or empirical evidence. Consequently, the text is useful as a model setup and a set of questions for further work, rather than as a demonstrated pricing method. Any answer would need to establish the pricing equations and examine which parameters can be distinguished from available bond-price data.

Key ideas

  • The short rate is modeled as mean-reverting toward a second stochastic state variable.
  • The second state variable follows its own mean-reverting process toward a long-run level.
  • The text asks whether bond prices under the model have an exponential-affine form.
  • The matrix drift representation motivates a calibration-identification question that the document leaves unanswered.

Tags

Fixed incomeStatisticsDerivatives pricing
Full text
# Is there a closed form solution to the following system of SDEs?


# Is there a closed form solution to the following system of SDEs?












Suppose we have the system \begin{align} dr_t=\alpha_r(x_t-r_t)dt+\sigma_rdW_t^r\\ dx_t=\alpha_x(\bar{x}-x_t)dt+\sigma_xdW_t^x\\ \end{align}

As this system is affine, I believe there should be an easy way to derive the closed-form solutions? Or at least numerically using ODEs. I know there are closed-form solutions for the two-factor Vasicek model. However, I am new to continuous time processes and still learning, and this setting does not exactly conform to the two-factor Vasicek system (note for instance no instantaneous correlation). Suppose, you wish to find the price of a bond, with the instantaneous short-term rate $r_t$, is their an exponential affine solution for such system? E.g. in the form of

\begin{equation} P(0,T)=\mathbb{E}^{\mathbb{Q}}\left[e^{-\int_0^Tr_sds}\right]=e^{a_t+b_tr} \end{equation} Assuming that $P(T,T)=1$ and both the system and expectation are under the risk neutral measure $\mathbb{Q}$.

Following, @kurtG. 's comment, and having looked at his response at the other post, how would that exactly look like in matrix form? You'd have: \begin{equation} d \begin{pmatrix} r_t\\ x_t \end{pmatrix}= \left( \begin{pmatrix} 0\\ \alpha_x\bar{x} \end{pmatrix} - \begin{pmatrix} \alpha_r & -\alpha_r\\ 0 & \alpha_x \end{pmatrix} \begin{pmatrix} r_t\\ x_t \end{pmatrix} \right) dt + \begin{pmatrix} \sigma_r &0\\ 0&\sigma_x \end{pmatrix} d \begin{pmatrix} W_t^r\\ W_t^x \end{pmatrix} \end{equation}

In that representation, wouldn't you have identification issue when calibrating?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.