1
$\begingroup$

I am looking for a good definition for the GSR model for short rate. As mentioned in the page of https://rkapl123.github.io/QLAnnotatedSource/db/dd8/class_quant_lib_1_1_gsr.html, this model is based on below parameters -

  1. Term structure
  2. Volatility step dates
  3. Volatilities
  4. Reversion
  5. Something called T

Can you please explain me the significance of those parameters for the GSR model? It appears that this is one type of Gaussian 1-dimensional model for rate. What are other such models other than GSR?

Any pointer is really appreciated.

$\endgroup$
3
  • 1
    $\begingroup$ I believe GSR stands for General Short Rate Model, and it is a generalization of Vasicek model. $\endgroup$
    – noob2
    Sep 13 '20 at 21:51
  • $\begingroup$ Thanks. Any idea of the parameters like Volatility step dates etc. $\endgroup$
    – Bogaso
    Sep 13 '20 at 22:02
  • $\begingroup$ In order to better fit the initial term structure, GSR assumes that volatility is time varying and will swtich to different values at some times in the future. I assume these are the "volatility step dates". That's all I know. $\endgroup$
    – noob2
    Sep 13 '20 at 22:58
3
$\begingroup$

GSR stands for Gaussian Short Rate model. It describes the short rate $r(t)$ dynamics under the Risk Neutral measure as:

$$ dr(t) = \kappa(t) \cdot (\theta(t) - r(t)) \cdot dt + \sigma(t) \cdot dW(t). $$

Please, note that this document describes the QuantLib implementation, which is also described in the Andersen and Piterbarg book: Interest Rate Modeling. I would recommend reading this book.

Let me extend my answer to be more helpful.

The GSR model is actually a sub-class of the Affine Short Rate models. These models describe the short rate dynamics using the following SDE:

$$ dr(t) = \kappa(t) \cdot (\theta(t) - r(t)) \cdot dt + \sigma(t) \cdot \sqrt{\alpha(t) + \beta(t) \cdot r(t)} \cdot dW(t). $$

Now you can see that the Vasicek, Hull-White, Cox–Ingersoll–Ross (CIR), GSR and many other models are just simplifications of Affine short rate models.

Until here I have only talked about one factor models. The theory can be extended to multi factor short rate models, where the dynamics of $N$ factors $x(t)$ are specified and then the short rate is given by a linear combination of those factors.

There is a lot to discuss about these subjects, I am just being as concise as posible. Please, let me know if there is anything else I can do to help you.

$\endgroup$
4
  • $\begingroup$ Thanks. The first equation explains the Volatilities and Volatility step dates. Also the Reversion parameter. But how to pass the $\kappa(t)$ in QuantLib implementation? Also, what is the $T$ in QuantLib implementation? $\endgroup$
    – Bogaso
    Sep 14 '20 at 7:13
  • $\begingroup$ Hi! Have you seen the document that I have referenced? I was looking at it and what they did was to perform a change of variable, from $r(t)$ to $x(t)$. Also, they fitted this model to the current term structure of zero coupon bonds $P(0, T)$. Under this circumstances, the GSR model has an analytical expression for $\theta(t)$. Then, you only need $\kappa(t)$ and $\sigma(t)$, which I believe are the parameters required in the input. Finally, $T$ is more difficult to guess. But, if I have to guess, it is the maximum maturity, but I am not sure. I recommend you section 10.1.2.2. Andersen book. $\endgroup$
    – rvignolo
    Sep 14 '20 at 13:08
  • $\begingroup$ That's really helpful. So, the Reversion parameter in the QuantLib's model corresponds to the $\kappa(t)$ , which is the speed of the reversion? Also what do you mean by the Maximum maturity? Maturity of what? $\endgroup$
    – Bogaso
    Sep 14 '20 at 13:22
  • $\begingroup$ Exactly, I think it is the mean reversion $\kappa(t)$ instead of speed reversion $\theta(t)$. I could be wrong but it seems reasonable. On the other hand, I would have to check the $T$ parameter. I think it is the maximum maturity for the zero coupon bond $P(t, T)$. A short rate model allows to compute these objects by solving a system of ODEs with terminal conditions at time $T$. I think this is the parameter $T$, but I am not sure because I don't know what QuantLib does for obtaining the $P(t, T)$ object. I would have to inspect the source code $\endgroup$
    – rvignolo
    Sep 14 '20 at 13:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.