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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.

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    $\begingroup$ I believe GSR stands for General Short Rate Model, and it is a generalization of Vasicek model. $\endgroup$
    – nbbo2
    Sep 13, 2020 at 21:51
  • $\begingroup$ Thanks. Any idea of the parameters like Volatility step dates etc. $\endgroup$
    – Bogaso
    Sep 13, 2020 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. $\endgroup$
    – nbbo2
    Sep 13, 2020 at 22:58

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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.

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  • $\begingroup$ 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, 2020 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, 2020 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, 2020 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, 2020 at 13:35

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