Calibrating a CIR Short-Rate Model to Bond Prices
Summary
The document asks whether the time-dependent drift level in a Cox–Ingersoll–Ross short-rate model can be chosen to fit observed zero-coupon bond prices or forward rates, analogously to the Hull–White model. It introduces the CIR short-rate process, the exponential-affine bond pricing form, and the relation that obtains forward rates from bond prices.
One reply outlines a possible route: derive forward-rate dynamics from the bond price representation and match their drift to the no-arbitrage drift required by the Heath–Jarrow–Morton framework, then solve for the drift level. However, the derivation is not completed and no explicit formula is provided. Another reply says the problem must be handled numerically. The discussion therefore identifies relevant model relationships but leaves the calibration method and its conditions unresolved.
Key ideas
- CIR bond prices have an exponential-affine form in the current short rate.
- Forward rates can be derived from the maturity derivative of log bond prices.
- Matching forward-rate drift conditions to the HJM framework is proposed as a route to calibrating the CIR drift level.
- The document supplies no completed formula and reports that numerical solution may be necessary.
Tags
Full text
# Calibrating CIR to bond prices
# Calibrating CIR to bond prices
Consider the Hull-White model - $$dr_t = (\theta_t - kr_t)dt + \sigma_tdw_t$$ We can/have to calibrate $\theta_t$ to the current bond prices $P(0,t)$ and make it consistent with the HJM framework. For Hull-White model, there is an explicit formula for $\theta_t$ in terms of $\sigma_t$, bond prices, forward rates and its derivatives. Can we get a similar formula for CIR model - $$dr_t = (\theta_t - kr_t)dt + \sigma_t\sqrt{r_t}dw_t$$
## Answer by wombat22 (score 3)
https://quant.stackexchange.com/a/79884
#### CIR Model
\begin{equation} dr_t = (\theta_t - k r_t)dt + \sigma \sqrt{r_t} \, dW_t \end{equation}
#### Consistency with HJM Framework
#### CIR Model Calibration
For the CIR model, we need to express $\theta_t$ in terms of observable quantities (like bond prices $P(0,t)$ or forward rates $f(0,t)$ and model parameters.
- Bond Price in CIR Model: The price of a zero-coupon bond $(P(t,T)$ under the CIR model can be expressed as: \begin{equation} P(t,T) = A(t,T) e^{-B(t,T) r_t} \end{equation} where $A(t,T)$ and $B(t,T)$ are functions derived from solving the CIR bond pricing PDE.
- Forward Rate $f(t,T)$: Given the bond price formula, the forward rate can be obtained as: \begin{equation} f(t,T) = -\frac{\partial \ln P(t,T)}{\partial T} \end{equation}
#### Deriving $\theta_t$
Using the bond price expression, we can derive the forward rate dynamics in the CIR framework. The key steps are:
- Drift of (r_t): The drift term in the CIR model is $(\theta_t - k r_t)$.
#### Step-by-Step Derivation
- Compute the Forward Rate Dynamics in CIR: \begin{equation} df(t,T) = \left(\frac{\partial f(t,T)}{\partial t} + \sigma \sqrt{r_t} \frac{\partial f(t,T)}{\partial r_t}\right) dt + \sigma \sqrt{r_t} \frac{\partial f(t,T)}{\partial r_t} dW_t \end{equation}
- Solve for $\theta_t$: By ensuring the drift terms match, we can isolate $\theta_t$ and express it in terms of observable quantities.
## Answer by achirikhin (score 0)
https://quant.stackexchange.com/a/79534
https://en.wikipedia.org/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model
Section "Bond Pricing".
Has to be solved numerically; this is as far as you can get.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.