Solving the CIR Bond Pricing Riccati Equation

Summary

The document sets up the Cox–Ingersoll–Ross short-rate model, with mean-reverting drift and volatility proportional to the square root of the rate. It applies an exponential-affine form to bond value and gives the resulting differential equation for the coefficient multiplying the short rate. The goal is to solve that equation and recover the stated closed-form expression, which depends on the time to maturity and the model parameters.

The text does not show the derivation or explain a solution method; it is a question asking for a starting point. It therefore provides no worked steps, numerical examples, or evidence validating the formula. A reader can identify the Riccati-type ODE and the target expression, but would need an additional explanation to learn how to transform or solve it. The discussion is limited to this bond-pricing coefficient and does not cover the other affine coefficient, calibration, or model assumptions.

Key ideas

• The CIR model uses square-root proportionality in its short-rate volatility.
• An exponential-affine bond value reduces pricing to ordinary differential equations for its coefficients.
• The coefficient multiplying the short rate satisfies a nonlinear Riccati-type equation.
• The document states a closed-form target for that coefficient but does not derive it.

Tags

# Derivation of CIR interest rate model


# Derivation of CIR interest rate model












I am trying to understand the derivation of the Cox-Ingersoll-Ross interest rate model. This has a stochastic differential equation of the form

$$dr=(\eta-\gamma r)dt + \sqrt{\alpha r} \space dX$$

With an affine solution of the form

$$V(r,t;T)=\exp\left[A(t) - rB(t)\right]$$

Putting this into the bond pricing equation and solving for $A(t;T)$ and $B(t;T)$ we arrive at a linear ODE for $B(t;T)$ in the form

$$\frac{d B(t;T)}{dt} = \frac{1}{2}\alpha (B(t;T))^2 + \gamma B(t;T) - 1$$

I need to solve this ODE to get the final solution for $B(t;T)$ in the form

$$B(t;T) = \frac{2\left(e^{\psi_1(T-t)}-1\right)}{(\gamma + \psi_1)(e^{\psi_1(T-t)}-1) + 2\psi_1}$$

Where

$$\psi_1=\sqrt{\gamma^2 + 2\alpha}$$

I can't think where to start solving this ODE. Could someone please give me a clue?

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