Interest Rate Models
CIR++ Interest Rate Simulator
CIR++ extends the Cox-Ingersoll-Ross short-rate model with a deterministic shift to fit the initial yield curve, making it practical for internal rate simulation and pricing workflows in QuantModels.ai.
Model Overview
A shifted short-rate framework for fixed-income analytics
CIR++ extends the Cox-Ingersoll-Ross short-rate model with a deterministic shift to fit the initial yield curve.
Mathematical Intuition
r(t) = x(t) + phi
dx(t) = kappa(theta - x(t))dt + sigma sqrt(x(t)) dW(t)
The deterministic shift phi lets the total short rate align with the market curve at inception, while x(t) retains the mean-reverting non-negative structure of CIR.
CIR++ Inputs
Terminal Output
Avg terminal short rate
0.0456
Simulated short-rate path table
| Step | Time | Avg x(t) | Avg r(t) |
|---|---|---|---|
| 0 | 0.0000 | 0.0300 | 0.0400 |
| 1 | 0.0833 | 0.0306 | 0.0406 |
| 2 | 0.1667 | 0.0311 | 0.0411 |
| 3 | 0.2500 | 0.0315 | 0.0415 |
| 4 | 0.3333 | 0.0321 | 0.0421 |
| 5 | 0.4167 | 0.0323 | 0.0423 |
| 6 | 0.5000 | 0.0330 | 0.0430 |
| 7 | 0.5833 | 0.0335 | 0.0435 |
| 8 | 0.6667 | 0.0339 | 0.0439 |
| 9 | 0.7500 | 0.0344 | 0.0444 |
| 10 | 0.8333 | 0.0348 | 0.0448 |
| 11 | 0.9167 | 0.0351 | 0.0451 |
| 12 | 1.0000 | 0.0356 | 0.0456 |
Use Cases
- interest-rate simulation
- yield curve fitting
- zero-coupon bond pricing
- interest-rate derivatives
- risk scenario generation
Simulation Workflow
- Set the short-rate state, mean-reversion, volatility, deterministic shift, and horizon assumptions.
- Simulate the CIR state process x(t) using a non-negative short-rate dynamic.
- Form the full short rate through r(t) = x(t) + phi.
- Review the average terminal rate and the simulated short-rate path table.
Pricing Applications
- Bond discounting and yield-curve aware valuation
- Interest-rate derivatives and structured rate scenarios
- Stress testing for treasury, ALM, and fixed-income risk workflows