Why CIR Short-Rate Paths Do Not Directly Give Forward Tenor Yields
Summary
The document asks how to turn simulated short-rate paths from the Cox–Ingersoll–Ross model into yields at longer maturities, and whether compounding short rates along each path produces the same price as the closed-form CIR zero-coupon bond formula. The proposed method compounds successive annual short rates and treats the resulting geometric average as a tenor rate, then repeats this across simulated dates to study curve shapes.
The reply cautions that a short rate cannot generally be mapped to a longer-tenor yield by simple compounding. The relationship depends on model dynamics such as mean reversion, and the simulated term structure must also be consistent with the observed curve. A model’s bond pricing formula is based on risk-neutral valuation, while a pathwise compounded average is not automatically the matching zero-coupon yield. The note recommends using an appropriate mapping, such as a calibrated tree, but provides no implementation details or numerical comparison.
Key ideas
- A simulated short rate alone does not determine longer-tenor yields through simple geometric averaging.
- The relation between short rates and tenor yields depends on model dynamics, including mean reversion.
- A sound interest-rate model should be consistent with the current yield curve.
- CIR zero-coupon bond prices come from model valuation and need not match a naive pathwise compounding calculation.
- A calibrated mapping, such as an interest-rate tree, can connect simulated short rates to tenor rates.
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# Simulating the Term Structure of Interest Rates in the CIR model # Simulating the Term Structure of Interest Rates in the CIR model I have successfully implemented the CIR model of the short rate, and now want to use these short rate paths to construct distributions of various tenors - 2y, 3y, 5y, 10y for example - across the curve for a custom simulation project of mine. By no-arbitrage, I can, at each time t and on each path n, calculate the geometric average of the short rates, up to the tenor desired, which will give me that particular tenor. For example, if my short rate were to be 1-year and I wanted the 3-year rate, I would take (1+r_0)(1+r_1)(1+r_2) ^ (1/3) = 3y rate over all of the paths. I can move time forward and do the same calculation at time t+1, t+2 ... t+n. The average slope of the simulated yield curve will depend on the difference between the starting rate r_0 and the long-run interet rate, as defined in the simulation. Is this correct? Another question comes from this. Let us say that I want to price a 10-year zero coupon bond. Should the closed-form CIR bond pricing formula give the same result as the one we would get if we simply compound the short rates, as I did above, to get the zero-coupon 10y rate, and discounted the cash flow at that 10y rate? By no-arbitrage it should. Looking to see if my intuition with these things is correct. Thanks ## Answer by Arshdeep (score 1) https://quant.stackexchange.com/a/78886 A theoretically sound IR process always reflects today's yield curve. Compounding short rates does not sound reasonable, say today's short rate is 1% and the 1 month spot is 2%, you are way off already. You should look into trees. Simulate the short rate but map it correctly to the tenor you want to approximate. The relationship between a short rate and a tenor rate is not just a time multiplier as you suggest, but it is determined by mean reversion etc.
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