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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

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  • $\begingroup$ Short rate is instantaneous, if you want to simulate the 1y rate directly, you won't be able to do much except just value instruments determined by exact multiples of 1y forwards. You also cannot calibrate vol of this rate as it is not the underlying of any option. $\endgroup$
    – Arshdeep
    Commented Apr 5 at 19:49
  • $\begingroup$ I'm just trying to simulate rates from which i may extract the rates of different tenors along the path at each time. I'm not trying to value an instrument directly; but instead I want to have an interest rate process which is theoretically sound. I'm using monthly time steps $\endgroup$
    – Wadstk
    Commented Apr 5 at 21:17

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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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  • $\begingroup$ If i were to select the mean reversion parameter, and the strength of mean reversion parameter, such that the average forward rates matched the market forward rates, I should have a well calibrated model, one which will price zero-coupon bonds correctly, I think $\endgroup$
    – Wadstk
    Commented Apr 5 at 22:01

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