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You can calculate the value of an option free bond or swap by using the spot curve and discounting cashflows accordingly. Alternatively, apparently you can use a single-factor short rate model in a binomial tree as you would for options.

Can someone explain why we refer to using short-rate models for pricing option-free instruments? For a swap, for example, I've seen the fixed side get valued by discounting cashflows, and the floating side estimated using the current curve and then discounted, to arrive at a value.

Is it that short-rate models allow you to add flavor to how the rates may move, and therefore assume that you're not stuck with symmetric volatility around each point on the curve? Is the short-rate approach superior to the estimate and discount cashflows with the current, static term structure?

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  • $\begingroup$ @ user1443 : I don't understand your example, where exactly do you use a model in this example ? Otherwise for convexity sensitive instruments (for example Libor futures) you might need a model to calculate a convextiy adjustment but there is no options involved in the product itself. $\endgroup$
    – TheBridge
    Commented Dec 2, 2011 at 10:28
  • $\begingroup$ Well, you certainly want to do it when calibrating the model. $\endgroup$
    – Brian B
    Commented Dec 2, 2011 at 14:44
  • $\begingroup$ Sorry if it wasn't clear. in this example, you use something like the hull-white model, and lay out a tree of possible rates, and discount probability weighted values back to the root node. I know this is useful for options because on the extremes of the tree, as the value of the instrument gets extreme, you may exercise optionality and adjust in a way that you can't with discounting cashflows with a SINGLE static curve. Why then do people take the tree approach with a short rate model also when there's no optionality? $\endgroup$
    – user1443
    Commented Dec 2, 2011 at 17:00

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A condition for correct calibration of the short rate model is that it exactly reproduce the present values of fixed (option-free) cashflows - that is, that it give the same answer as ordinary discounting at the spot rate. If it doesn't, you've done something wrong - sort of like using a model that violates put-call parity. (Actually, it's exactly like that.)

Ordinarily you wouldn't use a stochastic model to value an option-free security, if you knew ahead of time that it was really option-free. But if you have an option model lying around (e.g., a Hull-White implementation) and are pricing a general security, you might choose to use it and not worry about whether it happens to actually be option-free.

For example, you can show that a pure floater where the reference rate has the periodicity of the payments (e.g., 3m libor, pay in arrears) can be valued without using a stochastic model (I'm oversimplifying slightly). But if you want to value a quarterly-pay security where the interest rate references, say, 10 year CMS, you do need a stochastic model. So it's simpler just to use it and not worry.

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