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This thing confused me for a long time, since we can a have a curve (e.g. LIBOR 3M) bootstrapped from the market quotes of instruments (e.g. FRA, SWAP), and we can get the spot rates and also the forward rates for any time and any length, why we need the short rate model? (e.g. hull-white, HJM, etc).

What's the main difference between these two?

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I would like to make an analogy with equities. To price simple contacts like equity forwards or futures, you don't need a model, you use simple no arbitrage arguments. To price options and other more complex derivatives with non linear payoff you need a stochastic model. It is worth mentioning that when you use a stochastic model to price simple contract like forwards, you get the same price as with simple no arbitrage arguments. Therefore with such a model you can price virtually anything.

For interest rate derivatives, it's the same story. You only need the curve to price simple contracts like FRAs and SWAPs, but to price more complex derivatives like caps, floors, swaptions etc. you need a stochastic model like Hull-White. With stochastic model you can of course price SWAPs or FRAs, but no one does it because it is pointless.

Notice that HJM is not a short-rate model as it captures the dynamic of forwards rates and this is instantenous forward rate model.

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  • $\begingroup$ Thanks! Then what's the difference between using Black76 and Hull-White to price a swaption? If they will give the same result, why use one over the other? $\endgroup$
    – Parting
    Aug 20, 2021 at 20:31
  • $\begingroup$ This is a very good question! Remember that models does not only give price but also hedging deltas. Each model has its own assumptions, in Hull White model you have constant parameters that unfortunately change after recalibration each day (which should not!), moreover within short-rate models you assume that you can hedge any derivative of whatever maturity (say 5 years) with any bond/discount factor (say 10 years) which is not possible in practice because tenors are not perfectly correlated. Each model should be compard in terms of hedging effectiveness. You can google papers about that. $\endgroup$
    – emot
    Aug 20, 2021 at 21:43

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