When Interest Rate Derivatives Need a Stochastic Model
Summary
The document distinguishes using a bootstrapped interest rate curve from using a stochastic model of rate dynamics. A market curve built from instruments such as forward rate agreements and swaps is sufficient to price simpler contracts through no-arbitrage relationships. The answer compares this with equity forwards and futures, which can also be priced without a stochastic model.
For products with nonlinear payoffs, such as caps, floors, and swaptions, the answer says a stochastic model is needed to represent uncertainty in rates. Hull–White is given as an example of a short-rate model. The response notes that such models can also price simpler contracts, though that is generally unnecessary when curve-based pricing suffices. It further distinguishes HJM from a short-rate model, describing it as a model of forward-rate dynamics. The explanation is conceptual and does not specify calibration, model assumptions, or a particular pricing procedure.
Key ideas
- A bootstrapped yield curve supports no-arbitrage pricing of simpler contracts such as FRAs and swaps.
- Nonlinear rate products generally require a stochastic model of interest rate behavior.
- Hull–White is an example of a short-rate model used for complex interest rate derivatives.
- HJM models forward-rate dynamics rather than the short rate.
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Full text
# What's the difference between short rate and the bootstrapped interest rate? # What's the difference between short rate and the bootstrapped interest rate? 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? ## Answer by emot (score 2) https://quant.stackexchange.com/a/66534 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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