Why Interest Rate Options Use Forward Rates in Black Models

Summary

The document explains why Black-style models for interest rate options commonly use forward rates, rather than applying the Black–Scholes stock framework to a spot starting swap rate. The key practical issue is that a forward rate cannot generally be derived from one observed spot swap rate alone: it depends on multiple points along the yield curve. For example, a forward swap rate depends on rates associated with both the start and end of its forward period.

The answer also gives a modeling rationale. A diffusion model is convenient when its underlying is a martingale under an appropriate pricing measure. Stock prices are martingales under the money market measure, while a forward rate is a martingale under the forward annuity measure associated with the relevant swap period. This makes the forward rate a natural modeled variable for swaptions. The explanation is conceptual and brief; it does not derive pricing formulas, discuss calibration, or compare model performance across market conditions.

Key ideas

• A spot swap rate alone generally does not determine the forward swap rate needed for an interest rate option.
• The relevant forward rate depends on multiple parts of the interest rate curve.
• Black-style interest rate models use forward rates because they can be modeled as martingales under a suitable forward annuity measure.
• The document offers a modeling rationale but no formula derivation or empirical comparison.

Tags

# Black-Scholes vs Black equation


# Black-Scholes vs Black equation












Why is Black used for interest rate options pricing instead of Black-Scholes? Why are we more interested in Future rates instead of Spot rates when it comes to interest rate options? Basically, why can't we treat interest rates like stocks when pricing swaptions etc.?

## Answer by dm63 (score 7, accepted)

https://quant.stackexchange.com/a/37314

It's the forward rate which is fundamental to pricing for both stocks and interest rates. In the case of interest rates (unlike stocks) , it's difficult to compute the forward rate given the spot rate. Eg knowing the 10yr swap rate does not allow you to calculate the 1yr-10yr forward rate. The latter depends on the 11yr and 1yr parts of the curve for example. Hence in rates models the forward rate is used directly.

A point about modeling: in order to use the Brownian diffusion model, we need the underlying to be a martingale in some measure. For stocks , the stock price is s martingale in the money market measure. For interest rates, the forward rate is a martingale in the forward annuity measure (i.e. The value of a 1bp annuity for the forward period). So, modeling with the forward rate is 'nice'. As far as I know you cannot do that with the spot starting swap rate.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.