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.?
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.
1$\begingroup$ but isn't that a contradicting point? we're not calculating forward or spot rates, we're just using them to price options. Using the fact that it's hard to compute the forward rate, why would we use forward rates then? What I am asking is, why are we assuming forward rates brownian motion instead of spot rates doing so when it comes to interest rates? (Sorry if i misunderstood something) $\endgroup$ Dec 10, 2017 at 8:44
$\begingroup$ To clarify: if you have the whole swap curve, it is easy to compute the forward rate. It is difficult to write a formula for the forward rate in terms of the spot rate. $\endgroup$– dm63Dec 10, 2017 at 12:17
$\begingroup$ I added a paragraph to address the second part of your comment. $\endgroup$– dm63Dec 10, 2017 at 12:27