ABSTRACT: We show that, for the purpose of pricing swaptions, the swap rate and the corresponding forward rates
can be considered lognormal under a single martingale measure.
Swaptions can then be priced as options on a basket of lognormal
assets and an approximation formula is derived for such options.
This formula is centered around a Black-Scholes price with an
appropriate volatility, plus a correction term that can be
interpreted as the expected tracking error. The calibration
problem can then be solved very efficiently using semidefinite
programming.