  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.
