Interest rate model calibration using semidefinite Programming

It is shown 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 centred 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.