Notes on Convexity and Quanto Adjustments for Interest Rates and Related Options

We collect simple and pragmatic exact formulae for the convexity adjustment of irregular interest rate cash flows as Libor-in-arrears or payments of a swap rate (CMS rate) at an irregular date. The results are compared with the results of an approximative approach available in the popular literature. For options on Libor-in-arrears or CMS rates like caps or binaries we derive an additional new convexity adjustment for the volatility to be used in a standard Black & Scholes model. We study the quality of the adjustments comparing the results of the approximative Black & Scholes formula with the results of an exact valuation formula. Further we investigate options to exchange interest rates which are possibly set at different dates or admit different tenors. We collect general quanto adjustments formulae for variable interest rates to be paid in foreign currency and derive valuation formulae for standard options on interest rates paid in foreign currency.