Convexity meets replication: hedging of swap derivatives and annuity options

Convexity correction arises when one computes the expected value of an interest rate index under a probability measure other than its own natural martingale measure. As a typical example, the natural martingale measure of the swap rate is the swap measure with annuity as the numeraire. However, the evaluation of the discounted expectation of the payoff in a constant maturity swap (CMS) derivative is performed under the forward measure corresponding to the payment date. In this paper, we propose an extension of Carr-Madan’s static replication approach by exploring the linkage between replication, convexity correction and numeraire change. We illustrate how the static replication of a CMS caplet by a portfolio of payer swaptions is related to convexity correction associated with the bond-annuity numeraire ratio. We also demonstrate the use of the extended static replication approach for hedging in-arrears clean index principal swaps and annuity options.