The price of a derivative can be found by taking the expected payoff divided by the numeraire price and multiplying it by today's numeraire price1. A numeraire can be a single security or a portfolio. A dynamic portfolio that is rebalanced without adding outside funds also works, as long as its value is always positive2.

In interest rate models, the choices include the account that accumulates the short rate, the bond maturing at TT (the forward measure), and a portfolio of bonds maturing on the payment dates of a swap (the swap measure)3. Changing the numeraire changes the formula for the drift of the same forward rate.

Notes

  1. Martin Haugh, "Market Models" (Columbia University, Term Structure Models: IEOR E4710, 2010), page 9, equation (28). ↩

  2. Haugh, "Market Models", page 9, footnote 15. ↩

  3. Haugh, "Market Models", pages 5 to 6, "Numeraire-EMM Pairs". ↩