The Relative Valuation of Caps and Swaptions: Theory and Empirical Evidence

we showed that caps are simple portfolios of options on individual forward rates. In contrast, swaptions can be viewed as options on portfolios of forward rates. To see this, recall that a swaption is an option on the forward swap rate in the Black (1976) model. Furthermore, forward swap rates can be expressed as nearly linear functions of individual forward rates, where the weights are related to the durations of the cash flows from the fixed leg of the swap. From this, it follows that the swaption can be thought of as an option on a linear combination or portfolio of forward rates. Merton (1973) presents a number of no arbitrage propositions including the well-known result that the value of an option on a portfolio must be less than or equal to that of a corresponding portfolio of options. This inequality is strict if the assets underlying the individual options are not perfectly correlated. Although the forward swap rate is only approximately linear in the individual forward rates, the key implication of the Merton result, namely that the relative value of a portfolio of options and an option on a portfolio is determined by the correlations between the underlying assets, is directly applicable to caps and swaptions.

while both caps and swaptions are quoted in terms of the Black (1976) model, it should be recognized that the Black model is being actually used in different ways in these markets. In particular, the caps market uses the forward short-term Libor rate as the underlying state variable in the Black model, while the swaptions market uses longer-term forward swap rates. Since forward swap rates are nearly linear in individual forward rates, the lognormality assumption implicit in the Black model cannot hold simultaneously for both individual forward rates and forward swap rates, since a linear combination of lognormal variates is not lognormal. This is the sense in which the two markets use different models; the inputs used in the Black model differ across the two markets. In addition, since the volatilities used in the Black model are for fundamentally different rates, direct comparisons between the quoted implied volatilities of caps and swaptions are invalid. This has important implications for the risk management of portfolios of caps and swaptions.