Consider the simple convexity adjustment (Question 3 of Chapter 3) and apply it to the 10-year and 2-year CMS rates setting in 30 years’ time. (Suppose for this example that both swap rates are currently at 4%, lognormal volatility is 15% and that swap rates are annually compounded.) Thus, explain how a spread option will have value even if correlation is 100%.

Question 3,

For a cash-settled swaption, the cash-annuity is defined by  This is j ust some function of R_T. Do a second order Taylor-expansion of A_c(R_T) about Ro. Notice that under the T-forward measure, we get Et [A_c(R_T)] ≈ A_c (R₀). Further, assume that the swap rate follows a lognormal process, i.e. dR_t = σR_tdW_t. Hence, show that Et [R_T] ≈ R₀ −  gives the classical approximation for the CMS rate in the (skewless) lognormal world.

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Ac(RT) = 1 N 1 (1+ RT )ni n