凸状調整

最近利子による派生証券の値を計算しながら悩んだ。なぜLIBORとスワップレートに関係がある物にconvexity adjustmentが必要でしょうか。It seems that the yield calculation varies with the relative timing of the rate observation and payoff. For a 3-month yield observed 6 months from now where the payoff occurs in 9 months, the 6-month forward 3-month yield can be used as-is with no convexity adjustment. However, if the payoff is in 6 months on the observation date, a convexity adjustment must be made. Convexity adjustments when swap rates are involved seem less straightforward, as swap payments are made in arrears, it seems. In fact, nomenclature is inconsistent regarding plain vanilla interest rate swaps, it seems, and in fact payments need not be made in arrears. In addition, the forward swap rate equals the expected swap rate only if the timing of the derivative's playoffs mirror those of a swaption.
もう一つの面白い例があります。 If a derivative has a one-time payoff in 3 years based on the 3 year swap rate given a forward yield of y0, a function G mapping yield to bond price, and a forward yield volatility of sigmay, the convexity adjustment at year 3, based on the timing of the cash flow and the curvature of the swap curve, is

• (1/2)sigmay^2*y0^2*G"(y)/G'(y).