凸状調整、タイミング調整、クォント・オプション

最近ハル教授の利子とかわせ調整計算式を見直しといる。 Convexity adjustments are used with interest rate derivatives to express forward yields in terms of the forward rate at time 0, where the known forward rate assumes cash flows at time 2 would be based on rates the rate observed at time 1 and we want the rate at time 1 to determine the CF at time 1. Timing adjustments are the opposite, where we know the rate observed at time 1 for cash flows which would be paid at time 1 and want to move the cash flow to time 2. The timing adjustment can be expressed as a change of numeraire from P(t,T1) to P(t,T2) and uses the canonical change of numerator offset to the expected growth rate of p*sigmaF*sigmaG. Here sigmaF is the volatility of the interest rate, sigmaG is the volatility of the forward bond price (the numeraire ratio), and p is the correlation between the rate and the forward bond price (1 often used). The numeraire is the quantity that the derivative'so price is forward risk neutral with respect to. In this case, the initial numeraire us P(t,T1). The numeraire's volatility is the market price of risk of the derivative. An example of a timing adjustment is a derivative paying at time T based on the index value at T-1. The quanto adjustment is similar to timing, except that we are moving from risk neutrality with respect to a bond in currency x to RN with respect to a bond in currency y.
Hull's derivation of the convexity adjustment in terms of the timing adjustment is unclear, as it uses an m=1/tau assumption where tau=T2-T1, while also stating that G(y) =1/(1+y*tau) to relate price and yield of a bond (this is not the standard compounding formula).