(Based on Hull's Solutions Manual)
In the limiting case, the LMM formula converges to the HJM dF forward rate process, but the proof is non-intuitive.
Here, s(t) is the instantaneous absolute volatility of the forward rate, which (as L must be the percentage volatility of the forward rate) means lim(dk->0)FkLk = s(t)
Thus, the dz terms can be equated. The logic to equate the dt terms is somewhat more strained. 時間とともに動く因数のを導き出すのが簡単ではありません。The denominator within the sum of the LMM formula must be assumed to go to 1 as di goes to 0. then, the SIGMA factor becomes a summation of (di*Fi*Li) terms as di->0, which is equivalent to an integral over tau of s(tau). QED.
The next step is to determine how the above
is equivalent to
Here, v is the volatility of a zero-coupon bond price.
This requires expressing the forward rate as follows.
f(t,T1,T2) = (lnP(t,T1)-lnP(t,T2))/(T2-T1)
Taking the limit as T2-T1 -> 0, or for a rate holding from T to T+delta, we obtain the derivatives of v^2 and v.
Simplifying dv^2 yields
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