# ptoolis¤ÎÆüµ­

## 2017-03-17

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09:01

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(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.

The LIBOR formula is:

dFk = FkLkSIGMA(i=m(t) to k)*1dt + Lk*Fk*dz

Here, di is the time period (t(i+1)-ti). Firstly, the point is to demonstrate that the above is equivalent to the abstracted HJM formula dF(t) = s(t)*INTEGRAL(m(t) to k)(s(tau)dtau)*dt + s(t)dz

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. »þ´Ö¤È¤È¤â¤ËÆ°¤¯°ø¿ô¤Î¤òÆ³¤­½Ð¤¹¤Î¤¬´ÊÃ±¤Ç¤Ï¤¢¤ê¤Þ¤»¤ó¡£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

dF(t) = s(t)*INTEGRAL(m(t) to k)(s(tau)dtau)*dt + s(t)dz

is equivalent to

dF(t) = (dv(t,T)/dT)*v(t,T)*dt + (dv(t,T)/dT)dz

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)

Here, the rate is seen at time t and holds between T1 and T2. P is the price of the zero-coupon bond with maturities T1 or T2.

One can use Ito's Lemma to solve for dlnP(t,T1) and dln(t,T2), and the value for dlnf when df is known is a well-known case.

df(t,T1,T2) = ((v^2(t,T2)-v^2(t,T1)/(2(T2-T1)))dt + *2/(T2-T1))dz

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.

df(t,T) = (1/2)(dv^2(t,T)/dT)dt-(dv(t,T)/dT)dz

Simplifying dv^2 yields

df(t,T) = v(t,T)(dv(t,T)/dT)dt -(dv(t,T)/dT)dz QED

*1¡§di*Fi*Li)/(1+di*Fi

*2¡§v(t,T1)-v(t,T2