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ptoolisの日記

2017-03-17

利率模型の違い

09:01

最近ライボー・マーケット・モデルとヒース・ジャロー・モートンフレームワーク(HJM)の差を考えています。ライボー・マーケット・モデルは市場に現れる先物の利率を算出する。HJMは瞬間的なフォワードレートの模型です。

分かりにくかった点はライボーとヒース・ジャロー・モートンの瞬間的な相等性です。

(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. 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

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