債券値の微分方程式

最近再びヒース・ジャロー・モートン枠組を考えています。The Heath-Jarrow-Morton framework first formulates a diffusion process for the zero-coupon bond price increment dP and then expresses a process for the forward rate increment df. The Hull solutions manual reduces HJM to Ho-Lee or Hull-White by assuming a specific value for s, the volatility of P in HJM, and then using Ito's Lemma on the Ho-Lee or Hull-White bond price formulas and short rate processes to prove that the bond price volatility is the same in Ho-Lee or Hull White as that of HJM.
しかし予測利益率が最初に合わなかった。However, while the drift in HJM for the bond price is r, which is the expected return on a zero-coupon bond. using Ito's Lemma on the Ho-Lee or Hull-White P formulas, at first glance, yielded a different output for the drift. This may be because the author considered the A and B terms to be constant in time. I also treated r as constant in time, thus ignoring any dr/dr expression (this would be a theta expression in the Ho-Lee case?). It is unclear whether A should be differentiated with respect to t or with respect to (T-t), and in either case, thee derivative is confusing. It would seem to recursively include dP/dt, requiring some fairly involved algebra to facto the dP/dt on both sides of the equation.
The author encountered some difficulty in eliminating some terms in computing the drift due to confusion over the value of d/dt for the term: (T-t)dln(P)/dt. It is unclear whether a term to cancel the theta term in the Ito expression arises from this derivative. In addition the dr/dt terms in the Ito expression seem to be of the same sign and thus do not cancel.
In fact, upon making the simplifying assumption that the average rate over the span [0,t] is the same as the average over [t,T] and that dr/dt is also the same for the average r's over these two intervals, the terms cancel such that, if the further assumption that d^2r/dt^2 is zero holds, dP =Pr*dt+sigmaP*P*dz
Thus, the drift of the return is r, where r is a function of time.
To obtain the volatility of the bond price under HJM, Ito's Lemma is used on the dP process to derive dln(P), and the forward rate f is derived from this, as df=(dlnP(t,T1)-dlnP(t,T2))/(T2-T1)
This expression leads to a derivation of the volatility of the forward rate as dv/dt, or the time derivative of the volatility of the bond price. Integrating dv/dt with respect to t and substituting T into this expression to solve for the constant (given v(T,T) is zero) will enable a solution for v(t,T). This HJM v can be compared to the Ho-Lee or Hull-White v derived from Ito's Lemma.