ヒース・ジャロー・モートンの枠組み

ハル教授の派生証券の教科書にHJM モデルのフォワードレートの由来がわかったがショートレートの計算式が不明です。The confusing point is the coefficient of dz outside of the integral in the dr formula, (dv(tau,t,omega)/dt)(assign t to tau) is the expression,but it is unclear how this arose from taking d/dt of r(t). It would seem that this term is due either to the definition of a stochastic integral or of the derivative of a stochastic integral. やっぱり積分の微分の計算式の誤りによって混乱が発生した。The key bit of information is the formula for the derivative of the integral. d/dx(integral*1 = (integral(a to b)((df(x,t)/dx)dt) + f(b(x),t)(db(x)/dx)-f(a(x),t)(da(x)/dx). In the r(t) expression in Hull, a=t and b=0, so (da/dt)=1 and (db/dt)=0, resulting in the dr expression given in the text.

*1:a to b)(f(x,t)dt