Theory of finance

Problem 1 The original Vasicek model was dr(t) == (0 - ar(t) )dt + odw (t), where W is a Brownian motion, and 0, a and r are constants. Hull and White extended it to dr(t) = (0(t) - ar(t))di + adW(t), where now O(t) is a deterministic function. a) Can you give a reason why Hull and White made this extension? The Black-Karasinski model for the short term interest rate is given as dr (t) = (0(t) + - - alogr(t))r(t)at + or(t)dW(t). b) Let Y(t) = logr(t). What is the dynamics of Y (t)