Let be a probability space supporting a one dimensional Brownian motion W. (i) For a short-rate model having short-rate process r and risk-neutral measure Q, show that the time-t value of any...

Let

be a probability space supporting a one dimensional Brownian motion W.

(i) For a short-rate model having short-rate process r and risk-neutral measure Q, show that the time-t value of any attainable derivative with value

at maturity time T can be expressed as

Further, show that we can write

where F is a measure equivalent to Q on

which you should specify.

(ii) Ho–Lee model Suppose that under Q the short-rate process r satisfies the SDE

where θ_t
is a deterministic function of time and σ is a constant. Assuming that the model is calibrated to the initial discount curve, show that

Further, show that under F the short-rate process r satisfies

Where

is a standard Brownian motion under F.