(i) Consider a Hull–White model in which the short-rate process r satisfies the SDE   under the risk-neutral measure Q, where θ is a deterministic function of time and σ and a are constants. Let   be...

(i) Consider a Hull–White model in which the short-rate process r satisfies the SDE
  under the risk-neutral measure Q, where θ is a deterministic function of time and σ and a are constants.

Let
  be a sequence of regularly spaced times and set
  Show that the forward LIBOR
 satisfies the SDE

i being a standard Brownian motion under the EMM Fⁱ
corresponding to taking D·_Si
as numeraire.

(ii) Let   denote the value at time t of a European put option on a discount bond having payoff

  at time T_i
(not the more usual payment time S_i).

Show that the value at time t of a cap having payment dates S₁,...,S_n, and strike K can be written in the form

and find a closed-form formula for
  for the Hull–White model described in (i) above.