(i) For each i = 1,...,n, consider a caplet with start date Ti, cash flow at time Si and strike K. According to Black’s formula, the value at time t = 0 of the ith caplet is given by     where   is a...

(i) For each i = 1,...,n, consider a caplet with start date Ti, cash flow at time Si and strike K. According to Black’s formula, the value at time t = 0 of the ith caplet is given by

where
  is a positive constant and

Suppose an arbitrage-free term structure model has been defined which is consistent with Black’s formula at time t = 0 for all strikes for each of these caplets. Find the distribution of
  under an EMM corresponding to the numeraire D·S_i.

(ii) In a one-factor LIBOR market model, working in the measure F corresponding to the numeraire

  satisfies an SDE of the form

for constants
  a one-dimensional Brownian motion,
  for all
 and, for

Show that this model is consistent with Black’s formula as given in (i) for each of the caplets.