Exercises 3.2 Ho-Lee equity hybrid model The purpose of this exercise is to price exotic options depending on a single stock and characterized by a large maturity date. As a consequence, the assumption of deterministic rates should be relaxed. In this problem, we present a simple toy model which achieves this goal and we focus on calibration issues. It is assumed that the stock follows a Merton model dS S

= rtdt + (t)dWt where (t) is a time-dependent volatility. The instantaneous short rate rt is assumed to be driven by drt = (t)dt + dZt with a constant volatility , a time-dependent drift (t) and dZt a Brownian motion correlated to dWt: dWtdZt = dt. This is the short-rate Ho-Lee model

[7]. The next questions deal with how to calibrate these model parameters

(
), and (
).

1. Prove that the rate can be written as rt = (t) + xt with (t) a deterministic function that you will specify and xt a process following dxt = dZt 2. Prove that xt is a Brownian process and compute its mean and its variance at the time t.

3. By using the results of the question above, prove that the value of a bond PtT expiring at T and quoted at t < T is PtT = A(t; T)e????B(t;T )xt with A(t; T) and B(t; T) two functions that you will specify. A model such that the bond value can be written in this way is called an ane model.

4. Deduce that we can choose the function (t) in order to calibrate the initial yield curve P0T ; 8T.
5. From the previous questions, deduce that the bond PtT follows the SDE dPtT PtT = rtdt + P (t)dZt with P (t) = ???? (T ???? t).
6. Prove that the (equity) forward fT t = St PtT in the forward measure PT follows the SDE dfT t fT t = (t)dWt ???? P (t)dZt Why is it obvious that the forward is a local martingale?
7. As the forward obeys a log-normal process, prove that the implied volatility with maturity T is 2 BS;t(T ???? t) = Z T t ????
(s)2 + P (s)2 ???? 2(s)P (s)


ds 8. Deduce the Merton volatility (s) in order to calibrate the at-the-money market implied volatility.