Hi all, I am looking at chapter 27 in Hull (page 642) about exchange options. At first I found the earlier part of this section difficult to understand but I think I have now understood the risk-neutral case when a money market account is used as the numeraire (here the MPR is zero for both the accumulated money market account (the denominator) and the risky asset assuming a risk-neutral world. When we move to looking at forward rates between Times T1 and T2 in the future would I be right in thinking that the zero-coupon bond P(t, T2) is assumed to have zero volatility in a risk-neutral world (page 639 of Hull)? This would make sense to me seeing as we could lock into this today and are only interested in the resulting interest rate applying between time T1 and T2 in the future. My rationalle here was that if we assume the market is efficient the MPR should be zero today for the discounted bond P(t,T2). This brings me to my question. When we move to considering exchange options I wondered why the MPR of the numeraire (in this case a risky asset) would equal zero? I assumed that both the shares in a risk neutral would would have an MPR of zero (both have a numerator of R - R = 0) but that the volatility of the numeraire asset would likely not be zero. Does this invalidate the forward risk-neutral requirement? Hopefully somebody understands my question and can shed some light on this. Many thanks, Owen
No. Zero coupon bond has a volatility governed by the evolution of interest rates. P(t,T) = E_Q[exp(-{int from t to T} rs ds)] This is a function of t and rt differentiating this under Ito, dP(t,T) gives some dt term, drt term and (drt)^2 The drt term has a diffusion coefficient - (just sigma under Vasicek for eg). This multiplied by the coefficient that is along with rt in the P(t,T) equation will give the market price of risk. This MPR when multiplied again by the volatility of short rate gives the drift under forward risk neutral measure. This is the same principle under which LMM is derived too. Hope this makes sense. Sorry for the delayed reply.
The MPR of the numeraire will be zero only, as we will be considering a money market account as the numeraire asset, which has zero volatility. This choice of numeraire arises from the fact that the option has a continuous payoff (read the market price of this option), and hence the numeraire has to be the money market account. Otherwise the asset ratio St/Ft is just like any other asset with stochastic price evolution.
