The discussion in part one of this Learning Curve (DW, 2/2) has shown that it is possible to obtain a significant terminal de-correlation amongst rates even in the presence of perfect instantaneous correlation. Feasibility, however, does not automatically ensure desirability. Putting an inordinate burden on the shoulders the instantaneous volatility functions, for instance, can have effects as undesirable as the naive constant-instantaneous-volatility approach. The inescapable consequence, in fact, of any assumption about the instantaneous volatility (including the constant one) is that the future average volatility curve is completely determined by the choice of instantaneous volatility. By imposing a particular form for this function one must therefore have in mind not only the quality of the fit to a certain set of market prices, but also the implications about the future term structure of volatilities (TSV). It is true that little can be known about the latter, but probably most users would agree that, in the lack of precise information to the contrary, the assumption of a qualitatively unchanging shape of the average volatility curve is a reasonable one.
Apart from such subjective expectations or from historical estimations, certain market implied volatility functions must either be incompatible with the Black model, or imply a TSV that must change over time. It is in fact easy to show that, if the function s(t,T) that gives the average volatility at time t of a caplet expiring at time T does not change as a function of calendar time, then the quantity *(*)=s2 (*) * (*=T-t)) must be a strictly increasing function of maturity. Therefore, in a Black world, the average caplet volatility function cannot decline too sharply and, at the same time, be consistent with a time-homogeneous TSV. It is interesting to notice that the cap market values in most currencies imply volatilities that violate this condition, thereby implying that the market either rejects the Black formula, or prices in a change in the future shape or level of today's TSV.
Since, as discussed, an infinity of instantaneous volatility functions can be consistent with a given market TSV, the problem remains open of choosing a suitable shape for the instantaneous volatility function for each forward rate. This choice, in turn, will dictate the possible evolution of the TSV. More precisely, instantaneous and average volatilities (i.e the TSV) are linked by:
Notice that only today's term structure of volatilities, s(0,T) is known from the market. In general, one can easily show that:
Equation (2) is completely general and suggests an obvious link between instantaneous volatilities and the time derivative of the unconditional variance. Under specific assumptions, however, this equation specializes to more restrictive relationships between the quantities defined above. As mentioned above, one can impose that the TSV should not change over time. Alternatively, one might wish to assume that the instantaneous volatility of each individual forward rate should be constant over time; or that each forward rate should experience the same instantaneous volatility at each point in time as a function of maturity. It should be pointed out, however, that one of the most common assumptions, i.e. imposing a flat instantaneous volatility for each forward rate, gives rise to one of the most implausible evolutions for the term structure of volatilities, as shown in the figure below.
Whilst achieving a market fit to caplet prices using exactly the same instantaneous volatility function for all the forwards might be very difficult or even impossible, judicious choices of functional forms and parameters can make the average volatility 'almost' time homogeneous. Empirical evidence seems to point to the fact that, in general, this might be desirable, but it should be kept in mind that, sometimes, it might be more advisable to impose a deterministic change to the TSV. If it were indeed the case that a trader believed that a particular future TSV will prevail, significantly different instantaneous volatility functions for the different forward rates can be found to produce the desired evolution of the term structure of volatilities. These considerations become particularly important for the popular Brace-Gatarek-Musiela approaches, where the specification of the future instantaneous volatilities of the different forward rates (and therefore of the future average volatility) is under the direct control of the user.
The (approximate or exact) time-homegeneity condition makes the task of providing a joint description of the average and the instantaneous volatility curves even more daunting; on the other hand it can provide the constraints necessary to reduce in a sensible way the number of degrees of freedom at the user's disposal.
In order to price complex interest-rate derivatives the terminal correlation between rates must be correctly and coherently reproduced. When applied in conjunction, the traditional correlation approach, which assigns to rates imperfect instantaneous correlation, and the more recent instantaneous volatility framework can produce a parsimonious and realistic description of the dynamics of the yield curve.
This article contains parts which have been adapted from Chapter 4 of "Interest-Rate Option Models" by Riccardo Rebonato, 1998, (2nd Edition).