The consistent pricing of derivative products involving the joint realizations of a collection of forward rates requires the specification of the covariance matrix between the various underlying rates. A product as simple as a European swaption, if considered in terms of the underlying (spanning) forward rates, clearly shows the importance of this aspect. It is well known, in fact, that European swaptions trade at implied volatilities well below the average of the volatilities of the underlying forward rates (which would roughly correspond to perfect correlation between the rates themselves). In order to account for this well-known fact, imperfect (i.e. less than 1) instantaneous correlation between the forward rates has often been invoked. However, if one assumes, for each forward rate, a constant volatility from today to its expiry, one ends up estimating from traded prices of European swaptions correlations well below their econometrically observed values. Is the market offering the free gift of 'cheap' swaptions which could be bought with the proceeds of the sale of the 'expensive' underlying caplets? In reality, an alternative mechanism is available in order to reproduce the observed market prices, even with perfect instantaneous correlation. How this conjurer's trick can be carried out is best seen by analyzing the simplest possible case, i.e. a two-period European swaption.
Let us consider the case of a discrete trading horizon made up of two dates only, and constrain, for simplicity of expositions, all quantities, such as volatilities or correlations, to be piece-wise constant over each of the two possible 'time steps'. Let us then consider the case of two forward rates, f1 and f2, expiring at times t1 and t2, respectively, of average volatility of 20%, and displaying a correlation *. Finally, let the swaption spanned by these two forward rates, (i.e. the option to enter at time t1 a two-period swap starting at time t1 and maturing at time t3) have a market (Black) volatility of 18%. The underlying swap rate, SR(1,2), can be written as a linear combination of the two forward rates: SR(1,2) = w1 f1 + w2 f2. Therefore, noticing that the second forward rate can assume a possibly different instantaneous volatility from time t1 to time t2, one can write, with hopefully self-explanatory notation, for the average volatility of the swaption, *SR(1,2), the expression in equation 1.
Notice carefully that in equation 1 there only appear the instantaneous volatilities of the two forward rates from time t0 to time t1. Let us now require that both caplets and the swaption should be simultaneously exactly priced. As for the first caplet, no choice is left, but to ensure that its unconditional variance should indeed be equal to the value implied by the Black market volatility. This uniquely determines *1(t1) via the obvious relationship displayed in equation 2.
Remember, however, that not only do we want the swaption volatility to be recovered in a way consistent with the value *1(t1) which has already been fixed, but we also require the caplet expiring at time t2 to be correctly priced. The quantities still at our disposal in order to achieve these tasks are therefore the instantaneous volatility of the second forward rate from time t0 to time t1 (*2(t1)), its volatility from time t1 to time t2 (*2(t2)), and the correlation between the two forwards from time t0 to time t1, *. Looking at equation 3 one can readily see that there is an infinity of solutions, since one can, for instance, obtain *SR(1,2) 2 t1= (18%)2 t1 either by accepting a constant volatility for the second forward rate ( *2(t1)=*2(t2) = 20% ) and imposing a lower-than-one correlation, or by having a perfect correlation between the forward rates and a volatility for the second forward rate from time t1 to time t2 lower than 20%.
If route a) is chosen, i.e. with *<1, one can find a solution even with flat (20%) instantaneous volatilities for the second forward rate. In case b, on the other hand, one must decrease from its average level the instantaneous volatility of the second forward rate from t0 to t1 so that *SR(t1) = w1*1(t1) + w2*2(t1), and then consistently increase it above its average value as shown in equation 3.
In between these two extreme cases (i.e. perfect instantaneous correlation or flat constant volatility for the second forward rate) there obviously exists an infinity of possible solutions. Needless to say, whether or not it might indeed be desirable to force pricing of market swaptions by using the procedure described above within the context of a one-factor model is a completely different issue, that will be touched upon in a later learning curve. In the meantime, despite its simplicity, this example allows us to draw two important conclusions: first of all that imperfect correlation is strictly necessary in order to account for lower-than-weighted-average swaption volatilities only if the instantaneous volatilities of forward rates are assumed to be constant throughout the life of the forward itself. In addition the example shows that if the market volatility of swaptions is explained in terms of time-dependent instantaneous volatilities of the underlying forward rates, there are important implications about the evolution over time of the term structure of volatilities: for the simple example just considered, at time t0, in fact, the term structure of volatilities (i.e. the function that gives the average volatilities of forward rates of different maturities) was flat (at 20%) for both forwards. This term structure of volatilities remains unchanged for case a); in case b), however, the term structure at time t1 has to be different (higher) for the 'front' forward (the only one still 'alive' in this simple example) to attain its correct total variance, i.e. to price the time-2 caplet correctly.
Summarizing: pricing European swaptions in a manner consistent with the parallel cap market can be accomplished in general in an infinity of ways, as long as the user is prepared to allow for a time-dependent volatility for the underlying forward rates. Feasibility, however, does not automatically imply desirability. Part II of this Learning Curve will show how to narrow down the infinity of possible solutions by making the important connection between today's instantaneous volatility functions and the evolution of the term structure of volatilities.
This article contains parts which have been adapted from Chapter 4 of "Interest-Rate Option Models" by Riccardo Rebonato, 1998, (2nd Edition).
