## B

one can interpret a European swaption as an option to exchange a floating-rate bond (Float) for a fixed-coupon bond (XB). Since the value of the floating leg is given by P(t) — P(T), and, at each node, Pit) = 1, all that is required for the evaluation of the expectation at time i and state j of the value of the floating leg is the expectation Et\P(T)\rij\ contingent on the short rate at time i being in state j. This can be accomplished exactly as shown above for the caplet. Similar considerations apply to the fixed leg. Evaluation of these two quantities then allows computation of the payoff function given by Equation (12.13'); this, in turn, then simply has to be discounted to the origin to give the present value of the swaption.

Both caps and' swaptions can therefore be easily computed using BDT lattice trees. Their values, however, will obviously depend on the input volatility of the short rate. As will be shown, cap prices can be recovered exactiy, as can, under specific circumstances, some swaption prices. It will, however, become apparent that, due to the intrinsic one-factor nature of models such as the BDT, it is in general not possible to match at the same time both cap and swaption prices.

The price of a cap is the sum of the prices of the individual caplets. Let us assume that market prices for caps (of a given tenor) of all maturities are available; i.e. let us assume that the market prices of 6-month, 1-year, 18-month, etc., caps with 6-month tenors are available (for simplicity, the (n +• l)th cap is assumed to have exacdy the same reset dates as the nth cap, plus an additional one). For any one-factor model which exacdy fits the interest-rates term structure and which allows for a deterministically time-varying volatility, these market prices suggest the following procedure: from the first caplet one can obtain by a trial-and-error procedure1 the implied volatility of the suitable underlying variable, which, in the case of a BDT model, would be the future volatility of the short rate from time 0 to expiry (6 months in this example) of the first cap(let). By construction, given the tree and this constant volatility over the first 6-month period, the first cap(let) is therefore perfectiy priced. Moving on to the second cap price, one can construct the first 6 months of the tree using the previously determined flat volatility, and then search numerically for the constant volatility of the short rate to be used from 6 months to 1 year in order to reproduce the observed market price of the 1-year cap. By so doing, both the 6-month and the 1-year cap are therefore exactly priced by construction. Clearly the procedure can be extended to any cap maturity, since each new maturity always introduces a single new number to be determined, and does not 'spoil' the quantities previously determined. While these considerations are of general validity for any lattice model, for the case of BDT the procedure is even simpler, and the calibration can be accomplished almost instantaneously. Since market caps are priced using the Black formula, and since this, in turn, assumes log-normal forward rates, the market assumptions are very similar with the BDT framework, which displays 'almost' log-normal forward rates (see, however, Chapters 16 and 20

for a more precise discussion). Furthermore, by specifying a volatility a(t) in the BDT approach, one is imposing the unconditional variance of the short-rate distribution to be <y(t)2t\ but, since at option expiry the forward rate which enters the Black formula becomes a spot rate, the BDT volatility <r(t) is (almost) exactly the variance implied by the Black model price. Tree calibration to market prices of caps can therefore be carried out almost by inspection.

If one is not interested in pricing consistently a series of caplets, and is only interested in a single cap, a unique flat volatility value for all the caplets (rather than a time-varying function) can, of course, price that specific individual cap. However, one must be very careful in using these 'average volatilities': let us assume, to illustrate the dangers, that a four-period cap can be priced with an average (flat) volatility over the four periods of, say, 20%. Let us also assume that a five-period cap can be priced with, say, an 18% volatility. These two combined pieces of information have a strong implication for the implied forward volatility of the short rate from period 4 to period 5. If one priced the first four periods of the 5-year cap with the constant volatility of the four-period cap (i.e. 20%) one can easily compute the remaining variance added by the fifth period, and one can derive the implied forward volatility. But, for low enough values of the market price of the five-period cap, no solution might be found, i.e. only a zero or negative volatility could satisfy the constraints. (The situation is closely reminiscent of the possibility of negative forward rates often found when describing a yield curve in terms of gross redemption yields.) This implies that, for a given model, average volatilities cannot decline too sharply.

While the first procedure described above is therefore constructed to produce in most cases admissible short-rate volatilities when pricing consistently different caps, the 'average volatility' method is not guaranteed to yield meaningful results. The situation, once again, is very similar to discounting cash flows using a gross redemption yield or spot rates.

The issue of the time dependence of the volatility of the short rate in the BDT model warrants more careful analysis. As explained previously (see Section 11.1), the BDT model achieves mean reversion by allowing for the short-rate volatility to decline with time. One essential feature of mean-reverting models is that rates do not assume 'implausibly' large values over very long time horizons. In practical terms, they embody our expectation that the possible values explored by the short rate over, say, the next 30 years should not be significantly different from the range of values encountered over, say, a 40-year period. In general, when using a declining function for the future volatility of the short rate, care must be taken to ensure that the resulting model is viable. In particular, one's information about future financial quantities at a later time t" must always be less than the equivalent information regarding an earlier time t'. Our loss of information is translated in a diffusion model by an increase in the variance of the process. Therefore, the increase in variance from time t' to time f due to the passage of time must not be reduced to zero or allowed to become negative.

Since the volatility used at each time / in a BDT tree produces a process for the short-rate of total (unconditional) variance given by

the following relationship2 must hold:

Figure 12.8 illustrates what happens when the decrease in volatility is too sharp: the resulting BDT tree might imply that we know more about a future time t" than about an earlier time t'.

In practice one observes that by following the first procedure, the implied short-rate volatility curve obtained from the market is generally declining over time, possibly after an initial hump (see Figure 11.3). As pointed out above, a declining volatility of the short rate implies a mean-reverting BDT distribution of rates. This is paralleled by the market practice, in the context of the commonly adopted Black model, of using declining forward rates volatilities. Both 'stratagems' attempt to avoid in a simple way an excessive dispersion of the rates which would make

Unconditional Variance

Time (years)

Figure 12.8 The unconditional variance implied by a declining short-rate volatility of functional form <r(r) = <r0 exp[—vr], with <r = 20%, v = 0.0180 and v = 0.040. Notice that for the higher value of the decay constant v ('Decay 2' in the graph) the unconditional variance is not a monotonically increasing function of time, implying that, as of time 0, more is known about a future time t" than about an earlier time t'

Time (years)

Figure 12.8 The unconditional variance implied by a declining short-rate volatility of functional form <r(r) = <r0 exp[—vr], with <r = 20%, v = 0.0180 and v = 0.040. Notice that for the higher value of the decay constant v ('Decay 2' in the graph) the unconditional variance is not a monotonically increasing function of time, implying that, as of time 0, more is known about a future time t" than about an earlier time t'

long-dated caps 'too' expensive: a constant-volatility BDT tree might assign significant probabilities to implausibly high rates (see Figures 11.1 and 11.2), and a cap priced off this tree would therefore offer 'unnecessary' protection. The remarks made in Section 11.1 regarding Black volatilities and yield-curve-model volatilities in general apply very clearly to the BDT case. In particular, correct obtainment of today's prices of an arbitrary number of market caps should not disguise the fact that, for a declining implied (Black) volatility, the forward volatility (Equation (11.4)) will tend to smaller and smaller values, thus implying an asymptotically deterministic model.

As for European swaptions, as pointed out above, they can be indifferentiy regarded as options to pay or receive the fixed leg of a swap in exchange for the floating leg, or as options on the underlying forward swap rate. A forward swap rate, in turn, for a swap with tenor r, has been shown in Chapter 1 to be a linear combination of r-period forward rates. Hence, as stated before, a swaption can be priced as an option on a combination of forward rates.

Looking at a swaption in this light shows one of the biggest problems of swaption valuation in the framework of any one-factor model. In any such model, in fact, all rates are perfectiy correlated, and, if the short-rate process were assigned the time-dependent volatilities needed to price the market caps, the forward equilibrium swap rate volatility would therefore be overstated by such a model (see Chapter 1 again): roughly speaking, one-factor models do not allow for the possibility that one forward rate might be moving up while another might be declining. In addition, it was shown in Chapter 4 that imperfect terminal correlation between forward rates can be obtained even in the presence of perfect instantaneous correlation as long as the instantaneous volatilities of the forward rates themselves are sufficiendy different from being flat. Market prices of swaptions clearly reflect this imperfect terminal correlation between rates, as one can readily observe, at least in liquid markets, by comparing the observed cap and swaption volatility curves. In the context of the BDT model, it is not difficult, for an individual swaption, to adjust the overall volatility in such a way as to ensure that the BDT tree produces the same swaption price as observed in the market. Problems creep in, however, as one considers in conjunction caps and swaptions, or a series of swaptions, since the model is intrinsically inadequate to reproduce the overall variance-covariance matrix of the forward rates (see Chapter 3, Section 3.2).

As long as one is interested in pricing a single European swaption (and one is ready to misprice the 'underlying' caplets), then the distributional assumptions of the BDT model are, once again, of great help. The swap rate implied by the BDT tree is in fact approximately log-normal, and bounded by zero from below. Therefore the BDT distributional assumptions are very similar to the assumptions underlying the Black model, as used in the market for swaptions. Furthermore, since the volatilities of yields of different maturities display rather small variations, a very good guess to the swap rate volatility can be obtained by using for the desired unconditional variance of the short-rate, i.e. by using for a{t) the implied swap rate volatility obtainable from the Black swaption prices.

As pointed out above, a generic series of swaps cannot be consistently priced by any one BDT tree. There are situations, however, when the BDT model can still be reasonably calibrated to yield market results, even in the case of several swaptions. Let us consider a series of swaptions to enter swaps at different times, all terminating on the same calender date, e.g. a 6-month swaption into a 3.5-year swap, a 1-year swaption into a 3-year swap, etc. This is of greater relevance than one might initially surmise, since these are the European swaptions underlying a callable bond; or a fixed-maturity American swaption. In this situation, it is possible to describe the short-rate volatility as a time-varying function (for practical purposes a step-wise continuous function will do) such that, in the example above, the first constant value for the volatility spans the time steps between time 0 and 6 months, and the constant level of the volatility is chosen in such a way as to give the correct market price for the first swaption; the second step then goes from 6 months plus one time step to 1 year, and its level is determined so as to achieve a match of the second swaption market price; the procedure is continued until all the swaptions are priced. This approach allows the introduction of as many degrees of freedom as market values to be matched in such a way that each new segment does not 'spoil' the values obtained for swaptions of shorter maturity.

The positioning of the boundaries of the step function is crucial: if the first constant segment covered a period between time 0 and, say, 9 months, in general an exact match of the market values will not be obtained. This result is not as intuitively obvious as one might at first surmise: Figure 12.9 compares the volatility fitting for a cap and for a series of swaptions, and clearly shows that, while in the fitting of cap volatilities non-overlapping periods are used in turn, the same is not true for a series of swaptions.

If one had to price a generic set of swaptions, i.e. not necessarily swaptions maturing at the same calendar date, no procedure will produce, within a BDT approach, all the market prices. How important this might be depends on specific cases, but, in general, the effect cannot be safely neglected.

The possibility of adjusting the volatility curve in such a way as to 'force' the BDT model to price specific swaptions as the market does not obscure the fact that the model has intrinsic shortcomings for all those options where imperfect correlation between rates plays an important role. Under these circumstances users should therefore be particularly careful in interpreting the results. Needless to say, there is nothing special about the BDT model in so far as this aspect is concerned, since all one-factor models (e.g. the more complex Extended CIR or Vasicek) display the same deficiency.

Depending on what the most important shortcoming of the BDT approach might be perceived to be, several avenues are open for 'better' pricing: if the

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