## Statement Of The Problem

For a generic Wiener process of the form d In tit) = 9(t) dt + o-inst (i) dz(t) (21.1)

with 9(t) a deterministic drift, dz(f) the increment at time t of a Brownian process and o-inst (i) an instantaneous volatility (standard deviation per unit time), it is well known that the unconditional variance out to time T is given by

Notice that, by requiring 9(t) to be deterministic, we have explicitly excluded all the cases where the short rate enters the drift, as it does for mean-reverting models (see Equation (21.3) below). Therefore a priori we cannot apply result (21.2) to the BDT case. In general, in fact, for a log-normal mean-reverting process of the form dlnr(0 = [9(t)+k(xfr(t) - lnr(0)]di +o-i„st(i)<k(0 (21.3)

(with reversion speed k, reversion level i/r(i), and 9(t) a deterministic drift component) the unconditional variance will display a dependence not only on the (integral of) oinst (as given by Equation (21.2)), but also on the reversion speed. More specifically, the continuous-time equivalent of the BDT model can be written as (see Equation (12.2))

d In tit) = [9(t) - f'(t)(xj,(t) - In r(i))] dt + o"inst(i) dz(t) (21.4)

and with both &it) and o(t) deterministic functions of time. The subscript 'inst' (dropped in the following to lighten notation) has been appended to a to emphasise that it denotes the instantaneous volatility of the short rate. Equation (21.4) and its implications for the model behaviour have been explored at length in Chapter 12, where the link between the function 0(f) and the median of the short rate distribution has been highlighted. In the present context it will suffice to remind the reader that it is only in the presence of a time-decaying short rate volatility (9 In o(t)/dt < 0) that the resulting reversion speed (—/') is positive and the model displays mean reversion.

Whilst this is well known, it does seem to create a paradox, since it is 'empirically' known, and shown in the following, that in discrete time the unconditional variance of the short rate in the BDT model depends neither on the instantaneous volatility from time 0 to time T — At (as one would have been led to expect from Equation (21.2)) nor on the reversion speed —/' (as one might have surmised from Equation (21.4)). More precisely, one can easily show (see below) that

where a2(NAt) is the (square of the) instantaneous short rate volatility at time T = NAt. It is important to stress the crucial importance of Equation (21.6) for calibration purposes; it is only because the unconditional variance of the log-normal short rate is simply given by the expression above that calibration to caplet prices, routinely quoted on the basis of the log-normal Black (1976) model, is so easy. The reasoning goes as follows: the BDT forward induction construction implicitly carries out the Girsanov's drift transformation from the measure associated with the discount bond numeraire implied with the Black model to the equivalent measure associated with the (discretely compounded) money-market account implied by the BDT approach (see Chapter 12); as we know, this transformation modifies the drifts but not the variances of the various spot and forward rates; it follows that the Black implied market volatilities give direct information about the total unconditional variance of the relevant forward rates (spot rates at expiry). It therefore follows that, if Equation (21.6) is true, from the quoted Black implied volatilities of caplets of different expiries the user can almost exactly1 obtain their exact BDT pricing by assigning a value at expiry for the time-dependent short rate volatility equal to the Black implied volatilities.

It is widely known amongst practitioners that this is the case. What is often not appreciated is how this can be, since Equations (21.2) and (21.4) would in general suggest that both the whole instantaneous short rate volatility function from time 0 to time T and the reversion speed —/' should affect the unconditional variance from time 0 to time T that, in turn, determines the Black price of the T-expiry caplet. Sections 21.3 and 21.4 will therefore show that the 'empirically known' result mentioned above regarding the unconditional variance is indeed correct, and then (Section 21.4), moving to the continuous limit, shed light on the origin and resolution of the resulting apparent paradox.

21.3 THE UNCONDITIONAL VARIANCE OF THE SHORT RATE IN BDT — THE DISCRETE CASE

A calibrated BDT lattice is fully described by a vector r = {r;o}, i = 0, k, whose elements are the lowest values of the short rate at time step i, and by a vector a = {<r, }, i = 0, k, whose elements are the instantaneous volatilities of the short rate from time step i to time step i + 1. Every rate rij, in fact, can be obtained as m = no exp[2avj'VAf] where At, as usual, is the time step in years. Let us now define k random variables y\, yi,... ,yk by:

yk = 1 if an up move occurs at time (k — l)Ai yk = 0 if a down move occurs at time (k — l)Af.

For instance, for the path highlighted in Figure 21.1, y\ = 0, yi = 1, yz = 0, and y4 = 0. It will further be assumed (i) that the variables yj are independent, and (ii) that the probability P[yk = 1] = P[yk = 0] = In the terminology of Appendix A, Xk is therefore a symmetric unitary random walk. The variable Xk = Yl]=i,k yj. gives the 'level' of the short rate at time kAt, and the value of the short rate at time kAt in the state labelled by Xk is given by rk,x(.k) = no exp|2ovAWAi]. (21.7)

Our task is now to evaluate the expectation and variance (see also Rebonato and Kazziha (1997)) of the logarithm of this quantity, denoted by £[ln rktx(k)] and Var[ln rktx(k)], respectively. To this effect one can simply notice that, by construction, the probability of Xk assuming value j is given by the equiprobable binomial distribution, and therefore

with

But the probability of the short rate assuming the yth value at time k is also given by

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