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5 The Unconditional Variance of the Short Rate in BDT – the Discrete Case

5 The Unconditional Variance of the Short Rate in BDT – the Discrete Case

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18.5 THE UNCONDITIONAL VARIANCE OF THE SHORT RATE IN BDT



613



• by a vector r = {ri0 }, i = 0, k, whose elements are the lowest values of the short

rate at time-step i;

• by a vector σ = {σi }, i = 0, k, whose elements are the volatilities of the short rate

from time-step i to time-step i + 1.

This is all that is needed to characterize the BDT model. Every rate rij , in fact, can

be obtained as



rij = ri0 exp 2σi j

t

(18.24)

( t, as usual, is the time-step in years). Let us now define (see Figure 18.2) k random

variables y1 , y2 , . . . , yk by

yk = 1 if an up move occurs at time (k − 1) t



(18.25)



yk = 0 if a down move occurs at time (k − 1) t



(18.26)



For instance, for the path highlighted in Figure 18.2, y1 = 0, y2 = 1, y3 = 0 and

y4 = 0. Let us also assume, as is true in the BDT case, that the variables yj are independent

and that the probability P [yk = 1] = P [yk = 0] = 12 . Let us now define the variable

Xk =



yj



(18.27)



j =1,k



Xk therefore gives the ‘level’ of the short rate at time k t, and the value of the short

rate at time k t in the state labelled by Xk is given by



t]

(18.28)

rk,Xk = rk0 exp[2σi Xk

To lighten notation let us define rk,X0 = rk0 . The next task is then to evaluate the expectation, E[ln rk,Xk ], and the variance, var[ln rk,Xk ], of the logarithm of this quantity. To do



1



0



0

0



Figure 18.2 Values assumed by the random variables y1 , y2 , y3 and y4 for the down-up-downdown path highlighted.



614



CHAPTER 18 MEAN REVERSION IN INTEREST-RATE MODELS



this, we observe that the distribution of Xk is simply given by the Bernoulli (binomial)

distribution

j



C

P [Xk = j ] = kk

2



(18.29)



k!

(k − j )!j !



(18.30)



with

j



Ck =

Therefore



P [rk,Xk = rk0 exp[2σk j







j



t] = P [Xk = j ] =



Ck

2k



(18.31)



We can now evaluate E[ln rk,Xk ] (abbreviated as E[ln rk ]):

j



( 12 )k Ck (ln rk + 2σk j



E[ln rk ] =

j =0,k



= ln rkk ( 12 )k 2k + ( 12 )k 2σk











t)

j



t



j Ck



(18.32)



j =0,k



In arriving at Equation (18.32) use has been made of the fact that

j



Ck = 2k



(18.33)



j =0,k

j



Given, however, the definition of Ck , it is also true that

j



j −1



j Ck = kCk−1



(18.34)



Therefore, after substituting (18.34) into Equation (18.32) one obtains:



t

E[ln rk ] = ln rk + kσk



(18.35)



In order to calculate the variance

var[ln rk ] = E[(ln rk )2 ] − (E[ln rk ])2



(18.36)



we will also need the term E[(ln rk )2 ]. This can be evaluated as follows:

E[(ln rk )2 ] =



1

2



k



j



Ck ln rk + 2σk



j =0,k



= (ln rk )2 + 2kσk











tj



2



j



t ln rk + 4σk2 t



j 2 Ck

j =1,k



(18.37)



18.5 THE UNCONDITIONAL VARIANCE OF THE SHORT RATE IN BDT



615



But the term inside the summation sign is simply equal to

j



j −2



j −1



j 2 Ck = k(k − 1)Ck−2 + kCk−1



(18.38)



and therefore the last summation adds up to

j



j 2 Ck = k(k − 1)2k−2 + k2k−1



(18.39)



j =1,k



Putting the pieces together the unconditional variance is given by

var[ln rk ] = E[(ln rk )2 ] − (E[ln rk ])2





= (ln rk )2 + 2kσk ln rk

t + σk2 tk(k + 1) − (ln rk + kσk

t)2

= σk2 k t



(18.40)



Equation (18.40) therefore shows that Fact 3 is indeed true: despite the fact that the

continuous-time limit of the model displays both mean reversion and a non-constant short

rate volatility, the unconditional variance of the logarithm of the short rate in the BDT

model only depends on the final instantaneous volatility of the short rate (i.e. on the

volatility at time T = n t). Expression (18.40) therefore formally validates the ‘empirical’ procedure, well known among practitioners, to calibrate to caplet market prices.

Table 18.1 shows the results of calibrating the BDT tree using the Black implied volatilities as direct input to Equation (18.40).

Note, however, that Fact 3 has been proven to be true, but the paradox is still unexplained. To find an answer, let us move to Fact 2.

Table 18.1 Caplet prices per unit principal and at-themoney strikes for the GBP sterling curve of expiries

reported on the left-hand column, as evaluated using the

Black model (column Black), and the BDT model calibrated as described in the text (column BDT).

Expiry

01-Nov-95

31-Jan-96

01-May-96

31-Jul-96

31-Oct-96

30-Jan-97

01-May-97

01-Aug-97

31-Oct-97

30-Jan-98

02-May-98

01-Aug-98

31-Oct-98



Black

0.000443

0.000773

0.001148

0.001559

0.002002

0.002422

0.002746

0.003024

0.003265

0.003471

0.003449

0.003406



BDT

0.000431

0.000757

0.001133

0.001548

0.001994

0.002416

0.002742

0.003020

0.003263

0.003471

0.003452

0.003411



616



18.6



CHAPTER 18 MEAN REVERSION IN INTEREST-RATE MODELS



The Unconditional Variance of the Short Rate in

BDT – the Continuous-Time Equivalent



The above derivation has shown that, in discrete time, the unconditional variance of

the short rate is indeed given by expression (18.40) and therefore only depends on the

instantaneous value of the short rate at time T . What is still not apparent, however, is

why the reversion speed and/or the instantaneous short rate volatility from time 0 to time

T − t do not appear in the equation.

To see why this is the case it is more profitable to work in the continuous-time equivalent of the BDT model (Equation (18.19)), and to consider again the general expression

for the variance of a mean-reverting diffusion:

d ln rt = [a(t)(b(t) − ln rt )] dt + σ (t) dz(t)



(18.41)



I show in Appendix I that its variance is given by

T



var[ln rT ] = exp −2



T



a(s) ds



0



t



σ (t)2 exp 2



0



a(s) ds dt



(18.42)



0



As Equation (18.42) shows, the unconditional variance of the logarithm of the short rate

out to time T does indeed depend in general both on the reversion speed and on the values

of the instantaneous volatility σ (t) from time 0 to time T . This result is completely general,

but, if one specializes it to the case of the BDT model, a(t) = −f and f (t) = ln σ (t).

By direct substitution the unconditional variance of the log of the short rate out to time

T therefore becomes

T



var[ln r(T )] = exp[2f (T ) − f (0)]



σ (t)2 exp[−2(ft − f0 )] dt



0

T



= exp[2f (T )]



σ (t)2 exp[−2ft ] dt



(18.43)



0



Making use of the fact that f (t) = ln σ (t) in Equation (18.43), one can immediately

verify that, in the BDT case, the unconditional variance is indeed simply given by

var [ln σT ] = σT2



T

0



du = σT2 T



(18.44)



We have therefore reached an interesting conclusion. Take any mean-reverting process for

which the reversion speed is exactly equal to the negative of the logarithmic derivative of

the instantaneous volatility with respect to time (i.e. a(t) = − ∂ ln∂tσ (t) ). For such a process

neither the reversion speed nor the past instantaneous volatility affect the unconditional

variance, which only depends on the instantaneous short rate volatility at the final time.

This observation fully explains the BDT paradox, and sheds light on the reason why

a more satisfactory model like the Black and Karasinski (which displays ‘true’ mean

reversion) is considerably more difficult to calibrate. The reader might however still be

puzzled as to ‘what went wrong’ from the algorithmic point of view: by the end of the



18.7 RECOMBINING VS BUSHY TREES



617



BDT construction one has used all the degrees of freedom at one’s disposal and all of

today’s market inputs (bond and caplet prices) have been correctly recovered. How could

one have done anything differently and still retained a log-normal distribution for the

short rate? We address this question in the next section.



18.7



Mean Reversion in Short-Rate Lattices: Recombining

vs Bushy Trees



In order to understand what ‘went wrong’, let us look at the ‘algorithmic’ origin of the

result just obtained. More precisely, let us consider the BDT construction over two timesteps in the cases of a steeply decreasing and a steeply increasing short rate volatility

function. Figures 18.3–18.6 represent the first two steps of a non-recombining (‘bushy’)

tree and of a BDT tree with the same time-dependent volatility, and with the same

probabilities ( 12 ) for both jumps. On the y-axis one can read the logarithm of the short

rate. Given that an ‘up’ state is linked to its corresponding ‘down’ state by the relationship

rup = rdown exp 2σt







t



(18.45)



and that the volatility can depend on the time-step, but not on the state, all the ‘up’ and

‘down’ logarithms have the same separation (in log space) at a given time-step. Therefore,

both in the bushy tree and in the BDT tree, the√y-axis distance between any two states

t. It is important to point out that in

originating from the same node is given by 2σt

both trees the construction must recover not only the total unconditional variance from

the origin, but also the conditional variance from each node.

From the two couples of corresponding figures (i.e. from Figures 18.3 and 18.4 and

Figures 18.5 and 18.6) one can immediately appreciate that, in the bushy case, any drift

could have been assigned to the short rate, and the construction would still have been

possible. Looking at Figure 18.4, however, which refers to the case of sharply decreasing

volatility in the BDT construction, one can see that the only way to ensure that at each



node the condition rup = rdown exp 2σt

t is fulfilled and that the tree recombines is

to push the two ‘up’ nodes (labelled up/up and up/down in the figures) towards the two

‘down’ nodes (labelled down/down and down/up) in Figure 18.4. Similarly, looking at

Figure 18.5 one can see that, if the volatility of the short rate is increasing, the ‘up’ nodes

must be moved up even farther, and the ‘down’ nodes moved farther down.

Note that all the nodes can be moved up or down by the same amount (i.e. by adding

a purely time-dependent drift) without affecting the feasibility of the construction, as can

be appreciated by comparing Figures 18.6 and 18.7. Indeed, this is the ‘device’ used

to recover the exogenous bond prices exactly. The nodes cannot be moved in a statedependent way, however, without compromising the BDT recombining construction.

It is clear from the figures that the moving down of the ‘up’ nodes and up of the

‘down’ nodes shown in Figure 18.4 therefore introduces a mean-reverting component

to the process for the short rate. Similarly, the pushing apart of the nodes shown in

Figure 18.6, necessary to ensure recombination, introduces a mean-fleeing component.

In the BDT case the requirement of recombination therefore strongly limits the possible drifts, over and above the drift imposed by recombination, that can be specified by



618



CHAPTER 18 MEAN REVERSION IN INTEREST-RATE MODELS

up/up

up

up/down



down/up

down

down/down



Figure 18.3 A bushy-tree construction for the case of decreasing volatility.



up



up/up



up/down

down/up



down/down



down



Figure 18.4 The corresponding BDT construction for the same decreasing volatility.



the user. In particular, this ‘extra’ drift can at most be time dependent (the term µ(t)

in Equation (18.16), algorithmically accounted for by the construction in Figure 18.7),

and, therefore, cannot introduce ‘true’ mean reversion (since this would require the state

variable, r).

From these considerations one can see from yet another angle why a mean reversion

(of sorts) can only occur with the BDT algorithm if the volatility is time dependent (and,

more specifically, decaying). Note that the negative aspects introduced by the procedure

are more insidious than the usual limitations of one-factor or low-dimensionality models:

the binomial recombining-lattice geometry introduces an inextricable link between its



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