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.
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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
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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
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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