Unformatted text preview: Unconditional Variance, Mean
Reversion and Short Rate
Volatility in the Calibration of the
BlackDerman and Toy Model and
of TwoDimensional LogNormal
Short Rate Models
Soraya Kazziha
Riccardo Rebonato
Abstract
Calibration of the BDT model to cap prices is notoriously simple, since an almost
exact `guess’ of the correct timedependent volatility can be obtained from the market
implied volatility of caplets. This is a priori surprising, since the unconditional
variance of the BDTshort rate process should be expected to depend on the
(deterministic) history of the short rate volatility, and on the mean reversion of
logarithm of the short rate. This apparent paradox is resolved in the first part of the
paper, where general expressions (usable, for instance, to calibrate the BlackKarasinsky model) for the unconditional variance are obtained for a variety of onefactor models. The class of onefactor models for which the same type of relationship
holds true is also introduced.
The results are then extended to morethanonefactor models, with the introduction of
a class of stable and arbitragefree Generalized Brennan and Schwartz models. It is
shown that, if no arbitrage is to be enforced, this very large class of lognormalshort
rate models cannot be calibrated to cap prices using higherdimensional extensions of
the BDT procedure. Introduction
Of the several onefactor models used for pricing interest rate options, the Black
Derman and Toy (1990) (BDT in the following) is one of the best known, and of the
most widely used. Amongst its most appealing features are the capability to price
exactly an arbitrary set of received market discount bonds, the lognormal distribution
of the short rate, and the ease of calibration to cap prices. The first feature (exact
pricing of the yield curve) is shared by a variety of (nonequilibrium) models, such as
the Ho and Lee (1986) or the Hull and White (1990). The second (lognormal
distribution of rates) is also shared by the Black and Karasinski model. Only the BDT
approach, however, allows lognormal rates and calibration to caplet prices (in absence
of smile effects) that can be accomplished almost by inspection. This latter feature is, 2 Soraya Kazziha and Riccardo Rebonato
at the same time, the blessing and the bane of the BDT model, and directly stems
from the inflexible specification of the reversion speed, which is completely
determined by the future behaviour of the short rate volatility. This latter feature is
well known, and has already been amply criticized on theoretical grounds in the
literature. The decision as to whether, despite this rather artificial characteristic, the
BDT model can be profitably used in practical applications for option pricing depends
crucially on the type of option, and requires a considerable degree of experience and a
rather subtle understanding of the implications of the model. A discussion of the
related issues can be found, for instance, in Rebonato (1996).
In the light of the above, the present note
1. highlights the intimate connection, hitherto not fully appreciated, to the best
knowledge of the authors, between the ease of calibration to cap prices and the
particular link between the reversion speed and the logarithmic derivative of the
short rate volatility; 2. extends the results to different classes of onefactor models; 3. shows to what extent these findings affect the calibration of an important class of
noarbitrage twofactor models. Statement of the problem
For a generic Wiener process of the form
d ln r
t
tdt 'inst
tdz
t
1 with
t a determinstic drift, dz the increment of a Brownian process and 'inst
t an
instantaneous volatility (standard deviation per unit time) it is well known that the
unconditional variance out to time T is given by
T
'inst
u2 duX
2
Var
ln r
T
0 Furthermore, for a meanreverting process of the form
d ln r
t
t k
2
t À ln r
tdt 'inst
tdz
t
3 (with reversion speed k, reversion level 2
t, and
t a deterministic drift component)
the unconditional variance will, in general, depend on the reversion speed.
The continuoustime equivalent of the BDT model can be written as
d ln r
t
t À f H
t
2
t À ln r
tdt 'inst
tdz
t
4 f H
t d ln '
tad t
5 with and both
t and '
t deterministic functions of time. Equation (4) and its
implications as to the model behaviour are well known in the literature (see, e.g.
Rebonato (1996), where the link between the function
t and the median of the short
rate distribution is highlighted). For the present purposes it will suffice to say that it is Unconditional Variance, Mean Reversion and Short Rate Volatility 3 only in the presence of a time decaying short rate volatility
d ln '
tad t ` 0 that the
resulting reversion speed
Àf H 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 neither depends on the instantaneous volatility
from time 0 to time T À Át (as one would have been led to expect from Equation (2))
nor does it depend on the reversion speed Àf H (as one might have surmised from
Equation (4)). More precisely, one can easily show (see below) that
Var
ln r
N Át
N Át'2
N ÁtY
6 where '2
N Át is the (square of) the instantaneous short rate volatility at time
T N Át. It is important to stress the crucial importance of Equation (6) for
calibration purposes; it is only because the instantaneous volatility of the lognormal
short rate is simply given by the expression above that calibration to caplet prices is
so easy: market prices are in fact routinely quoted on the basis of the lognormal
Black (1976) model, and 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 asociated with the
(discretelycompounded) moneymarket account implied by the BDT approach (see
Rebonato (1996)). Since the market Black implied volatilities give direct information
about the unconditional variance of the relevant forward rates (spot rates at expiry),
from the quoted implied Black volatilities of caplets of different expiries the user can
almost exactly1 obtain their exact BDT pricing by assigning a timedependent short
rate volatility matching the implied Black volatilities: see Equation (6) above.
It is rather well known amongst practitioners that this is the case. What is not
generally appreciated is how this can be, since Equations (2) and (4) would in general
suggest that both the instantaneous short rate volatility from time 0 to time T and the
reversion speed Àf H should affect the unconditional variance from time 0 to time T,
that, in turn, determines, the Black price of the Texpiry caplet. The first part of this
paper (Sections 3 and 4) will first show that the ‘empirically known’ result mentioned
above regarding the unconditional variance is indeed correct, and then (Section 4),
moving to the continuous limit, shed light on the origin and resolution of the resulting
apparent paradox. The unconditional variance of the short rate in BDT ± the discrete case
A calibrated BDT lattice is fully described by a vector r fri0 gY
i 0Y k whose
elements are the lowest values of the short rate at time step i, and by a vector
' f'i gY
i 0Y k , whose elements are the volatilities of the short rate from timestep
p
i to timestep i 1. Every rate rij , in fact, can be obtained as rij ri0 exp 2'i j Át.
(Át, as usual is the time step in years). Let us now define k random variables
y1 Y y2 Y F F F Y yk by 1 Since the short rate enters the expression of the drift, the unconditional variance is not exactly equal
to '2 ( (where ( is the time to expiry and 'Black the market implied volatility). The
Black
approximation is however excellent (see Rebonato (1998)) for a discussion of this point. 4 Soraya Kazziha and Riccardo Rebonato = 1
0 0 = = 0 = Figure 1 Values assumed by the random variables y1 , y2 , y3 and y4 for the
downupdowndown path highlighted
&
yk 1 if an up move occurs at time (k  1) D t
0 if a down move occurs at time (k  1) D t. For instance, for the path highlighted in Figure 1, y1 0, y2 1, y3 0, y4 0. It
will further be assumed i) that the variables yj are independent, and ii) that the
probability Pyk 1 Pyk 0 1. The variable Xk j1Yk yj therefore gives the
2
‘level’ of the short rate at time k Át, and the value of the short rate at time k Át in the
state labelled by Xk is given by
p
7
rk YX
k rk 0 exp 2'i Xk ÁtX
Our task is now to evaluate the expectation and variance of the logarithm of this
quantity, denoted by Eln rk YX
k and Varln rk YX
k , respectively. To this effect one
must first find the distribution of Xk . By evaluating its characteristic function one can
easily show that the probability of Xk assuming value j is given by
j
PXk j Ck a2k Y
8 j
Ck k 3a
k À j3 j 3X
9 with Therefore p
j
Prk YX
k rk 0 exp 2'k j Át PXk j Ck a2k X
10 We are now in a position to evaluate Eln rk YX
k (rk YX
k will be abbreviated as rk in
the following to lighten notation):
Eln rk 1k p
pj
1k
1k
j
Ck
ln rk 2'k Átj ln rk 2k 2'k Át
jC k X
2
2
2
j0Yk
j1Yk Given, however, the definition of C jk ,
jC jk kC jkÀ1 Y
À1
11 Unconditional Variance, Mean Reversion and Short Rate Volatility 5 and, after substituting in (11), one obtains p
Eln rk ln rk k 'k ÁtX
12 Similarly for the variance
E
ln rk 2 1k
j0Yk 2 p
C jk
ln rk 2'k Átj2
p
ln rk 2 2k 'k
ln rk Át 4'2 Át
j2 C jk X
k
13 j1Yk But the last term is simply equal to
j2 C jk k
k À 1C jkÀ2 kC jkÀ1 Y
À2
À1
and therefore the last summation adds up to
j2 C jk k
k À 12k À2 k 2k À1 X
14
15 j0Yk It follows that the unconditional variance is given by
Varln rk E
ln rk 2 À
Eln rk 2
p
p
ln rk 2 2k 'k
ln rk Át '2 Átk
k 1 À
ln rk k 'k Át2
k
'2 k ÁtX
k
16 The expression above therefore shows that 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, despite the continuoustime limit of the model displaying both meanTable 1 Caplet prices per unit notionals and ATM strikes for the GBP sterling
curve of expiries reported on the lefthand column, as evaluated using the Black
model (column Black), and the BDT model calibrated as described in the text
(column BDT)
Expiry
01Nov95
31Jan96
01May96
31Jul96
31Oct96
30Jan97
01May97
01Aug97
31Oct97
30Jan98
02May98
01Aug98
31Oct98 Sterling Curve Nov 1995
Black
BDT
0.000443
0.000773
0.001148
0.001559
0.002002
0.002422
0.002746
0.003024
0.003265
0.003471
0.003449
0.003406 0.000431
0.000757
0.001133
0.001548
0.001994
0.002416
0.002742
0.003020
0.003263
0.003471
0.003452
0.003411 6 Soraya Kazziha and Riccardo Rebonato
reversion and a nonconstant short rate volatility, and therefore validates the
‘empirical’ procedure, wellknown amongst practitioners, to calibrate to caplet market
prices. The table above shows the results of calibrating the BDT tree using the Black
implied volatilities. The unconditional variance of the short rate in BDT ± the continuoustime equivalent
The above derivation has shown that, in discrete time, the unconditional variance of
the short rate is indeed given by expression (6). What is not apparent, however, is why
the reversion speed and/or the instantaneous short rate volatility from time 0 to time
T À Át does not appear in the equation. To see why this is the case it is more
profitable to work in the continuoustime equivalent of the BDT model, Equation (4).
This can be rewritten as a diffusion of the general form:
d lnr
t a
t
b
t À ln r
tdt '
tdz
tY
17 where a
t, b
t and '
t are deterministic functions of time. The SDE (*) can easily
be solved (see Appendix I) giving
t
t
t
2
a
sds
'
t exp2
a
sdsdtX
18
Varln r
T expÀ2
0 0 0 As it can be appreciated from this result, the unconditional variance of the logarithm
of the short rate out to time T does in general indeed depend on the reversion speed,
and on the values of the instantaneous volatility '
t from time 0 to time T. This
result is completely general, but it is instructive to specialize it to the case of the BDT
model. In this case a
t Àf H , and therefore, recalling that f
t ln '
t, the
unconditional variance of the short rate out to time T becomes
t
Varln r
T exp2f
T À f
0
'
t2 expÀ2
f
t À f
0dt
0
exp2f
T T '
t2 expÀ2f
tdtX
19 0 Finally, recalling that f
t ln '
t, one can immediately verify that, in the BDT
case, the unconditional variance is indeed simply given by
T
2
du '
T 2 T Y
20
Var
ln '
T '
T
0 i.e. for any meanreverting 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 Àd ln '
tad t neither the reversion speed nor the past instantaneous
volatilty enter the expression for the unconditional variance, which only depends on
the instantaneous short rate volatility at the final time. This result fully resolves the
‘BDT paradox’, and indicates the necessary and sufficient conditions under which any
locally lognormal model can be simply calibrated to cap prices by using its terminal
instantaneous volatility. Unconditional Variance, Mean Reversion and Short Rate Volatility 7 Extensions to twofactor approaches
The importance of using morethanone factor models is widely recognized in the
financial community, especially for pricing options that depend in an important way
on the imperfect correlation amongst rates. From the discussion above, on the other
hand, it is easy to see that retaining a lognormal distribution of rates is very important
if one is to achieve easy calibration of any model to market cap prices across several
strikes. The impact of nonlognormal distributions on model cap prices has been
clearly shown, for instance, for the normalshortrate Hull and White Generalized
Vasicek model (see Rebonato (1996)).
In order to retain the lognormality of the short rate, and to extend the analysis to
more than one factor, one is naturally led to consider the general framework
introduced by Brennan and Schwartz (1982, 1983), who showed that, if one of the two
state variables is chosen to be the consol price or yield, the accompanying market
price of risk can be made to disappear from the resulting parabolic partial differential
equation which describes the ‘realworld’ evolution of the price of a generic security.
It would be very useful for practical applications if one could specify a twofactor lognormal short rate model along the same conceptual lines, so that the model calibration
could be accomplished as readily as in the BDT case. In the following it is shown that
this is not possible, if arbitrage is to be prevented, but approximate expressions for the
unconditional variance of the short rate for this class of models are currently under
study.
The specific model proposed by Brennan and Schwartz has been shown (Hogan(1993))
to suffer from instability of the long yield. This feature, however, stems from the
arbitrary ‘realworld’ specification of the dynamics of the state variables chosen by
Brennan and Schwartz. Their central insight regarding the market price of long yield
risk remains valid, and is made use of in the following in the context of the riskneutral (as opposed to ‘realworld’) measure. More precisely, the derivations presented
in the following apply to the measure Q (often referred to as ‘riskneutral’) under
which asset prices divided by the rolledup moneymarket account are martingales. It
will be recalled at this point that, in any ndimensional treebased nnomial
methodology where, with obvious extension of the BDT algorithm, payoffs are first
averaged and then discounted to the ‘originating’ node by the short rate corresponding
to that node, one is effectively discounting final payoffs by the (discretely) rolledup
money market account (see Rebonato (1996)). Therefore the riskneutral measure
defined above is indeed the appropriate measure to consider for latticebased
methodologies.
If one wants to retain, at the same time, the Brennan and Schwartz general approach
and local lognormality for the short rate, one is naturally led to choose (see Rebonato
(1997)) as state variables the consol yield, L, and the ratio, K, of the short rate, r, to
the consol yield:
r KL
21 If, in addition both K and L are assumed to be lognormally distributed, not only
would positivity of the short rate be automatically ensured, but its distribution would
also turn out to be lognormal. Therefore 8 Soraya Kazziha and Riccardo Rebonato dK aK "K
tY K Y Ldt 'K dzK
22H dLaL "L
tY K Y Ldt 'L dzL
22HH EdzK Y dzL &
22HHH The expressions above contain the unknown drifts for the consol yield and for the
relative spread K. Following the spirit of the Brennan and Shwartz approach, however,
the noarbitrage riskneutral dynamics for the consol bond, C 1aL, can then be
obtained by imposing that, since the consol bond is an asset, it must grow in Q at
r À L, i.e. at the short rate minus its dividend yield, if discounting is effected using
the moneymarket account. Applying Ito’s lemma one therefore obtains:
dLaL L
1 À K '2 dt 'L dzL
L
23 dL L
L À R '2 Ldt L'L dzL X
L
24 or, equivalently, Notice that, from this noarbitrage condition, the long (consol) yield flees the short
rate with fleeing speed L; this would seem to imply an intrinsically unstable
behaviour for the joint dynamics of the state variables. This is, however, not
necessarily the case, as can be see in the following. Ito’s lemma, in conjunction with
the above equation and the definitions i) and ii), in fact gives for the SDE for R
drar aH
t
L À r "K dt 'r dzr Y
25 with '2 '2 '2 Y 2'K 'L &, i.e. the short rate is lognormally distributed and it
r
K
L
reverts to the long yield with reversion speed 1. It is important to notice that this
condition directly follows from nothing else but the distributional assumptions and the
noarbitrage requirement. Notice also that Equation (25) implies a reversion of the
short rate to the long rate with reversion speed equal to 1. This is particularly
significant, since Hogan (1993) shows that the reversion speed above must be ! 1 for
the coupled system of equations describing the evolution of r and L to be stable.
The equations obtained up to this point have been fully determined by the noarbitrage
conditions and the distributional assumptions. Nothing, however, has been said about
the drift of K. At this point one could want to impose that the drift of K should be
equal to an arbitrary function f time only and a linear function of K and L only:
"K
tY K Y L b0
t b1 K b2 LY
26 with b1 ` 0 so as to ensure the reversion of K to a constant level. Whatever the merits
of this choice, Equation (26) can always be seen as obtained by retaining the firstorder term of the expansion of the (unknown) true drift. Different choices of
functional dependence of the drift of K on the state variables give rise to different
models, which can therefore aptly described as belonging to the Generalized Brennan
and Schwartz family.
Whatever the choice for "K might be, Equation (25) is crucial to the following
argument: the system of SDEs (22)–(25) has been shown to stem from the very
conditions of noarbitrage, and, therefore, for the chosen numeraire and distributional
assumptions, these results are inescapable for any viable model. This fact, however,
poses a grave problem insofar as ease of calibration is concerned. On the one hand, in Unconditional Variance, Mean Reversion and Short Rate Volatility 9 fact, it has been observed above that necessary condition for the variance of a meanreverting process to depend only on the (terminal value of) the instantaneous volatility
and not on the reversion speed is that the latter should be identical to the negative of
the logarithmic time derivative of the instantaneous volatility (See Eq. (20)). We then
argued that it was because of this very feature that calibration to cap prices in the
BDT model is so straightforward. On the other hand, even if the drift of K were
assumed to be purely a function of time, there is no way to retain stability of the
dynamics of r and L and avoiding arbitrage without the term r
L À r, which implies a
reversion speed for the short rate (rather than its logarithm) that is not related to the
logarithmic derivative of the instantaneous volatility. Therefore the unconditional
variance of the short rate out to time T cannot be simply related to the instantaneous
volatility of the short rate at time T, as it is the case for the BDT model. In other
words, it is not possible to extend the BDT model to more than one state variable as
proposed above in such a way that one of its most important features (the ease of
calibration) is preserved. Conclusions
The inflexible nature of the reversion speed in the BDT model is, at the same time, its
blessing and its bane. The positive features, connected with the ease of calibration, are
too well known to be dwelt upon, and are, to a large extent, responsible for the wide
acceptance of the model amongst practitioners. The negative aspects, however, should
not be underestimated. These are more pernicious than the usual limitations of onefactor or lowdimensionality models (see, e.g. Rebonato and Cooper (1996), for a
discussion of the latter): one of the distinctive features of the BDT model is in fact the
inextricable link it implies between its reversion speed and the logarithmic derivative
of the short rate volatility. The time decaying volatility needed in the BDT model in
order to ‘contain’ an excessive dispersion of rates can, in fact, well succeed in
obtaining an unconditional distribution of rates consistent with the one implied by the
cap market; but, since an explicit deterministic mean reversion is absent from the
model for any nondecaying behaviour of the short rate volatility, this is obtained at
the expenses of a lower and lower forwardforward volatility. This undesirable feature
can have a limited impact for relatively short maturity options, but must always be
born in mind by users who extend their analyses well beyond the common ‘volatility
hump’ observed in most cap markets. Unfortunately, this paper has shown that it is
impossible to remove this undesirable feature and to retain at the same time the
original ease of calibration.
In an attempt to obviate those shortcomings of the BDT model shared by all onefactor models, an obvious extension along the lines of the Brennanand Schwartz
approach was introduced in the second part of the paper, and a class of arbitragefree
lognormal short rate models which do not display the Hogan instability was obtained
in the second part of the paper. It was shown, however, that, despite the lognormality
of the short rate, if arbitrage is to be avoided these models cannot have the same type
of unconditional variance displayed by the BDT model. On the one hand this allows
for ‘true’ mean reversion to occur in the SDE for the short rate even in the presence
of constant volatility; on the other hand, however, the resulting calibration to cap
prices is prima facie considerably more arduous. Research in the validity of
approximate expressions that could make the cap calibration almost as easy as for the
BDT approach is currently under way. 10 Soraya Kazziha and Riccardo Rebonato Appendix: Evaluation of the variance of the logarithm of the
instantaneous short rate
From (*) one can write
d lnr
t a
t ln r
tdt a
tb
tdt '
tdz
tX
This implies that
t
t
expÀ
a
sdsd ln r
t exp a
sds a
tb
tdt '
tdz
tX
0 0 t But the quantity ln r
T exp 0 a
sds can be written as
T
T
t
ln r
T exp
a
s ds ln r
0
a
tb
t exp a
s dsdt
0 0 T 0 t '
t exp 0 a
s dsdz
tY 0 and, therefore
ln r
T expÀ T
a
s ds ln r
0 expÀ 0
0
a
s ds 0
T expÀ T T a
s ds
'
t exp 0 t T t a
tb
t exp 0 a
s dsdt 0 a
s dsdz
tX 0 Remembering that, for any deterministic function f
t, Var
it then follows that t 0 f
u dz
u t
0 f
s2 dt, Varln r
T E
ln r
T 2 À
Eln r
T 2
T expÀ2
0 T a
s ds
0 t 2 '
t exp2 a
s dsdtX 0 References
Black F., Derman E., Toy W., (1990) A onefactor model of interest rates and its
application to Treasury bond options, Fin. An. Jour., 1990, 33–339.
Black F., Karasinski P., (1991) Bond and option pricing when short rates are
lognormal, Fin. An. Jour., July–August.
Brennan M.J., Schwartz E.S. (1982) An equilibrium model of bond pricing and a test
of market efficiency, Jour. Fin. Quan. An., 17, 301–329.
Brennan M.J., Schwartz E.S., (1983) Alternative methods for valuing debt options,
Finance, 4, 119–138.
Ho T.S.Y., Lee S.B. (1986) Term structure movements and pricing interest rate
contingent claims, Jour. Fin., 41, 1011–1028. Unconditional Variance, Mean Reversion and Short Rate Volatility 11 Hogan M., Problems in certain twofactor term structure models, Ann. Appl. Prob., 3,
2.
Hull J., White A. (1990a) Pricing interestrate derivative securities, Rev. Fin. Stud., 3.
Rebonato R. (1996) Interest Rate Option Models, John Wiley.
Rebonato R., Cooper I.A., (1996) The limitations of simple twofactor interest rate
models, Journ. Fin. Engin., March.
Rebonato R. (1997) A class of arbitragefree lognormalshortrate twofactor models,
accepted for publication in Applied Mathematical Finance.
Rebonato R. (1998) On the pricing implications of the joint lognormal assumptions
for the swaption and caps markets, submitted to Net Exposure. ß 1998 Soraya Kazziha and Riccardo Rebonato
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