9 Appendix I: Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate
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18.9 VARIANCE OF THE LOGARITHM OF THE INSTANTANEOUS SHORT RATE 623
T
But the quantity ln r(T ) exp
a(s) ds can be written as
0
T
ln r(T )
T
a(s) ds = ln r(0) +
0
t
a(t)b(t) exp
0
T
+
a(s) ds] dt
0
t
σ (t) exp
0
a(s) ds dz(t)
(18.48)
0
and, therefore
T
ln r(T ) = exp −
a(s) ds ln r(0)
0
T
+ exp −
0
0
a(s) ds dt
0
T
a(s) ds
t
a(t)b(t) exp −
0
T
+ exp −
T
a(s) ds
t
σ (t) exp
0
a(s) ds dz(t)
0
Remembering that, for any deterministic function f (t),
t
var
t
f (u) dz(u) =
0
f (s)2 dt
(18.49)
0
it then follows that
var[ln r(T )] = E[(ln r(T ))2 ] − (E[ln r(T )])2
T
= exp −2
0
T
a(s) ds
0
t
σ (t)2 exp 2
0
a(s) ds dt
(18.50)
Chapter 19
Volatility and Correlation in the
LIBOR Market Model
19.1
Introduction
Until the mid-1990s no clear consensus had emerged among practitioners or academics
as to the ‘favourite’ or most widely used interest-rate model.1 Trade-offs always had to be
made between ease of calibration (or, more often than not, lack thereof), availability of
closed-form solutions, realism of the distributional assumptions, etc. The common feature
of almost all these earlier models was that the yield curve was assumed to be driven
by the unobservable short rate (plus, sometimes, another equally unobservable variable,
such as the yield of the consol, or the variance of the short rate). In all cases, the trader
had to perform, explicitly or implicitly, a transformation between the input values of the
unobservable state variables, and those quantities, such as the term structure of implied
volatilities of forward or swap rates, that she could observe in the market. The model,
in this respect, acted as the black box that transformed what the user could input (e.g.
the volatility of the short rate) into what the user would have liked to be able to input
(e.g. caplet or swap volatilities).
The Heath–Jarrow–Morton (1989) (HJM in what follows) approach radically changed
the set of driving state variables by focusing attention on the inﬁnite number of (instantaneous) forward rates that describe the yield curve. These instantaneous forward rates
were still, strictly speaking, unobservable, but they were somewhat closer to what the
user had direct access to (i.e. discrete-tenor forward LIBOR rates).
In its original form, however, the HJM model is hardly more user-friendly, when it
comes to calibration to market data, than traditional short-rate-based models. A number of
papers appeared in these early years proposing more or less cumbersome methodologies
to specify the volatility functions in such a way that cap prices could be approximately
recovered. Furthermore, the fact that no non-exploding solutions exist for log-normally
1I
have surveyed the evolution of interest-rate models in Rebonato (2004).
625
626
CHAPTER 19 THE LIBOR MARKET MODEL
distributed instantaneous forward rates2 did not make matters any simpler. Practitioners
decided to avert their eyes from the problem, and pretend that it did not exist; academics
proposed several partial and, one has to say, highly artiﬁcial solutions.
The LIBOR market model approach (LMM in what follows), pioneered by Brace et al.
(1995), Jamshidian (1997), Musiela and Rutkowski (1997), Rutkowski (1998) and others,
radically changed the situation. Now direct market observables (i.e. LIBOR–forward or
swap–rates and their volatilities) became the building blocks of the new methodology,
earning the new approach the title of ‘the market model’. The LMM formalism quickly
proved to be ideally suited for discrete-look, path-dependent derivatives products in a
multi-factor framework. Acceptance of the LMM approach was facilitated by the fact
that it shared with the HJM model one fundamental insight, which had by then become
familiar among practitioners, namely that the no-arbitrage conditions describing the deterministic part of the evolution of forward rates could be expressed purely in terms of their
correlations and volatilities. As a result, the research emphasis became focused on the
correct speciﬁcation of the time-dependent instantaneous volatility of the underlying state
variables, i.e. the discrete LIBOR rates.
At the same time, it became progressively clear that in order to achieve de-correlation
among forward rates introducing a non-ﬂat volatility for the forward rates could provide
a more important and often more realistic mechanism than invoking a large number of
driving factors (see the discussion in Chapter 5). In other words, the attention gradually
shifted from the instantaneous to the terminal correlation amongst rates. In a nutshell,
the simultaneous speciﬁcation of these time-dependent volatilities and correlations became
not one of the problems, but the problem, in the speciﬁcation of the LMM model.
Calibrating the LMM model simply to caplet or to European swaption prices (if the
forward-rate-based or the swap-rate-based implementation is chosen, respectively) is very
simple indeed, and, as shown below, can be accomplished exactly in an inﬁnity of ways.
The fact that so many possible solutions exist creates however a problem: each speciﬁcation of the time-dependent volatilities will give rise to a different degree of terminal
correlation amongst the forward rates, and to a different evolution of the term structure
of volatilities. The question therefore arises as to whether it is in some way possible
to choose among these exact (or quasi-exact) calibrations to caplet prices so that other
desirable properties are recovered. This is one of the questions addressed in this chapter.
19.2
Specifying the Forward-Rate Dynamics
in the LIBOR Market Model
19.2.1 First Formulation: Each Forward Rate in Isolation
In order to see how the forward rates that describe the yield curve can be evolved in an
arbitrage-free way one can proceed as follows. Let us assume ﬁrst that, perhaps on the
basis of the discussion presented in the following chapters, the functional dependence on
calendar time, t, and on the forward-rate expiry, Ti , of the instantaneous volatility, σi , of
2 ‘Explosion’ in this context means that a log-normally-distributed instantaneous forward rate will reach
inﬁnity in a ﬁnite time with probability one.
19.2 FORWARD-RATE DYNAMICS IN THE LIBOR MARKET MODEL
627
the ith forward rate has been chosen:
σi = σ (t, Ti )
(19.1)
Similarly, we will also assume that the trader has chosen a correlation function, ρ:
ρij = ρ(t, Ti , Tj )
(19.2)
These two important topics are dealt with in Chapters 21 and 22. Let then σ (Ti ) (or,
sometimes, more brieﬂy, just σi ) be the implied Black volatility of the forward rate of
maturity Ti , fi (Ti ) (often abbreviated in what follows as fi ). Since we are, for the moment,
ignoring the possibility of smiles, the instantaneous and the implied Black volatilities are
linked by the relationship
σi2 Ti =
Ti
σ (u, Ti )2 du
(19.3)
0
If Equation (19.3) is satisﬁed, the Black price of the ith caplet is exactly recovered.
Condition (19.3) is therefore referred to in what follows as the caplet-pricing condition.
If we consider one forward rate at a time and impose that its process should be an
arbitrage-free diffusion with deterministic volatility, its evolution can be most simply
described by a SDE of the form
dfi
= µi ({f }, t) dt + σi dwi
fi
(19.4)
where the drift term µi ({f }, t) is derived by invoking absence of arbitrage. The formal
solution of Equation (19.4) (see Chapter 5), given today’s value, fi (0), for the forward
rate, is given by
t
fi (t) = fi (0) exp
0
µi ({fu }, u) − 12 σ (u, Ti )2 du exp
t
σ (u, Ti ) dwu
(19.5)
0
As mentioned, in the equation above the drift µi ({fu }, u) reﬂects the no-arbitrage condition, and is, in general, a function of the instantaneous volatilities of the forward rates, of
the correlation amongst them, and of the forward rates themselves. Once the volatility and correlation functions have been speciﬁed, these drifts are therefore uniquely
determined. The expression for the drift depends on the chosen numeraire. Obtaining
it would at this point entail a rather long detour. The interested reader could see, for
example, Jamshidian (1997), or Rebonato (2002) for a simpler derivation. Since this
book is about volatility and correlation, for our purposes the most important feature of
these drifts is that they are purely a function of forward-rate volatilities and correlations. Once these functions are chosen, the drifts follow automatically, and therefore
they are no longer explicitly dealt with in this chapter. In a way, despite the fact that
so much emphasis is often devoted to obtaining expressions for the no-arbitrage drifts,
the ‘real action’ with the LMM lies in the choice of the input volatility and correlation
functions.
628
CHAPTER 19 THE LIBOR MARKET MODEL
For future reference, if n forward rates describe the yield curve, Equation 19.4 can be
written in matrix form as:
df
= µ(f, t) dt + S dw
f
(19.6)
where the vector dff is to be understood in a component-wise fashion (i.e. its ﬁrst, second,
df2
dfn
1
. . . , nth elements are df
f1 , f2 , . . . , fn ), and the [n × n] matrix S is a time-dependent
diagonal matrix whose iith element is given by the instantaneous volatility of the ith
forward rate:
σ1 . . . . . .
0
0
σ2
0
0
(19.7)
S=
··· ··· ··· ···
0
0
. . . σn
An exogenous correlation structure between forward rates can be recovered by requiring
that
ρ dt = dw dwT
(19.8)
where ρ is the desired [n × n] correlation matrix.
Equations (19.4)–(19.8) give a perfectly adequate description of the LMM forwardrate dynamics if as many factors as forward rates are retained, but provide no obvious
indication as to how the dimensionality of the problem can be reduced. This problem can
be addressed as follows.
19.2.2 Second Formulation: The Covariance Matrix
Let us now assume that only m independent Brownian shocks (factors) describe the
evolution of the forward rates. Let us rewrite Equation (19.6) as
df
= µ(f, t) dt + σ dz
f
(19.9)
where now dz is an [m × 1] vector whose elements are the increments of m orthogonal
Brownian motions:
dz dzT = I dt
(19.10)
I is the identity matrix:
1
0
I =
...
0
0
1
...
0
...
...
1
...
0
0
...
1
(19.11)
19.2 FORWARD-RATE DYNAMICS IN THE LIBOR MARKET MODEL
and the (j, k)th element of the [n × m]
σ11
σ21
σ =
...
σn1
629
matrix σ ,
σ12
σ21
...
σn2
...
...
...
...
σ1m
σ2m
...
σnm
(19.12)
contains the responsiveness of the j th forward rate to a random shock from the kth factor.
So, for instance, if the ﬁrst factor had been chosen to represent shocks to the level of the
yield curve, the column vector
σ11
σ21
(19.13)
...
σn1
would give the responsiveness of the ﬁrst, second, . . . , nth forward rate to a level change;
if the second factor represented shocks to the slope of the yield curve, the column vector
σ12
σ22
(19.14)
...
σn2
would give the responsiveness of the ﬁrst, second, . . ., nth forward rate to a slope change;
etc.
We have seen in Chapter 5 that the covariance matrix between the forward rates is
the crucial quantity that determines the stochastic part of their evolution. I have shown
in Rebonato (2002) that the same covariance matrix elements also constitute the most
important component of the deterministic part of the forward-rate evolution. The formulation presented in this section is therefore particularly useful, because it can be shown by
straightforward matrix multiplication that the covariance matrix, , between the forward
rates is given by
= σσT
Exercise 1 Obtain Equation (19.15) by remembering that covar
and explicitly calculating the terms
dzj dzk = δj k .
dfj dfk
dfj df k
(19.15)
dfj
fj
k
, df
fk = E
dfj dfk
fj fk
,
using Ito’s rules: dt dt = 0, dzj dt = 0 and
Another useful feature of the formulation (19.9) is that the caplet-pricing condition
(19.3) can be automatically satisﬁed (and, for m > 1, in inﬁnitely many ways) as long as
σj2k = σj2
k=1,m
Exercise 2 Prove Equation (19.16).
(19.16)
630
CHAPTER 19 THE LIBOR MARKET MODEL
19.2.3 Third Formulation: Separating the Correlation
from the Volatility Term
The second formulation just presented does not distinguish between volatility and correlation information, because it directly uses the forward-rate covariance elements. I discuss
in Rebonato (2002) why it may be desirable to model separately the volatility component
(about which some almost-direct3 information can be obtained from the market prices of
caplets), and the correlation component, which I believe can be more proﬁtably described
on the basis of statistical information.4 This can be accomplished as follows.
Let us start from Equation (19.9), which can be rewritten term-by-term as
dfi
= µi ({f }, t) dt +
fi
σik dzk
(19.17)
k=1,m
and multiply and divide each loading σik by the volatility, σi , of the ith forward rate:
dfi
= µi ({f }, t) dt + σi
fi
σik
dzk
σi
k=1,m
(19.18)
Making use of the caplet-pricing condition (19.16), this can be rewritten as
dfi
= µi ({f }, t) dt + σi
fi
σik
2
k=1,m σik
k=1,m
dzk
(19.19)
If now we deﬁne the quantity bik as
bik ≡
σik
(19.20)
2
k=1,m σik
Equation (19.19) can be rewritten in a more compact way as
dfi
= µi ({f }, t) dt + σi
fi
bik dzk
(19.21)
k=1,m
Expression (19.21) is very useful because, if we denote by b the [n × m] matrix of
elements bj k , it can be readily shown that
bbT = ρ
(19.22)
3 The information is ‘almost’ direct because from the market prices of caplets we only have information
about the root-mean-squared instantaneous volatility. See the discussion in Sections 19.4 and 19.5.
4 Swaption prices do contain some information about the correlation function. However, I have argued in
Rebonato (2002) and Rebonato (2004) why it might not be a good idea to make use of this ‘implied’ information.
See also de Jong et al. (1999). It must be said, however, that my views are not universally accepted (see, for
example, Alexander (2003) and Schoenbucher (2003)). The topic is also discussed at length in Section 1.4 of
Chapter 1.
19.3 LINK WITH THE PRINCIPAL COMPONENT ANALYSIS
631
Exercise 3 Prove Equation (19.22). (Hint: Look at the treatment for the two-factor case
discussed in the next chapter.)
Finally, if the root mean square of the chosen instantaneous volatility is equal to the
Black volatility, the caplet-pricing condition
σj2k = σj2
(19.23)
bj2k = 1
(19.24)
k=1,m
simply becomes
k=1,m
But Equation (19.24) is certainly always satisﬁed as long as we have constructed the
coefﬁcients {bik } using Equation (19.20). We have therefore achieved the desired task of
decomposing the stochastic dynamics for the forward rates in terms of coefﬁcients, σi ,
purely dependent on their volatilities, and coefﬁcients, b, which solely depend on the
correlation structure. I will show in the next chapters how this decomposition can be
proﬁtably made use of in the calibration of the LMM. Before moving on to this task, it
is useful to highlight the links between the treatment presented so far and the result of
the Principal Component Analysis (PCA) of the correlation matrix. This is undertaken in
the next section.
19.3
Link with the Principal Component Analysis
The formulations presented above allow for a transparent link with the results of the PCA.
If as many factors as forwards rates are retained, one particular set of loadings, say, {bik },
is linked in fact to the eigenvectors, {aik }, and eigenvalues, λk , of the correlation matrix
by the relationship
bik = aik λk
(19.25)
For this particular choice, in addition to the constraint (19.24), also the orthogonality
relationships
bik bij = δj k
(19.26)
i=1,n
will be satisﬁed. (Note carefully that the sum is now over forward rates, not over the
independent Brownian motions.) If we are using a full-factor version of the LMM, there
is no special merit in this particular rotation of the axes, apart, perhaps, from being able
to use the intuition afforded by the well-known interpretation of the PCA eigenvectors as
level, slope, curvature, etc. However, if we had to use fewer factors than forward rates, it
is important that the modes of deformation retained should capture as much as possible of
the variability across forward rates. Clearly, if one simply ‘threw away’ the eigenvectors
of order higher than m (where m is the number of retained factors), the caplet pricing
632
CHAPTER 19 THE LIBOR MARKET MODEL
condition would no longer be satisﬁed. What is required therefore is a reallocation of the
overall variance from higher-frequency modes of deformation to the retained factors. In
what follows I will present a technique to carry out this reallocation in an ‘optimal’ way
(where ‘optimal’ means ‘in such a way that an exogenous correlation matrix can best be
recovered’).
The three formulations presented so far provide an ‘empty receptacle’ into which the
volatilities and correlation functions can be ‘poured’. We have not said anything yet as to
how these functions might be chosen. This is the topic of the next three chapters. In order
to put this treatment into context, and to provide some intuition, I conclude this chapter
with two case studies that
• highlight the link between the current term structure of volatilities and the future
instantaneous volatilities;
• show (again) the difference between instantaneous and terminal correlation;
• display the intrinsic incompleteness of the caplet and swaption markets.
19.4
Worked-Out Example 1: Caplets and a Two-Period
Swaption
Let us consider the case of two forward rates, f1 and f2 , and the swap rate SR12 expiring
at the same time as the ﬁrst forward rate and maturing at the same time as the second:
f1 = f1 (t, T1 , T2 ), f2 = f2 (t, T2 , T3 ), SR12 = SR12 (t, T1 , T3 ). See Figure 19.1. The
underlying swap rate, SR12 (SR in what follows), can be written as a linear combination
of the two forward rates:
SR12 = SR = w1 f1 + w2 f2
(19.27)
with w1 and w2 the weights given in Chapter 20, Equation (20.32), or derived, for
example, in Rebonato (2002).
Let us consider a discrete trading horizon made up of two dates only (the reset dates
of the caplets), and, for the sake of simplicity, let us constrain all quantities, such as
volatilities or correlations, to be piecewise constant over each of the two possible ‘timesteps’. Let us further assume that the two caplets associated with the forward rates f1
SR12
f1
f2
t
T1
T2
T3
Figure 19.1 The reset and expiry times of the two forward rates, f1 and f2 , and of the swap rate
SR12 . The thick arrows indicate where the relative rate expires. The thin arrow shows the period
covered by that rate; the arrow ends at maturity time.
19.4 WORKED-OUT EXAMPLE 1: CAPLETS AND A TWO-PERIOD SWAPTION 633
and f2 trade in the market with implied Black volatilities, σ1 and σ2 , of 20%, and that
the two rates display an instantaneous correlation ρ. Finally, let the swaption spanned by
these two forward rates (i.e. the option to enter at time T1 a two-period swap starting at
time T1 and maturing at time T3 ) have a market implied (Black) volatility, σSR , of 18%.
We want to allow for the possibility that the second forward rate might assume a
different instantaneous volatility from time T1 to time T2 . To do this, I will assign two
volatilities for the second forward rate, the ﬁrst prevailing from time T0 to time T1 :
σ2,1 ≡ σ (t, T2 ),
T0 ≤ t ≤ T 1
(19.28)
T1 < t ≤ T 2
(19.29)
and the second from time T1 to time T2 :
σ2,2 ≡ σ (t, T2 ),
For symmetry of notation, I shall also denote the volatility of the ﬁrst forward rate over the
ﬁrst time interval by σ1,1 ; however, the fuller notation is in this case redundant, because
there is only one period before the expiry of the ﬁrst forward rate, and we have assumed
the volatilities to be piecewise constant.
I will show in the next chapter (see also Jaeckel and Rebonato (2003) and references
therein) that one can write a reasonably accurate approximation for the volatility of the
swaption, σSR , from time T0 to its expiry (time T1 ) in terms of the volatility of and the
correlation between the underlying forward rates as
2
σSR
SR 2
2
2
= w12 f12 σ1,1
+ w22 f22 σ2,1
+ w1 w2 f1 f2 σ1,1 σ2,1 ρ
(19.30)
See Section 20.5 of Chapter 20 for a discussion of the approximations required to obtain
Equation (19.30). Note carefully that in Equation (19.30) there appears the instantaneous
volatility of the second forward rate only from time T0 to time T1 .
Let us now require that both caplets and the swaption should be simultaneously exactly
priced. As for the ﬁrst caplet, no choice is left but to ensure that its unconditional variance
should indeed be equal to the value implied by the Black market volatility. This uniquely
determines σ1,1 via the relationship
σ12 (T1 − T0 ) =
T1
T0
2
2
σ1,1
du = σ1,1
(T1 − T0 ) → σ1 = σ1,1
(19.31)
This relationship establishes one term in the RHS of Equation (19.30). Remember, however, that we do want the swaption volatility to be recovered in a way compatible with
the value σ1,1 which has already been ﬁxed, and we also require the caplet expiring at
time T2 to be correctly priced. The pricing of the second caplet will require that
σ22 (T2 − T0 ) =
T2
T0
2
2
σ22 du = σ2,1
(T1 − T0 ) + σ2,2
(T2 − T1 )
(19.32)
The quantities still at our disposal in order to achieve these tasks are therefore the instantaneous volatility of the second forward rate from time T0 to time T1 , σ2,1 , its volatility