4 Worked-Out Example 1: Caplets and a Two-Period Swaption
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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
634
CHAPTER 19 THE LIBOR MARKET MODEL
from time T1 to time T2 , σ2,2 , and the correlation between the two forwards from time T0
to time T1 , ρ. Looking at Equation (19.30), however, one can readily see that there are
2 T = (18%)2 T , for instance, by accepting
inﬁnitely many solutions. One can obtain σSR
1
1
a constant volatility for the second forward rate (σ2,1 = σ2,2 = 20%) and imposing a
lower-than-one correlation (case (a)); or perhaps by having a perfect correlation between
the forward rates and a volatility for the second forward rate from time T1 to time T2 ,
σ2,2 , lower than 20% (case (b)); or in many other ways. Let us explore more carefully
what each choice entails.
If the ﬁrst route is chosen, i.e. if one chooses ρ < 1, one can ﬁnd a solution even
with ﬂat (20%) instantaneous volatilities for the second forward rate. In the second case
(ρ = 1), on the other hand, ﬁnding a solution requires a two-step procedure. First, one
must decrease,5 from its average level, the instantaneous volatility of the second forward
rate from T0 to T1 so as to recover the volatility of the swap rate:
2
SR 2 = w1 f1 σ1,1 + w2 f2 σ2,1
σSR
σ2,1 =
2
→
σSR SR − w1 f1 σ1,1
w2 f2
(19.33)
Having done that, one must consistently increase the volatility during the second period
above its average value in such a way that the second caplet is correctly priced:
2
2
σ22 (T2 − T0 ) = σ2,1
(T1 − T0 ) + σ2,2
(T2 − T1 )
2
σ2,2
=
2 (T − T )
σ22 (T2 − T0 ) − σ2,1
1
0
T2 − T1
→
(19.34)
In between these two extreme cases (i.e. perfect instantaneous correlation or ﬂat constant
volatility for the second forward rate throughout its life) there obviously exist inﬁnitely
many intermediate possible solutions.
This example allows one to draw two important conclusions:
1. an imperfect correlation is strictly necessary in order to account for lower-thanweighted-average swaption volatilities only if the instantaneous volatilities of forward rates are assumed to be constant throughout the life of the forward rate itself;
2. if a convincing explanation of the observed market volatility of swaptions requires
signiﬁcantly non-constant, time-dependent instantaneous volatilities of the underlying forward rates, there are important implications about the evolution over time of
the term structure of volatilities.
Let us look at the second point in more detail. Note that, for the simple example just
considered, at time T0 the term structure of volatilities (i.e. the function that gives the
average volatilities of forward rates of different maturities) was ﬂat (at 20%) for both
forwards. This term structure of volatilities remains unchanged for case (a); in case (b),
5 In order to say with certainty that the volatility will have to be decreased one would have to know the
exact values of w1 , w2 , f1 and f2 . However, in practice, most reasonable values for these quantities give rise
to a lower level of the volatility of the second forward rate from time T0 to time T1 .
19.5 WORKED-OUT EXAMPLE 2: SERIAL OPTIONS
635
however, the term structure at time T1 has to be different (higher) for the ‘front’ forward
(the only one still ‘alive’ in this simple example) to attain its correct total variance, i.e.
to price the time-2 caplet correctly.
This observation allows us to draw a more general conclusion: any choice of the
apportioning of the volatility of the second forward rate throughout its life will uniquely
determine the evolution of the term structure of volatilities. If we choose as our criterion
for a ‘good’ time dependence of the volatility of the forward rates the time homogeneity
of the resulting term structure of volatilities, we might obtain one result. If we require that
market swaption prices should be recovered as well as possible (for a given correlation
function), we might obtain a different one. Luckily, I discuss in Rebonato (2002) that the
‘solutions’ for the time dependence of the forward-rate volatility functions arrived at via
these two conceptually very different routes, while by no means identical, do display a
reassuring degree of internal coherence.
A second important example can further illustrate the importance of the time dependence of the volatility of each forward rate, namely the case of serial options presented
below.
19.5
Worked-Out Example 2: Serial Options
Let us consider the case of an option expiring at time T1 , whose payoff depends on the
value at time T1 of the forward rate f (t, T2 , T3 ), spanning the period [T2 T3 ]. Such an
option is sometimes called a ‘serial option’. See Figure 19.2. Note that the expiry time of
the option (T1 ) is before the reset time of the forward rate (T2 ), i.e. T2 > T1 . Therefore at
time T1 the forward rate that determines the payoff has not yet reset. In order to determine
the option payoff, the buyer and the seller of the option have to obtain a number of market
quotes of the forward rate and ‘distill’ from these a value for f (T1 , T2 , T3 ).
Recall that via the market price of a caplet expiring at time T2 one obtains information
about the total variance of the forward rate f (t, T2 , T3 ) from the trade date (‘today’) to
the forward expiry (T2 ), i.e. the value of the integral
T2
0
σT2 (u)2 du
(19.35)
where σT2 (u) indicates the instantaneous volatility at time u of the forward rate expiring
at time T2 . In order to price the serial option one would need the value of the integral
T1
0
σT2 (u)2 du
(19.36)
f (t, T2, T3)
t (0)
T1
T2
T3
Figure 19.2 The important event times for a serial option: the thick arrow indicates the period
spanned by the forward rate, i.e. the period [T2 T3 ]; the option expiry (at time T1 ) is indicated by
the thin arrow.
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CHAPTER 19 THE LIBOR MARKET MODEL
Unfortunately, there is no plain-vanilla instrument whose value directly depends on this
quantity. The price of the caplet expiring at time T1 , in fact, does give information about
T1
0
σT1 (u)2 du
(19.37)
but, as discussed in Section 3.5 of Chapter 3, this is the variance of a different instrument,
which has no direct bearing on the value of Equation (19.35) above.
What if the expiry of the serial option coincided with the expiry of a two-period
swaption, spanning the forward rates f (t, T1 , T2 ) and f (t, T2 , T3 ), i.e. the forward rates
that reset at times T1 and T2 ? (To lighten notation I will, in what follows, sometimes refer
to these forward rates as f1 and f2 , respectively.) Could the combined information from
the two caplets and this swaption be enough to determine the variance of f (t, T2 , T3 ) from
today to the serial option expiry? Unfortunately this is not the case. This can be seen as
follows. If one knew the correlation between f1 and f2 from time t0 to time T1 , then,
equating the implied volatility of the swaption with the expression given in the previous
worked-out example involving the volatilities of the two forward rates, one would obtain:
2
T1
σSR1,2 (T1 )2 SR1,2
w12 f12 σ1 (T1 )2 T1 + w22 f22 σ2 (T1 )2 T12 + 2f1 f2 w1 σ1 (T1 )T1 w2 σ2 (T1 )T1 ρ
(19.38)
If we knew the correlation, the only unknown would be the quantity σ2 (T1 ), which could
therefore be extracted from the price of the swaption and of the T1 -expiry caplet. This
quantity would then permit the unambiguous pricing of the serial option described above.
Unfortunately, the whole construction is predicated on the correlation between the two
forwards being known with certainty. In reality, for any choice of the (time-dependent)
quantity ρ there corresponds a different possible choice for σ2 (T1 ). The value of serial
options cannot therefore be uniquely determined by no-arbitrage arguments from the prices
of the liquid plain-vanilla instruments. It is easy to show that introducing more caplets or
swaptions would not help.
Similar considerations can be extended to the case of forward-starting swaptions (i.e.
swap options that expire before the start of the underlying swap). A moment’s reﬂection
shows in fact that serial options are just a particular case of forward-starting swaptions
(i.e. one-period forward-starting swaptions).
19.6
Plan of the Work Ahead
In the present chapter I have presented some formulations for the evolution of the forward rates that are useful for recovering exactly the current market prices of caplets. I
have stressed that this market information constrains but does not pin down uniquely the
coefﬁcients of the model. It would be tempting to make use of additional market information from the traded prices of swaptions. Unfortunately these instruments bring into play
an additional unknown quantity, i.e. the instantaneous correlation. If we could observe
reliable prices for serial options (and/or forward-starting swaptions) of a great number of
maturities and expiries we could close the circle, but unfortunately this is not the case.
19.6 PLAN OF THE WORK AHEAD
637
So, the plan of the work ahead is, in a nutshell, how to choose in a desirable way the
time dependence of the volatility functions (i.e. how to choose the functions σi (t)), and
how to apportion this time-dependent volatility to the different modes of deformation of
the yield curve (i.e. how to choose the coefﬁcients {bj k }). In a way, stating our goal this
way does not say much, because, once these quantities are given, there is absolutely nothing else to a fully speciﬁed LIBOR market model. Any choice we might dream up recovers
the market prices of the hedging instruments (assumed to be caplets) and corresponds to
an arbitrage-free evolution for the forward rates. We want to choose these input quantities, however, in such a way that the caplets are correctly priced, and ‘something else’
ends up having desirable properties. The ‘something else’ might be the observed prices
of swaptions; or a statistically determined correlation matrix; or the time-homogeneity
desideratum that the future, as seen by the model, should look at least approximately like
the present. The art of calibrating the LIBOR market model boils down to obtaining this
‘something else’ in a coherent and ﬁnancially meaningful way. This is the topic succinctly
addressed in the remaining chapters of Part III. A more detailed and in-depth treatment
can be found in Rebonato (2002).
Chapter 20
Calibration Strategies for the
LIBOR Market Model
20.1
Plan of the Chapter
Three important calibration problems are dealt with in this chapter. The general assumption underlying the treatment presented below is that we have a set of market and/or
historical data that we would like our implementation of the LIBOR market model to
recover, either exactly or as best we can. This ‘target set’ need not be today’s statistical
or market data, because we might want to impose exogenous requirements (trading views)
on future prices, as we implicitly do when we require a certain behaviour for the evolution
of the term structure of volatilities. This exercise must be undertaken with care, but it is
none the less possible: for instance, many (actually, inﬁnitely many) future term structures
of volatilities are compatible with today’s market prices, as discussed in Section 19.2 of
Chapter 19.
The question I address in this chapter therefore is: How can we implement the LIBOR
market model in such a way that it is compatible with this target information set?
Note that I am not addressing the conceptually separate question: to what set of current
prices (caplets?, all swaptions? co-terminal swaptions?) should I simultaneously try to
calibrate the model? I will assume that this choice has been made elsewhere, perhaps on
the basis of the discussion in Chapter 1.
A full treatment of the topics in this chapter is to be found in Rebonato (2002).
20.2
The Setting
From a computational point of view, the type of problem for which the LIBOR market
model constitutes an ideal tool is the pricing of discrete-look, path-dependent derivatives
in a multi-factor framework. The payoff of the derivative product should only depend
on a ﬁnite number of price-sensitive events, because the LIBOR market model evolves
a set of discrete LIBOR rates in continuous time. Ideally, it should be path dependent,
because the forward-rate process implied by the LIBOR market model is in general
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CHAPTER 20 CALIBRATION STRATEGIES FOR THE LMM
non-Markovian and therefore does not lend itself to straightforward mapping onto recombining trees. Backward induction is therefore difﬁcult with the LIBOR market model,
while a forward-induction (Monte Carlo) evolution is very easy. Finally, the approach
can easily be multi-factor because one does not have to (or, rather, cannot) build recombining trees that pay an exponential price in the number of state variables (the so-called
‘curse of dimensionality’). The computational burden of carrying out a Monte Carlo
simulation, on the other hand, grows in an approximately linear fashion with the number of evolved state variables. Examples of securities that readily lend themselves to
pricing using the combined LIBOR market model/Monte Carlo approach are therefore
trigger swaps, indexed-principal swaps, knock-out caps, one-way ﬂoaters, ratchet caps,
discrete-sampling average-rate caps, obligation ﬂexi caps, etc.
We have seen in the previous chapter that, for all these applications, it is very easy
to calibrate the LIBOR market model to caplet volatilities. Let us take, in fact, the third
formulation of the forward-rate dynamics when m factors shock the yield curve:
dfit
= µi ({f t }, t) dt + σit
fit
t
bik
dzk
(20.1)
k=1,m
We have seen that if the square of the chosen instantaneous volatility functions σit ≡
σ (t, Ti ) properly integrate to
Ti
σ (Ti )2 Ti =
σ (u, Ti )2 du
(20.2)
0
and if
2
bik
=1
(20.3)
k=1,m
t
, they
then all the caplets will be correctly priced. As for the quantities (‘loadings’) bik
can be interpreted as the sensitivities at time t of the ith forward rate to the kth shocks.
t
Note that the loadings bik
can have a calendar-time dependence, and are speciﬁc to each
individual forward rate via the ﬁrst index. They cannot, however, be of the form bik (fi , t)
and preserve the log-normal distributional feature for the forward rate fi . Therefore the
third formulation truly represents the most general speciﬁcation of any m-factor model
consistent with log-normal forward rates.1
20.2.1 A Geometric Construction: The Two-Factor Case
Can one gain some intuitive (or geometric) understanding for condition (20.3)? Let us
consider the case when only two factors are allowed to shock the yield curve (m = 2).
Then, ignoring the drifts, which are irrelevant for the discussion, one can write
dfit
t
t
= σit bi1
dz1 + bi2
dz2
fit
(20.4)
1 For a given choice of numeraire, the drift of a given forward rate is in general non-zero, and its distribution
is therefore not exactly log-normal. The caplet prices, however, are exactly consistent with the Black prices
obtained from a log-normal density in the appropriate terminal measure. See Rebonato (2002) for a discussion
of this subtle point.
20.2 THE SETTING
641
and condition (20.3) simply becomes
2
2
bi1
+ bi2
=1
(20.5)
Equation (20.5) is quite interesting. Recall, in fact, that, for any angle θ , it is always
true that
(20.6)
sin2 (θ ) + cos2 (θ ) = 1
Therefore any angle θ , 0 ≤ θ < π, speciﬁes a possible set of coefﬁcients, bi1 , bi2 , and
therefore a possible allocation of the loadings onto the two Brownian motions compatible
with the recovery of the caplet prices. How can we choose among this inﬁnity of solutions?
One possible way is to look at the correlation function implied by a given choice of θ .
What we would like to do is to relate the angles θ to the model-implied correlation
function, thereby constraining the inﬁnity of solutions. To calculate the model correlation
between forward rate j and forward rate k, ρj k , we must evaluate
t
dfkt dfj
E
fkt fjt
ρj k =
dfjt dfjt
E
fjt fjt
Let us start from the denominator. For the two-factor case, the quantity E
becomes
E
dfkt dfkt
fkt fkt
(20.7)
dfkt dfkt
E
fkt fkt
dfkt dfkt
fkt fkt
simply
= σk2 [bk1 dz1 + bk2 dz2 ] [bk1 dz1 + bk2 dz2 ]
2
2
= σk2 (bk1
+ bk2
) dt = σk2 (sin2 θk + cos2 θk ) dt = σk2 dt
(20.8)
where use has been made of the fact that we have chosen to work with orthogonal
Brownian increments, and therefore
E[dz1 dz2 ] = 0
(20.9)
E[dz1 dz1 ] = E[dz2 dz2 ] = dt
(20.10)
Similarly,
E
dfjt dfjt
fjt fjt
= σj2 (bj21 + bj22 ) dt = σj2 (sin2 θj + cos2 θj ) dt = σj2 dt
(20.11)
Therefore the denominator is just equal to
E
dfjt dfjt
fjt fjt
E
dfkt dfkt
fkt fkt
= σk σj dt
(20.12)
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CHAPTER 20 CALIBRATION STRATEGIES FOR THE LMM
As for the numerator, by again making use of the orthogonality relationships (20.9) and
(20.10), one readily computes
E
t
dfkt dfj
fkt fjt
= σk [bk1 dz1 + bk2 dz2 ] σj bj 1 dz1 + bj 2 dz2
= σk [sin θk dz1 + cos θk dz2 ] σj sin θj dz1 + cos θj dz2
= σk σj [sin θk sin θj 1 + cos θk cos θj ] dt
= σk σj [cos(θk − θj )] dt
(20.13)
Putting all the pieces together we obtain
ρj k = [cos(θk − θj )]
(20.14)
This equation shows, that, in the two-factor case, the correlation between two forward rates
is purely a function of the difference between the ‘angles’ that we saw were associated
with the loadings. So, for any of the inﬁnite choices of angles such that the caplets are
correctly priced, there corresponds a particular correlation function. I will show in what
follows that this simple observation can provide a useful tool to calibrate the LIBOR
market model to a set of caplet prices and to an exogenous correlation matrix.
20.2.2 Generalization to Many Factors
Can we systematically generalize the results just obtained to more than two factors? This
can be easily achieved by noting that, when we are dealing with m factors, the capletpricing condition (20.3) simply deﬁnes the co-ordinates of a point on the surface of a
hypersphere of radius 1. Therefore, by recalling the expressions for the polar co-ordinates
of a point on the surface of a unit-radius hypersphere, one can immediately generalize to
m factors by writing:
k−1
bik = cos θik
sin θij ,
k = 1, 2, . . . , m − 1
(20.15)
j =1
k−1
bik =
sin θij ,
k=m
(20.16)
j =1
At the moment, being able to express the caplet-pricing condition in terms of arbitrary
angles might seem to provide little, if any, advantage. We will show later in the chapter
that the introduction of these new variables {θ } is actually computationally very useful,
since it allows us to cast a constrained-optimization problem in terms of an equivalent
unconstrained one.
20.2.3 Re-Introducing the Covariance Matrix
Let us go back to the case when as many factors as forward rates are retained. Let us
consider any path-dependent pricing problem such that the expiries and maturities of a
20.3 FITTING AN EXOGENOUS CORRELATION FUNCTION
643
set of n forward rates constitutes all the dates when price-sensitive events occur. There is
therefore a direct correspondence between the number of price-sensitive events and the
number of forward rates necessary to describe a given LIBOR problem. After each of
these price-sensitive events takes place, the number of forward rates left in the problem
is therefore reduced by one. Let then h(i) be the number of forward rates ‘alive’ at timestep i. (This coincides with the number of residual price-sensitive events still to be set
at time-step i.) As discussed in detail in Sections 5.3 and 5.4 of Chapter 5, what any
model (implicitly or explicitly) produces for this type of problem is a series of discrete
covariance elements of the type
Covij k =
ti+1
ti
σ (u, Tj )σ (u, Tk )ρj k du
(20.17)
If, at each time-step, the chosen number of factors, m, is equal to the number of forward
rates necessary to describe the yield curve, h(i), the problem of ﬁnding a formulation
of the LIBOR market model perfectly consistent with an arbitrary set of instantaneous
volatilities and the correlation matrix is conceptually straightforward. This task can in fact
always be exactly accomplished simply by orthogonalizing the time-dependent covariance
matrix above and by working with the associated eigenvectors and eigenvalues (Principal
Component Analysis). These covariance matrices, obtained from our choice of volatility
and correlation functions, can be labelled ‘desired’ covariances, and should be distinguished from the covariance matrices that we could obtain if we had to use fewer factors
than forward rates.
In the context of the present discussion, how the volatilities and correlation might
have been chosen (i.e. on the basis of market information, or of statistical analysis, or
of a combination of the two) is irrelevant. The important point is that if we work with a
full-factor LIBOR market model we have just enough degrees of freedom to recover any
exogenous set of covariance elements we might want. Perhaps our choice of covariance
elements might be ‘poor’, in the sense that it will imply an unsatisfactory dynamics for the
yield curve (indeed, the greatest danger of the LIBOR market model lies in its ﬂexibility);
or perhaps no set of covariance elements will satisfactorily recover the market prices of
caplets and swaptions. However, we only have to make ‘hard choices’ (i.e. we only have
to use a parameterization of the LIBOR market model that will fail to recover whatever
desired covariance elements we might have chosen to use) if we have to work with fewer
factors than forward rates. This will rarely be the case for path-dependent problems, where
straightforward Monte Carlo simulations can be used, and using a very large number of
factors is a feasible computational proposition. The need to reduce (sometimes drastically)
the dimensionality of the problem occurs routinely, however, when the option problem
at hand is compound in nature, as is the case for Bermudan swaptions. We therefore
tackle in the next sections the problem of how to reduce the dimensionality of a LIBOR
implementation in a systematic way. We will see that this problem can be directly linked
to the calibration of the LIBOR market model to a correlation function when the number
of factors is smaller than the number of forward rates alive.
20.3
Fitting an Exogenous Correlation Function
We have seen that if one retained m(i) = h(i) factors at each time-step there would be
just enough degrees of freedom in order to specify any feasible exogenously speciﬁed