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9 Appendix I: Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate

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



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