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4 Worked-Out Example 1: Caplets and a Two-Period Swaption

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.



636



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

639



640



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)



642



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



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