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6 Problems with the Predictor–Corrector Approximation for the LMM

6 Problems with the Predictor–Corrector Approximation for the LMM

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748



CHAPTER 24 CEV PROCESSES IN THE CONTEXT OF THE LMM



The first problem is the following. So far we have failed to mention the drifts (i.e.

we have always implicitly assumed that we were working in the terminal measure). This

is perfectly adequate for the pricing of, say, a single caplet, but for this purpose we

would not use a Monte Carlo evolution anyhow. For the pricing of a complex derivative,

several forward rates will have to be evolved at the same time, and no numeraire will

make them all simultaneously driftless. See Rebonato (2002) on this point. Unfortunately,

the drifts contain the forward rates themselves. Therefore, when applying the formulae

proposed above for a given forward rate, its drift will contain forward rates that might

not have been already ‘updated’ by the predictor–corrector procedure. This would create

a problem, because the Hunter et al. (1999) formula for the j th forward rate can contain,

depending on the choice of numeraire, both the initial and the approximate final values

of some or even all the forward rates in the yield curve. It would therefore seem that, in

order to obtain an approximate expression for the terminal value of the j th forward rate,

the correct terminal values of the other rates would be required. Simply using the initial

(‘frozen’) values of the forward rates in the drift for the j th forward rate would create an

inconsistency and, for long jumps, noticeable numerical errors.

This problem can be solved by choosing, as common numeraire for all the forward

rates, the discount bond associated with the terminal measure of the last forward rate.6

When this choice is made, the last forward rate is driftless, the treatment above applies to

this forward rate with no changes, and therefore we can obtain its (approximate) terminal

value. The drift, µ, of the second-, third-, . . . , nth-to-last forward rate contains the last,

last two, . . . , last n − 1 forward rates only:

µfi = −σi (t)

k=i+1,j



σk (t)ρik (t)fk (t)τk

1 + fk (t)τk



(24.49)



Therefore, by updating the last forward rate first, and all the others in descending order,

the drift of the current forward rate contains forward rates that have already been correctly evolved, and for which the initial and (approximate) terminal values are therefore

available. The formulae above can therefore be applied recursively.

The second problem is that the predictor–corrector method presented above was cast

in a single-factor framework. With the LMM, however, we would almost certainly want

to work with many factors. The formulae presented above do extend to several factors,

but soon become unwieldy and numerically very slow. The solution to this problem is to

recast the original LMM problem, expressed, say, in the form

dfi

= µi ({f }, t) dt +

fi



σik dzk = µi ({f }, t) dt + σi

k=1,m



bik dzk



(24.50)



k=1,m



(with dzk the increments of independent Brownian motions) in the equivalent form

dfi

= µi ({f }, t) dt + σi dwk

fi



(24.51)



6 Incidentally, this is also the recommended choice for reasons linked to the Hunter et al. method. See

Rebonato (2002) on this point.



24.6 APPROXIMATION FOR THE LMM



749



with dwk the increments of correlated Brownian motions, and

σi2 =



2

σik



(24.52)



k=1,m



With this formulation each forward rate is shocked by a single Brownian shock, and

the Kloeden and Platen approximations above can therefore be directly applied. The

increments dwk are now no longer orthogonal to each other (as the dzk were), but are

chosen in such a way that the original correlation matrix is still recovered:

E dwj , dwk = ρj k = bbT



jk



(24.53)



This can always be done using a simple modification of the Choleski decomposition (see,

for example, Kreyszig (1993)).

Summarizing: by choosing the numeraire carefully, and by using a system of nonorthogonal axes, the predictor–corrector method can be still be applied to many-forwardrates LMM problems.



Chapter 25



Stochastic-Volatility Extensions

of the LMM

25.1



Plan of the Chapter



I have shown in Chapter 23 that, if one assumes forward rates to be log-normally distributed in their own terminal measure, there exists a strong dependence between changes

in the level of rates and changes in the level of the implied volatility. I have also shown

that if we move from a geometric diffusion to a CEV description of the dynamics of the

forward rates, most of this correlation disappears.

When this transformation of co-ordinates is carried out, the conclusions that one can

draw about the degree of variability in the volatility of forward rates can change dramatically. See Figure 25.1, which I show again for ease of reference. This is particularly true

in periods when the level of rates changes (in percentage terms) dramatically, as indeed

happened during the months of 2002 displayed in this figure. What I am trying to model

in this chapter is therefore the (relatively more limited) variability of the lower curve, not

the dramatic changes of the upper one. I intend to model this relatively modest variability

using a mean-reverting diffusion process for the volatility of the new state variables in

the new co-ordinate system.

Before embarking on this project, though, and looking at the same figure again, one

might wonder whether such an apparently small degree of variability is really an essential

part of the description of the volatility process. The answer is not straightforward. To begin

with, I show in the next chapter that occasionally the rather ‘tame’ picture conveyed by

Figure 25.1 ceases to be valid, and large and sudden changes in the level of volatility

do occur. I will argue in the next two chapters, however, that modelling this feature

using a Brownian diffusion is very unsatisfactory, and that a two-state regime switch

(Markov-chain process), whereby the instantaneous volatility ‘oscillates’ between two

easily identifiable regimes, accounts much better for the empirical evidence. So, even if

bursts of volatility do occur, within each regime the variability of the volatility might still

be rather limited, and the large sudden changes should probably not be described by a

diffusive process.

751



752



CHAPTER 25 STOCHASTIC-VOLATILITY EXTENSIONS OF THE LMM

2.5



2



1.5



1



0.5

Rescaled

Percentage

0

25-Mar-02



14-May-02



03-Jul-02



22-Aug-02



11-Oct-02



Figure 25.1 Rescaled (lower curve) and percentage (upper curve) volatilities (1 × 1 swaption

series) rebased at 1 on 1-Apr-2002. See Chapter 23 for a discussion.



A description in terms of Brownian processes can, however, still make sense within

each regime, and, despite the picture conveyed by Figure 25.1, be rather important. This is

because it is the expectation of future volatility that matters, coupled with the aversion to

volatility risk. Since in the presence of stochastic volatility perfect hedging is no longer

possible,1 a volatility risk premium enters the drift of the volatility process, and the

volatility ‘distilled’ from the traded price need not coincide with the historically observed

one. I have discussed the volatility risk premium in Chapter 13.

The shapes of market smile surfaces provide some indirect corroboration of this view.

I will show in Chapter 27, in fact, that, after the change of variables discussed above,

just positing a two-regime volatility process provides an intriguingly good qualitative

description of the observed smile surface. However, it is only after adding a diffusive

component to the variability of the volatility that a (time-homogeneous) fit of trading

quality can be obtained. It is true that I have made the point throughout this book that

the quality of the fit to the current smile surface is, by itself, a very poor indicator of

the quality of a modelling approach. However, in this case I have systematically built

the different components of the volatility description (CEV change of variables, two

regimes and diffusive variability) on the basis of a clear financial motivation, and the final

agreement between the theoretical and observed smile therefore can provide a meaningful

corroboration of the overall approach.2

1 At least as long as one is hedging purely with the underlying. If one allows for plain-vanilla options to be

added to the set of hedging instruments, one would still need to know their full process, not just their prices

today. See Chapter 13.

2 Needless to say, in order to make this statement I still have to rely on some degree of market efficiency,

i.e. I must assume that the salient features of the ‘true’ volatility dynamics are reflected in the option prices.



25.2 WHAT IS THE DOG AND WHAT IS THE TAIL?



25.2



753



What is the Dog and What is the Tail?



Before starting the description of this approach another observation is in order. If, as

proposed, a strong correlation (or, indeed a functional dependence) between the forward

rates and the level of volatilities is assumed, whenever a large (Brownian) shock to the

forward rates occurs a large change in the volatility will take place as well. Furthermore,

if the change in the level of rates and volatility is large enough, the modelling approach

proposed in the next chapters will suggest that this can give rise to a switch in volatility

regime from a normal to an excited state. Typically, the expected lifetime in the excited

state turns out to be rather short, and the combined effect of a functional dependence with

the forward rates and of the volatility shift is to produce sudden but short-lived bursts in

volatility. In this picture, it is therefore the changes in the level of rates that bring about

the changes in the volatility.

With GARCH-type processes (for which typically the volatility will be some function

of the past squared returns in the underlying) the volatility can also change rapidly,

and display localized bursts (‘clusters’). Furthermore, GARCH processes also display

a volatility that is functionally dependent on (the history of) the underlying. And also

in the GARCH case the driver of a large change in volatility is the realization of a

large return in the underlying. So, in both approaches the ‘dog’ is the level of rates

and the ‘tail’ is the level of volatilities. However, this need not necessarily be the case.

Consider, in fact, the case of a diffusive (or otherwise) process for the volatility which

is weakly (or not at all) dependent on the process of the forward rates. Also in this case

the magnitudes of the interest-rate and volatility changes are linked: following a volatility

‘spike’, subsequent moves in forward rates will, on average, be larger than ‘usual’. In this

description, however, it is a large change in the volatility (the dog) that wags the forwardrate tail. Two questions therefore naturally arise: ‘Is a GARCH description ultimately

equivalent to the approach described in these chapters?’; and, if not, ‘What description is

the correct one for the phenomenon at hand?’.

The picture that emerges by assuming an independent process for the volatility can be

easily ruled out: a large increase in volatility will in fact increase the probability of a large

change in the forward rates of either sign. However, we have seen in previous chapters

that, apart from the regime switch, large increases in the forward rates are associated

with large decreases in the volatility and vice versa. So, empirically there appears to be a

link not only between the levels of the two processes, but also between the signs of the

changes. An uncorrelated volatility process is therefore not the dog.

A similar argument, however, also shows that a GARCH-type process is unlikely be

responsible for the observed behaviour. The increase in volatility produced by a GARCH

process in fact depends on the square of the past returns, and is therefore sign-insensitive:

a large increase in the level of rates can produce an increase in volatility just as effectively as a large fall. This, again, is against the empirical evidence presented in previous

chapters.

All of this ties in well with financial intuition: in most situations, the arrival of new

information will have a first-order impact on the level of rates; the subsequent increase in

uncertainty will then cause volatility to increase. On the other hand, it is difficult to think

of a financial mechanism that might give rise to a large increase in volatility first, that

then feeds into larger-than-normal moves in the yield curve. In a nutshell, in the picture

I am presenting, the forward rates are the dog, and the volatility is definitely the tail.



754



25.3



CHAPTER 25 STOCHASTIC-VOLATILITY EXTENSIONS OF THE LMM



Displaced Diffusion vs CEV



I have discussed in the previous chapter that, for suitable values of the exponent, the

CEV process can prevent forward rates from becoming negative. In this respect is it

therefore more desirable, but also more difficult to implement, than the displaced-diffusion

approach. However, for computational purposes and for simplicity of exposition I will

assume in this chapter that, given a preferred exponent for the CEV process, the equivalent

displacement coefficient has been determined, and I will work directly with displaced

diffusions. In most cases this should give rise to few problems. When the level of rates

is particularly low (at this writing, short-term interest rates in USD stand at 1.00%), the

displaced-diffusion solution might become questionable. In these cases, the CEV approach

could be used instead, possibly coupled with the numerical techniques described in the

previous chapter. In either case, the approach presented in what follows would not change,

and only the interpretation of the volatility that assumes a diffusive behaviour does: it

would be in one case the volatility of (f + a), and in other the term σβ in

df = µf dt + f β σβ (t) dzt



25.4



(25.1)



The Approach



The method here proposed is discussed at greater length in Joshi and Rebonato (2003)

and Rebonato (2002). The reader is therefore referred to these works for more details.

The broad outline of the approach is as follows.

In order to extend the standard LIBOR market model (LMM) one can start from

its usual deterministic-volatility formulation, with the instantaneous volatility function

discussed in Chapter 21:

dfT (t)

= µfT dt + σ (t, T ) dzT (t)

fT (t)



(25.2)



σ (t, T ) = kT g(T − t)



(25.3)



g(T − t) = [a + b(T − t)] exp[−c(T − t)] + d



(25.4)



where σ (t, T ) is the instantaneous volatility at time t of the T -maturity forward rate,

and kT is a forward-rate-specific constant needed in order to ensure correct pricing of the

associated caplet. Ignoring smiles for the moment, we have seen (Equation (21.11)) that

the caplet pricing condition is ensured in the deterministic-volatility setting by imposing

that

σ (T )2 T

T



= kT2



(25.5)



g(u, T )2 du

0



The reason for proposing this separable functional form for the instantaneous volatility

is that, when the latter is deterministic, this function lends itself very readily to finding



25.4 THE APPROACH



755



the most time-homogeneous evolution of the term structure of volatilities and of the

swaption matrix consistent with a given family of parameterized functions g(T − t). As

we saw in the Chapter 19, this can be achieved simply by imposing that the idiosyncratic

terms, kT , should be as constant as possible across forward rates. Why do I insist on

the time homogeneity of the term structure of volatilities? Because of the link between

future re-hedging costs and the future smile surface discussed in Chapter 1. When perfect

hedging is not possible, the success of an approximate (parameter-) hedging strategy will

be based on how well the model ‘knows’ about the future re-hedging costs. For a detailed

discussion of this point, see also Rebonato (2002) or Brigo and Mercurio (2001).

Once this volatility function has been chosen, the arbitrage-free stochastic differential

equation for the evolution of the Ti -expiry forward rate in the Q-measure associated with

the chosen numeraire is given by

dfTi (t)

= µQ ({fTj (t)}, t) dt + σ (t, Ti )

fTi (t)



bik dzkQ (t)



(25.6)



k=1,m



In this expression dzkQ are orthogonal increments of standard Q-Brownian motions,

µQ ({fTj (u)}, u) is the measure-, forward-rate- and time-dependent drift that reflects the

conditions of no arbitrage, and the coefficients {b}, linked by the caplet-pricing condition

2

k=1,m bik = 1, fully describe the correlation structure given the chosen number, m, of

driving factors (see Chapter 19 and Rebonato (1999a, 2002)).

In order to account for smiles, this standard deterministic-volatility formulation can be

extended in two ways:

1. by positing a displaced-diffusion evolution of the forward rates according to

d fTi (t) + α

= µQ

α {fTj (t)}, t dt + σα (t, Ti )

fTi (t) + α



Q



bik dzk (t)



(25.7)



k=1,m



and

2. by making the instantaneous volatility non-deterministic via the following stochastic

mean-reverting behaviour for the coefficients a, b, c and d, or their logarithm, as

appropriate:

dat = RSa (RLa − at ) dt + σa (t) dzta



(25.8)



dbt = RSb (RLb − bt ) dt + σb (t) dztb



(25.9)



d ln ct = RSc (RLc − ln ct ) dt + σc (t) dztc



(25.10)



d ln dt = RSd (RLd − ln dt ) dt + σd (t) dztd



(25.11)



Note that in Equation (25.7) both the drifts and the volatilities refer to the quantity f + a.

Also, in Equations (25.8)–(25.11) all the Brownian increments are uncorrelated with each

other and with all the Brownian increments dzkQ (t) and the symbols RSa , RSb , RSc , RSd ,



756



CHAPTER 25 STOCHASTIC-VOLATILITY EXTENSIONS OF THE LMM



RLa , RLb , RLc and RLd denote the reversion speeds and reversion levels, respectively,

of the relative coefficients, or of their logarithms, as appropriate.

It is important to comment on the assumption of independence of the increments of

the Brownian processes shocking the volatility on the one hand, and the forward rates on

the other. This independence is crucial to the practical implementation presented below,

and is financially motivated by the change of variables behind a displaced-diffusion or

CEV description. Recall, in fact, from Chapter 23 that a CEV exponent close to zero was

shown to produce changes in implied volatilities almost independent of changes in the

underlying, in good (if not perfect) agreement with empirical observations. In practice

this de-coupling of the volatility and interest-rate dynamics is achieved by means of the

displacement coefficient α (see Rubinstein (1983) and Marris (1999) for the link with the

CEV model), which is introduced to account for the deviation from exact proportionality

with the level of the basis point move of the forward rates. It is this feature that produces

a monotonically decaying (with strike) component of the smile surface. In addition, the

stochastic behaviour of the (coefficients of) the instantaneous volatility is invoked in order

to account for the residual variability displayed in Figure 25.1. I show below that this

feature can also account for the more recently observed ‘hockey-stick’ shape of the smile

curves.

What is the financial meaning of Equations (25.8)–(25.11)? Given the econometric

interpretation that can be given to a, b, c and d (see the discussion in Chapter 21),

Equations (25.8)–(25.11) allow the initial slope, the long-term level and the location of

the maximum of the instantaneous volatility functions to fluctuate stochastically around

some long-term levels. The resulting changes in the instantaneous volatility are shown in

Figures 25.2–25.5.



25.5



Implementing and Calibrating the

Stochastic-Volatility LMM



One of the most appealing features of the LMM is its ease of calibration to market

prices. For the stochastic-volatility extension to remain popular and useful it is necessary

to provide similarly practical and financially well motivated calibration procedures. An

overview of how this can be accomplished is given in this section. Again, Rebonato

(2002) and Joshi and Rebonato (2003) provide more details.

The general strategy that can be followed to calibrate the stochastic-volatility extension

of the LMM presented above in an efficient way rests on three simple observations.

1. Given the posited independence between the forward rates and the stochastic volatilities, conditional on a particular volatility path having been realized, the problem

looks exactly like a standard (deterministic-volatility) LMM problem.

2. The Black formula is, at-the-money, almost exactly linear in the root-mean-square

volatility.

3. Joshi and Rebonato (2003) show that surprisingly few volatility paths are sufficient

for an accurate sampling of the volatility probability density.



25.5 CALIBRATING THE STOCHASTIC-VOLATILITY LMM



757



0.2

0.18

0.16

0.14

0.18-0.2

0.16-0.18



0.12



c



0.14-0.16



0.1



0.12-0.14

0.1-0.12



1.9

0

9.6



8.8



8



7.2



6.4



5.6



4.8



4



3.2



2.4



1.6



0.08-0.1

0.8



0



0.08



Figure 25.2 The instantaneous volatility curve when a = −0.02, b = 0.1, c = 1, d(0) = 0.14,

and d is stochastic.



0.18

0.17

0.16

0.15



0.17-0.18

0.16-0.17



0.14



0.15-0.16

0.13



0.14-0.15

c



0.12



0.13-0.14

0.12-0.13



0.11



0.11-0.12

0.1-0.11



0

9.6



8.8



8



1.9

7.2



6.4



5.6



4.8



4



3.2



1.6



2.4



0



0.8



0.1



Figure 25.3 The instantaneous volatility curve when a = −0.02, b = 0.1, c = 1, d(0) = 0.14,

and c is stochastic.



758



CHAPTER 25 STOCHASTIC-VOLATILITY EXTENSIONS OF THE LMM

0.2



0.19

0.18

0.19-0.2



0.17



0.18-0.19



0.16



0.17-0.18



0.15



0.16-0.17



0.14



0.15-0.16



0.13



0.14-0.15



c



0.13-0.14



0.12



0.12-0.13



0.11



0.11-0.12



0

9.6



8.8



8



1.9

7.2



6.4



5.6



4.8



4



3.2



1.6



0.1-0.11

2.4



0



0.8



0.1



Figure 25.4 The instantaneous volatility curve when a = −0.02, b = 0.1, c = 1, d(0) = 0.14,

and b is stochastic.



0.19

0.18

0.18-0.19



0.17



0.17-0.18

0.16



0.16-0.17

0.15-0.16



0.15



0.14-0.15

0.14



0.13-0.14



c



0.12-0.13



0.13



0.11-0.12



0.12



0.1-0.11



0.11

3.4



0

9.6



9



8.4



7.8



7.2



6.6



6



5.4



4.8



4.2



3.6



3



1.7

2.4



1.8



1.2



0.6



0



0.1



Figure 25.5 The instantaneous volatility curve when a = −0.02, b = 0.1, c = 1, d(0) = 0.14,

and a is stochastic.



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