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.
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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.