5 Discussion, Conclusions and Suggestions for Future Work
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CHAPTER 26 THE DYNAMICS OF THE SWAPTION MATRIX
have been proposed in the literature.10 In the next chapter I will show that a modelling
approach based on a discrete, two-state Markov chain for the instantaneous volatility can
provide a better answer to the modelling of these features. I will also show that these
regime transitions, far from being a second-order effect, are likely to be very important
in accounting for the observed market smiles.
Despite these shortcomings, even in its present form the proposed modelling approach
displays several important encouraging features. First, I have shown in this chapter that,
after orthogonalizing the model and empirical covariance matrices of the changes in
implied volatilities, the qualitative shape of the ﬁrst eigenvector turned out to bear a close
resemblance to the corresponding empirical quantity. In particular, the same periodicity
was observed in the real and model data.
Second, the decaying behaviour of the ﬁrst principal component as a function of
increasing expiry, observed in the real data when the covariance matrix is orthogonalized,
was found to be naturally recoverable and explainable by the mean-reverting behaviour for
the instantaneous volatility. This feature in turn constitutes the most salient characteristic
of the stochastic-volatility extension of the LIBOR market model proposed in Chapter 25.
Furthermore, the values for the mean reversion that had been previously and independently obtained using static information (i.e. by ﬁtting to the smile surface) turned out to
be adequate to explain in a satisfactory way the qualitative features of such intertemporal
features as the shape of the eigenvectors (obtained from time-series analysis).
It therefore appears fair to say that, despite the obvious shortcomings, the extension
of the standard deterministic-volatility LIBOR market model presented in the previous
chapter appears to be a useful ﬁrst step in the right direction. How this model can be
enriched further is the topic of the next chapter.
10 Linking
the volatility in a deterministic manner to the stochastic forward rates could produce sharp moves
in the level of the swaption matrix, if the forward rates displayed a discontinuous behaviour (as in Glasserman
and Kou (2000) and Glasserman and Merener (2001)). It is difﬁcult, however, to see how a deterministic
functional dependence on the forward rates could give rise to a sudden change in the shape of the swaption
matrix.
Chapter 27
Stochastic-Volatility Extension
of the LMM: Two-Regime
Instantaneous Volatility
27.1
The Relevance of the Proposed Approach
One of the recurring messages from this book has been that many approaches provide a
ﬁt to current plain-vanilla option prices of similar quality, and that therefore additional
criteria must be used in order to choose a ‘good’ model. I have argued in Chapter 1
that the joint practices of the theoretically inconsistent vega (re-)hedging and of daily recalibrating the model expose the trader to future realizations of the smile surface (i.e. to
the future prices of the re-hedging options). From the perspective of a complex-derivatives
trader, an important criterion for the success of a model should therefore be its ability to
predict, either in a deterministic or in a stochastic manner, the current and future prices
of the vega-hedging instruments (i.e. the future smile). The main feature of the modelling
approach presented in this chapter is therefore not aimed at obtaining a more accurate
recovery of the empirical smile surface today, but a more convincing description of the
evolution of the future smile surface. The extension presented below of the stochasticvolatility version of the LIBOR market model (LMM) discussed in the previous chapters
should be seen as a description of the smile surface dynamics that, by being more closely
aligned with empirical evidence, will provide a better pricing and hedging tool for traders.
27.2
The Proposed Extension
How could one improve upon the approach presented in the previous chapter in such a
way as to retain its desirable features and to take into account the empirical evidence
discussed above?1
1 Points
of this chapter have been adapted from Rebonato and Kainth (2004).
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CHAPTER 27 STOCHASTIC-VOLATILITY EXTENSION OF THE LMM
The most salient missing features are probably
• the ability to reproduce rapid transitions of the swaption matrix from one ‘mode’ to
another;
• the ability to return, after one such transition has taken place, to a shape similar to
the original one;
• the recovery of fatter tails in the distribution of changes of implied volatilities (in
agreement with empirical data); and
• a better apportioning of the total variance among the eigenvectors obtained from
orthogonalizing the changes in swaption implied volatilities.
A simple and natural way to model these features, while retaining the simplicity and
intuition behind the approach described above, is the following. Let us posit the existence
of a latent variable, y, that follows a two-state Markov-chain process between two states,
x and n, with transition probabilities:
λx→x
λx→n
λn→x
λn→n
(27.1)
and which can only take the values 1 (if state n prevails) or 0 (if state x prevails). One
could then proceed as follows.
1. Choose a simple criterion to determine whether the swaption matrix is currently in
the normal or excited state. Looking at the ﬁgures in Chapters 7 and 26, one such
criterion could be whether the n-year-into-1-year swaption series displays a hump
or not. Given today’s shape for this curve, we therefore know with certainty which
state we are in.
2. Posit the existence of two instantaneous volatility functions for each forward rate.
Using the notation introduced in the previous chapters, these can described by the
following functional forms:
σin (t, Ti ) = [atn + btn (T − t)] exp(−ctn (T − t)) + dtn
(27.2)
σix (t, Ti ) = [atx + btx (T − t)] exp(−ctx (T − t)) + dtx
(27.3)
with different coefﬁcients {a n , bn , cn , d n } and {a x , bx , cx , d x } associated with the
normal (superscript n) and excited state (superscript x).
3. At any point in time the instantaneous volatility for forward rate i, σi (t, Ti ), is
given by
σi (t, Ti ) = yt σin (t, Ti ) + (1 − yt )σix (t, Ti )
(27.4)
4. All the coefﬁcients {a n , bn , cn , d n } and {a x , bx , cx , d x } are stochastic, and follow
the same Ornstein–Uhlenbeck process described in the stochastic-volatility model
presented in Chapter 25. Recall that their processes are all uncorrelated, between
themselves and with the forward rates.
27.3 AN ASIDE: SOME SIMPLE PROPERTIES OF MARKOV CHAINS
785
5. The transition of the instantaneous volatility from the normal to the excited state
occurs with frequency λn→x , and the transition from the excited state to the normal
state with frequency λx→n . The latent variable y remains in the excited or normal
state with probabilities per unit time, λx→x and λn→n , respectively. Note that if we
tried to ‘imply’ these frequencies from the prices of traded options, they would all
be risk-adjusted and not real-world frequencies.
6. Since the same assumption of independence between the volatility processes and
the forward rate processes is enforced, once again along each volatility path the
problem is exactly equivalent to the deterministic case, apart from the fact that, at
random times, the coefﬁcients would switch from one state to the other.
7. Because of point 6, the evaluation of the variances or covariances along each path
proceeds exactly as described in Chapter 25, with possibly different coefﬁcients
‘half-way through’ some of the paths if a transition has occurred. The evaluation of
caplets and European swaptions would be virtually unaltered.
Since the properties of Markov chains, which are a central ingredient of this description,
are not as widely known as the properties of, say, Brownian diffusions, they are brieﬂy
reviewed in the following section.
27.3
An Aside: Some Simple Properties of Markov Chains
Markov chains are frequently used to model time series that undergo regime changes.
(See, for example, Hamilton (1994), Ross (1997) or Gourieroux and Jasiak (2001) for
their main properties.) Following Hamilton (1994), let us assume that we have a stochastic
process, y, which at each point in time can be in any of N possible discrete states. For
simplicity, I will place myself in a discrete-time setting. From one time-step to the next,
y can either remain in the prevailing state, or migrate to any of the possible states. The
probability of remaining in the prevailing state is denoted by λi→i (t), or, more simply,
λii (t). The transition probability for the process y to move at time t from state i to state
j is denoted by λi→j (t), or, more simply, λij (t). Since at every point in time the process
must ‘do something’ (i.e. either undergo a transition or remain in the current state), it
must be the case that
λi1 + λi2 + . . . + λiN = 1
It is often convenient to arrange the probabilities in
matrix, :
λ11 λ21 . . .
λ12 λ22 . . .
=
...
...
...
λ1N λ2N . . .
(27.5)
a matrix, known as the transition
λN1
λN2
...
λN1
(27.6)
Because of (27.6), each column of the transition matrix adds up to 1. Note in passing
the conventional ordering of the indices: since the index of the ‘departure’ state is ﬁrst, the
element of, say, the ﬁrst row and second column is denoted by λ21 . This convention is not
always followed, and to preserve a more matrix-like indexing, in some ﬁelds (quantum
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CHAPTER 27 STOCHASTIC-VOLATILITY EXTENSION OF THE LMM
mechanics in particular) the departure state is sometimes associated with the second index,
and the arrival state with the ﬁrst. To conform to ﬁnancial notation I retain the ‘awkward’
matrix notation, but preserve the natural ordering of arrival and departure states.
In general, the transition probabilities could depend on the whole history of y up to
time t:
P (yt+1 = j |yt = i, yt−1 = k, . . .) = λij (t)
(27.7)
If, however, the transition probability only depends on the current state,
P (yt+1 = j |yt = i, yt−1 = k, . . .) = P (yt+1 = j |yt = i) = λij (t)
(27.8)
the process is said to be a Markov chain.
In some contexts, notably in credit default modelling (see, for example, Schoenbucher
(1996)), some states might be such that, if they are ever reached, the process remains in
that state for ever. If this is the case, these states are called absorbing, the Markov chain
reducible, and the associated transition matrix can be rearranged in such a way that all
its non-zero elements are on or above the main diagonal. In the present chapter we will
always work with irreducible transition matrices, i.e. with processes for which no state
is absorbing. Because of Equation (27.6), for a two-state Markov chain to be irreducible
it is necessary and sufﬁcient that λii < 1, i = 1, 2.
If we have a discrete number of possible states (as will always be assumed to be the
case in this chapter) it is convenient to denote the state at the current time t by a vector,
ξt , which contains zeros in the unoccupied states, and a 1 in the prevailing state:
ξt =
0
...
1
...
0
(27.9)
When we are looking at the current time this column vector (whose elements trivially add
up to 1) can be regarded as a discrete probability density, with a Dirac-delta distribution
localized around the current state. (Of course, since we are dealing with discrete states,
we should be dealing with a Kronecker delta, δij . Thinking of a delta distribution as the
limit of an ‘inﬁnitely narrow’ Gaussian distribution, however, helps the intuition, and I
will retain the continuous description.)
As of today we can be interested in how the expectation of the occupancy of future
states of the world will change, i.e. we would like to know how this delta function spreads
out over time. It is easy to guess that the transition matrix will ‘smear out’ today’s deltalike density into a more diffuse distribution. More precisely, using the notation introduced
above, conditional on the current state occupancy being equal to i, the expectation of the
state vector at the next time-step is given by the ith column of the transition matrix:
λi1
λi2
E ξt+1 |yt = i =
(27.10)
...
λiN