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5 Discussion, Conclusions and Suggestions for Future Work

5 Discussion, Conclusions and Suggestions for Future Work

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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


Chapter 27

Stochastic-Volatility Extension

of the LMM: Two-Regime

Instantaneous Volatility


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.


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).




The most salient missing features are probably

• the ability to reproduce rapid transitions of the swaption matrix from one ‘mode’ to


• 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:






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


σix (t, Ti ) = [atx + btx (T − t)] exp(−ctx (T − t)) + dtx


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 )


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.



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.


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 . . .


a matrix, known as the transition


λN2 

... 



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



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)


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)


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 = 







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:


 λi2 

E ξt+1 |yt = i = 


 ... 


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