3 An Aside: Some Simple Properties of Markov Chains
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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
27.3 AN ASIDE: SOME SIMPLE PROPERTIES OF MARKOV CHAINS
787
or, equivalently, by
E ξt+1 |yt = i =
ξt
(27.11)
More generally, it is straightforward to show that the expectation of the occupation state
m time-steps from today, ξt+m , is given by
E ξt+m |yt = i =
m
ξt
(27.12)
where the symbol m denotes the repeated (m times) application of the transition matrix.
These deﬁnitions and results can be brought together to produce some interesting
results. Assume that a Markov chain is irreducible (i.e. that it contains no absorbing
states). Since every column of the transition matrix adds up to unity, it is easy to show
(Hamilton (1994)) that unity is always an eigenvalue of the transition matrix. If all the
other eigenvalues are inside the unit circle (i.e. have magnitude smaller than 1), then
the Markov chain is said to be ergodic, and the associated eigenvector, denoted by π, is
called the vector of ergodic probabilities. It enjoys the properties
π =π
(27.13)
and
m
lim
m→∞
= π1T
(27.14)
where 1 is an (N ×1) vector of 1s, and the superscript T denotes matrix transposition. From
these properties it follows (see again Hamilton (1994)) that, if the Markov chain is ergodic,
the long-run forecast of the occupation vector ξ , i.e. limm→∞ ξt+m , is given by the vector
of ergodic probabilities, π, and is independent of the current state. Therefore this vector
of ergodic probabilities can be regarded as providing the unconditional expectation of the
occupation probability of each state. Finally, an ergodic Markov chain is a (covariance-)
stationary process.
27.3.1 The Case of Two-State Markov Chains
The ﬁnancial modelling approach proposed above allows for two states only, the normal
and the excited. In this particularly simple situation the general results and deﬁnitions
above can be specialized as follows. If the Markov chain is irreducible, the two eigenvalues
of the transition matrix, 1 and 2 , are given by
=1
(27.15)
= −1 + λ11 + λ22
(27.16)
1
and
2
The second eigenvalue will be inside the unit circle, and the Markov chain therefore
ergodic, provided that 0 < λ11 + λ22 < 2. In this case, the vector of ergodic probabilities
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CHAPTER 27 STOCHASTIC-VOLATILITY EXTENSION OF THE LMM
is given by
1 − λ22
2 − λ11 − λ22
π =
1 − λ11
2 − λ11 − λ22
(27.17)
and the unconditional occupation probability of states 1 and 2, P {y = 1} and P {y = 2},
are given by
1 − λ22
2 − λ11 − λ22
1 − λ11
P {y = 2} =
2 − λ11 − λ22
P {y = 1} =
(27.18)
(27.19)
respectively.
It is also interesting to observe that, if λ11 + λ22 > 1, the process is more likely to
remain in the current state than to migrate, and the variable ξ would be positively serially
correlated. If λ11 + λ22 < 1, the process is more likely to migrate from the current state,
and the variable ξ would be negatively serially correlated.
27.4
Empirical Tests
27.4.1 Description of the Test Methodology
In order to test the effectiveness of the model extension proposed above one can begin
by studying the qualitative behaviour of a two-regime stochastic-volatility LMM by using
the two instantaneous volatility functions (normal and excited) depicted in Figure 27.1.
The swaption data described in detail in Chapters 26 and 27 were chosen to carry
out the test. The reason for using swaption rather than caplet data is also explained
in Chapter 26. Once again we want to create a time series of model-implied swaption
matrices so that they can be analysed by PCA, or by other means. The following extension
of the algorithm described in Section 26.4 can be used to determine the eigenvalues and
eigenvectors of the model implied volatility covariance matrix.
• The parameters of the normal and excited volatility curves were determined so as to
be consistent with the ﬁnancial interpretation of the two volatility states as a normal
and an excited state, and to provide a good ﬁt to caplet prices. See the discussion
in point 1 below. The parameters so obtained are reported in Table 27.1
• On the basis of the current swaption matrix, the decision was made as to the current value of the latent variable y (0 or 1). In general, deciding in which state a
Markov-chain process currently ﬁnds itself is not a trivial matter. However, given
the discussion in Section 26.3.2, the normal and excited states can be convincingly characterized by the humped or monotonically decaying shape of the 1 × 1
implied volatility series, respectively. Therefore I choose the shape of these swaption
volatilities to decide unambiguously whether today’s state for y is 0 or 1.
27.4 EMPIRICAL TESTS
789
0.3
Normal
Excited
0.25
0.2
0.15
0.1
0.05
0
0
Figure 27.1
0.5
1
1.5
2
The normal and excited instantaneous volatility curves chosen for the study.
Table 27.1 The parameters describing the normal and excited volatility curves.
Initial value
Volatility of
the volatility
Reversion
speeds
Reversion
levels
Displacement
a
b
c
d
lower
lower
lower
lower
0.08
0.27
3.41
0.03
0.67
0.03
0.77
0.18
0.55
0.35
0.11
0.64
0.08
0.27
3.41
0.03
0.029
a
b
c
d
upper
upper
upper
upper
0.20
0.08
5.34
0.07
0.45
0.01
0.77
0.19
0.33
0.35
0.13
0.63
0.20
0.08
5.34
0.07
0.020
• The a n(x) , bn(x) , cn(x) and d n(x) coefﬁcients were evolved from today’s state over a
simulation period of one week.
• Conditional on this evolution of the instantaneous volatilities and on the prevailing
value for y, the forward rates were also evolved over the same one-week period.
• Given this state of the world the at-the-money swaptions were priced and their
implied volatilities obtained.
• At this point (i.e. one time interval t from today) the prevailing value of the latent
variable y was updated on the basis of the transition probability using a random
draw.
• The procedure was repeated over a large number of time-steps, thereby creating a
time series of model implied volatilities for the swaptions.
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CHAPTER 27 STOCHASTIC-VOLATILITY EXTENSION OF THE LMM
• These quantities were then analysed (e.g. the covariance matrix of the volatility
changes were orthogonalized, the skew and kurtosis analysed, etc.) and compared
with the market data.
The following observations are in order.
1. In order to limit the number of degrees of freedom, the a n(x) , bn(x) , cn(x) and d n(x)
coefﬁcients were not treated as fully free-ﬁtting parameters; instead, we started from
shapes for the ‘normal’ and ‘excited’ instantaneous volatility functions consistent
with the ﬁnancial model discussed above, and locally optimized the parameters
around these initial guesses.
2. The test was run by ﬁtting to caplet data and then exploring swaption data. No best
ﬁt to swaption-related quantities was attempted in the choice of a n(x) , bn(x) , cn(x)
and d n(x) . The test is therefore quite demanding, in that it requires a satisfactory
description of the evolution of swap rates using parameters estimated on the basis
of forward-rate information alone.
3. The levels and the shapes of the normal and excited volatility curves refer to the
risk-neutral world and not to the real-world measure. The same consideration applies
to the frequency of transition from one state to the other. Therefore no immediate
conclusions can be drawn from these values. Despite this, I present below an orderof-magnitude comparison of the transition frequency.
4. Fits to caplet prices of very similar (and very good) quality can be obtained with
different parameters for the normal and excited coefﬁcients. Therefore recovery
of the caplet prices is a poor criterion to choose between different instantaneous
volatility curves. I suggest below that an analysis of the kurtosis of the changes in
implied volatilities and of the eigenvalues’ behaviour can provide better information.
5. Statistical estimates of kurtosis are in general very noisy, and, anyhow, because of
point 3 above, cannot be directly compared with the model values. This is because
the model kurtosis will depend on the transition probability matrix, and this matrix,
once implied from the market prices, is risk adjusted (i.e. pertains to the pricing
measure, not the real-world measure). Therefore we did not attempt a ﬁt to the
kurtoses for the various swaption series. However, we present below the real and
risk-adjusted values for a qualitative comparison.
27.4.2 Results
The Real and Model Path of Implied Volatilities
Assigning a regime switch to the instantaneous volatility does not automatically ensure
that the implied volatility (which was linked to the root-mean-squared integral of the
former) will also display a noticeable discontinuity. Since the empirical evidence refers
to implied volatilities, it is important to check that the desired effect is indeed produced.
To this effect, Figures 27.2 and 27.3 display time series of the changes in instantaneous
and implied volatilities for a 5×5 constant-maturity swaption obtained using the procedure
described above and the best-ﬁt parameters in Table 27.1. It is clear that the proposed
27.4 EMPIRICAL TESTS
791
0.08
0.06
0.04
0.02
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
−0.02
−0.04
−0.06
−0.08
Figure 27.2
data).
The instantaneous volatility path: model data (5 × 5 swaption time series, ﬁt to USD
8
6
4
2
0
0
200
400
600
800
1000
1200
−2
−4
−6
−8
Figure 27.3
The implied volatility path: model data (5 × 5 swaption time series, ﬁt to USD data).
792
CHAPTER 27 STOCHASTIC-VOLATILITY EXTENSION OF THE LMM
process does produce regime shifts also in the implied volatility changes, with most
changes being ‘small’, and a relatively smaller fraction of changes very high.
Figures 27.4 and 27.5 display two of the empirical time series of implied volatilities,
showing the qualitative similarity between the model and real-world data. Also, in this
case the time spent in the normal and excited states in the real and risk-adjusted world
cannot be directly compared because of the risk-adjusted nature of the quantities (the level
of the volatility curves and the frequency of the jump) that enter the pricing of caplets.
Recovery of the Market Smile Surface
The ﬁt to the market smile surface (USD data Mar2003) is shown in Figure 27.6. It is
clearly of very high quality, the more so if we recall that it was obtained by choosing
beforehand values of the transition probabilities that would produce acceptable ratios
for the eigenvalues obtained by orthogonalizing the implied volatility model covariance matrix. See, in this regard, the discussion below. (Recall that the purely diffusive
stochastic-volatility model presented in Chapter 25 loaded more than 90% of the explanatory power onto the ﬁrst eigenvector.) With these transition probabilities the coefﬁcients
for the normal and excited volatility curves were then obtained as described above.
The ﬁt was obtained using the same time-independent coefﬁcients for the whole surface, and therefore the resulting swaption matrix displays a desirable time-homogeneous
behaviour.
This ﬁt should be compared with the ﬁt obtainable with a regime switch between purely
deterministic volatilities. This important feature is discussed in Section 27.5.
2
1.5
1
0.5
0
19-Jun97
−0.5
05-Jan- 24-Jul-98 09-Feb98
99
28-Aug99
15-Mar00
01-Oct00
19-Apr01
−1
−1.5
−2
Figure 27.4 Empirical implied volatility path (10 × 10 USD).
05-Nov01
27.4 EMPIRICAL TESTS
793
2
1.5
1
0.5
0
27-Sep-97 15-Apr-98 01-Nov-9820-May-9906-Dec-99 23-Jun-00 09-Jan-01 28-Jul-01
−0.5
−1
−1.5
−2
−2.5
Figure 27.5
Empirical implied volatility path (1 × 3 EUR).
Kurtosis
Table 27.2 shows the kurtosis of the distribution of the implied volatility changes for
the two-regime model. This quantity is displayed for several swaptions series and was
obtained using the parameters listed above for the two instantaneous-volatility states and
the jump intensities. Note that these values of kurtosis are substantially larger than the
kurtosis that would be obtained using a single-regime stochastic-volatility LMM, and
much closer to the values estimated using market data. The usual caveats about the
change of measure apply.2 Since the true distribution of the changes in implied volatility
is not known a priori it is not possible to associate statistical error bars to the experimental
values. However, in order to give an idea of the possible estimation uncertainty Table 27.3
displays the real-world estimates obtained using the ﬁrst and second halves of the available
data.
Skew
No explicit attempt was made to reproduce the skew of the distribution of the change in
implied volatilities. None the less Table 27.4 shows good agreement between the model
(theoretical) and real-world quantities. The same observations about the statistical error
bars and the change of measure hold, and again the estimates obtained using the ﬁrst and
second halves of the data are presented (see Table 27.5).
2 It might seem surprising that also the kurtosis (and the skew) should be measure dependent (they are
measure invariant in the purely diffusion approach of Chapter 25). The reason is that all the quantities affected
by the transition frequency (which is certainly risk-adjusted) change in moving between measures. These
quantities are, for instance, the eigenvalues, the kurtosis and the skew.
794
CHAPTER 27 STOCHASTIC-VOLATILITY EXTENSION OF THE LMM
0.4
0.4
t = 1 year
t = 2 years
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.1
0.02
0.065
0.35
0.03
0.04
0.05
0.06
t = 5 years
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.04
0.06
0.08
0.1
0.1
0.02
0.12
0.3
0.04
0.06
0.08
0.1
0.12
0.3
t = 12 years
t = 14 years
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.02
0.08
0.35
t = 4 years
0.1
0.02
0.07
0.04
0.06
0.08
0.1
0.12
0.1
0.02
0.04
0.06
0.08
0.1
0.12
Figure 27.6 The quality of the recovery to the market smile surface. USD data, option expiries
from 1 to 14 years.
Table 27.2
Model and empirical kurtosis.
Kurtosis
Real
1×2
1×5
1 × 10
3×3
3×5
14.3935
10.10832
20.11422
15.93261
11.70314
Model
11.43534
15.01118
17.20576
17.2059
21.18686
27.4 EMPIRICAL TESTS
795
Table 27.3 As for Table 27.2, using the ﬁrst and second
halves of the available data.
Kurtosis
1×2
1×5
1 × 10
3×3
3×5
Real
Real(I)
Real(II)
Model
14.3935
10.10832
20.11422
15.93261
11.70314
16.09731
10.27761
11.99224
12.06961
8.457682
4.821323
2.640147
28.02998
4.364113
3.049932
11.43534
15.01118
17.20576
17.2059
21.18686
Table 27.4 Model and empirical kurtosis.
Skew
Real
1×2
1×5
1 × 10
3×3
3×5
Model
0.767421
0.529616
0.868207
1.421207
0.443754
0.202036
0.386044
0.296187
0.352619
0.327331
Table 27.5 As for Table 27.4, using the ﬁrst and second
halves of the available data.
Skew
1×2
1×5
1 × 10
3×3
3×5
Real
Real(I)
Real(II)
0.767421
0.529616
0.868207
1.421207
0.443754
1.056023
0.515964
1.260718
1.36261
0.375826
−0.03201
0.322293
0.501672
0.294108
0.140795
Model
0.202036
0.386044
0.296187
0.352619
0.327331
Eigenvalues and Eigenvectors Again
I have argued in Chapter 26 (see also Rebonato and Joshi (2002)) that a comparison
between the eigenvectors and eigenvalues estimated from real-world data and simulated
by the model is a useful tool to assess the quality of a model, and to overcome the difﬁculties in comparing real-world and risk-adjusted quantities. One of the main conclusions
from this type of analysis was that the simple (i.e. purely diffusive) stochastic-volatility
model produced a good qualitative shape for the eigenvectors obtained from the orthogonalization of the covariance matrix of the changes in implied volatilities. The relative
size of the eigenvalues in the original model, however, was at variance with what was
observed in reality. As pointed out in footnote 1, however, this comparison is, strictly
speaking, no longer warranted in the presence of a regime change. In order to understand
the nature of the transformation brought about by the measure change, one can reason
as follows. The eigenvectors decompose the possible changes in the implied volatilities
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CHAPTER 27 STOCHASTIC-VOLATILITY EXTENSION OF THE LMM
into orthogonal modes of deformation. If all the eigenvectors are retained they provide a
possibly more efﬁcient, but otherwise totally equivalent, set of co-ordinates. A change in
measure, as usual, can change the probability of different events happening, but cannot
make impossible events possible, and vice versa. See the discussion in Section 6.7.3 of
Chapter 6. Therefore what risk aversion can change is the relative importance of different modes of deformation, i.e. the relative magnitude of the eigenvalues. If risk-averse
traders ﬁnd it more difﬁcult to hedge their complex-option positions when, say, tilts in
the swaption matrix occur, they might assign a greater probability to the second mode of
deformation than to the ﬁrst (the ‘parallel’ eigenvector). If, however, tilts were ‘impossible’ in the real-world measure, traders could not ‘invent them’ and ask for compensation
for the undiversiﬁable risk associated with them. It is therefore plausible that risk aversion
should not change the shape of the eigenvectors, but should alter the relative magnitudes
of the eigenvalues. Note in passing that, since the relative importance of different modes
of deformation can change across measures, a given eigenvector, say the second, in the
real world can assume a different index, say the third, in the pricing measure.
Can we say something a bit more precise? Let us look at the problem from a slightly
different angle, i.e. by focusing on the transition probability rather than on the eigenmodes.
If λn→x is the risk-adjusted transition probability from the normal to the excited state
(similar considerations apply to the probability λx→n ) , one can write
λn→x = λrw
n→x +
λn→x
(27.20)
where λrw
λn→x is the change in transin→x is the real-world transition probability and
tion probability in switching between measures. We can therefore regard the change in
eigenvalues in moving from the purely diffusive state to the Markov-chain description
as being due to two distinct contributions, the ﬁrst coming from the real-world transition
probability, and the second from risk aversion. The principle of absolute continuity, however, guarantees that if the real-world process were indeed a two-state Markov chain, so
would the risk-adjusted process. See again the discussion in Section 6.6.3. If the effect
of risk aversion were to make investors ‘imply’ a higher transition probability from the
normal to the excited state, then the results (e.g. the kurtosis) obtained from the ﬁt to the
market data would display an ‘overshoot’ in moving from the purely diffusive case to
the Markov-chain case. In particular, if the kurtosis found in the diffusive-volatility case
were too low compared with the empirical data, and the model were correctly speciﬁed,
the risk-adjusted kurtosis would turn out to be higher than the econometric one.
Can we expect λn→x to be positive? To answer this question we have to put ourselves
in the shoes of a trader who hedges a complex product following a delta and vega strategy,
much as described in Chapter 1. We have seen in Chapter 13 that hedging against a simple
level mis-speciﬁcation of the volatility by means of vega hedging is relatively easy. It
is changes, and especially sudden changes, in the shape of the smile surface that tax the
hedging ability of the trader. Therefore one can expect that the distribution of slippages3
should display a greater variance if a regime switch has occurred during the life of
the complex option. As a consequence, a risk-averse trader would ‘fear’ the occurrence
of regime switches, and incorporate in her prices a higher (risk-adjusted) frequency of
transition. If this is the case the risk-adjusted distribution of volatility changes would
3 Recall that in Chapter 4 we deﬁned ‘slippages’ as the differences between the payoff of an option, and the
payoff of the hedging strategy.