Tải bản đầy đủ - 0 (trang)
3 An Aside: Some Simple Properties of Markov Chains

# 3 An Aside: Some Simple Properties of Markov Chains

Tải bản đầy đủ - 0trang

786

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

788

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.

790

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

796

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.

### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

3 An Aside: Some Simple Properties of Markov Chains

Tải bản đầy đủ ngay(0 tr)

×