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4 Empirical Study I: Transforming the Log-Normal Co-ordinates

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CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES

Table 23.1 The exponent required to produce a zero correlation

between the implied volatilities and the swap rates for the various

swaption series. The column on the right displays the average

exponent for the 1×, 3×, and 5× series.

1

1

1

1

×1

×3

×5

× 10

1.37

1.44

1.45

1.37

1× series

1.41

3

3

3

3

×1

×3

×5

× 10

1.12

1.17

1.14

1.06

3× series

1.12

5

5

5

5

×1

×3

×5

× 10

0.80

0.88

0.88

0.91

5× series

0.87

Table 23.2 Empirical correlation coefﬁcient between the implied

volatilities and the swap rates. The column on the right displays the

average exponent for the 1×, 3×, and 5× series.

1

1

1

1

×1

×3

×5

× 10

−0 .86

−0 .83

−0 .79

−0 .70

1× series

−0.80

3

3

3

3

×1

×3

×5

× 10

−0 .73

−0 .68

−0 .63

−0 .54

3× series

−0.65

5

5

5

5

×1

×3

×5

× 10

−0 .51

−0 .53

−0 .51

−0 .47

5× series

−0.51

series during the period leading to October 2002. See Figure 23.1. This ﬁgure assumes

that the percentage volatility is the appropriate quantity to track, and that therefore lognormal co-ordinates are the appropriate ones. On the basis of this assumption, and looking

at the time series depicted in Figure 23.1, one would conclude that the period under

consideration has witnessed an exceptionally high level of uncertainty in rates (much

higher than what was experienced during the Russia crisis). The picture becomes radically

different if one changes co-ordinates and looks at the ‘rescaled volatility’ y(α) = σimpl S α

(with α chosen as in Table 23.2). See Figure 23.18. The same message is conveyed

23.4 TRANSFORMING THE LOG-NORMAL CO-ORDINATES

717

350

300

250

200

150

100

50

0

15-Apr-98 01-Nov-98 20-May-99 06-Dec-99 23-Jun-00 09-Jan-01 28-Jul-01 13-Feb-02 01-Sep-02

Figure 23.18 Time series of the quantity y(α) = σimpl S α for the 1 × 1 swaption volatility.

2.5

2

1.5

Rescaled

Percentage

1

0.5

0

25-Mar-02

14-May-02

03-Jul-02

22-Aug-02

11-Oct-02

Figure 23.19 The percentage (upper curve) and the rescaled 1 × 1 swaption volatilities, after

rescaling to 1 at the beginning of the period.

with even greater clarity by Figure 23.19, where both the percentage and the rescaled

volatilities are shown over the same period, after rebasing each for ease of comparison

to the value of 1 at the beginning of the period. The quantity y displays a much more

regular behaviour, and actually declines over the same period during which the implied

volatility reaches unprecedented heights. (During the same period the swap rate attained

similarly exceptionally low values – see again Figure 23.2.)

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CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES

These empirical ﬁndings motivate the question of whether, and to what extent, the

quantity y, which refers to implied volatilities, is related to the instantaneous volatility

of a CEV process. Answering this question would provide a useful piece of information

in the modelling of interest rates. This is accomplished in the next section.

23.5

The Computational Experiments

It might be possible, but it is certainly not straightforward, to answer the question

formulated at the end of Section 23.4 by following a semi-analytic route. The difﬁculty stems from the fact that the concept of implied volatility is totally based on a

log-normal co-ordinate system that becomes analytically cumbersome as soon as one

leaves the Black setting (the implied volatility simply becomes ‘the wrong number to

put in the wrong formula to get the right price’). I therefore follow a computational

approach.

I start by assuming that the ‘true’ process for the underlying (the swap rate) is

indeed of the form (23.1). For simplicity, I choose to carry out the valuation of a payer

swaption (call on the swap rate) in the measure under which the swap rate itself is a

martingale. This implies using as numeraire the ﬁxed leg of the annuity of the associated swap. Therefore, ignoring the present-valuing, irrelevant for the discussion, the

problem reduces to the evaluation of a K-strike call on the swap rate with payoff, P ,

given by

P = max[SRT − K]+

(23.13)

I assume that the swap rate follows a process as in Equation (23.1), and choose several

values of the CEV exponent β (say, 1, corresponding to the log-normal case, 0.75 and

0.5, corresponding to the square-root process). To each value of β I associate a different

volatility coefﬁcient4 σ = σβ , chosen so as to give the same price today for the at-themoney call. To do so for an arbitrary strike one would require a numerical inversion of the

Black formula (see, for example, Marris, 1999). However, if one is simply working with

at-the-money options, an extension of the discussion presented in Chapter 16 suggests

that a very good approximation can be obtained by ‘guessing’ the value of σβ to be

given by

σβ SRβ = σβ SRβ

(23.14)

In particular, if, say, β = 1 (corresponding to the log-normal case), and the associated

percentage volatility is simply denoted by σ , then Equation (23.14) gives

σβ = σ

SR

SRβ

(23.15)

We will check in Section 23.6 that this choice does produce the correct at-the-money

price. For the moment we will take this result on faith.

4 To be precise, the quantity σ that appears in Equation (23.1) should be called the deterministic component

of the instantaneous volatility (which is given by σ S β ). For simplicity, however, I will refer to σ simply as

the volatility.

23.6 THE COMPUTATIONAL RESULTS

719

Having chosen the volatility using Equation (23.15), I ﬁrst calculate the value of atthe-money call options of different maturities using ‘today’s’ value of the swap rate. I

then impart various instantaneous changes to its level and recalculate the prices of at-themoney swaptions. Clearly, since the underlying swap rate has moved, the absolute value

of the strike is different for each swaption. What remains constant is only their at-themoneyness. Finally, I convert these prices into implied percentage volatilities, σimpl , and

I plot these quantities as a function of the at-the-money swap rate level, SR: σimpl (SR). I

stress that the procedure differs from the more usual investigation of the strike dependence

of the implied volatility, which is a well known, but, in this context, irrelevant topic.

23.6

The Computational Results

The results of the tests described in Section 23.5 are shown in Figures 23.20 and 23.21

for the exponent β in Equation (23.1) equal to 0.75 and 0.5. I conducted the test using

ﬁve maturities, ranging from 1 to 5 years. See Tables 23.3 and 23.4. The ﬁrst ﬁve data

columns in these tables display the at-the-money implied volatilities when the swap rate

is moved. Incidentally, the prices and implied volatilities obtained for the reference level

show that Equations (23.14) and (23.15) were highly effective in producing the correct

at-the-money call prices.

From these ﬁgures qualitatively it appears that, indeed, if the underlying process is a

CEV diffusion, the implied volatility changes as a function of the at-the-money volatility

as an inverse power law. Furthermore, within numerical error, the same curve is traced

by the implied volatilities irrespective of the maturity of the option. The average of

these implied volatilities for each swap-rate level is shown in the sixth column of both

tables. Finally, the seventh column shows the implied volatility that would be obtained

by imposing that the implied volatility should behave as

σimpl (SR) = σimpl (SR0 )

SR

SR0

1−β

(23.16)

As one can see the ﬁt is virtually perfect. This allows one to conclude that:

1. if the process for the underlying is a β-CEV diffusion, as in Equation (23.1), the

associated at-the-money implied volatilities for all maturities also change with the

swap rate following an inverse power law; and

2. the exponents β and γ of the CEV process and of the implied volatilities, respectively, are linked by the simple relationship γ = 1 − β. Therefore

σimpl (SR) = kSR1−β = kSRγ

(23.17)

for a constant k given by

k=

σimpl (SR0 )

(SR0 )1−β

(23.18)

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CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES

21.5%

ATM implied volatility

21.0%

20.5%

20.0%

19.5%

19.0%

18.5%

3.00%

3.50%

4.00%

4.50%

5.00%

5.50%

6.00%

6.50%

ATM swap rate

Figure 23.20 At-the-money implied volatilities for different maturities as the swap rate is moved

from its initial value for the exponent β = 0.75.

23.0%

22.5%

ATM implied volatility

22.0%

21.5%

21.0%

20.5%

20.0%

19.5%

19.0%

18.5%

18.0%

3.50%

4.00%

4.50%

5.00%

5.50%

6.00%

6.50%

ATM swap rate

Figure 23.21 At-the-money implied volatilities for different maturities as the swap rate is moved

from its initial value for the exponent β = 0.50.

It is reassuring that, if β = 1, we are in a log-normal world (γ = 1 − β = 0), and one

recovers the independence of the implied volatility from the swap-rate level implied by

the homogeneity of degree one of the now-correct Black formula.

So far we have established a link from the process to the exponent: if the process for

the underlying is a CEV diffusion of the type (23.1), then the implied volatility is linked to

the level of the at-the-money swap rate by a relationship of the type (23.16). Clearly, the

23.7 EMPIRICAL STUDY II: THE LOG-LINEAR EXPONENT

721

Table 23.3 At-the-money implied volatilities for different maturities as the

swap rate is moved from its initial value for the exponent β = 0.75. The

average of the implied volatilities across maturities is also displayed, together

with the theoretical at-the-money implied volatility.

4.00%

4.50%

5.00%

5.50%

6.00%

1

2

3

4

5

Average

21.1%

20.5%

20.0%

19.5%

19.1%

21.1%

20.5%

19.9%

19.6%

19.1%

21.2%

20.4%

20.0%

19.4%

19.1%

21.1%

20.6%

19.9%

19.4%

19.1%

21.2%

20.4%

20.0%

19.5%

19.2%

21.2%

20.5%

20.0%

19.5%

19.1%

Analytic

21.1%

20.5%

20.0%

19.5%

19.1%

Table 23.4 At-the-money implied volatilities for different maturities as the

swap rate is moved from its initial value for the exponent β = 0.50. The

average of the implied volatilities across maturities is also displayed, together

with the theoretical at-the-money implied volatility.

4.00%

4.50%

5.00%

5.50%

6.00%

1

2

3

4

5

Average

22.3%

21.3%

20.0%

19.2%

18.2%

22.3%

21.1%

20.0%

19.0%

18.3%

22.4%

21.1%

19.9%

19.0%

18.2%

22.3%

21.2%

20.0%

19.1%

18.2%

22.3%

21.0%

19.9%

19.0%

18.3%

22.3%

21.1%

20.0%

19.1%

18.2%

Analytic

22.3%

21.0%

20.0%

19.0%

18.2%

reverse need not be true: the relationship between implied volatilities and at-the-money

swap rates could be of the type (23.16), and yet the underlying process might not be a

CEV diffusion. Can we say something stronger?

23.7

Empirical Study II: The Log-Linear Exponent

In order to examine further the relationship between CEV processes and powerlaw implied volatilities, let us look again at the transformed variable y(α), deﬁned in

Equation (23.12). Recall that the maturity-dependent exponent α had been numerically

determined so that the linear correlation coefﬁcient between the variables {y} and the

at-the-money swap-rate levels {SR} would be zero. Clearly, when the variable y(α) is

deﬁned as in Equation (23.12), such an exponent can always be found. This does not

automatically mean, however, that there is no dependence left between SR and y. The

relationship between the variables SR and y could give a value of zero to the coefﬁcient

of linear correlation, but could still be such as to display a lot of structure. I therefore

undertake a more direct test of the power-law hypothesis. More precisely, I investigate

how well a linear relationship between the variables ln σimpl and x = ln(kSRδ ) describes

the empirical data.5 If, after taking logs, the linear relationship held exactly one could

5 Because

of the proliferation of exponents, the following brief summary should help the reader:

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CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES

Table 23.5 The zero-correlation exponent α

and the (negative of the) slope δ.

Series

1 ×1

1 ×3

1 ×5

1 × 10

ZeroCorr

1.37

1.44

1.45

1.37

Slope

1.40

1.46

1.46

1.39

3 ×1

3 ×3

3 ×5

3 × 10

5 ×1

5 ×3

5 ×5

5 × 10

1.12

1.17

1.14

1.06

0.80

0.88

0.88

0.91

1.12

1.14

1.09

0.94

0.74

0.82

0.82

0.83

write

ln σimpl = ln k + δ ln SR

(23.19)

If Equation (23.19) were the correct description of the relationship between σimpl and

ln SR, then the residuals would have zero correlation with ln SR. How well are the data

explained by a relationship like (23.19)? And what is the link between δ and α? The

answer is provided by Table 23.5 which shows the zero-correlation exponent α and the

(negative of the) slope δ. It is interesting to observe how similar the slopes {δ} turned out to

be to the (negative of the) zero-correlation exponents {α}. Summary results of the regressions for the various swaption series are presented in Table 23.6. Figures 23.22–23.24

show the graphs of ln σimpl vs ln SR for a few swaption series.

As Table 23.6 indicates, the explanatory power of the linear relationship becomes

increasingly poorer as the expiry of the option increases. However, I discuss in the next

section how this might be related to the existence of distinct volatility regimes, each one

possibly still of a power-law type. In other words, a plot such as Figure 23.24 suggests

that the strong linear relationship might still hold for each temporally connected section of

the graph, but that different clusters might have different slopes and/or intercepts. Shifts

in the volatility regimes are the topic of Chapters 26 and 27.

• α denotes the empirical exponent that gives zero correlation between the quantity y(α) ≡ σimpl SRα and

the swap rate, SR;

• β denotes the theoretical exponent in the candidate (CEV) model for the evolution of the swap rate;

• γ denotes the exponent that best ﬁts the theoretically obtained behaviour between levels of swap rates

and of implied volatilities under the assumption that the swap rate follows a CEV behaviour;

• δ denotes the empirical exponent derived from the slope of the log-linear regression of the observed

implied volatility against KSRδ .

0.151

1165

5.381

5.323

5.440

179.752

−1.402

−1.436

−1.367

−79.726

0.845

1×1

0.150

1165

5.527

5.442

5.613

126.971

−1.465

−1.513

−1.417

−59.612

0.753

1×3

0.151

1165

5.524

5.417

5.631

101.155

−1.461

−1.520

−1.401

−48.318

0.667

1×5

Summary of the regression statistics.

Standard error

Observations

Intercept

Lower 95%

Upper 95%

t-statistic

Slope

Lower 95%

Upper 95%

t-statistic

R2

Table 23.6

0.154

1165

5.319

5.175

5.464

72.322

−1.350

−1.428

−1.272

−33.964

0.498

1 × 10

0.138

1165

4.987

4.878

5.095

89.890

−1.130

−1.190

−1.070

−37.155

0.543

3×1

0.140

1165

4.972

4.842

5.102

75.026

−1.140

−1.210

−1.070

−31.825

0.465

3×3

0.141

1165

4.853

4.708

4.999

65.578

−1.089

−1.167

−1.011

−27.463

0.393

3×5

0.141

1165

4.616

4.444

4.788

52.637

−0.985

−1.075

−0.894

−21.241

0.280

5×1

0.126

1165

4.342

4.194

4.490

57.532

−0.819

−0.897

−0.740

−20.471

0.265

5×3

0.125

1165

4.306

4.151

4.461

54.623

−0.820

−0.902

−0.739

−19.740

0.251

5×5

0.125

1165

4.260

4.080

4.441

46.382

−0.828

−0.922

−0.734

−17.265

0.204

5 × 10

724

CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES

4.25

4.05

3.85

3.65

3.45

3.25

3.05

2.85

2.65

2.45

2.25

0.9

1.1

1.3

1.5

1.7

1.9

2.1

Figure 23.22 Graph of ln σimpl = ln k + δ ln SR for the 1 × 1 series.

3.35

3.25

3.15

3.05

2.95

2.85

2.75

2.65

2.55

2.45

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

2.05

Figure 23.23 Graph of ln σimpl = ln k + δ ln SR for the 5 × 1 series.

2.1

23.9 WHERE DO WE GO FROM HERE?

725

3.15

3.05

2.95

2.85

2.75

2.65

2.55

2.45

2.35

1.7

1.75

Figure 23.24

23.8

1.8

1.85

1.9

1.95

2

2.05

2.1

Graph of ln σimpl = ln k + δ ln SR for the 5 × 10 series.

Combining the Theoretical and Experimental Results

We are ﬁnally in a position to put the theoretical results and the empirical data together.

We have established that a CEV diffusion produces implied volatilities that follow the

law (23.16). The empirical data also indicate that one cannot reject at the 95% conﬁdence

level the hypothesis that the implied volatilities and the swap-rate levels should be linked

by a power law, and that the exponent δ should assume the values given in Table 23.5.

If this behaviour had indeed been generated by a CEV process, the coefﬁcients γ , β and

δ should be linked by

γ =1−β =δ ⇒β =1−δ

(23.20)

Unfortunately, for most of the swaption series analysed the β-CEV exponent produced

by Equation (23.20) turn out to be negative, which is clearly not compatible with a CEV

explanation. For those series (mainly the 5× series) where β is positive, and where a

CEV diffusion could therefore have produced the data, the exponent β is very small (in

the range 0.1–0.2), suggesting an almost normal behaviour. These are, however, the same

series where the volatility regime switch mentioned above appears more evident. It might

therefore be the case that the lower exponent is simply arising from the inappropriateness

of a simple linear model.

23.9

Where Do We Go From Here?

The empirical analysis presented in this chapter has shown that a large part of the variability of the at-the-money implied volatility can be accounted for by the variability of

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CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES

the underlying swap rate, and that the dependence is of a power-law type. When this

contribution to the variability of the implied volatility is stripped out, there remains (a

considerably smaller) degree of variability that can arguably be described by a stochasticvolatility approach (as described in Chapter 25 and in Rebonato and Joshi (2002)). The

ﬁrst conclusion is therefore that it could be a misguided attempt to account for all the

variability displayed by swaption percentage volatilities in terms of log-normal swap rates

with a stochastic volatility. As Figures 23.1, 23.2 and 23.21 clearly show, in fact, a high

degree of functional dependence exists between the volatilities and the swap rates, and,

as a consequence, a simple transformation of variables can be effective in de-coupling

the dynamics of the swap rates and their volatility (once appropriately chosen).

However, when one moves from these qualitative features to the quantitative analysis

of the data, it becomes apparent that, if the underlying process was truly of the CEV

type, the exponent β derived from Equation (23.20) would have to be negative in order

to account for the empirical behaviour of the implied volatilities. This is unfortunately

not compatible with a CEV description as the sole mechanism for producing the negative

slope in the implied volatility plots. One is therefore left with three logical alternatives:

1. the changes in implied volatilities are compatible with a well-speciﬁed stochastic

process for the underlying, but this is not (exactly, or exclusively) of the CEV type;

2. the changes in implied volatilities are compatible with a well-speciﬁed stochastic

process for the underlying, this is of the CEV type, but the exponent γ has been

mis-estimated in the empirical analysis;

3. the changes in implied volatilities, which are, after all, simply prices quoted by

traders, are not derived from a coherent arbitrage-free process for the underlying.

As for the ﬁrst possibility, it might be the case that the exponent itself is a function of

the level of swap rates: for instance, when swap rates are high, they might behave almost

log-normally, but when they are low they might approach a more normal behaviour.

Alternatively, the true process could be a diffusion (perhaps of the CEV type) but with the

coefﬁcient σ in Equation (23.1) now stochastic. An ‘extra’ negative skew in the implied

volatility plot could be generated if this stochastic coefﬁcient followed a diffusion with

Brownian increments negatively correlated with the increments of the forward (or swap)

rates.

The third possibility is unpalatable from a theoretical point of view, because it should in

theory expose the trader who made prices in this manner to the risk of being arbitraged. It

should not, however, be discarded a priori if the arbitrage that could punish the inconsistent

trader were very difﬁcult to put in place in practice. See again the discussion in Chapter 1.

Finally, visual inspection of the chronologically linked empirical data suggested that a

volatility regime switch might be at play. This qualitative ﬁnding is in agreement with the

empirical results in Rebonato (2002), who has shown the existence of regime switches in

the shape of the swaption matrix for USD and EUR/DEM. Work to translate this insight

into a coherent modelling framework is presented in Chapter 27. (See also Rebonato and

Kainth (2004)).

The most important message from the analysis presented in this chapter is therefore that,

while insufﬁcient by itself to account for the observed market behaviour, a CEV process

is likely to be very important in accounting for a good part of the observed dependence

of implied volatilities on the level of swap rates. If one recalls the discussion presented

## Volatility correlation, rebonato

## 6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches

## 4 Hedging Options: Volatility of Spot and Forward Processes

## 9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again

## 1 Correlation, Co-Integration and Multi-Factor Models

## 3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options

## 6 Conclusions (or, Limitations of Quadratic Variation)

## 4 Green’s Functions (Arrow–Debreu Prices) in the DK Construction

## 2 The Financial Model: Smile Tale 2 Revisited

## 10 Jump–Diffusion Processes and Market Completeness Revisited

## 1 A Worked-Out Example: Pricing Continuous Double Barriers

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4 Empirical Study I: Transforming the Log-Normal Co-ordinates