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1 What Do We Want to Capture? A Hierarchy of Smile-Producing Mechanisms

1 What Do We Want to Capture? A Hierarchy of Smile-Producing Mechanisms

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704



CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES



If this analysis is correct, one can conclude that capturing component 1 is the most

important modelling feature, and that 2 might be as, if not more, important than 3. This

chapter and the following will deal with part 1 of this programme.



23.2



Are Log-Normal Co-Ordinates the Most

Appropriate?



In this chapter I discuss both empirical and theoretical results about swaption implied

volatilities (see also Rebonato (2003a)).1 The theoretical analysis is based on some properties of CEV processes. These are discussed in some detail in the next chapter. For the

moment I simply make use of the well-known observation that CEV models of the form

dx = µ dt + x β σβ dw



(23.1)



produce call prices whose implied volatilities display, as function of strike, an inverse

power-law behaviour. However, it is not obvious what behaviour for the implied volatility

as a function of the underlying these CEV model predicts. The first (theoretical) question

that I address is therefore: what behaviour for the at-the-money implied volatility as

a function of an instantaneous change in the underlying would be generated by a CEV

process for the underlying? I then move to the complementary (empirical) question: given

this information about the CEV-induced implied volatility dynamics, could the empirically

observed behaviour of the implied volatilities as a function of changes in the underlying

swap rates be explained by a CEV process?

Let me put the relevance of these questions into perspective. First of all, if the underlying followed a geometric diffusion with constant or time-dependent volatility, inverting

same-maturity prices to obtain an implied volatility would always give, by definition, the

same value for the implied volatility. Furthermore, if we lived in a log-normal world, as

the underlying moves, the price of an at-the-money call would exactly scale as the ratio

of the old to the new level of the swap rate. This immediately follows because of the

homogeneity of degree one of the Black formula.2 If we live in a CEV world, this is

no longer true. To begin with, the implied volatility as a function of strike is not flat.

Therefore, if we assume that the process for the at-the-money forward swap rate is given

by an equation such as (23.1), the percentage implied volatility (which is obtained by

inverting the now ‘inappropriate’ Black formula) will in general not remain the same

across strikes. In addition, a change in the level of the underlying causes a change in the

at-the-money call price that does not scale in a proportional fashion.

With these considerations in mind, the theoretical questions in the first paragraph of

this section can be rephrased more precisely as follows:

• if we observe today’s at-the-money implied volatility and swap rate to be σ0ATM

and SR0 , respectively, and the process for the swap rate is of the CEV type with

an exponent β, can we say that tomorrow’s implied volatility, σATM (SR), when the

1 Parts



of this chapter have been adapted from Rebonato (2003a).

that a function, f (x), is homogeneous of degree one in x means that f (kx) = kf (x), with k a

constant. More generally, a function f (x) is said to be homogeneous of degree n in x if f (kx) = x n f (x).

2 Saying



23.2 ARE LOG-NORMAL CO-ORDINATES THE MOST APPROPRIATE?



705



swap rate has moved to level SR, is given by

σATM (SR) = σ0ATM



SR

SR0



γ



(23.2)



for some exponent γ ?

• If this is the case, how are the exponents γ and β related?

As for the empirical questions addressed in this chapter, they can be formulated as

follows.

• Is there is a correlation between changes in the level of forward rates and changes

in the implied volatility?

• If such a correlation exists, is there a transformation (a change of co-ordinates) that

can be applied to the forward rates, such that this correlation disappears?

• Would a log-linear regression between changes in the underlying forward rates and

changes in implied volatility be statistically significant in accounting for the empirical

data?

• How is the slope from this log-linear regression linked to the exponent that gives

zero correlation between the changes in implied volatilities and the power-lawtransformed underlying forward rates?

Before tackling these issues, let me define precisely in the next sub-section what I

mean by ‘appropriate co-ordinates’.



23.2.1 Defining Appropriate Co-ordinates

Ideally, one would like to be able to find some function, f , of the state variable x,

such that the volatility of f , σf , would not display any dependence on x itself. The

transformation from x to f (x) can be seen as a change of co-ordinates. In terms of these

transformed co-ordinates, one would like to be able to express the SDE for f in the form

df (x) = σf (t) dz



(23.3)



with σf (t) a deterministic function of time. In a log-normal (Black) world the function

f is clearly

f (x) = ln(x)



(23.4)



d ln x = − 12 σf (t)2 dt + σf (t) dz



(23.5)



because we know that



Similarly, if the process for x was a normal diffusion, the function would simply be

f (x) = x



(23.6)



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CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES



Consider now the case when the function f (x) is f (x) = x ξ . Then

df = d(x ξ ) = ξ x ξ −1 dx



(23.7)



If we want this transformation to be the right change of co-ordinates then it must be the

case that

df (x) = d(x ξ ) = σf (t) dz



(23.8)



From (23.7) it follows that

ξ x ξ −1 dx = σf (t) dz

Therefore

dx =



σf (t) 1−ξ

dz

x

ξ



(23.9)



and, by setting

σf (t)

= σβ

ξ

1−ξ =β



(23.10)

(23.11)



one can see that the change of co-ordinates from x to f (x) = x ξ is appropriate if the

process for x was a CEV diffusion with exponent β = 1−ξ (at least as long as ξ < 1 – see

below).



23.3



Description of the Market Data



The market data used for the study presented in this chapter consist of time series of daily

at-the-money swaption implied volatilities and of the associated at-the-money forward

swap rates. (USD, period 5-Jun-1998/22-Nov-2002 for a total of 1166 days.) The data are

particularly significant because they span the turbulent period of the Russia/LTCM crisis.

The swaptions and swap rates analysed belonged to the n × m series,3 with n = 1, 3, 5,

and m = 1, 3, 5, 10. Figure 23.1 displays the time series of the volatilities. Figure 23.2

shows the time series of one particular swap rate and of the associated percentage swaption

volatility (1×1). Figures 23.3–23.14 display the scatter plots of the implied volatilities and

of the swap rates for the various available combinations of option expiries and underlying

swap lengths.

3 The



n × m swaption is the option to enter in n years’ time a swap with m years of residual maturity.



23.3 DESCRIPTION OF THE MARKET DATA



707



70

US SWAPTIONVOL 1Y fwd 1Y

US SWAPTIONVOL 1Y fwd 3Y

US SWAPTIONVOL 1Y fwd 5Y

US SWAPTIONVOL 1Y fwd 10Y

US SWAPTIONVOL 3Y fwd 1Y

US SWAPTIONVOL 3Y fwd 3Y

US SWAPTIONVOL 3Y fwd 5Y

US SWAPTIONVOL 3Y fwd 10Y

US SWAPTIONVOL 5Y fwd 1Y

US SWAPTIONVOL 5Y fwd 3Y

US SWAPTIONVOL 5Y fwd 5Y

US SWAPTIONVOL 5Y fwd 10Y



60



50



40



30



20



10



0

10-Jun-98 27-Dec-98 15-Jul-99 31-Jan-00 18-Aug-00 06-Mar-01 22-Sep-01 10-Apr-02 27-Oct-02



Figure 23.1 Time series of different at-the-money percentage-implied-volatility swaption series

(USD market). Note how the shortest-maturity, shortest-expiry (1 × 1) series (top line) reaches 60%

in 2002.

90

80

70

60

50



1Y fwd 1Y

Vol



40

30

20

10

0

15-Apr-98



Figure 23.2



28-Aug-99



09-Jan-01



24-May-02



Implied volatility of the 1×1 swaption series vs forward rate (rates multiplied by 10).



708



CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES

70

60

50

40

30

20

10

0

2



3



4



5



6



7



8



9



Figure 23.3 Scatter plot of implied volatilities and swap rates (1 × 1 swaption series).

50

45

40

35

30

25

20

15

10

5

0

2



3



4



5



6



7



8



9



Figure 23.4 Scatter plot of implied volatilities and swap rates (1 × 3 swaption series).



Figures 23.15, 23.16 and 23.17 show again the scatter plots of selected implied volatilities vs the corresponding swap rate levels. The points, however, are now displayed with

a chronological link by joining with a continuous line the realizations corresponding to

consecutive business days. The features displayed by these three graphs are discussed

in more detail at the end of the chapter (Section 23.8) and in Chapter 27. Already at

this stage, however, one can say that the points in the scatter plot appear clustered into



23.3 DESCRIPTION OF THE MARKET DATA



709



45

40

35

30

25

20

15

10

5

0

3



4



5



6



7



8



9



Figure 23.5 Scatter plot of implied volatilities and swap rates (1 × 5 swaption series).



35

30

25

20

15

10

5

0

4



Figure 23.6



4.5



5



5.5



6



6.5



7



7.5



8



Scatter plot of implied volatilities and swap rates (1 × 10 swaption series).



8.5



710



CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES



40

35

30

25

20

15

10

5

0

3



4



5



6



7



8



9



Figure 23.7 Scatter plot of implied volatilities and swap rates (3 × 1 swaption series).



35

30

25

20

15

10

5

0

4



4.5



5



5.5



6



6.5



7



7.5



8



8.5



Figure 23.8 Scatter plot of implied volatilities and swap rates (3 × 3 swaption series).



23.3 DESCRIPTION OF THE MARKET DATA



711



35

30

25

20

15

10

5

0

4



4.5



5



5.5



6



6.5



7



7.5



8



8.5



Figure 23.9 Scatter plot of implied volatilities and swap rates (3 × 5 swaption series).



30



25



20



15



10



5



0

5



5.5



6



6.5



7



7.5



8



Figure 23.10 Scatter plot of implied volatilities and swap rates (3 × 10 swaption series).



8.5



712



CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES



30



25



20



15



10



5



0

5



5.5



6



6.5



7



7.5



8



8.5



Figure 23.11 Scatter plot of implied volatilities and swap rates (5 × 1 swaption series).



30



25



20



15



10



5



0

5



5.5



6



6.5



7



7.5



8



Figure 23.12 Scatter plot of implied volatilities and swap rates (5 × 3 swaption series).



8.5



23.3 DESCRIPTION OF THE MARKET DATA



713



30



25



20



15



10



5



0

5



5.5



6



6.5



7



7.5



8



8.5



Figure 23.13 Scatter plot of implied volatilities and swap rates (5 × 5 swaption series).



24

22

20

18

16

14

12

10

5.5



6



6.5



7



7.5



8



Figure 23.14 Scatter plot of implied volatilities and swap rates (5 × 10 swaption series).



8.5



714



CHAPTER 23 HOW TO MODEL INTEREST-RATE SMILES



70

60

50

40

30

20

10

2.25



3.25



4.25



5.25



6.25



7.25



8.25



Figure 23.15 Scatter plot of implied volatilities and swap rates (1 × 1 swaption series). The

realizations corresponding to consecutive business days are linked by a continuous line.

30

28

26

24

22

20

18

16

14

12

10

4.75



5.25



5.75



6.25



6.75



7.25



7.75



8.25



Figure 23.16 Scatter plot of implied volatilities and swap rates (3 × 5 swaption series). The

realizations corresponding to consecutive business days are linked by a continuous line.



subdomains within which an inverse power-law behaviour appears to prevail (although,

possibly, with a different exponent). Even more interesting is that these clusters appear

to be arrived at via a single entry or exit point. This would seem to indicate the existence

of distinct implied volatility regimes, with sudden switches between one regime and the

other.



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