1 What Do We Want to Capture? A Hierarchy of Smile-Producing Mechanisms
Tải bản đầy đủ - 0trang
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)
706
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.