5 Fitting the Instantaneous Volatility Function: Imposing Time Homogeneity of the Term Structure of Volatilities
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678
CHAPTER 21 THE INSTANTANEOUS VOLATILITY OF FORWARD RATES
40.00%
Implied volatility
35.00%
30.00%
35.00%-40.00%
30.00%-35.00%
25.00%-30.00%
20.00%-25.00%
15.00%-20.00%
10.00%-15.00%
25.00%
20.00%
10
29-May-96
8
9
9.5
Option expiry
8.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
10.00%
15-May-96
02-May-96
18-Apr-96
03-Apr-96
21-Mar-96
07-Mar-96
15.00%
Trade date
Figure 21.10 Implied volatilities for different maturities collected over a period of approximately
three months in 1996 for FRF.
approximately 1.5 and 2.5 years for the GBP data, seem to be preserved, despite the fact
that the plateau itself might change level (see Figure 21.9).
As for the second regime, this is associated with periods where the uncertainty about
the short end of the yield curve can dramatically increase. As can be appreciated by
looking at the far side of Figure 21.10, the change from the humped to the decaying state
can occur very abruptly (in a matter of days), and tends to disappear somewhat more
slowly (over many days or a few weeks).
If only the ﬁrst regime existed, a good criterion for the choice of the instantaneous
volatility parameters would be to recover in a reasonably time-homogeneous manner the
average (over trading days), rather than the current, term structure of volatilities. The
resulting average curves are shown for GPB and pre-Euro DEM in Figure 21.11. One can
readily notice the qualitative similarity between the obtained average curve and the sample
of curves presented in Figure 21.10. The motivation for this procedure would be to make
the model produce, as its guess for the future term structure of volatilities, something
very close to what has been observed in the past. Given the discussion in Section 1.3 of
Chapter 1 about future re-hedging costs, this is certainly plausible and desirable.
The same procedure can, of course, be followed by including in the average also the
‘unstable’ periods, but in this case looking at the average of radically different regimes can
be much less meaningful. I discuss in Chapter 27 how this problem can be tackled in a more
satisfactory manner (i.e. by introducing a shift in volatility regimes). More generally, if the
trader had extra information (or simply different views) about the evolution of the implied
volatility curve, she could always choose a set of parameters that embody these views.
This way of looking at the volatility parameterization actually constitutes one of the
greatest strengths of the LMM approach: by choosing a particular instantaneous-volatility
function, and therefore a future term structure of volatilities, the exotic trader directly
expresses a view about the quantities (the instantaneous volatilities of forward rates) that
constitute the difﬁcult-to-hedge component of the present value of exotic trades. This state
21.6 IS A TIME-HOMOGENEOUS SOLUTION ALWAYS POSSIBLE?
679
27.00%
25.00%
Implied volatility
23.00%
21.00%
19.00%
DEM
GBP
FRF
17.00%
15.00%
13.00%
11.00%
9.00%
7.00%
0
1
2
3
4
5
6
Option expiry
7
8
9
10
Figure 21.11 The average (over approximately two years of trading dates – 1996, 1997) of the
implied volatilities at different maturities observed in the market for GBP, DEM and FRF.
of affairs can be compared with the predicament of a trader who has to use one of the
many models that take as input much more opaque quantities, such as the volatility of
the all-driving short rate.
Note carefully that roughly parallel shifts in the term structure of volatilities (which
can often be adequately captured also by the black-box models) are unlikely to hurt the
trader signiﬁcantly, as long as she has put in place a reasonable vega hedge (see again the
discussion in Chapter 1). But, as the discussion in the previous chapter has shown, there
are not enough liquid instruments to ‘lock-in’, with suitable hedges, the contribution to the
value of an exotic trade originating from the apportioning of the instantaneous volatility
over the option life given a ﬁxed root-mean-squared volatility. In other words, the exotic
trader will ﬁnd it most difﬁcult to hedge against changes in the value of an exotic product
arising from changes in the shape (not the level) of the term structure of volatilities. It is
therefore exactly in this latter dimension that the superiority of the LMM approach can
be most easily appreciated.
21.6
Is a Time-Homogeneous Solution Always Possible?
A further important comment regarding this proposed approach is that today’s term structure of volatilities can sometimes indicate the extent to which the market ‘believes’ that
it can remain unchanged in shape and level over time. More precisely, one can easily
show (see, for example, Rebonato (2002) for a proof) that, if the market operates in an
arbitrage-free log-normal Black world and the quantity
ξ(T ) ≡ σT2 T
(21.13)
is not a strictly increasing function of maturity, then the future term structure of volatilities
cannot have the same shape and level as today’s. In other words, if ξ(T1 ) > ξ(T2 ), with
T1 < T2 , then, even if we changed the functional form of the instantaneous volatility and
added as many parameters as we want, we would still not be able to ﬁnd a normalization
vector {kT } exactly equal to 1 for all the forward rates. The converse is also true: if
the term structure of volatilities is such that the quantity ξ(T ) is strictly increasing in
680
CHAPTER 21 THE INSTANTANEOUS VOLATILITY OF FORWARD RATES
T , an instantaneous volatility of the form g(T − t) that prices the caplet market exactly
can always be found. Such an instantaneous-volatility function would generate an exactly
time-homogeneous evolution of the term structure of volatilities. See Rebonato (2002) for
a proof.
Summarizing: as discussed in Section 3.6.2 of Chapter 3, the failure of the quantity
ξ(T ) to be strictly increasing does not imply an arbitrage possibility (even in a perfect
Black world). It does imply, however, either that the market does not believe in the
forward rates being log-normally distributed, or that it believes that the level and/or the
shape of the implied volatility curve will change in the future.
Interestingly enough, particularly strong and systematic violations of the condition
dξ(T )
dT > 0 were observed in recent years for European currencies in the run-up to the Euro
conversion. During this period observation of caplet pricing practice for the less liquid
currencies (say, ITL and ESP) and comparison with the same-maturity quotes available
for DEM indicated that traders tended to subscribe to a more normal/square-root process
for the forward rates, than to a log-normal process. If one couples this observation with the
fact that the rates in most currencies were expected to move down towards convergence,
one can readily explain why term structures of volatilities with locally decreasing ξ(T )
were observed.
21.7
Fitting the Instantaneous Volatility Function:
The Information from the Swaption Market
In the ﬁrst part of this chapter I proposed one possible criterion (i.e. imposing that the
vector {kT } should be as constant as possible) in order to choose the functional form
and pin down the parameters of the instantaneous volatility function. As mentioned in
the ﬁrst section, however, if one couples this approach with a plausible choice for the
instantaneous correlation function, one can bring into play the combined information from
the cap and swaption markets. This can be done by making use of the relationship between
forward-rate and swap-rate volatilities discussed in Section 20.5 (see Equation (20.41) in
particular). These equations provide a convenient link between the swap-rate volatilities
implied by a set of chosen instantaneous volatilities and correlations on the one hand, and
the corresponding market prices of swaptions on the other.
If we assumed to know with certainty the forward-rate instantaneous volatility functions, and that the cap/swaption markets were perfectly coherent, then the swaption implied
volatility matrix would uniquely pin down the unknown correlation function (more precisely, would allow us to determine the covariance elements discussed in Section 5.4 of
Chapter 5). Instantaneous volatility functions are, however, not God-given; nor should one
automatically assume that the cap and swaption markets should be perfectly ‘in line’ with
each other (see Chapter 1). Therefore a combined approach, where considerations about the
time-homogeneity of the term structure of volatilities are analysed together with the resulting
model swaption implied volatility matrix, is in practice probably the most proﬁtable.
If this route is taken, the trader would not expect to match the market and model
swaption volatility matrices almost perfectly everywhere, but would try to modify the
choices of instantaneous volatility and/or correlation whenever the discrepancies showed a
particularly strong or systematic bias. Tables 21.1–21.3 and Figures 21.12–21.15 provide
some examples of good and obviously poor matches.
24-May-99
24-Nov-99
24-May-00
24-Nov-00
24-May-01
24-Nov-01
24-May-02
24-Nov-02
24-May-03
24-Nov-03
24-May-04
24-Nov-04
24-May-05
24-Nov-05
24-May-06
24-Nov-06
24-May-07
24-Nov-07
24-May-08
24-Nov-08
19.95%
21.97%
22.00%
22.00%
21.76%
21.50%
21.50%
21.50%
21.01%
20.51%
20.07%
19.63%
19.20%
18.76%
18.22%
17.68%
17.14%
16.59%
16.05%
15.51%
12
20.19%
21.48%
21.50%
21.50%
21.25%
21.01%
20.87%
20.76%
20.26%
19.76%
19.28%
18.83%
18.35%
17.89%
17.44%
17.00%
16.54%
16.10%
15.65%
15.21%
18
20.43%
20.99%
21.00%
21.00%
20.76%
20.50%
20.26%
20.00%
19.51%
19.01%
18.51%
18.01%
17.51%
17.01%
16.66%
16.31%
15.96%
15.61%
15.26%
14.91%
24
19.95%
20.50%
20.50%
20.50%
20.13%
19.76%
19.50%
19.26%
18.69%
18.14%
17.72%
17.32%
16.91%
16.51%
16.20%
15.89%
15.58%
15.28%
14.96%
14.66%
30
19.47%
19.99%
20.00%
20.00%
19.51%
19.01%
18.76%
18.50%
17.89%
17.26%
16.94%
16.63%
16.32%
16.01%
15.74%
15.47%
15.21%
14.94%
14.67%
14.40%
36
18.98%
19.74%
19.63%
19.50%
19.01%
18.51%
18.19%
17.89%
17.26%
16.64%
16.35%
16.07%
15.78%
15.51%
15.28%
15.06%
14.83%
14.61%
14.38%
14.16%
42
18.50%
19.48%
19.25%
19.00%
18.51%
18.01%
17.63%
17.26%
16.64%
16.01%
15.76%
15.50%
15.26%
15.00%
14.82%
14.64%
14.46%
14.27%
14.09%
13.90%
48
18.38%
19.24%
19.00%
18.76%
18.14%
17.51%
17.09%
16.69%
16.12%
15.56%
15.29%
15.03%
14.76%
14.51%
14.36%
14.22%
14.08%
13.94%
13.79%
13.65%
54
18.26%
18.99%
18.75%
18.50%
17.77%
17.01%
16.56%
16.11%
15.61%
15.11%
14.83%
14.55%
14.28%
14.00%
13.90%
13.80%
13.70%
13.60%
13.50%
13.40%
60
18.20%
18.74%
18.44%
18.13%
17.39%
16.64%
16.20%
15.77%
15.33%
14.90%
14.62%
14.36%
14.09%
13.82%
13.71%
13.60%
13.49%
13.39%
13.28%
13.18%
66
18.14%
18.49%
18.13%
17.76%
17.02%
16.26%
15.85%
15.43%
15.06%
14.68%
14.42%
14.15%
13.89%
13.63%
13.52%
13.40%
13.29%
13.18%
13.07%
12.95%
72
18.08%
18.25%
17.82%
17.39%
16.64%
15.89%
15.49%
15.10%
14.78%
14.47%
14.21%
13.96%
13.70%
13.44%
13.32%
13.20%
13.08%
12.97%
12.84%
12.73%
78
18.02%
18.00%
17.51%
17.01%
16.27%
15.51%
15.13%
14.76%
14.51%
14.25%
14.01%
13.75%
13.51%
13.25%
13.13%
13.00%
12.88%
12.75%
12.63%
12.50%
84
17.94%
17.75%
17.24%
16.72%
16.04%
15.35%
14.99%
14.63%
14.38%
14.13%
13.88%
13.62%
13.37%
13.11%
12.97%
12.84%
12.70%
12.57%
12.43%
12.30%
90
17.86%
17.51%
16.97%
16.43%
15.81%
15.18%
14.84%
14.51%
14.26%
14.00%
13.75%
13.49%
13.23%
12.97%
12.82%
12.67%
12.53%
12.38%
12.23%
12.09%
96
17.78%
17.26%
16.69%
16.14%
15.57%
15.01%
14.69%
14.38%
14.13%
13.88%
13.62%
13.35%
13.09%
12.83%
12.67%
12.51%
12.35%
12.20%
12.04%
11.88%
102
17.71%
17.01%
16.43%
15.84%
15.35%
14.84%
14.55%
14.25%
14.01%
13.75%
13.49%
13.22%
12.96%
12.69%
12.52%
12.35%
12.18%
12.01%
11.84%
11.67%
108
Table 21.1 The implied swaption volatilities for GBP observed in the market on 24 November 1998. On the x-axis one can read the maturity (in
months) of the swap into which the option can be exercised; on the y-axis one can read the expiry date.