7 Fitting the Instantaneous Volatility Function: The Information from the Swaption Market
Tải bản đầy đủ - 0trang
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.
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
19.42%
21.54%
21.69%
21.70%
21.45%
21.17%
21.13%
21.08%
20.60%
20.11%
19.70%
19.29%
18.88%
18.45%
17.94%
17.43%
16.92%
16.40%
15.89%
15.39%
12
19.78%
21.15%
21.25%
21.24%
20.98%
20.72%
20.55%
20.40%
19.91%
19.44%
19.00%
18.56%
18.11%
17.67%
17.23%
16.81%
16.37%
15.95%
15.52%
15.11%
18
20.06%
20.74%
20.79%
20.77%
20.52%
20.25%
19.98%
19.72%
19.24%
18.77%
18.30%
17.83%
17.36%
16.88%
16.53%
16.18%
15.84%
15.50%
15.16%
14.83%
24
19.69%
20.31%
20.33%
20.32%
19.95%
19.57%
19.30%
19.05%
18.51%
17.99%
17.59%
17.21%
16.81%
16.42%
16.10%
15.80%
15.49%
15.19%
14.89%
14.61%
30
19.29%
19.86%
19.88%
19.85%
19.38%
18.89%
18.63%
18.37%
17.79%
17.20%
16.89%
16.58%
16.27%
15.96%
15.68%
15.41%
15.15%
14.89%
14.63%
14.37%
36
18.87%
19.62%
19.53%
19.39%
18.92%
18.44%
18.11%
17.81%
17.22%
16.64%
16.35%
16.07%
15.78%
15.50%
15.26%
15.04%
14.80%
14.58%
14.36%
14.14%
42
18.45%
19.38%
19.18%
18.93%
18.46%
17.98%
17.61%
17.24%
16.66%
16.07%
15.81%
15.56%
15.30%
15.04%
14.85%
14.65%
14.46%
14.28%
14.09%
13.91%
48
18.34%
19.15%
18.94%
18.69%
18.11%
17.52%
17.12%
16.72%
16.19%
15.66%
15.39%
15.12%
14.85%
14.59%
14.43%
14.28%
14.12%
13.98%
13.83%
13.68%
54
18.22%
18.91%
18.70%
18.45%
17.77%
17.07%
16.63%
16.19%
15.72%
15.24%
14.96%
14.69%
14.41%
14.14%
14.02%
13.90%
13.78%
13.67%
13.56%
13.45%
60
18.16%
18.68%
18.41%
18.11%
17.42%
16.72%
16.30%
15.88%
15.46%
15.04%
14.76%
14.49%
14.23%
13.96%
13.83%
13.71%
13.59%
13.47%
13.36%
13.25%
66
18.09%
18.44%
18.12%
17.76%
17.07%
16.37%
15.97%
15.56%
15.20%
14.83%
14.57%
14.30%
14.04%
13.78%
13.65%
13.52%
13.40%
13.28%
13.16%
13.04%
72
18.03%
18.21%
17.82%
17.41%
16.72%
16.02%
15.63%
15.25%
14.93%
14.63%
14.37%
14.11%
13.86%
13.60%
13.47%
13.34%
13.21%
13.08%
12.95%
12.83%
78
17.96%
17.98%
17.53%
17.06%
16.38%
15.67%
15.30%
14.93%
14.68%
14.42%
14.17%
13.92%
13.67%
13.43%
13.29%
13.15%
13.02%
12.88%
12.75%
12.63%
84
17.88%
17.74%
17.28%
16.79%
16.15%
15.51%
15.16%
14.81%
14.55%
14.30%
14.04%
13.79%
13.54%
13.29%
13.14%
13.00%
12.85%
12.71%
12.57%
12.44%
90
17.80%
17.51%
17.02%
16.51%
15.93%
15.34%
15.01%
14.68%
14.43%
14.17%
13.92%
13.66%
13.41%
13.15%
13.00%
12.84%
12.69%
12.54%
12.39%
12.24%
96
17.71%
17.27%
16.76%
16.24%
15.71%
15.18%
14.86%
14.55%
14.30%
14.05%
13.79%
13.53%
13.27%
13.02%
12.85%
12.69%
12.53%
12.37%
12.21%
12.06%
102
17.63%
17.04%
16.51%
15.97%
15.49%
15.01%
14.72%
14.43%
14.18%
13.93%
13.66%
13.40%
13.14%
12.88%
12.71%
12.53%
12.37%
12.20%
12.03%
11.86%
108
Table 21.2 The model swaption volatilities for 24 November 1998 obtained using the {a, b, c, d} parameters that gave rise to the vector of
normalization factors kT shown in Figure 21.5.
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
0.53%
0.43%
0.31%
0.30%
0.31%
0.33%
0.37%
0.42%
0.42%
0.39%
0.37%
0.34%
0.32%
0.30%
0.28%
0.25%
0.22%
0.20%
0.17%
0.12%
12
0.42%
0.33%
0.25%
0.27%
0.27%
0.29%
0.32%
0.36%
0.34%
0.32%
0.29%
0.26%
0.24%
0.22%
0.20%
0.19%
0.17%
0.16%
0.13%
0.10%
18
0.37%
0.25%
0.21%
0.23%
0.23%
0.26%
0.27%
0.29%
0.27%
0.24%
0.21%
0.18%
0.15%
0.13%
0.13%
0.12%
0.12%
0.11%
0.09%
0.07%
24
0.26%
0.19%
0.16%
0.19%
0.18%
0.19%
0.20%
0.21%
0.18%
0.15%
0.13%
0.11%
0.10%
0.09%
0.09%
0.09%
0.09%
0.08%
0.07%
0.05%
30
0.18%
0.14%
0.12%
0.15%
0.13%
0.12%
0.13%
0.14%
0.10%
0.06%
0.05%
0.05%
0.05%
0.05%
0.06%
0.06%
0.06%
0.05%
0.04%
0.03%
36
0.11%
0.12%
0.09%
0.11%
0.09%
0.08%
0.08%
0.08%
0.04%
0.00%
0.00%
0.00%
0.00%
0.01%
0.02%
0.02%
0.02%
0.02%
0.02%
0.01%
42
0.05%
0.10%
0.07%
0.07%
0.05%
0.03%
0.03%
0.02%
−0.02%
−0.06%
−0.06%
−0.05%
−0.04%
−0.04%
−0.03%
−0.02%
−0.01%
−0.01%
−0.01%
−0.01%
48
0.04%
0.09%
0.06%
0.06%
0.02%
−0.01%
−0.02%
−0.03%
−0.07%
−0.10%
−0.10%
−0.09%
−0.09%
−0.08%
−0.07%
−0.06%
−0.05%
−0.04%
−0.03%
−0.03%
54
0.03%
0.08%
0.05%
0.05%
0.00%
−0.05%
−0.07%
−0.09%
−0.11%
−0.14%
−0.13%
−0.13%
−0.13%
−0.13%
−0.11%
−0.10%
−0.08%
−0.07%
−0.06%
−0.05%
60
0.04%
0.06%
0.03%
0.03%
−0.03%
−0.08%
−0.10%
−0.11%
−0.12%
−0.14%
−0.14%
−0.14%
−0.14%
−0.14%
−0.12%
−0.11%
−0.09%
−0.08%
−0.08%
−0.07%
66
0.04%
0.05%
0.01%
0.00%
−0.06%
−0.11%
−0.12%
−0.13%
−0.14%
−0.15%
−0.15%
−0.15%
−0.15%
−0.15%
−0.13%
−0.12%
−0.11%
−0.10%
−0.09%
−0.09%
72
Table 21.3 The difference between the market data and model data in Tables 21.1 and 21.2.
0.05%
0.04%
−0.01%
−0.03%
−0.08%
−0.13%
−0.14%
−0.15%
−0.15%
−0.16%
−0.16%
−0.16%
−0.16%
−0.16%
−0.15%
−0.13%
−0.12%
−0.12%
−0.11%
−0.10%
78
0.06%
0.02%
−0.03%
−0.05%
−0.11%
−0.16%
−0.17%
−0.18%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.16%
−0.15%
−0.14%
−0.13%
−0.13%
−0.12%
84
0.06%
0.01%
−0.04%
−0.07%
−0.12%
−0.16%
−0.17%
−0.18%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.18%
−0.17%
−0.16%
−0.15%
−0.14%
−0.14%
−0.14%
90
0.07%
0.00%
−0.05%
−0.09%
−0.13%
−0.16%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.18%
−0.18%
−0.17%
−0.17%
−0.16%
−0.16%
−0.16%
−0.16%
96
0.07%
−0.02%
−0.07%
−0.10%
−0.14%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.18%
−0.18%
−0.19%
−0.18%
−0.18%
−0.17%
−0.17%
−0.17%
−0.18%
102
0.08%
−0.03%
−0.08%
−0.12%
−0.15%
−0.17%
−0.17%
−0.17%
−0.17%
−0.17%
−0.18%
−0.18%
−0.19%
−0.19%
−0.19%
−0.19%
−0.19%
−0.19%
−0.19%
−0.19%
108
684
CHAPTER 21 THE INSTANTANEOUS VOLATILITY OF FORWARD RATES
23.00%
21.00%-23.00%
19.00%-21.00%
17.00%-19.00%
15.00%-17.00%
13.00%-15.00%
11.00%-13.00%
19.00%
17.00%
15.00%
60
48
36
24
24-May-08
12
24-Nov-06
24-May-05
Expiry date
24-Nov-03
24-Nov-00
24-May-02
24-May-99
96
84
11.00%
72
13.00%
108
Implied volatility
21.00%
Maturity (months)
Figure 21.12 The data presented in Table 21.1.
Model implied volatility
23.00%
21.00%-23.00%
19.00%-21.00%
17.00%-19.00%
15.00%-17.00%
13.00%-15.00%
11.00%-13.00%
21.00%
19.00%
17.00%
102
84
66
48 Maturity (months)
15.00%
13.00%
11.00%
24-May-08
12
24-May-07
24-May-06
24-May-05
24-May-03
Expiry date
24-May-04
24-May-01
24-May-02
24-May-99
24-May-00
30
Figure 21.13 The data presented in Table 21.2.
In particular, Figures 21.14 and 21.15 show that the effect of moving from a ﬂat to a
time-dependent instantaneous volatility is very noticeable. It is also important to point out
that one can empirically observe (see, for example, Rebonato (2002)) that allowing for a
time-dependent volatility can have a stronger impact on the resulting swaption volatility
matrix than varying (within reasonable limits) the instantaneous correlation. Therefore
taking caplet and swaption prices together is a blunt instrument in order to pin down the
21.7 INFORMATION FROM THE SWAPTION MARKET
685
0.50%-0.60%
0.40%-0.50%
0.30%-0.40%
0.20%-0.30%
0.10%-0.20%
0.00%-0.10%
−0.10%-0.00%
−0.20%--0.10%
0.60%
Volatility error
0.50%
0.40%
0.30%
0.20%
102
84
0.10%
0.00%
66
−0.10%
48
−0.20%
Maturity (months)
24-May-08
24-May-06
24-May-07
24-May-04
Expiry date
24-May-05
24-May-03
24-May-01
24-May-02
24-May-99
24-May-00
30
12
Figure 21.14 The data presented in Table 21.3.
Volatility error
1.00%
0.00%-1.00%
−1.00%-0.00%
−2.00%--1.00%
−3.00%--2.00%
−4.00%--3.00%
−5.00%--4.00%
0.00%
−1.00%
−2.00%
−3.00%
−4.00%
24-May-99
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
−5.00%
24-Nov-01
24-May-04
Expiry date
24-Nov-06
102
108
Maturity (months)
Figure 21.15 The difference between the market and model data obtained using the same correlation function used to produce Table 21.2 and Figure 21.14, but with time-independent volatilities
for the forward rates. Note the different scale for the z-axis in Figures 21.14 and 21.15. See the
discussion in the text.
correlation function (see also Sidenius (1999) and De Jong et al. (1999) who concur about
this point). If the trader attempted this implied route to estimating the correlation between
forward rates she would not only be faced with the usual problems of having to rely on
the market’s informational efﬁciency (see Section 1.4.3 of Chapter 1), but also with the
speciﬁc problem that the dependence of the correlation on the liquid market instruments
686
CHAPTER 21 THE INSTANTANEOUS VOLATILITY OF FORWARD RATES
is relatively weak. This might prompt the trader to conclude that, if she had to choose
between a one-factor model with realistic instantaneous volatilities, and a multi-factor
approach with ﬂat instantaneous volatilities, she would probably opt for the former. I
would concur with this choice, but, at the same time, this conclusion should not make us
conclude that ‘correlation hardly matters at all’: what appears to be the case for swaptions
might not be the case at all for other exotic instruments,2 and the ‘neglected’ correlation
might turn out to be much more important for the pricing of a particular exotic instrument.
The trader should always check, when pricing exotic products, whether the correlation
impact is indeed as mild and benign as it appears to be in the case of swaptions.
Finally, the analysis of the model and market implied swaption volatility matrices
clearly indicates that the swaption market (at least in all the major currencies) seems to
trade strongly at odds with a ﬂat-instantaneous-volatility assumption. See, for instance,
Figure 21.14. This observation suggests that the market concurs with the analysis presented in the previous section, and implies a signiﬁcant dependence of the instantaneous
volatility functions on the residual time to maturity.
21.8
Conclusions
In this chapter I have highlighted the link between the (non-directly-observable) time
dependence of the instantaneous volatility of forward rates and the evolution of today’s
(observable) term structure of volatilities. I have argued that this way of looking at option
pricing is both powerful and helpful, and constitutes one of the strongest points in favour
of the LMM approach.
I then proposed a speciﬁc separable functional form for the instantaneous volatility function. I have shown how the range of acceptable parameters characterizing this
description can be narrowed down (if not really pinned down uniquely) by the joint
requirements of a good ﬁt to the swaption implied volatility function, and of an approximately time-stationary behaviour for the term structure of caplet volatilities. Statistical
data can also supply useful indications by giving an idea of the most likely ranges for the
various parameters. Furthermore, the functional form for the instantaneous volatility suggested in Section 21.2 has the advantage of allowing for a very transparent link between
statistically- and market-accessible quantities, and some of the parameters.
I have also presented market examples from the swaption markets, and I have made
the case that (within a range of ‘reasonable’ values) the instantaneous correlation plays a
relatively minor role in determining the implied volatility of a swaption. I have therefore
argued that this market does not provide a very accurate tool to discriminate between
different possible functional forms or parameterizations for the correlation function. This
does not mean, unfortunately, that instantaneous correlation ‘does not matter’ for any
exotic instrument. The topic of how to choose a desirable correlation function is therefore
tackled in the next chapter.
2 The
examples of ﬂexi caps and of Total Accrual Redemption Notes (TARNs) spring to mind.
Chapter 22
Specifying the Instantaneous
Correlation Among Forward
Rates
22.1
Why Is Estimating Correlation So Difﬁcult?
When we deal with same-currency interest rates we are in a regime of relatively high
correlation among the state variables. This is just the situation where the time dependence
of the instantaneous volatility has the strongest impact on terminal de-correlation (see,
in this respect, the discussion in Chapter 5). It is easy to see why this is the case. If the
instantaneous correlation among a set of variables is already close to zero, there is little
further ‘scrambling’ that can be produced by time-dependent volatilities. However, if the
instantaneous correlation is close to one, volatilities ‘out of phase’ with each other (i.e.
one volatility being high when the other is low) can produce a signiﬁcant decrease in
terminal correlation.
So, the precise time behaviour of forward-rate volatilities can be very important in
order to understand their terminal de-correlation. Speaking of the time dependence of the
volatility tout court can however be rather delicate. This is because there are two distinct
possible sources of time dependence for the volatility. One arises because the underlying
phenomenon is intrinsically not time-homogeneous (the future is different from today). In
a world without smiles, this is the informational content that one is tempted to draw from
a stock-declining implied volatility. See, in this respect, the discussion in Chapter 3.1
The second type of time dependence arises because, as time goes by, a certain asset or
the associated rate changes its nature; for instance, its residual maturity decreases. This
applies to bonds and forward rates. So, we can still have time homogeneity if the volatility
depends on calendar time, as long as the dependence is of the form f (T − t).
Why does this matter in the context of the estimation of interest-rate correlation from
option prices? Because the terminal correlation, ρij (t, T ), which, as we saw in Chapter 5,
1 Recall, however, that one should always ask the questions: ‘Is the market informationally efﬁcient?’ and
‘Is it really trying to tell us something about future volatility, or is there a supply-and-demand effect at play?’.
687
688
CHAPTER 22 INSTANTANEOUS CORRELATION AMONG FORWARD RATES
one can more or less directly impute from observed market prices,
T
t
ρij (t, T ) =
T
t
σi (u)σj (u)ρij (u) du
σi (u)2 du
T
t
(22.1)
σj (u)2 du
(with T ≤ min(Ti , Tj )) depends both on the time dependence of the volatility and on
the instantaneous correlation. Therefore, if we wanted to extract information about the
correlation among forward rates from, say, swaption prices, this would only be possible if
our assumption about the time dependence of the instantaneous volatility of the forward
rates was correct. But, when dealing with same-currency forward and swap rates, we
are just in the regime of relatively high instantaneous correlation alluded to above. In
this regime the nature of the time dependence of the volatility can signiﬁcantly affect
the terminal de-correlation, and relatively small differences in the speciﬁcation of the
instantaneous volatility functions can have a big impact on the ‘implied’ correlation.
There are also more fundamental problems. Extracting the correlation among forward
rates from swaption data implies a great faith in the joint informational efﬁciency of
the caplet and swaption markets. I have expressed my doubts about this in Chapter 1,
and therefore I will not repeat the argument here. See, however, Rebonato (2002) for a
discussion of the systematic and similar biases observed in the instantaneous volatility
curves distilled from caplet and swaption data in several currencies. Similar reservations
are also expressed in Fan et al. (2003).
In view of the above, my ﬁrst, and perhaps most important, message in modelling
correlation is to keep the treatment simple and transparent. Above all, I believe that the
trader should resist the temptation to use the O(n2 ) elements of a correlation matrix in
order to recover ‘at all costs’ the market prices of swaptions.
22.2
What Shape Should We Expect for the Correlation
Surface?
In a time-homogeneous world, the de-correlation between two forward rates depends on
(at least) two quantities: how ‘distant’ (in expiry times) the two forward rates are; and
the expiry of, say, the ﬁrst of the two.2 The dependence of the de-correlation on the ﬁrst
quantity is intuitively easy to guess: the farther apart the two forward rates are, the less we
should expect them to move ‘in step’. The second dependence is not so straightforward to
guess a priori. On the one hand, it is reasonable to expect that the de-correlation between
two forward rates expiring, say, in nine and 10 years’ time should be smaller than the decorrelation between forward rates expiring in one and two years’ time (despite the fact that
the distance in expiries is the same). It is not obvious, however, whether this dependence
of the rate of de-correlation on the expiry of the ﬁrst forward rate should be monotonic.
Should we take as a necessary desideratum for a ‘good’ model correlation function that
the de-correlation between same-distance forward rates should be the stronger the shorter
2 Needless to say, if the future looks different than the past, there would also be a separate dependence on
calendar time.
22.3 FEATURES OF THE SIMPLE EXPONENTIAL CORRELATION FUNCTION 689
the expiry of the ﬁrst rate, irrespective of how short this expiry might be? Intuitively,
this requirement makes sense, and, for instance, Schoenmakers and Coffey (2000) (see
below) directly build this requirement into their construction of their correlation surface.
The empirical evidence, however, is not so clear. I review in Rebonato (2002) some
studies (e.g. Longstaff et al. (2000a)) that suggest that imposing this requirement might
not be supported by market data. I therefore keep an open mind on the matter at this stage.
22.3
Features of the Simple Exponential
Correlation Function
The simplest functional form for a correlation function is possibly the following:
ρij (t) = exp[−β|Ti − Tj |],
t ≤ min(Ti , Tj )
(22.2)
with Ti and Tj the expiries of the ith and j th forward rates, and β a positive constant.
Note that, at the risk of being pedantic, one should perhaps write
ρij (t) = exp −β|(Ti − t) − (Tj − t)|
(22.3)
which obviously is equivalent to (22.2). The full expression (22.3) will come in handy,
however, when we deal with more general (non-linear) functions, g, of the residual time
to expiry, of the form
ρij (t) = exp[−β|g(Ti − t) − g(Tj − t)|]
(22.4)
Equation (22.2) clearly satisﬁes the requirements of the ﬁrst type of dependence: the
farther apart two forward rates are, the more de-correlated they are. Furthermore, for
any positive β one can rest assured that the corresponding matrix ρ will always be
an admissible correlation matrix (i.e. a real, symmetric matrix with positive eigenvalues).
What expression (22.2) does not handle well is the second desideratum: two forward rates,
separated by the same ‘distance’, Ti − Tj , will de-correlate just as much irrespective of
whether the ﬁrst forward rate expires in three months or 30 years. See Figure 22.1.
This ﬁnancially undesirable feature is directly reﬂected in the absence of an explicit
time dependence in Equation (22.2). The ﬁnancial blemish, however, has a desirable
computational effect: in the LIBOR market model, in fact, the central quantities that
drive both the deterministic and the stochastic parts of the evolution are the covariance
elements
C(i, j, k) =
Tk+1
Tk
σi (u)σj (u)ρij (u) du
(22.5)
If the correlation function ρij is of the form (22.2), however, the absence of an explicit
time dependence allows one to write
C(i, j, k) = ρij
Tk+1
Tk
σi (u)σj (u) du
(22.6)
690
CHAPTER 22 INSTANTANEOUS CORRELATION AMONG FORWARD RATES
1
0.95
0.9
0.95-1
0.9-0.95
0.85
0.85-0.9
0.8-0.85
0.8
0.75-0.8
0.7-0.75
0.75
9
6
6
4.5
3
1.5
0
0
9
7.5
0.7
3
Figure 22.1 The correlation surface in the case of the simple exponential function
(Equation (22.2)).
and, with the functional forms for the instantaneous volatility discussed in Chapter 21,
the integral (22.6) can be pre-calculated analytically, thereby lightening the computational
burden. This advantage is not crucial and, by itself, would not justify the use of a crude
correlation function, if this had a seriously negative pricing impact. However, I show
later in this chapter that, as long as the same degree of de-correlation is, on average,
correctly recovered, the details of the shape of the correlation function are relatively
unimportant. Therefore, one does not have to pay too high a price for simplicity and
ease of computation. Furthermore, Joshi (2001), (unpublished result, quoted in Rebonato
(2002)), argues that the functional form (22.2) is more general and less ad hoc than one
might at ﬁrst surmise. Joshi in fact shows the following. Take a very simple yield curve
described by three forward rates. If
• the correlation function between the forward rates is of the form
ρ = ρ(|Ti − Tj |)
(22.7)
and
• the part of the responsiveness to shocks of forward rate f1 that is uncorrelated with
the responsiveness of f2 is also uncorrelated with changes in f3 , then the correlation
function must be of the form
ρ(Ti − Tj ) = exp(−β|Ti − Tj |)
(22.8)
where β is a constant. Rebonato (2002) discusses an extension of this simple
example.
22.4 MODIFIED EXPONENTIAL CORRELATION
691
Finally, a very simple and very useful generalization of the functional form (22.2) can
always be used at ‘no extra cost’: we can easily impose that the asymptotic de-correlation
among forward rates should not go asymptotically to zero with increasing ‘distance’, but
to some ﬁnite level, LongCorr, simply by rewriting Equation (22.2) in the form
ρij (t) = LongCorr + (1 − LongCorr) exp[−β|Ti − Tj |]
(22.9)
Also, in this case the matrix is always real, symmetric and has positive eigenvalues (and
therefore it is a possible correlation matrix). This extension is so simple and so useful
that in what follows I will implicitly assume that it is always carried out, also for the
more complex forms presented below.
22.4
Features of the Modiﬁed Exponential
Correlation Function
In order to obviate the shortcomings of the simple exponential functional form (22.2) I
have suggested elsewhere (Rebonato (1999c)) the simple modiﬁcation
ρij (t) = exp[−βmin(Ti ,Tj ) |Ti − Tj |]
(22.10)
with βmin(Ti ,Tj ) no longer a constant, but a function of the expiry of the earliest-expiring
forward rate. If the function were chosen to produce a decay constant that became smaller
and smaller as the ﬁrst expiry decreased, the second requirement of a desirable correlation
function would be fulﬁlled automatically. However, as Schoenmakers and Coffey (2000)
have correctly pointed out, for an arbitrary function βmin(Ti ,Tj ) it cannot be guaranteed that
all the eigenvectors of the associated matrix ρ will be positive (in the ﬁrst edition of this
book I had chosen a polynomial dependence on min(Ti , Tj ) for the function βmin(Ti ,Tj ) ).
When this happens, the resulting matrix ρ may fail to represent a possible correlation
matrix.
Fortunately, this problem can be easily ﬁxed, because, if one chooses
β min(Ti , Tj ) = β0 exp[−γ min(Ti , Tj )]
(22.11)
one can show that the eigenvalues of ρij , now deﬁned by
ρij (t) = exp { − β0 exp[−γ min(Ti , Tj )]|Ti − Tj |}
(22.12)
are always all positive. Furthermore, expression (22.12) preserves the computationally
desirable feature of not having an explicit dependence on time, t. Therefore, also in this
case the correlation function can be ‘pulled out’ of the covariance integral, making its
analytic evaluation possible. The shape of the modiﬁed exponential correlation function
is displayed in Figure 22.2.
Let us compare the qualitative behaviours of the modiﬁed exponential and of the simple
exponential correlation functions. To carry out the comparison in a meaningful way, let