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7 Fitting the Instantaneous Volatility Function: The Information from the Swaption Market

7 Fitting the Instantaneous Volatility Function: The Information from the Swaption Market

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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



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