13.15.1.2 Volatility Curve Approach

This approach matches parameters to the historical relative volatilities of bond yields at different maturities. This approach worked moderately well.

Many market participants use parameter estimates that are consistent with the historical relative degree of volatility of longer-term “yields” about the short rate. Market participants calculate observable variances for bond yields and then select values of s and α that best fit historical “volatility.” The findings for the Canadian Government Bond market were based on the use of par bond coupon rates as proxies for zero-coupon bond yields. The results of this analysis are summarized in the following table:

Example: Variance in Canadian Government Interest Rates January 2, 1987, to March 6, 1996.

Table 13-11 Variance in Canadian Government Interest Rates from January 2, 1987, to March 6, 1996

	Canadian Treasury Bills				Canadian	Government	Bonds
	1 mth	2 mth	3 mth	6 mth	1 yr	2 yr	3 yr	4 yr	5 yr	7 yr	10 yr	25 yr
Actual Variance	7.677	8.346	7.433	6.842	6.529	3.88	3.044	2.623	2.237	1.909	1.471	0.006
Estimated Variance	7.943	7.745	7.552	7.008	6.051	4.564	3.496	2.717	2.143	1.389	0.794	0.14
Error	-0.266	0.602	-0.119	-0.166	0.478	-0.684	-0.452	-0.095	0.094	0.52	0.677	0.856
Squared Error	0.071	0.362	0.014	0.028	0.228	0.468	0.204	0.009	0.009	0.27	0.458	0.732
Best Fitting Parameter Values:
α	0.305172
s	2.854318%
Maturity	0.083	0.167	0.25	0.5	1	2	3	4	5	7	10	25

Note:

Interest Rates and Variances are in %.

The best-fitting α value was 0.305. This is a fairly large speed of mean reversion and reflects the relatively large variation in short term Canadian rates, relative to long rates, over the sample period. Interest rate volatility was also high at .0285. At the 1996 low levels of Canadian interest rates, this volatility level would be too high. Comparable figures for the United States swap market, based on a fit to 100 swaptions prices, reflected a speed of mean reversion of 0.05 and an interest rate volatility of 0.013.