13.15.1.4 Single-Day Yield Curve Fitting

This approach is the yield curve equivalent of "implied volatility" using Black-Scholes. Most market participants feel more comfortable basing the analysis on parameter values implied from observable securities prices than on historical data when observable prices are sufficient for this task. For example, if the only observable data is the yield curve itself, we can still attempt to fit the actual data to the theory by maximizing the goodness of fit from the theoretical model.

We arbitrarily set the market price of risk to zero and the long-run expected value of the short rate to equal the ten-year bond yield. They then find the best fitting α and s. The result is generally of marginal acceptability. This is a common conclusion, as pointed out by the former head of derivatives research at Merrill Lynch, and one of the reasons why market participants often feel compelled to supplement current yield curve data with historical parameter data.

To illustrate the yield curve fitting approach, we took yield curve data for the beginning, middle, and end of the data set and picked the days for which the ten-year Canadian government bond yield reached its highest and lowest points. The following maturities have been used: one-month, six-months, two, three, four, five, seven, and ten years. The results of this analysis, using simple spreadsheet software to obtain parameters, were as follows:

• Best Fitting Parameters from Selected Yield
• Canadian Government Bond Market
• Extended Vasicek Model
• Using Common Spreadsheet Non-Linear Equation Solver

Table 13-13 Maturities dating: one-month, six-months, two, three, four, five, seven, and ten years

Environment	Date Beginning	Highest Rates	Date Mid-Point	Lowest Rates	Date Ending6
Date	January 2,1987	April 19, 1980	August 1, 1991	January 28, 1994	March 6, 1996
Mean Reversion	0.01462	0.25540	0.62661	0.70964	0.58000
Volatility	0.00000	0.05266	0.00000	0.00000	0.00100
Mkt Price of Risk	0.00000	0.00000	0.00000	0.00000	0.00000
Long Run Rate	0.08730	0.11950	0.09885	0.06335	0.07600
Estimate Quality	Low	Medium	Low	Low	Low

Note:

Spreadsheet solver capabilities are limited. The market price of risk and the long-run rate was arbitrarily set to displayed values with optimization speed of mean reversion and volatility.

The results were consistent with other approaches in generally showing a high degree of mean reversion. The lack of power in spreadsheet non-linear equation solving is reflected in the low or zero values for interest rate volatility and illustrates the need for other data (caps, floors, swaptions, bond options prices, and so on) and more powerful techniques for obtaining these parameters.