13.15.1.5 The Best Approach

Given these results, we think it is essential to use parameters estimated from observable caps, floors, and swaptions data (or other option-related securities prices) to the extent it is available. To illustrate the power of this approach, consider now the U.S. dollar data on European swaption prices observable in August 1995. At the time the data were obtained, there were 54 observable swaption prices. A swaption gives the holder the right to initiate a swap of a predetermined maturity and fixed-rate level on an exercise date in the future. We estimated Extended Vasicek model parameters by choosing the speed of mean reversion (α) and interest rate volatility (s), which minimized the sum of the squared errors in pricing these 54 swaptions. The “price” of the swaption was obtained by converting the Black-Scholes volatility quotation for the swaption price to the percentage of notional principal that the equivalent dollar swaption price represented. The exercise periods on the swaptions were 0.5, one, two, three, four, and five years. The underlying swap maturities were 0.5, one, two, three, four, five, six, seven, and ten years.

Overall, the Extended Vasicek model's performance was extraordinary. The average model error was 0 basis points with a mean absolute error of five basis points of notional principal, even though only two parameters (in addition to the current yield curve) were used to price 54 securities. Compare this to the Black model for commodity futures, which is often used for swaptions and caps and floor pricing. The Black model required 54 different implied volatility values to match actual market prices, even though the model, in theory, assumes that one volatility parameter should correctly price all 54 swaptions. Volatilities in the Black model ranged from 0.13 to 0.226, a very wide range that should indicate to swaption market participants the need for caution.

In summary, the extended version of the Vasicek model, when applied to swaption prices, proved two things:

• Swaptions provide a rich data set with excellent convergence properties that enable market participants to use even common spreadsheet software to obtain high-quality term structure parameter estimates.
• The accuracy of the Extended Vasicek model using only two parameters held constant over 54 swaptions, is far superior to that of the Black commodity futures model in predicting actual market prices.

In estimating term structure parameters, the lesson is clear. A rich data set of current prices of securities with significant optionality is necessary to provide an easy-to-locate global optimum for almost any popular term structure model.