CHAPTER 9
New Convexity Bias Series
The purpose of this note is to introduce a new convexity bias series that corrects an apparent shortcoming in the original Carr Futures convexity bias estimates. The original series worked well when both the yield curve and volatility curve were comparatively flat. This past year, however, the Fed’s aggressive campaign to cut interest rates produced an unusually steep Eurodollar futures rate curve and a steeply inverted implied volatility curve. (See the upper and lower charts in Exhibit 9.1.) As a result, our estimates of the convexity at key maturities began to fall well below market values. (See Exhibit 9.2.)
EXHIBIT 9.1
Eurodollar Futures Rates
January 4, 2001, and January 4, 2002
EXHIBIT 9.2
Convexity Bias Values for 5-Year Swaps
January 26, 1996, through December 31, 2001
The new series, which employs implied cap and floor volatilities together with estimated correlations between Eurodollar futures rates and corresponding zero-coupon rates, has been designed to be more robust in the face of unusual changes either in the slope or curvature of the Eurodollar rate curve or in the shape of the implied volatility curve.
The effect of our improved approach on estimates of the bias for 5-year and 10-year swaps can be seen in Exhibit 9.2, which plots both the new series and the original series against the market value of the bias for the past 6 years. For most of these 6 years, the two series produced roughly the same estimates. Over the past year, however, the new series does a much better job of tracking the market. While the original series actually showed declining bias values over the past year, the new series not only catches the change in direction of the market bias but tracks the market values as well.
Eurodollar Futures Implied Volatilities
January 4, 2001, and January 4, 2002
Convexity Bias Values for 10-Year Swaps
January 26, 1996, through December 31, 2001
