Convexity adjustment

The volatility value used can be estimated in two ways. We can estimate volatility separately and then use this to calculate what the approximate convexity adjustment should be, or we may observe the convexity bias directly and derive a volatility value from this. This would require an examination of market swap rates and bond yields, and use these to estimate the volatility implied by these rates. 

For example, for the UK gilt principal strip, the convexity adjustment to the percentage price change based on duration if the yield increases from 4.435% to 5.435% is... 

The approximate percentage price change based on duration and the convexity adjustment is found by summing the two estimates. So, for example, if yields change from 4.435% to 5.435%, the estimated percentage price change would be... 

Estimated change using duration alone = -17.63 Convexity adjustment = +1.65... 

The convexity adjustment is —1.2% and therefore the bond exhibits the negative convexity property illustrated in Exhibit 4.18. The approximate percentage price change after adjusting for convexity is... 

Which is correct, 165.35 or 330.68 The answer is both. The reason is that the corresponding equation for computing the convexity adjustment would not be given by equation (4.5) if the convexity measure is obtained from equation (4.6). Instead, the corresponding convexity adjustment formula would be... 

Convexity adjustment to percentage price change = (Convexity measnre/2) x (Ay) x 100... 

Equation (4.7) differs from equation (4.5) in that the convexity measure is divided by 2. Thus, the convexity adjustment will be the same whether one uses equation (4.4) to get the convexity measure and equation (4.5) to get the convexity adjustment or one uses equation (4.6) to compute the convexity measure and equation (4.7) to determine the convexity adjnstment. 

Equation (4.8) differs from equation (4.4) by the inclusion of 100 in the denominator. In our illustration, the convexity measure would be 1.6535 rather than 165.35 nsing equation (4.4). The convexity adjustment formula corresponding to the convexity measure given by equation (4.8) is then... 

Consequently, the convexity measure (or just simply convexity as it is referred to by some market participants) that could be reported for this UK strip are 165.35, 330.68, 1.6535, or 3.3068. All of these values are correct, but they mean nothing in isolation. To use them to obtain the convexity adjustment to the price change estimated by duration requires knowing how they are computed so that the correct convexity adjustment formula is used. It is the convexity adjustment that is important—not the convexity measure in isolation. 

The technique for constructing the swap term structure, as constructed by market participants for marking to market purposes, divides the curve into three term buckets. The short end of the swap term structure is derived using interbank deposit rates. The middle area of the swap curve is derived from either forward rate agreements (FRAs) or interest rate futures contracts. The latter requires a convexity adjustment to render it equivalent to FRAs. The long end of the term structure is constructed using swap par rates derived from the swap market. 

The estimated or assumed values of the parameters a and o can be used in combination with the Hull-White convexity adjustment term to estimate the convexity bias embedded in futures rates. 

If the yield rises to 7.50 percent, a change of 200 basis points, the convexity adjustment that would be made to the price change calculated using modified duration and equation (2.21) is... 

Assume that the yield of the hypothetical 5 percent 2015 bond rises to 8.50 percent, a change of 300 basis points. The percentage convexity adjustment is... 

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