The Price-Yield Curve descripts the relationship between a bond’s price and yield, and they are normally negatively correlated.
We here consider the Bond’s price as a function of the bond’s yield, \(P(Y)\), and study the approximation of the bond’s price.
We firstly apply the Taylor Expansion at \( (Y_0,P_0 \),
$$ P(Y)\approx P(Y_0)+\frac{dP}{dY}(Y-Y_0)+\frac{d^2P}{dY^2}\frac{(Y-Y_0)^2}{2!}+O((Y-Y_0)^3) $$
$$ P(Y)-P_0 \approx +\frac{dP}{dY}(Y-Y_0)+\frac{d^2P}{dY^2}\frac{(Y-Y_0)^2}{2!}+O((Y-Y_0)^3) $$
$$ \triangle P \approx \frac{dP}{dY}\triangle Y+\frac{d^2P}{dY^2}\frac{(\triangle Y)^2}{2!}+O((\triangle Y)^3) $$
Devided by P from both side, then the LHS means the percentage change of Price.
$$\frac{ \triangle P}{P} \approx \frac{dP}{dY}\triangle Y \frac{1}{P}+\frac{d^2P}{dY^2}\frac{(\triangle Y)^2}{2!}\frac{1}{P}+O((\triangle Y)^3) $$
By definition, the Modified Duration \( D=\frac{\triangle P / P}{\triangle Y} = \frac{dP}{dY}\frac{1}{P}\), and convexity \( C=\frac{d^2 P}{d Y^2}\frac{1}{P} \). We replace them into the expansion function. and drop the last term.
$$\% \triangle P \approx D\cdot \triangle Y+\frac{C}{2}\cdot (\triangle Y)^2$$
Finally, we get the approximated bond price curve. Second order Taylor Expansion is applied, and the Duration and Convexity, two important properties of bonds are included. For more accurate approximation, more terms need to be expanded.