Taylor series

Replacing a function by a polynomial

Near a point, a smooth function looks like a polynomial built from its derivatives there. This is the Taylor series: starting at $x$ and moving a small step $h$, the function changes by its slope times $h$, plus a correction from its curvature, plus smaller corrections from higher derivatives.

Four series are worth knowing near 0: $e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots$, $\sin x=x-\frac{x^3}{6}+\cdots$, $\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$ and $\frac{1}{1-x}=1+x+x^2+\cdots$. Truncating after a few terms gives a cheap approximation whose error is about the first missing term.

Finance uses this daily. A bond price $P$ depends on the yield $y$; to second order $\Delta P\approx -D\,P\,\Delta y+\tfrac{1}{2}C\,P\,(\Delta y)^2$, where $D$ is the modified duration (the slope term) and $C$ the convexity (the curvature term). The same shape appears as delta and gamma for an option: $\Delta V\approx \Delta\cdot\Delta S+\tfrac{1}{2}\Gamma(\Delta S)^2$.

Lab · what's graded

• The series of the exponential15 pts
• Approximating the sine15 pts
• The size of the error15 pts
• A bond price change20 pts
• Which term is the curvature10 pts
• A series by hand15 pts
• Functions and their series10 pts