Financial Mathematics · Lecture 4 of 12

Taylor Series: Local Approximations That Run Finance

Duration, convexity, delta, gamma and Itô's lemma are all Taylor expansions in disguise. Here is the tool, its error term, and why the second-order term is the one that matters.

November 15, 2026By Diganta Bhattacharya5 min read

The idea

Near a point $a$, a smooth function looks like a polynomial:

$$f(a+h)=f(a)+f'(a)h+\frac{f''(a)}{2!}h^2+\frac{f^{(3)}(a)}{3!}h^3+\cdots$$

Truncate after $n$ terms and you get the Taylor polynomial $T_n$. The error is controlled by the next derivative (Lagrange form):

$$f(a+h)-T_n(a+h)=\frac{f^{(n+1)}(\xi)}{(n+1)!}h^{n+1}\quad\text{for some }\xi\text{ between }a\text{ and }a+h.$$

So for small $h$ each extra term buys a whole extra power of $h$ in accuracy.

Expansions you will use

$$e^x = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots,\qquad \ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots\;(|x| \lt 1),\qquad \sqrt{1+x}\approx 1+\frac x2.$$

Simple vs log returns: $\ln(1+R)\approx R-\tfrac12R^2$. Averaging this over many periods, $\mathbb{E}[R^2]\approx\sigma^2$, gives the −σ²/2 from Lecture 2 again.
$(1+r/m)^m=e^{m\ln(1+r/m)}=e^{r-r^2/(2m)+\cdots}\to e^r$, which is continuous compounding.

Two variables

For $f(t,x)$ with small changes $\Delta t,\Delta x$:

$$\Delta f\approx f_t\,\Delta t+f_x\,\Delta x+\tfrac12 f_{xx}(\Delta x)^2+f_{tx}\,\Delta t\,\Delta x+\tfrac12 f_{tt}(\Delta t)^2.$$

In ordinary calculus we keep only first-order terms, because $(\Delta x)^2$ is negligible next to $\Delta x$ when $x$ is smooth. Hold that thought. When $x$ is a Brownian path, $(\Delta x)^2$ is of the same order as $\Delta t$, so the $\tfrac12 f_{xx}(\Delta x)^2$ term survives. That single surviving term is Itô's lemma (Lecture 8).

Finance as Taylor series

Object	First order	Second order
Bond price $P(y)$	duration $-\frac{1}{P}\frac{dP}{dy}$	convexity $\frac1P\frac{d^2P}{dy^2}$
Option value $V(S)$	delta $\Delta=\partial V/\partial S$	gamma $\Gamma=\partial^2V/\partial S^2$
Option value $V(t)$	theta $\Theta=\partial V/\partial t$	—

A delta-hedged option book has P&L over a small step of roughly

$$\Delta \Pi \approx \Theta\,\Delta t+\tfrac12\Gamma(\Delta S)^2.$$

Gamma earns money when the stock moves, theta bleeds money as time passes, and Black–Scholes is the statement that these balance exactly when $(\Delta S)^2$ averages $\sigma^2S^2\Delta t$. Lectures 11–12 make this precise.

When Taylor failsExpansions are local. A 20% gap move is not "small", and higher-order terms (and jumps) matter. Risk systems that rely only on delta and gamma underestimate tail losses for exactly this reason.

Key takeaways

Taylor: f(a+h) ≈ Σ f⁽ᵏ⁾(a)hᵏ/k!, with error of order h^{n+1}.
Duration/convexity and delta/gamma are first and second Taylor coefficients.
In stochastic calculus (Δx)² ~ Δt, so the second-order term survives. That is Itô's lemma.

Exercises

Expand e^{−rT} to second order and compare with the exact value for r = 5%, T = 2.
Using ln(1+R) ≈ R − R²/2, show E[ln(1+R)] ≈ μ − σ²/2 when E R = μ and Var R = σ².
A position has Δ = 0, Γ = 0.05 per ₹ and Θ = −2 per day. How big a daily move breaks even?