Mathematics × Financial engineering

Math for Finance

10 sections

18 interactive plots

Self-contained

01 — Derivatives

First and second derivatives, drawn

f(x) = x³ − 3x and its two derivatives

f(x) = x³ − 3x f′(x) = 3x² − 3 f″(x) = 6x Tangent at x₀

First derivative = slope

Second derivative = rate of change of the slope

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02 — Calculus

The limit, the rules, and the integral

The limit that defines a derivative

Step h

1.000

The integral is an accumulated sum

Rectangles n

8

The rules you actually use

Rule	Form	Why it matters here

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03 — Function shapes

Eight shapes that cover most of finance

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04 — Taylor series

Any smooth curve is a polynomial in disguise

Truncated expansion vs the true function

True function Taylor approximation |error|

The one expansion every options trader carries

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05 — Linear algebra

A matrix is a thing that moves space

2×2 transform of the unit square

Before (unit square) After (A·x) Eigenvectors

The operations, and what each one means

Operation	Geometric meaning	Finance use

Portfolio variance is one matrix product

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06 — Trigonometry

Circles unrolled into waves

Unit circle → sine and cosine

sin θ cos θ Radius vector

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07 — Fourier

Every signal is a stack of sine waves

Partial sums of a Fourier series

Target wave Partial sum Harmonic amplitudes

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08 — Probability

Normal, lognormal, and the random walk

Normal → lognormal

σ (annualised vol)

45%

Horizon (days)

30

Geometric Brownian motion

Paths

40

New random seed

Itô's lemma — the chain rule when the input is random

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09 — Kelly criterion

How much to bet, derived rather than guessed

Long-run growth rate vs bet fraction

Expected log growth g(f) Full Kelly f* Half Kelly Simulated wealth paths

Where the formula comes from

Continuous form, and why nobody bets full Kelly

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10 — Applied

Which piece of maths does which job

Mathematical tool	The financial object	What it actually does for you