Mathematics of Finance

The mathematics of money and markets — interest and time value, bond valuation, portfolio risk, CAPM, utility, and growth-optimal betting.

Money & Interest

Interest & Compounding — Interest that earns interest: from simple growth to the continuous limit.
Time Value of Money — Move every cash flow to the same date: forward multiplies, backward divides.
Annuities & Loan Amortization — Value a stream of equal payments with one geometric series — pensions, perpetuities, and loan payments.
NPV & IRR — Discount a project's dated cash flows to one number today — and learn when the rate-based shortcut lies.
Bond Pricing & Yield — A bond's price is the present value of its promised cash — and it seesaws against the yield.

Portfolio Return & Risk — Means blend, risks partially cancel — correlation decides how much risk diversifies away.
The Efficient Frontier — Sweep the mix between two assets and a curve appears — only its upper limb is worth holding.
CAPM & Beta — Diversifiable risk earns nothing — beta measures the risk that remains, and the CAPM line prices it.
Expected Utility & Risk Aversion — Why decline a fair bet? Concave utility makes the average of utilities less than the utility of the average.
Kelly Criterion — Bet everything and ruin is certain; bet nothing and the edge is wasted — the log finds the fraction in between.

Arbitrage & the Law of One Price — Identical payoffs must carry identical prices, or free money exists.
Forwards & Futures — Lock a price today with no forecast: the fair forward is the cost of carry.
Options & Put–Call Parity — Calls and puts are rights, and one identity ties their prices together with no model of the stock.
Binomial Option Pricing — Replicate an option with shares and a bond, and its price falls out — with no forecast of the stock.
Risk-Neutral Pricing — Price any claim as its expected payoff under pretend probabilities, discounted at the risk-free rate.
Black–Scholes — One formula for a call: C = S_0 (d_1) - Ke^-rT (d_2), from a single Gaussian integral.
The Greeks — Delta, gamma, theta, and vega: the partial derivatives that tell a trader how an option moves.