Options Pricing
Options Basics taught you to read a payoff. Now learn the machine that prices it — replication and risk-neutral valuation, the binomial tree, Black–Scholes and its d₁/d₂, the Greeks, and the volatility smile the market actually trades.
How an option's fair premium is actually computed — from no-arbitrage and replication, through the risk-neutral binomial tree, to the Black–Scholes formula, the Greeks (delta, gamma, theta, vega, rho), and implied volatility with its smile and skew.
You can already read an option’s hockey-stick payoff — but you pay the premium now, before you know which future shows up. This topic answers the harder question: what is the fair price today? The mind-bending punchline: you can price an option without ever guessing where the stock is going.
We build that pricing machine from the ground up:
- Why a payoff isn’t a price — naive “expected value” fails because it discounts at a rate nobody can agree on.
- No-arbitrage & replication — rebuild the payoff from stock and cash, and the option must cost exactly what the recipe costs, or free money appears.
- Risk-neutral valuation & the binomial tree — a model simple enough to run with a pen, yet sliced fine it becomes Black–Scholes.
- The Black–Scholes formula — where it comes from, what its assumptions buy and cost, and the meaning hiding in
d₁andd₂(spoiler:N(d₂)is a probability). - The Greeks — delta, gamma, theta, vega and rho: how a position breathes as the world moves, and how to hedge it.
- Implied volatility, smile & skew — run the formula backwards, find a different vol at every strike, and see why that curve admits Black–Scholes is only an approximation.
This is the bridge from “I can read an option” to “I can price, hedge and trade one.” It’s the most quantitative course on the ladder so far — every section earns its formula with an analogy, a worked number, and a chart you can poke.
In this topic
- 1 Why a Payoff Isn't a Price A payoff diagram is the value at expiry; the premium is paid today. Why naive expected-value pricing fails, the law of one price, no-arbitrage, replicating a call with stock and cash, and the one-step binomial leap to risk-neutral valuation. 9 min
- 2 The Binomial Model in Full The full Cox–Ross–Rubinstein tree — up/down factors, the risk-neutral probability q, backward induction over many steps, calibrating u and d to volatility, convergence to Black–Scholes, and early exercise. 10 min
- 3 The Black–Scholes Formula The closed-form European call price C = S·N(d₁) − K·e^(−rT)·N(d₂): its lognormal, constant-vol, constant-rate assumptions, what d₁ and d₂ mean, why N(d₂) is the risk-neutral probability of finishing in the money, and a full worked example. 10 min
- 4 The Greeks An option price's sensitivities — delta (hedge ratio), gamma (curvature), theta (time decay), vega (volatility), rho (rates): their signs, shapes, intuition, and how delta-hedging works. 10 min
- 5 Implied Volatility & the Smile Run Black–Scholes backwards to extract the market's implied volatility; implied vs realized vol; why every strike quotes a different vol — the volatility smile, equity skew, term structure, and the VIX. 9 min
- 6 Final Exam: Pricing the Option A graded, locked capstone across all of options pricing — no-arbitrage and replication, the binomial tree, the Black–Scholes formula, the Greeks, and implied volatility with the smile. 15 min
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