Guide
Lessons meant to be read in order. Each one only uses what came before it.
Derivatives pricing
From payoffs, through arbitrage-free pricing and the binomial tree, to the Black–Scholes formula.
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- Forwards, calls and putsWhat a derivative is; payoffs of forwards, calls and puts at maturity; long and short positions.
- Combining positionsPayoffs add up: spreads, straddles, strangles, butterflies and the synthetic forward.
- Arbitrage and the forward priceThe law of one price, the cash-and-carry trade and the fair forward price.
- Put–call parityTwo portfolios with the same payoff give a model-free link between call and put prices.
- The one-step binomial modelThe stock moves up or down; replicating a call with stock and a bond.
- Risk-neutral probabilitiesThe replication price as a discounted expectation with weights q; the roles of P and Q.
- Multi-step trees and backward inductionRecombining trees, the CRR parametrisation, pricing node by node and convergence to Black–Scholes.
- American options and early exerciseEarly exercise in a tree, why calls without dividends are never exercised early, the early-exercise premium of a put and the exercise boundary.
- Delta and hedgingDelta as a sensitivity and as a hedge ratio; gamma and the error from discrete rebalancing.
- Geometric Brownian motionThe standard model for stock prices: the log-normal distribution, its mean and its median.
- The Black–Scholes formulaHow the formula is obtained, what d₁ and d₂ mean, and what the model leaves out.
- The Greeks: gamma, vega, theta and rhoHow an option's value responds to the stock, volatility, time and rates; gamma and theta as two sides of hedging.
- Implied volatility and the volatility smileInverting Black–Scholes for σ, vega, and what the smile and the term structure say about the model.
- Stochastic volatility and the Heston modelThe Heston variance process, mean reversion, vol-of-vol, the Feller condition, and how correlation and vol-of-vol shape the smile.
- Static replication: Breeden–Litzenberger and Carr–MadanThe risk-neutral density from call prices, and building any payoff from bonds, forwards and options.
- Exotic options: barriers, Asians and digitalsPath-dependent payoffs, the effect of a knock-out barrier and of averaging, and why Monte Carlo is the natural tool.
Check yourself
Questions from the lessons of this series. Open a lesson's group whenever you want to test yourself.