Notes¶
These notes are a compact “story” of the standard option-pricing toolkit used throughout the library: no-arbitrage → risk-neutral pricing → Brownian motion/Itô → GBM → Black–Scholes → implied vol → numerical methods.
They are written to be readable linearly, but each note also stands on its own.
Suggested reading order¶
- Risk-neutral pricing — why we price under \(\mathbb{Q}\) and why the drift becomes \(r\).
- Brownian motion and Itô — the noise and calculus behind continuous-time models.
- Geometric Brownian motion — the lognormal stock model and distribution of returns.
- Black–Scholes pricing — PDE and closed-form European option prices, plus intuition.
- Implied volatility — “vol as the price”; inversion, bounds, and practical solver notes.
- Monte Carlo — simulation pricing, error bars, and variance reduction.
- Binomial CRR — discrete-time replication and convergence to Black–Scholes.
Conventions and notation¶
- Time \(t\) is measured in years.
- Rates are continuously compounded unless stated otherwise.
- \(S_t\): underlying price at time \(t\).
- \(r\): continuously-compounded risk-free rate (constant in basic models).
- \(B_t = e^{rt}\): money-market account (numéraire in the basic setting).
- \(W_t\): Brownian motion.
- \(\mathbb{P}\): real-world (physical) probability measure.
- \(\mathbb{Q}\): risk-neutral measure under which discounted traded prices are martingales.
When a continuous dividend yield \(q\) is relevant, the risk-neutral drift becomes \(r-q\).