Black-Scholes Options Pricing
Black-Scholes Options Pricing
Comprehensive reference on the Black-Scholes model, derivation, assumptions, and practical applications for options pricing.
Open PDF in new tab →Key Concepts
Covered Calls
BeginnerSell a call option against stock you own. You collect the premium as income, but cap your upside at the strike price. Best in sideways-to-slightly-bullish markets.
Implied Volatility (IV)
BeginnerThe market's expectation of future price movement, embedded in option prices. Higher IV = higher premiums = more income for sellers, but also higher risk of large moves.
Black-Scholes Model
IntermediateThe foundational formula for pricing European options. Inputs: stock price, strike, time to expiry, risk-free rate, and volatility. Assumes log-normal distribution and continuous hedging.
The Greeks
IntermediateSensitivity measures: Delta (price sensitivity), Theta (time decay), Vega (volatility sensitivity), Gamma (delta acceleration). For covered call sellers, Theta is your friend.
Put-Call Parity
IntermediateThe relationship between call and put prices: C - P = S - K×e^(-rT). Violations create arbitrage opportunities. Understanding this helps you evaluate whether to sell calls or puts.
Volatility Smile & Skew
AdvancedIV varies by strike price. OTM puts typically have higher IV than OTM calls (skew). When ATM calls have higher IV than puts, the market expects upside — ideal for covered call selling.
Risk-Neutral Pricing
AdvancedOptions are priced assuming investors are risk-neutral — the expected return of every asset equals the risk-free rate. This is why you can price options without knowing the stock's expected return.
Wash Sale Rules
TaxIf you sell a stock at a loss and repurchase it (or a substantially identical security) within 30 days, the loss is disallowed for tax purposes. Critical for active covered call strategies.
Recommended Reading
Options, Futures, and Other Derivatives
Textbookby John C. Hull
The definitive textbook on derivatives pricing. Covers Black-Scholes, binomial models, Greeks, and exotic options.
Option Volatility and Pricing
Tradingby Sheldon Natenberg
The practical guide to options trading. Focuses on volatility, strategies, and risk management from a trader's perspective.
The Concepts and Practice of Mathematical Finance
Quantitativeby Mark S. Joshi
Bridges the gap between theory and practice. Covers stochastic calculus, Monte Carlo simulation, and model implementation.
Dynamic Hedging
Advancedby Nassim Nicholas Taleb
Managing vanilla and exotic options from the perspective of a practitioner. Heavy focus on real-world risks that models miss.
The Options Playbook
Beginnerby Brian Overby
Visual, strategy-by-strategy guide to options. Great starting point for understanding covered calls, spreads, and risk profiles.
Online Resources
MIT OpenCourseWare: Topics in Mathematics with Applications in Finance →
Full MIT lecture series covering stochastic calculus, Black-Scholes, options pricing, and portfolio theory.
CBOE Options Education →
Free courses from the Chicago Board Options Exchange covering strategies, Greeks, and market mechanics.
Options Industry Council (OIC) →
Free options education from the industry council. Webcasts, courses, and strategy guides.
tastytrade Learn Center →
Practical, research-backed options education focused on probability and premium selling strategies.