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Black-Scholes Options Pricing

Black-Scholes Options Pricing

Comprehensive reference on the Black-Scholes model, derivation, assumptions, and practical applications for options pricing.

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Key Concepts

Covered Calls

Beginner

Sell 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)

Beginner

The 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

Intermediate

The 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

Intermediate

Sensitivity measures: Delta (price sensitivity), Theta (time decay), Vega (volatility sensitivity), Gamma (delta acceleration). For covered call sellers, Theta is your friend.

Put-Call Parity

Intermediate

The 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

Advanced

IV 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

Advanced

Options 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

Tax

If 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

Textbook

by John C. Hull

The definitive textbook on derivatives pricing. Covers Black-Scholes, binomial models, Greeks, and exotic options.

Option Volatility and Pricing

Trading

by 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

Quantitative

by Mark S. Joshi

Bridges the gap between theory and practice. Covers stochastic calculus, Monte Carlo simulation, and model implementation.

Dynamic Hedging

Advanced

by 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

Beginner

by Brian Overby

Visual, strategy-by-strategy guide to options. Great starting point for understanding covered calls, spreads, and risk profiles.

Online Resources

OptionHive Documentation