An option is a right without an obligation. A call gives its buyer the right to buy an asset at a fixed strike price until or on an expiry date; a put gives the right to sell. The buyer pays a premium up front. The seller, or writer, keeps the premium and takes on the obligation. That asymmetry makes options the instrument through which most of this knowledge base's systems are expressed: insurance, lottery tickets, income strategies, volatility trades and market making. The modern listed market began when the Chicago Board Options Exchange opened on 26 Apr 1973 with calls on 16 stocks; 911 contracts traded on the first day (OCC 50th-anniversary history; Cboe 40th-anniversary release). A month later the Journal of Political Economy published Fischer Black and Myron Scholes' pricing formula, and Robert Merton's companion paper appeared in the Bell Journal of Economics and Management Science the same year (vol. 4, no. 1, Spring 1973, 141–183); Scholes and Merton received the 1997 Nobel prize "for a new method to determine the value of derivatives"; Black had died in 1995 and was not eligible (FRBSF Economic Letter, Feb 1998; NBER). Half a century later US listed options trade on an unprecedented scale: ~15.2bn contracts in 2025 (OCC, Jan 2026), and same-day-expiry contracts reached a record 66.2% of SPX volume in Jul 2026 (Cboe; both Shared facts). Every US equity or index option contract covers 100 units of the underlying, so a quoted premium of $2.45 costs $245 per contract. This section explains payoffs and moneyness, American and European exercise and assignment (including early exercise around dividends), put–call parity, the three pricing approaches that matter (Bachelier 1900, Black–Scholes–Merton 1973, and the Cox–Ross–Rubinstein binomial tree of 1979), the Greeks and how they trade off, implied volatility and the skew that appeared after the 1987 crash, and the contract details (SPX versus SPY versus XSP, cash versus physical settlement, AM versus PM settlement) that decide what a trade actually does. The companion worked example delta-hedges a stylized option for ten days to show where an options trader's profit really comes from: the race between gamma and theta.
What the study covers
- 1Overview: insurance, leverage and a bet on movement
- 2Units and definitions
- 3Payoffs and moneyness
- 4Exercise, assignment and settlement
- 5Put–call parity
- 6Pricing models
- 7The Greeks
- 8Implied volatility, the smile and the skew
- 9Contract specifications that change outcomes
- 10Europe and the rest of the world
- 11How to read options claims
- 12Key numbers
- 13Landmark firms and cases
- 14Pitfalls
- 15How an expert thinks about it
- 16Glossary
Technical exhibits
Worked examples, checklists and case studies included with this study.
Delta Hedging Gamma vs Theta 10 Days Worked Example
A stylized one-month at-the-money call on a $100 index priced at 20% implied volatility, bought and delta-hedged daily for ten days along three price paths with realised volatility of 30%, 20% and 12%. Day-by-day gamma gains, theta costs and total P&L per 10 contracts; the break-even daily move; hedging costs; the unhedged comparison; and what a change in implied volatility would add (computed in Python)