6 Options: Rights, Obligations, and Payoffs
6.1 The contract before the model
A call gives its holder the right, but not the obligation, to buy the underlying asset at strike \(K\) under the contract’s terms. A put confers a right to sell. A short option position creates an obligation when exercised or assigned.
At expiration, the payoffs per share are
\[\text{Call payoff}=\max(S_T-K,0),\qquad \text{Put payoff}=\max(K-S_T,0).\]
Profit also accounts for the premium paid or received and, for a real position, commissions and other frictions.
6.2 The dimensions of an option
Option value depends on the underlying price, strike, time to expiration, rates, expected distributions, and the market’s assessment of future volatility. Contract size, exercise style, and settlement rules also matter operationally.
6.3 A first payoff table
For a long call with strike \(100\) and premium \(6\) per share:
| Expiration price | Option payoff | Profit before costs |
|---|---|---|
| $90 | $0 | -$6 |
| $100 | $0 | -$6 |
| $110 | $10 | $4 |
| $120 | $20 | $14 |
Important distinction: A position closed before expiration depends on option market value, not just this expiration diagram.
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.