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Introduction to Options

An option gives the right, not the obligation, to trade at a fixed price. A call is the right to buy at strike KK; a put is the right to sell at KK.

Call payoff=max⁡(S−K, 0)Put payoff=max⁡(K−S, 0)\text{Call payoff} = \max(S - K,\, 0) \qquad \text{Put payoff} = \max(K - S,\, 0)

That max⁡\max is the whole instrument. You exercise only when it helps, so the payoff is bent rather than straight, and everything distinctive about options follows from that bend.

The kink sits at the strike. To the left of it a call is flat, so you have no exposure at all; to the right the line runs at 45 degrees, so you have the full exposure of the underlying. That single change of slope is where the Greeks come from.

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