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Calls, puts and payoffs
The two basic contracts and the one thing they are always worth at expiry.
An option is the right, but not the obligation, to trade an underlying at a fixed strike price \(K\) by an expiry date. A call is the right to buy at \(K\); a put is the right to sell at \(K\).
At expiry an option is worth only its intrinsic payoff. You exercise a call only if the stock \(S_T\) is above the strike, and a put only if it is below:
\[ \text{call payoff} = \max(S_T - K,\ 0), \qquad \text{put payoff} = \max(K - S_T,\ 0). \]
Why it matters
Every other idea in options pricing builds on this hockey-stick payoff. A market maker is really quoting the present value of this uncertain final payoff, so you need to read it instantly.
Try it
Option type
Strike K
100
Spot at expiry
120
Payoff
20
Worked example
With the stock at \(S_T = 112\), the \(100\) call pays \( \max(112 - 100, 0) = 12 \) and the \(100\) put pays \( \max(100 - 112, 0) = 0 \). The \(120\) call pays \(0\); the \(120\) put pays \(8\).
In the interview
What they are testing
That you can write down and read an option payoff at expiry without hesitation.
How to narrate it
Say the formula, then the number: "the call is max of stock minus strike and zero, so 112 minus 100 is 12."
Common mistake
Forgetting the floor at zero, or flipping the call and put directions. A call wins when the stock rises, a put when it falls.
Quick check
The stock finishes at 95. What does the 90 put pay at expiry?