Payoffs, moneyness and the bounds

An option is the right, not the obligation, to trade at a fixed price. Everything downstream — parity, the Greeks, the whole surface — is built on that one asymmetry, and on the fact that a payoff floored at zero can never be worth less than nothing.

Equation 1.1

Payoff at expiry

The floor at zero is the entire point: you walk away rather than exercise into a loss.

C_T = \max(S_T - K,\ 0), \qquad P_T = \max(K - S_T,\ 0)

$S_T$: The price of the underlying at expiry.
$K$: The strike.

Common trap. Confusing payoff with profit. A long call has a payoff that is never negative, but its *profit* is the payoff minus the premium, and that is negative whenever the option expires below the strike plus the premium. Instead. Say which you are drawing. Interviewers ask for payoff diagrams far more often than profit diagrams, and answering the wrong one looks like you do not know the difference.

Figure 1.2 · Payoff against profit. The

\$45

call bought for

\$7

. The payoff is the same shape sitting

\$7

higher and is never negative; the profit is, up to the premium, and does not turn positive until

\$52

. Confusing the two is the mistake above, and the diagram makes it impossible.

Position	Call	Put	Intrinsic value
In the money	$S > K$	$S < K$	Positive
At the money	$S = K$	$S = K$	Zero
Out of the money	$S < K$	$S > K$	Zero

Table 1.3 · Moneyness and value. Extrinsic value — the rest of the premium — is what you are paying for the time and uncertainty remaining. It is zero at expiry, by definition.

Definition 1.4

Splitting the premium

Intrinsic and extrinsic value, $C = \underbrace{\max(S-K,0)}_{\text{intrinsic}} + \underbrace{(C - \max(S-K,0))}_{\text{extrinsic}}$ — Intrinsic is what the option is worth if nothing moves again; extrinsic is everything you are paying for the possibility that it does. Extrinsic value is largest at the money and decays to zero at expiry.

Proposition 1.5

No-arbitrage bounds

A European call is worth at least $S_0 - Ke^{-rT}$ and never more than $S_0$ . A European put is worth at least $Ke^{-rT} - S_0$ and never more than $Ke^{-rT}$ . Each bound is enforced by a trade you can actually put on if it is violated.

Holds when

A call above the spot would let you sell the option, buy the stock and keep the difference risk-free.
A call below $S_0 - Ke^{-rT}$ lets you buy it, short the stock and invest the strike — a guaranteed profit.

Why the bounds are about carry, not about volatility. A call is worth at least $S_0 - Ke^{-rT}$ because owning it and putting the discounted strike in the bank replicates the stock without ever costing more than the stock does. Nothing in that sentence mentions how the stock moves. Every no-arbitrage bound is the same kind of statement — a comparison of cash flows — which is why they hold in a crash and a model-based price does not.

Payoffs, moneyness and the bounds

After this lesson you should be able to

Payoff at expiry

Splitting the premium

No-arbitrage bounds

The rest of this lesson is in Premium