3.1.2 Options Mechanics & Payoff Diagrams

Basic Prerequisites: 01 · What Is a Derivative.

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1. Intuition & Practical Objective

An option contract is described by exactly five things: the underlying, the type (call = right to buy, put = right to sell), the strike $K$, the expiry $T$, and the style (European = exercise only at $T$; American = any time up to $T$). Everything else - payoff, diagram, moneyness, value decomposition - is a consequence of those five. The practical objective of this page is to make you fluent in reading an option from its payoff diagram and in decomposing its price into intrinsic value + time value, which is the vocabulary a trading desk uses all day.

The reason payoff diagrams matter so much: an option’s price is hard, but its payoff is trivial - it is a kinked line you can draw by hand. Almost every insight in the pillar is visible in the picture first and formal later. A long call is flat, then rises. A short put is flat, then falls. Add them and you get any shape you like - which is exactly how §6’s trading strategies are built.

The one-sentence essence. “A call pays $\max (S_T-K,0)$ and a put pays $\max (K-S_T,0)$; the kink at the strike is the only non-linearity in the contract, and the option’s price is the market’s price for that kink - split into intrinsic value (what you’d get if you exercised now) and time value (what the kink is still worth).“

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2. Mathematical Ground Truth & Derivations

2.1 The four positions and their payoff diagrams (Hull Ch 10.2)

Position	Payoff at expiry	Diagram shape
Long call	$\max (S_T-K,0)$	flat on $[0,K]$, then $45°$ up
Short call	$-\max (S_T-K,0)=\min (K-S_T,0)$	flat, then $45°$ down
Long put	$\max (K-S_T,0)$	$45°$ down to $K$, then flat
Short put	$-\max (K-S_T,0)=\min (S_T-K,0)$	$45°$ up to $K$, then flat

The buyer’s maximum loss is the premium; the seller’s maximum loss is unbounded for calls ($K-\text{premium}$ for puts). This asymmetry is not a flaw in the picture - it is the contract.

2.2 Moneyness, intrinsic value, and time value (Hull Ch 10.4)

• In the money (ITM): $S>K$ for a call, $S<K$ for a put.
• At the money (ATM): $S\approx K$.
• Out of the money (OTM): the reverse of ITM.
• Intrinsic value $=\max (S-K,0)$ (call) or $\max (K-S,0)$ (put) - what exercise pays right now.
• Time value $=$ option price $-$ intrinsic value - what you pay for the possibility that things improve before expiry. Time value is $\ge 0$ and decays to zero at expiry (the Greek Theta, see The Greeks & Dynamic Hedging).

2.3 European vs American, and the “free option on an option”

An American holder may exercise at any $t\le T$; a European holder only at $T$. Hence

$$C\ge c,P\ge p,\text{and more sharply}P\ge \max (K-S_0,0)>\max (K{e}^{-rT}-S_0,0)\text{when deep ITM.}$$

The extra value of the American right is the early-exercise premium. It is zero for a call on a non-dividend-paying stock (exercise early only throws away insurance and the time value of money - Hull Ch 11.5; Shreve §7), but strictly positive for a put (you can bank the strike and earn interest). This single asymmetry - calls never early-exercised, puts sometimes - explains why the BSM closed form prices American calls exactly but not American puts.

2.4 Structure of the payoff and its sensitivity

For a European call, the price is $c(S_0,K,\sigma ,r,T)$ and its sensitivity to the underlying is the delta $\Delta =\partial c/\partial S\in (0,1)$; for a put $\Delta \in (-1,0)$. At expiry the payoff’s kink is a delta discontinuity. In between, delta is smooth, and the curvature (gamma) is largest ATM - which is precisely where one pays the most time value. The payoff diagram’s kink is the expiry limit of a smooth, expensive curve.

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3. Computational Implementation - drawing the kink and splitting the price

Standard library only. This computes the four payoffs, classifies moneyness across strikes, decomposes the ATM price into intrinsic $+$ time value (the BSM numbers are Haug-verified), and renders a text payoff diagram.

Two things to notice. First, the ATM call is pure time value ($10.4506$ of intrinsic-free premium) - a contract that pays nothing if you exercised today but is worth $10$ because the kink still has a year of uncertainty left. Second, at $S_T=110$ the call is in the money yet the position loses $0.45$: being right about direction is not the same as making money, because you paid $10.45$ for the kink. The breakeven is $K+\text{premium}=110.45$.

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4. Failure Modes & First-Principles Breakdowns

1. Confusing “in the money” with “profitable.” At expiry $S_T>K$ means the payoff is positive, but the position profits only above $K+\text{premium}$. As the diagram shows, ITM at $110$ still loses money when the premium was $10.45$.
2. Reading the wrong kink for the style. A European call diagram at $t<T$ is smooth and above the kinked payoff - the vertical distance is time value. Plotting the expiry kink and calling it “the value today” underprices by the time value (which at ATM is $100\%$ of the price).
3. Assuming American $\Rightarrow$ worth exercising. For a call on a non-dividend stock, early exercise is never optimal - the American price equals the European (Hull 11.5). For a deep-ITM put it often is. Applying the call intuition to puts is a classic error.
4. Dividend/split blindness. Cash dividends do not normally adjust the strike, but splits do ($K\to K\cdot m/n$, shares $\to n/m$; Hull Ch 10.4). An unadjusted split makes the payoff diagram wrong by a factor.
5. Ignoring the multiplier. The price is per share; the position is $100\times$ per contract (Hull Ch 10.3). A “$3 premium” is $300 per contract - see 03 · Markets & Products.

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5. References

• Hull, Options, Futures, and Other Derivatives
• Shreve, Stochastic Calculus for Finance I
• Haug, The Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Back: 01 · What Is a Derivative · Index Hub
• Next: 03 · Markets & Products · 04 · No-Arbitrage & Bounds
• Strategies built from these payoffs: 06 · Advanced Extensions
• Forward topic-pages: The Greeks & Dynamic Hedging · Black–Scholes–Merton