Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Option Strategies and Payoff Combinations › Payoff Diagrams as Building Blocks

Michael BrenndoerferNovember 13, 202510 min read

Quantitative FinanceAll chapters

Part 3 of 15Forwards, Futures, and Options

Chapter 19Option Strategies and Payoff Combinations › Payoff Diagrams as Building Blocks

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Before constructing complex strategies, you need to internalize how individual option payoffs work as building blocks. Every option strategy is simply a linear combination of basic payoff functions. This insight simplifies option analysis. Rather than memorizing each strategy, you can understand it by decomposing it into its constituent parts.

Payoff combinations are linear. When you hold multiple options simultaneously, their individual payoffs simply add together at every possible stock price. If one option gains $5 at a particular price while another loses $3, your combined position gains $2. This additive property means you can analyze complex strategies by first understanding each component, then summing their contributions across all possible outcomes.

Payoff vs. Profit

The payoff of an option at expiration is its intrinsic value: $\max(S_T - K, 0)$ for a call and $\max(K - S_T, 0)$ for a put. The profit subtracts the premium paid (or adds the premium received) from the payoff. Payoff diagrams show the value at expiration; profit diagrams account for the initial cost.

The four basic building blocks form the foundation of all option strategies. Each building block has a distinctive shape that reflects the economic nature of the position. Understanding these shapes, particularly where they bend and what slopes they exhibit, enables you to mentally construct any combination before committing capital.

Long call: Payoff is:

\max(S_T - K, 0)

where $S_T$ is the stock price at expiration, $K$ is the strike price of the option, and $\max(S_T - K, 0)$ is the intrinsic value of the call (positive when stock exceeds strike, zero otherwise). This formula shows that call options are asymmetric. The max function ensures that the payoff can never be negative, which means your downside is limited to the premium paid. When the stock price rises above the strike, the call's value increases dollar-for-dollar with the stock, capturing all the upside without limit. The payoff is zero below the strike and then rises linearly above it. You pay premium upfront, so profit starts negative.

Short call: Payoff is:

-\max(S_T - K, 0)

where $S_T$ is the stock price at expiration, $K$ is the strike price of the option, and $-\max(S_T - K, 0)$ is the negative of call intrinsic value (obligation to sell stock at strike if exercised). The negative sign shows how short positions work: you are on the opposite side of the long call holder. Whatever they gain, you lose. This creates a mirror-image payoff structure.

The payoff is flat at zero below the strike, then falls linearly. Since you receive a premium, profit starts positive but decreases as the price rises.

Long put: Payoff is:

\max(K - S_T, 0)

where $K$ is the strike price of the option, $S_T$ is the stock price at expiration, and $\max(K - S_T, 0)$ is the intrinsic value of the put (positive when stock falls below strike, zero otherwise). Notice that the terms inside the max function are reversed compared to the call: the strike price comes first, then the stock price. This reversal reflects the put's role as downside protection. As the stock price falls further below the strike, the put becomes more valuable because it guarantees you can sell at a higher price than the market offers.

The payoff rises linearly as the price drops below the strike and is zero above it. Since you pay a premium, profit starts negative.

Short put: Payoff is:

-\max(K - S_T, 0)

where $K$ is the strike price of the option, $S_T$ is the stock price at expiration, and $-\max(K - S_T, 0)$ is the negative of put intrinsic value (obligation to buy stock at strike if exercised). Selling a put obligates you to purchase shares at the strike price if the option is exercised. This obligation becomes costly when the stock falls significantly below the strike, as you must buy shares at above-market prices.

The payoff falls as the price drops below the strike and is zero above it. Since you receive a premium, profit starts positive.

The key insight is that these payoffs are linear in different regions. When you add them together, the slopes add. This is how complex payoff structures emerge from simple components. Consider what happens when we combine a long call and a long put at the same strike: below the strike, the call contributes slope 0 while the put contributes slope -1, yielding a combined slope of -1. Above the strike, the call contributes slope +1 while the put contributes slope 0, yielding a combined slope of +1. The result is a V-shaped payoff that profits from movement in either direction.

In[2]:

Code

import warnings

import numpy as np

warnings.filterwarnings("ignore")


def call_payoff(S, K, premium=0, position="long"):
    """Calculate call option payoff/profit."""
    payoff = np.maximum(S - K, 0)
    if position == "short":
        payoff = -payoff
        return payoff + premium
    return payoff - premium


def put_payoff(S, K, premium=0, position="long"):
    """Calculate put option payoff/profit."""
    payoff = np.maximum(K - S, 0)
    if position == "short":
        payoff = -payoff
        return payoff + premium
    return payoff - premium


def stock_payoff(S, entry_price):
    """Calculate stock position payoff."""
    return S - entry_price


# Define price range and basic parameters for visualizations
S = np.linspace(50, 150, 200)
K = 100
premium = 5

Let's visualize these four building blocks to see how they combine. Examining the visual representation helps build intuition for how each component contributes to a combined strategy.

Out[3]:

Visualization

Long call profit and loss diagram showing zero payoff below the strike and rising profit above it. — Profit and loss profile of a long call option across a range of stock prices. The position exhibits limited downside risk defined by the premium paid, with unlimited upside potential once the stock price rises above the strike price.

Long put profit and loss diagram showing rising profit as the stock price falls below the strike. — Profit and loss profile of a long put option across a range of stock prices. The position gains value as the stock price falls below the strike, while risk is strictly limited to the initial premium paid if the stock price rises.

Out[4]:

Visualization

Short call profit and loss diagram showing collected premium eroding as the stock price rises above the strike. — Profit and loss profile of a short call option across a range of stock prices. The position generates immediate premium income but faces theoretically unlimited risk if the stock price rises significantly above the strike.

Short put profit and loss diagram showing collected premium eroding as the stock price falls below the strike. — Profit and loss profile of a short put option across a range of stock prices. The position generates immediate premium income but faces substantial downside risk as the stock price falls toward zero.

Notice the symmetry: long and short positions are mirror images of each other across the horizontal axis. Also observe that calls have their "action" above the strike price, while puts have their "action" below it. These visual patterns become essential when you start combining positions.

Out[5]:

Visualization

Long call payoff diagram annotated with slope regions showing zero slope below the strike and plus one slope above it. — Slope regions for a long call option payoff across stock price movements. The slope remains at zero below the strike and transitions to positive one above it, creating a profile that limits downside while capturing gains.

Long put payoff diagram annotated with slope regions showing minus one slope below the strike and zero slope above it. — Slope regions for a long put option payoff across stock price movements. The slope is negative one below the strike and zero above it, and gives downside exposure that terminates once the stock price exceeds the strike.