Payoff Diagrams for Options

1. From Linear to Non-Linear: Payoff vs. Profit in Options

Unlike forwards, where you have an obligation to transact, an option gives the buyer the right, but not the obligation, to do so. This choice creates a "kink" in the payoff profile. Let's distinguish two key concepts:

• Payoff (or Intrinsic Value): This is the value of the option at expiration, ignoring the initial cost (premium). It can never be negative, because the holder will simply not exercise the option if it's not profitable to do so.
• Profit/Loss: This is the net financial outcome after accounting for the premium. For the buyer, Profit = Payoff - Premium Paid. For the seller, Profit = Payoff (a loss to them) + Premium Received.

The diagram below illustrates the pure payoff (intrinsic value) for a buyer of a call or put option. Notice the characteristic "hockey stick" shape, which comes from the choice to not exercise when the option is "out-of-the-money".

Let's look at the formulas that produce these shapes:

• Call Option Payoff: $\max (0,S_T-K)$
• Put Option Payoff: $\max (0,K-S_T)$

Here, $S_T$ is the spot price of the underlying at expiration, and $K$ is the fixed strike price. The $\max (0,...)$ function is the mathematical representation of the holder's choice: if the exercise value is negative, they abandon the option, and the payoff is zero.

When we adjust for the premium, we get the full profit/loss diagram. This is what we will focus on constructing for the rest of the lesson.

2. The Four Building Blocks of Option Strategies

Every complex option strategy is built from four basic transactions. We will now explore the profit/loss diagram for each one. The following video provides an excellent step-by-step walkthrough with clear numerical examples. We will watch it in segments as we discuss each position.

2.1 The Long Call: A Bet on the Upside

When you buy (go "long") a call option, you are expressing a bullish view. You believe the price of the underlying asset will rise significantly. You pay a premium for the right to buy the asset at the strike price $K$.

Your risk is limited to the premium you paid, while your potential profit is, in theory, unlimited.

Now, let's see how this is constructed.

As the video demonstrated, the key features of the long call profit/loss diagram are:

• Maximum Loss: Limited to the premium paid. This occurs if $S_T\le K$.
• Break-even Point: The stock price at which you recover your premium. It is calculated as: Strike Price + Premium.
• Profit Zone: Occurs when $S_T>\text{Break-even Point}$. The profit potential is unlimited.

2.2 The Short Call: A Bet on Stagnation or a Fall

When you sell (or "write") a call option, you take the other side of the trade. You receive a premium in exchange for taking on the obligation to sell the asset at the strike price if the buyer chooses to exercise. This is a neutral to bearish position.

Your potential profit is limited to the premium you received, but your potential loss is unlimited. This makes selling "naked" calls a very risky strategy.

The short call profit/loss diagram is characterized by:

• Maximum Profit: Limited to the premium received. This is achieved if $S_T\le K$.
• Break-even Point: The same as for the long call: Strike Price + Premium.
• Loss Zone: Occurs when $S_T>\text{Break-even Point}$. The potential loss is unlimited, accelerating as the stock price rises.

2.3 The Long Put: A Bet on the Downside

If you are bearish and believe an asset's price will fall, you can buy a put option. This gives you the right to sell the asset at the strike price $K$.

Similar to a long call, your risk is limited to the premium paid. Your profit potential is substantial but capped, as the asset's price cannot fall below zero.

The key features of the long put profit/loss diagram are:

• Maximum Loss: Limited to the premium paid. This occurs if $S_T\ge K$.
• Break-even Point: The stock price at which you recover your premium. It is calculated as: Strike Price - Premium.
• Profit Zone: Occurs when $S_T<\text{Break-even Point}$. The maximum possible profit is $K-\text{Premium}$, which would occur if the stock price fell to zero.

2.4 The Short Put: A Bet on Stability or a Rise

Finally, selling a put option is a neutral to bullish strategy. You receive a premium and accept the obligation to buy the asset at the strike price if the holder exercises their right.

Your profit is limited to the premium, while your potential loss is substantial (though capped, as the price can only fall to zero).

The short put profit/loss diagram is characterized by:

• Maximum Profit: Limited to the premium received, achieved if $S_T\ge K$.
• Break-even Point: The same as for the long put: Strike Price - Premium.
• Loss Zone: Occurs when $S_T<\text{Break-even Point}$. The maximum possible loss is substantial.

3. A Consolidated View and Key Terminology

We have now constructed the diagrams for all four basic positions. The image below brings them all together for easy comparison.

To precisely interpret these diagrams, especially for exam purposes, it's helpful to use standard terminology for an option's status. This is known as its "moneyness".

In short:

• In-the-Money (ITM): The option has a positive intrinsic value. A call is ITM if $S_T>K$; a put is ITM if $S_T<K$.
• Out-of-the-Money (OTM): The option has zero intrinsic value. A call is OTM if $S_T<K$; a put is OTM if $S_T>K$.
• At-the-Money (ATM): The stock price equals the strike price, $S_T=K$.

Notice that an option being in-the-money does not guarantee a profit. For a buyer, the option must be sufficiently deep in-the-money to overcome the initial premium paid.

Conclusion

In this lesson, we have constructed and interpreted the profit/loss diagrams for the four fundamental option positions. These visual tools are indispensable for understanding the risk and reward of any option strategy.

Key Takeaways:

• Unlike the linear payoffs of forwards, options have asymmetric, non-linear payoffs due to the holder's right, but not obligation, to exercise.
• Option buyers (long positions) have limited risk (the premium paid) and significant profit potential.
• Option sellers (short positions) have limited profit (the premium received) and significant, often unlimited, risk.
• The break-even point for a call is $K+\text{premium}$, while for a put it is $K-\text{premium}$.
• An option's moneyness (ITM, ATM, OTM) describes its intrinsic value relative to the strike price at any given moment.

In our next lesson, we will put this visual framework into practice. We will focus on applying the profit and loss formulas to perform detailed calculations for various scenarios, solidifying your ability to quantify the financial outcomes of these positions.