Introduction to Options
Payoffs and profits from buying and selling call options and put options
1 Introduction
Imagine you’re considering buying a house, but you’re not quite ready to commit. A builder offers you a deal: pay $5,000 today, and you’ll lock in the right to buy a specific home at $400,000 anytime in the next six months. You’re not obligated to buy—if housing prices drop or your circumstances change, you simply walk away, losing only your $5,000. But if prices rise to $450,000, you can still purchase at the agreed $400,000, capturing a significant gain. This arrangement captures the essence of what options are in financial markets.
Options belong to a broader category of financial instruments called derivative securities. The term “derivative” reflects that these instruments derive their value from some other asset—called the underlying asset. Unlike stocks, where value stems from a company’s future earnings, or bonds, where value comes from promised interest payments, derivatives have no intrinsic value on their own. Their worth is entirely dependent on what happens to the underlying asset. This relationship is what makes derivatives both powerful and, at times, complex.
Derivatives serve several important functions in financial markets. First, they enable hedging, allowing investors and corporations to protect themselves against adverse price movements. A farmer worried about falling wheat prices can use derivatives to lock in a selling price today, eliminating uncertainty about future revenue. Second, derivatives facilitate speculation, letting traders express views about future price movements with potentially higher returns (and higher risks) than trading the underlying asset directly. A small investment in options can control exposure to a much larger position in stock. Third, derivatives support price discovery and market efficiency by incorporating information about future expectations into current prices. Finally, derivatives enable sophisticated risk management strategies that would be impossible or prohibitively expensive to implement by trading underlying assets alone.
The most common types of derivative securities include options (the focus of this lecture), futures and forward contracts (agreements to buy or sell an asset at a future date), and swaps (agreements to exchange cash flows). Each serves different purposes, but all share the fundamental characteristic of deriving value from an underlying asset.
This lecture focuses on options—specifically, call and put options on stocks. We’ll explore what these instruments are, how they work from both the buyer’s and seller’s perspectives, and how to calculate the payoffs and profits from various option positions. Understanding options provides a foundation for more advanced derivatives topics and for implementing sophisticated investment strategies.
2 Call Options
A call option is a contract that gives its holder the right—but not the obligation—to purchase a specific stock at a predetermined price on or before a specified date. This definition contains several crucial elements that deserve careful attention.
The stock that can be purchased is called the underlying asset or simply the underlying. The predetermined purchase price is called the strike price (also known as the exercise price). The specified date by which the option must be used is the expiration date (or maturity date). When you actually use your option to buy the stock, you are said to exercise the option. The price you pay to acquire the option contract itself is called the premium.
The distinction between American and European options matters for understanding when exercise can occur. American options can be exercised at any time up to and including the expiration date, giving the holder maximum flexibility. European options can only be exercised on the expiration date itself—not before. Despite the geographic names, both types trade worldwide. In practice, most exchange-traded stock options in the United States are American-style. Throughout this course, unless stated otherwise, assume we are dealing with American options.
When does it make sense to exercise a call option? The answer becomes clear when you consider what exercise means: you’re buying the stock at the strike price. If the stock is currently trading in the market at $80 and your call option has a strike price of $75, exercising means you can buy at $75 what’s worth $80—an immediate $5 advantage. But if the stock is trading at $70, why would you exercise your right to buy at $75 when you could simply purchase shares in the market at $70? You wouldn’t. This intuition leads to a fundamental principle: a call option should only be exercised when the stock price exceeds the strike price.
On October 14, 2014, a call option on Facebook (FB) with an expiration date of November 14, 2014 and a strike price of $75 sold for a premium of $5. The stock price of FB on October 14 was $76.
Questions:
Identify the underlying asset, strike price, expiration date, and premium.
What is your profit if you exercise this option immediately after buying it?
Suppose on October 21, 2014, FB is trading at $74 per share. Would you exercise the option?
Suppose right before expiration (November 14, 2014), FB is trading at $76 per share. Would you exercise the option?
Underlying asset: Facebook (FB) stock
Strike price: $75
Expiration date: November 14, 2014
Premium: $5If you exercise immediately when FB is at $76, you buy stock worth $76 for the strike price of $75, gaining $1 per share. However, you paid $5 for the option itself. Your net profit is $1 − $5 = −$4 (a loss).
When FB trades at $74, the stock price is below the strike price of $75. If you exercised, you would pay $75 for something worth only $74—a pointless transaction. You would not exercise the option at this price. Note that the option still has time value since it hasn’t expired yet; the stock price could rise before November 14.
At expiration with FB at $76, the stock price exceeds the strike price. If you exercise, you buy at $75 what’s worth $76, capturing $1 of value. Since the option is about to expire (no more time for the stock to move higher), you would exercise to capture this $1. Your total profit would be $1 − $5 = −$4, still a loss, but better than letting the option expire worthless and losing the entire $5 premium.
2.1 Call Option Payoff and Profit
To analyze option positions systematically, we distinguish between payoff and profit. The payoff represents the gross value received at exercise—what you gain from the exercise transaction itself, ignoring what you originally paid for the option. The profit accounts for the initial cost of acquiring the option.
For a call option you own (a “long call” position), the payoff depends on the relationship between the stock price \(S\) at expiration and the strike price \(X\). If the stock price exceeds the strike price, you exercise and receive the difference. If the stock price is at or below the strike price, you don’t exercise and receive nothing. Mathematically:
\[\text{Payoff} = \max(S - X, 0)\]
The \(\max\) function simply means “take the larger of these two values.” When \(S > X\), the payoff is \(S - X\) (a positive number). When \(S \leq X\), the payoff is 0 (since \(S - X\) would be negative or zero, and 0 is larger).
Profit incorporates the cost of acquiring the option:
\[\text{Profit} = \max(S - X, 0) - \text{Premium}\]
Notice that the profit can be negative—you can lose money on a call option even if you make the optimal exercise decision. The maximum loss occurs when the option expires worthless (when \(S \leq X\)), in which case you lose your entire premium. This limited downside is one of the attractive features of owning options: your loss is capped at what you paid, but your potential gain is theoretically unlimited as the stock price rises.
You buy a call option on Stock A for a premium of $14. The strike price is $80 and the option expires in one month.
Questions:
What is the payoff from this call option if the stock price at expiration is $60? $70? $80? $90? $100?
What is the profit in each scenario?
At what stock price do you break even?
Using the payoff formula \(\text{Payoff} = \max(S - X, 0)\) where \(X = 80\), and profit formula \(\text{Profit} = \text{Payoff} - 14\):
| Stock Price (\(S\)) | Payoff Calculation | Payoff | Profit Calculation | Profit |
|---|---|---|---|---|
| $60 | \(\max(60 - 80, 0) = \max(-20, 0)\) | $0 | \(0 - 14\) | −$14 |
| $70 | \(\max(70 - 80, 0) = \max(-10, 0)\) | $0 | \(0 - 14\) | −$14 |
| $80 | \(\max(80 - 80, 0) = \max(0, 0)\) | $0 | \(0 - 14\) | −$14 |
| $90 | \(\max(90 - 80, 0) = \max(10, 0)\) | $10 | \(10 - 14\) | −$4 |
| $100 | \(\max(100 - 80, 0) = \max(20, 0)\) | $20 | \(20 - 14\) | $6 |
Break-even analysis: You break even when profit equals zero. Setting \(\max(S - 80, 0) - 14 = 0\), we need \(S - 80 = 14\), so \(S = 94\). At a stock price of $94, your $14 payoff exactly offsets your $14 premium.
Notice that even when the stock rises to $90, you still have a loss of $4. The stock must rise high enough above the strike price to cover your initial premium before you earn any profit.
2.2 Selling (Writing) Call Options
So far we’ve considered the perspective of someone who buys a call option. But every option transaction has two parties: a buyer and a seller. The seller of an option is said to “write” the option, and takes on a very different set of risks and rewards.
When you write a call option, you grant someone else the right to buy stock from you at the strike price. You receive the premium upfront—that’s your compensation for taking on this obligation. If the stock price stays below the strike price, the buyer won’t exercise, and you keep the premium as pure profit. But if the stock price rises above the strike price, the buyer will exercise, and you must sell shares at the strike price—potentially well below the market price. If you don’t already own the shares, you’ll need to buy them at market price and sell at the strike price, suffering a loss.
The payoff and profit formulas for writing a call are simply the negatives of the buyer’s formulas:
\[\text{Payoff} = -\max(S - X, 0)\]
\[\text{Profit} = -\max(S - X, 0) + \text{Premium}\]
This creates an asymmetric risk profile: the most you can gain is the premium (when the option expires worthless), but your potential loss is theoretically unlimited as the stock price rises. Writing options therefore requires careful consideration of the risks involved.
You sell (write) a call option on Stock A for a premium of $10. The strike price is $50 and the option expires in one month.
Questions:
What is your payoff if the stock price at expiration is $40? $50? $60? $70?
What is your profit in each scenario?
At what stock price do you break even?
Using the seller’s payoff formula \(\text{Payoff} = -\max(S - X, 0)\) where \(X = 50\), and profit formula \(\text{Profit} = -\max(S - X, 0) + 10\):
| Stock Price (\(S\)) | Payoff Calculation | Payoff | Profit Calculation | Profit |
|---|---|---|---|---|
| $40 | \(-\max(40 - 50, 0) = -\max(-10, 0)\) | $0 | \(0 + 10\) | $10 |
| $50 | \(-\max(50 - 50, 0) = -\max(0, 0)\) | $0 | \(0 + 10\) | $10 |
| $60 | \(-\max(60 - 50, 0) = -\max(10, 0)\) | −$10 | \(-10 + 10\) | $0 |
| $70 | \(-\max(70 - 50, 0) = -\max(20, 0)\) | −$20 | \(-20 + 10\) | −$10 |
Break-even analysis: You break even when profit equals zero. From the table, this occurs at \(S = 60\), where your $10 loss on the exercise is exactly offset by the $10 premium you received. At any price above $60, you lose money—and those losses grow dollar-for-dollar with the stock price.
Notice the seller’s profit profile is the mirror image of the buyer’s: the seller’s maximum gain equals the buyer’s maximum loss (the premium), and the seller’s potential losses are unlimited, matching the buyer’s unlimited gain potential.
3 Put Options
While call options give the right to buy, put options give the right to sell. A put option grants its holder the right—but not the obligation—to sell a specific stock at a predetermined strike price on or before a specified expiration date.
Why would someone want the right to sell? Consider an investor who owns shares of a stock currently trading at $100 but worries about a potential decline. By purchasing a put option with a strike price of $95, the investor establishes a floor on potential losses. Even if the stock crashes to $50, the put holder can sell at $95 per share. The put acts as insurance against price declines.
The terminology for puts mirrors that of calls: the put has an underlying asset, a strike price, an expiration date, and a premium. Exercise decisions follow opposite logic: you exercise a put when the stock price is below the strike price. If you have the right to sell at $75 and the stock is trading at $70, exercising lets you sell at $75 something worth only $70—a $5 advantage. But if the stock is trading at $80, why sell at $75 when the market offers $80? You wouldn’t exercise.
On October 14, 2014, you buy a put option on Facebook (FB) for $5. The option expires on November 14, 2014 and has a strike price of $75. The stock price of FB on October 14 was $74.
Questions:
Identify the underlying asset, strike price, expiration date, and premium.
What is your profit if you exercise this option immediately after buying it?
Suppose on October 21, 2014, FB is trading at $76 per share. Would you exercise the option?
Suppose right before expiration (November 14, 2014), FB is trading at $74 per share. Would you exercise the option?
Underlying asset: Facebook (FB) stock
Strike price: $75
Expiration date: November 14, 2014
Premium: $5If you exercise immediately when FB is at $74, you sell stock at the strike price of $75 that’s worth $74, gaining $1 per share. However, you paid $5 for the option. Your net profit is $1 − $5 = −$4 (a loss).
When FB trades at $76, the stock price exceeds the strike price of $75. If you exercised, you would sell at $75 something worth $76—that makes no sense. You would not exercise the option at this price.
At expiration with FB at $74, the stock price is below the strike price. If you exercise, you sell at $75 what’s worth only $74, capturing $1 of value. Since the option is about to expire, you would exercise to capture this $1. Your total profit would be $1 − $5 = −$4, but this is better than letting the option expire worthless and losing the entire $5 premium.
3.1 Put Option Payoff and Profit
The payoff structure for puts reverses the relationship between strike price and stock price. For a put option you own (a “long put”), you profit when the stock price falls below the strike price. The payoff formula captures this:
\[\text{Payoff} = \max(X - S, 0)\]
When the stock price \(S\) is below the strike price \(X\), you exercise and receive \(X - S\). When the stock price equals or exceeds the strike price, you don’t exercise and receive nothing.
Profit accounts for the premium paid:
\[\text{Profit} = \max(X - S, 0) - \text{Premium}\]
Like calls, puts have limited downside for the buyer—the maximum loss is the premium paid. The maximum gain for a put, however, is not unlimited. Since a stock price cannot fall below zero, the most a put can be worth at expiration is the strike price itself (which would occur if the stock became worthless).
You buy a put option on Stock A for a premium of $10. The strike price is $40 and the option expires in one month.
Questions:
What is the payoff from this put option if the stock price at expiration is $60? $40? $20? $0?
What is the profit in each scenario?
At what stock price do you break even?
Using the payoff formula \(\text{Payoff} = \max(X - S, 0)\) where \(X = 40\), and profit formula \(\text{Profit} = \text{Payoff} - 10\):
| Stock Price (\(S\)) | Payoff Calculation | Payoff | Profit Calculation | Profit |
|---|---|---|---|---|
| $60 | \(\max(40 - 60, 0) = \max(-20, 0)\) | $0 | \(0 - 10\) | −$10 |
| $40 | \(\max(40 - 40, 0) = \max(0, 0)\) | $0 | \(0 - 10\) | −$10 |
| $20 | \(\max(40 - 20, 0) = \max(20, 0)\) | $20 | \(20 - 10\) | $10 |
| $0 | \(\max(40 - 0, 0) = \max(40, 0)\) | $40 | \(40 - 10\) | $30 |
Break-even analysis: You break even when profit equals zero. Setting \(\max(40 - S, 0) - 10 = 0\), we need \(40 - S = 10\), so \(S = 30\). At a stock price of $30, your $10 payoff exactly offsets your $10 premium.
Notice the put buyer benefits from falling prices. The maximum profit of $30 occurs if the stock goes to zero (payoff of $40 minus $10 premium). The maximum loss of $10 occurs at any price of $40 or above.
3.2 Selling (Writing) Put Options
Just as with calls, every put option has both a buyer and a seller. When you write a put option, you give someone else the right to sell stock to you at the strike price. You receive the premium upfront, but you’re obligated to buy the shares if the option holder exercises.
Writing puts can be attractive when you’re willing to buy a stock but prefer to wait for a lower price. If you write a put with a $50 strike price and collect a $3 premium, you’re essentially saying “I’ll buy this stock if it drops to $50, and someone will pay me $3 for making that commitment.” If the stock stays above $50, you keep the premium without having to buy anything.
The risk arises if the stock plummets. You’ll be forced to buy at the strike price, regardless of how far the market price has fallen. The payoff and profit formulas for put writers are:
\[\text{Payoff} = -\max(X - S, 0)\]
\[\text{Profit} = -\max(X - S, 0) + \text{Premium}\]
The maximum gain is limited to the premium received, while the maximum loss occurs if the stock becomes worthless—you’d pay the full strike price for something worth nothing.
You sell (write) a put option on Stock A for a premium of $10. The strike price is $60 and the option expires in one month.
Questions:
What is your payoff if the stock price at expiration is $1? $50? $60? $70? $80?
What is your profit in each scenario?
At what stock price do you break even?
Using the seller’s payoff formula \(\text{Payoff} = -\max(X - S, 0)\) where \(X = 60\), and profit formula \(\text{Profit} = -\max(X - S, 0) + 10\):
| Stock Price (\(S\)) | Payoff Calculation | Payoff | Profit Calculation | Profit |
|---|---|---|---|---|
| $1 | \(-\max(60 - 1, 0) = -\max(59, 0)\) | −$59 | \(-59 + 10\) | −$49 |
| $50 | \(-\max(60 - 50, 0) = -\max(10, 0)\) | −$10 | \(-10 + 10\) | $0 |
| $60 | \(-\max(60 - 60, 0) = -\max(0, 0)\) | $0 | \(0 + 10\) | $10 |
| $70 | \(-\max(60 - 70, 0) = -\max(-10, 0)\) | $0 | \(0 + 10\) | $10 |
| $80 | \(-\max(60 - 80, 0) = -\max(-20, 0)\) | $0 | \(0 + 10\) | $10 |
Break-even analysis: You break even when profit equals zero. From the table, this occurs at \(S = 50\), where your $10 loss on the exercise is exactly offset by the $10 premium you received.
Notice the risk profile: the seller’s maximum gain is $10 (the premium), achieved at any price of $60 or above. But as the stock falls, losses mount. In the extreme case where the stock becomes nearly worthless ($1), the seller loses $49—nearly five times the premium received.
4 Comparing Calls and Puts
The following table summarizes the key differences between call and put options from the buyer’s perspective:
| Feature | Call Option | Put Option |
|---|---|---|
| Right granted | Buy the underlying | Sell the underlying |
| Exercise when | Stock price > Strike price | Stock price < Strike price |
| Payoff formula | \(\max(S - X, 0)\) | \(\max(X - S, 0)\) |
| Benefits from | Rising prices | Falling prices |
| Maximum loss (buyer) | Premium paid | Premium paid |
| Maximum gain (buyer) | Unlimited | Strike price − Premium |
For option sellers (writers), all the payoffs and profits are reversed. Sellers benefit when options expire worthless and face losses when buyers exercise.
5 Key Takeaways
Options are derivative securities whose value depends on an underlying asset—typically a stock in the context of equity options. Unlike owning shares outright, options provide conditional exposure: you benefit from price movements in one direction while limiting your downside risk to the premium paid. This asymmetric payoff profile makes options valuable for hedging, speculation, and implementing nuanced investment strategies.
Call options give you the right to buy at the strike price, becoming valuable when the stock rises above that level. Put options give you the right to sell at the strike price, becoming valuable when the stock falls below that level. In both cases, the option buyer’s maximum loss is limited to the premium, while the gain depends on how far prices move in the favorable direction. Option sellers take the opposite side of these transactions, earning premiums upfront but accepting potentially significant obligations if prices move against them.
The payoff and profit formulas for options all derive from a simple principle: exercise only when it’s advantageous, which means when the option is “in the money.” For calls, that means \(S > X\); for puts, \(X > S\). The \(\max\) function in our formulas captures this logic mathematically, ensuring that payoffs are never negative from the holder’s perspective.
Understanding these basic building blocks—the mechanics of calls and puts from both buyer and seller perspectives—provides the foundation for analyzing more complex option strategies and understanding how options contribute to overall portfolio management.
6 Key Formulas Summary
| Concept | Formula | When to Use |
|---|---|---|
| Long call payoff | \(\max(S - X, 0)\) | Calculate what a call option is worth at expiration to the buyer |
| Long call profit | \(\max(S - X, 0) - \text{Premium}\) | Calculate net profit/loss from buying a call option |
| Short call payoff | \(-\max(S - X, 0)\) | Calculate what a call option costs the seller at expiration |
| Short call profit | \(-\max(S - X, 0) + \text{Premium}\) | Calculate net profit/loss from selling (writing) a call option |
| Long put payoff | \(\max(X - S, 0)\) | Calculate what a put option is worth at expiration to the buyer |
| Long put profit | \(\max(X - S, 0) - \text{Premium}\) | Calculate net profit/loss from buying a put option |
| Short put payoff | \(-\max(X - S, 0)\) | Calculate what a put option costs the seller at expiration |
| Short put profit | \(-\max(X - S, 0) + \text{Premium}\) | Calculate net profit/loss from selling (writing) a put option |
In all formulas: \(S\) = stock price at expiration, \(X\) = strike price, Premium = price paid (or received) for the option.
7 Practice Problems
On March 1, 2025, a call option on Apple (AAPL) with an expiration date of April 1, 2025 and a strike price of $180 sells for $8. The stock price of AAPL on March 1 is $185.
Questions:
Identify the underlying asset, strike price, expiration date, and premium.
What is your profit if you exercise this option immediately after buying it?
If AAPL is trading at $175 on March 15, would you exercise the option?
If AAPL is trading at $190 at expiration, would you exercise? What is your profit?
Underlying asset: Apple (AAPL) stock
Strike price: $180
Expiration date: April 1, 2025
Premium: $8If you exercise immediately when AAPL is at $185, you buy stock worth $185 for $180, gaining $5 per share. Subtracting the $8 premium: Profit = $5 − $8 = −$3 (a loss).
When AAPL trades at $175, the stock price is below the strike price of $180. Exercising would mean paying $180 for something worth only $175. You would not exercise. The option still has time remaining until April 1, so you would hold it hoping for a price increase.
At expiration with AAPL at $190: Yes, you would exercise since $190 > $180.
Payoff = $(190 - 180, 0) = $10
Profit = $10 − $8 = $2
You buy a call option on Stock B for a premium of $6. The strike price is $55 and the option expires in one month.
Questions:
What is the payoff from this call option if the stock price at expiration is $45? $55? $65? $75?
What is the profit in each scenario?
At what stock price do you break even?
Using the payoff formula \(\text{Payoff} = \max(S - X, 0)\) where \(X = 55\), and profit formula \(\text{Profit} = \text{Payoff} - 6\):
| Stock Price (\(S\)) | Payoff | Profit |
|---|---|---|
| $45 | $(45 - 55, 0) = $0 | $0 - 6 = −$6 |
| $55 | $(55 - 55, 0) = $0 | $0 - 6 = −$6 |
| $65 | $(65 - 55, 0) = $10 | $10 - 6 = $4 |
| $75 | $(75 - 55, 0) = $20 | $20 - 6 = $14 |
Break-even: Setting profit to zero: \(S - 55 - 6 = 0\), so $S = $61.
You write a call option on Stock C for a premium of $7. The strike price is $40 and the option expires in one month.
Questions:
What is your payoff if the stock price at expiration is $35? $40? $50? $60?
What is your profit in each scenario?
At what stock price do you break even?
Using the seller’s payoff formula \(\text{Payoff} = -\max(S - X, 0)\) where \(X = 40\), and profit formula \(\text{Profit} = -\max(S - X, 0) + 7\):
| Stock Price (\(S\)) | Payoff | Profit |
|---|---|---|
| $35 | $-(35 - 40, 0) = $0 | $0 + 7 = $7 |
| $40 | $-(40 - 40, 0) = $0 | $0 + 7 = $7 |
| $50 | $-(50 - 40, 0) = −$10 | $-10 + 7 = −$3 |
| $60 | $-(60 - 40, 0) = −$20 | $-20 + 7 = −$13 |
Break-even: Setting profit to zero: \(-(S - 40) + 7 = 0\), so \(S - 40 = 7\), thus $S = $47.
On March 1, 2025, you buy a put option on Microsoft (MSFT) for $6. The option expires on April 1, 2025 and has a strike price of $400. The stock price of MSFT on March 1 is $395.
Questions:
Identify the underlying asset, strike price, expiration date, and premium.
What is your profit if you exercise this option immediately after buying it?
If MSFT is trading at $410 on March 15, would you exercise the option?
If MSFT is trading at $380 at expiration, would you exercise? What is your profit?
Underlying asset: Microsoft (MSFT) stock
Strike price: $400
Expiration date: April 1, 2025
Premium: $6If you exercise immediately when MSFT is at $395, you sell stock at $400 that’s worth $395, gaining $5 per share. Subtracting the $6 premium: Profit = $5 − $6 = −$1 (a loss).
When MSFT trades at $410, the stock price exceeds the strike price of $400. Exercising would mean selling at $400 something worth $410—pointless. You would not exercise.
At expiration with MSFT at $380: Yes, you would exercise since $380 < $400.
Payoff = $(400 - 380, 0) = $20
Profit = $20 − $6 = $14
You buy a put option on Stock D for a premium of $8. The strike price is $50 and the option expires in one month.
Questions:
What is the payoff from this put option if the stock price at expiration is $70? $50? $30? $10?
What is the profit in each scenario?
At what stock price do you break even?
Using the payoff formula \(\text{Payoff} = \max(X - S, 0)\) where \(X = 50\), and profit formula \(\text{Profit} = \text{Payoff} - 8\):
| Stock Price (\(S\)) | Payoff | Profit |
|---|---|---|
| $70 | $(50 - 70, 0) = $0 | $0 - 8 = −$8 |
| $50 | $(50 - 50, 0) = $0 | $0 - 8 = −$8 |
| $30 | $(50 - 30, 0) = $20 | $20 - 8 = $12 |
| $10 | $(50 - 10, 0) = $40 | $40 - 8 = $32 |
Break-even: Setting profit to zero: \(50 - S - 8 = 0\), so $S = $42.
You write a put option on Stock E for a premium of $5. The strike price is $45 and the option expires in one month.
Questions:
What is your payoff if the stock price at expiration is $5? $35? $45? $55?
What is your profit in each scenario?
At what stock price do you break even?
Using the seller’s payoff formula \(\text{Payoff} = -\max(X - S, 0)\) where \(X = 45\), and profit formula \(\text{Profit} = -\max(X - S, 0) + 5\):
| Stock Price (\(S\)) | Payoff | Profit |
|---|---|---|
| $5 | $-(45 - 5, 0) = −$40 | $-40 + 5 = −$35 |
| $35 | $-(45 - 35, 0) = −$10 | $-10 + 5 = −$5 |
| $45 | $-(45 - 45, 0) = $0 | $0 + 5 = $5 |
| $55 | $-(45 - 55, 0) = $0 | $0 + 5 = $5 |
Break-even: Setting profit to zero: \(-(45 - S) + 5 = 0\), so \(45 - S = 5\), thus $S = $40.
8 Ask an LLM
Here are three questions you might ask an AI assistant to deepen your understanding of these concepts:
- What happens to the value of a call option as it approaches its expiration date, and why do options have “time value” in addition to their payoff value?
- Can you walk me through a real-world example where a company might use put options to hedge against a decline in its stock portfolio, including how they would choose the strike price and expiration date?
- How would I construct a payoff diagram that combines both buying a stock and buying a put option on that same stock, and what real-world protection strategy does this represent?
- What is the relationship between call and put option prices for the same underlying asset, strike price, and expiration date (put-call parity), and why must this relationship hold?
- If I’m bearish on a stock, should I buy a put option or write a call option? What are the trade-offs between these two strategies in terms of risk and reward?