Cash flows and characteristics of puts and calls

Q: What is the fundamental difference between an option and an underlying asset trade in terms of obligation?

An option gives the buyer the right , but not the obligation , to buy or sell the underlying asset at a fixed price.

Q: Define the 'Strike Price' and 'Premium' in the context of an option contract.?

The Strike Price ($K$) is the fixed price at which the asset can be traded, and the Premium is the upfront cost paid to purchase the option.

Q: What are the maximum loss and maximum gain for a Long Call position?

The maximum loss is the premium paid (plus interest), and the maximum gain is theoretically unlimited.

Q: How does the cash flow of an option seller (writer) relate to the cash flow of an option buyer?

The seller's cash flow is the exact negative of the buyer's cash flow; it is a zero-sum game.

Q: What does it mean for an option to be 'In-the-Money' (ITM)?

An option is In-the-Money if exercising it right now would result in a positive payoff.

Q: Why is it necessary to accumulate the premium when calculating option profit?

To compare the initial cost with the final payoff at the same point in time (expiration) using the risk-free rate.

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 5 min read

Welcome to Option Pricing Fundamentals!

Hey there, future actuary! If you’ve ever wished you could "lock in" a price today for something you want to buy or sell in the future, then you already understand the heart of options. In this chapter, we’re diving into the cash flows and characteristics of puts and calls. This is a foundational part of Exam FAM, and while the math might look intimidating at first, it’s really just a way of keeping score in a high-stakes game of "What If?"

By the end of these notes, you'll be able to calculate exactly how much money an option holder makes (or loses) and understand why these financial tools are so vital for managing risk. Let’s get started!

1. What is an Option?

An option is a contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset (like a stock) at a fixed price within a specific timeframe.

Think of it like a coupon. If you have a coupon for a \$10 pizza, you have the right to buy it for \$10. If the pizza normally costs \$15, you’ll definitely use the coupon! But if the pizza is on sale for \$8, you’ll just throw the coupon away and buy the pizza at the cheaper price. That’s exactly how options work.

Key Terms to Know:

Underlying Asset: The thing being traded (usually denoted as $S$).
Strike Price ($K$): The fixed price set in the contract.
Expiration Date ($T$): The date the option expires.
Premium: The upfront cost paid to buy the option. Think of this as the "insurance premium."
$S_T$: The price of the stock at the time of expiration.

2. Call Options: The Right to Buy

A Call Option gives the holder the right to buy an asset at the strike price $K$. You want the stock price ($S_T$) to go up so you can buy it cheaply and potentially sell it for a profit.

Payoff vs. Profit

Don't worry if these two terms seem similar—they are related but different! Payoff is the amount of money you receive at expiration. Profit is your payoff minus the premium you paid at the start.

Call Payoff Formula:
$ \text{Payoff} = \max(0, S_T - K) $

Call Profit Formula:
$ \text{Profit} = \max(0, S_T - K) - \text{Future Value of Premium} $

Note: For Exam FAM, we usually look at the profit at the time of expiration, so we have to grow the initial premium using the risk-free interest rate $r$.

Analogy: The Concert Ticket

Imagine you have a "call option" to buy a concert ticket for \$100 ($K$). On the day of the show, tickets are selling for \$150 ($S_T$). You exercise your right to buy for \$100, and your payoff is \$50. However, if you paid \$10 for that "call option" originally, your actual profit is \$40.

Quick Review: The Long Call
- Who: The Buyer.
- Outlook: Bullish (expects price to rise).
- Max Loss: The premium paid.
- Max Gain: Theoretically unlimited!

3. Put Options: The Right to Sell

A Put Option gives the holder the right to sell an asset at the strike price $K$. You want the stock price ($S_T$) to go down. It’s essentially insurance against a price drop.

Put Payoff Formula:
$ \text{Payoff} = \max(0, K - S_T) $

Put Profit Formula:
$ \text{Profit} = \max(0, K - S_T) - \text{Future Value of Premium} $

Analogy: Car Insurance

A put option is very similar to car insurance. You pay a premium to have the right to "sell" your car back to the insurance company for its total value ($K$) if it gets wrecked ($S_T$ drops to zero). If the car stays in perfect condition, you don't use the insurance, and you lose only the premium.

Quick Review: The Long Put
- Who: The Buyer.
- Outlook: Bearish (expects price to fall).
- Max Loss: The premium paid.
- Max Gain: Significant (limited only because a stock price can't go below zero).

4. The Seller's Perspective (Short Positions)

For every buyer, there is a seller (also called the Writer). The seller’s cash flow is the exact opposite of the buyer’s.

Short Call: You receive the premium upfront but are obligated to sell the stock if the buyer wants it. You hope the price stays below the strike price.
Short Put: You receive the premium upfront but are obligated to buy the stock if the buyer wants to sell it. You hope the price stays above the strike price.

Memory Trick:
The buyer is "Long." The seller is "Short."
Long Call: "I call the stock to me" (Buy).
Long Put: "I put the stock away to someone else" (Sell).

5. Moneyness: Where do we stand?

Moneyness describes the relationship between the current stock price and the strike price. It tells us if exercising the option would result in a positive payoff right now.

In-the-Money (ITM): The option has a positive payoff. (Exercise it!)
- Call: $S_T > K$
- Put: $S_T < K$

At-the-Money (ATM): The stock price equals the strike price.
- $S_T = K$

Out-of-the-Money (OTM): The option has a zero payoff. (Don't exercise it!)
- Call: $S_T < K$
- Put: $S_T > K$

Did you know? Even if an option is ITM, you might still have a net loss if the payoff is smaller than the premium you paid!

6. Summary and Key Takeaways

Understanding these cash flows is the "bread and butter" of actuarial financial mathematics. Here is a quick wrap-up of the essential points:

Calls are for buyers who think the price will go up.
Puts are for buyers who think the price will go down.
Payoff ignores the cost of the option; Profit includes the cost (premium) plus interest.
The buyer of an option has limited risk (just the premium) and potentially high reward.
The seller of an option has limited reward (the premium) and potentially high risk.

Common Mistake to Avoid:

When calculating Profit, students often forget to accumulate the premium to the expiration date. Always check if the problem provides an interest rate. If it does, you usually need to use $ \text{Premium} \times e^{rT} $ to find the cost at time $T$.

Don't worry if the formulas for "Short" positions feel confusing—just calculate the buyer's profit and flip the sign (multiply by -1). If the buyer makes \$20, the seller must have lost \$20. It's a zero-sum game!

You’ve got this! Keep practicing those payoff diagrams!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the fundamental difference between an option and an underlying asset trade in terms of obligation?

An option gives the buyer the right, but not the obligation, to buy or sell the underlying asset at a fixed price.

Define the 'Strike Price' and 'Premium' in the context of an option contract.

The Strike Price ($K$) is the fixed price at which the asset can be traded, and the Premium is the upfront cost paid to purchase the option.

What is the mathematical formula for the payoff of a Call Option at expiration?

$ \text{Payoff} = \max(0, S_T - K) $

How is the profit of a Call Option calculated, taking into account the time value of money?

$ \text{Profit} = \max(0, S_T - K) - (\text{Premium} \times e^{rT}) $

What are the maximum loss and maximum gain for a Long Call position?

The maximum loss is the premium paid (plus interest), and the maximum gain is theoretically unlimited.

What is the mathematical formula for the payoff of a Put Option at expiration?

$ \text{Payoff} = \max(0, K - S_T) $

What is the maximum gain for a Long Put position?

The maximum gain is $K - (\text{Premium} \times e^{rT})$, occurring if the stock price falls to zero.

How does the cash flow of an option seller (writer) relate to the cash flow of an option buyer?

The seller's cash flow is the exact negative of the buyer's cash flow; it is a zero-sum game.

What does it mean for an option to be 'In-the-Money' (ITM)?

An option is In-the-Money if exercising it right now would result in a positive payoff.

Define 'At-the-Money' (ATM) and 'Out-of-the-Money' (OTM) for a Call option.

ATM: $S_T = K$; OTM: $S_T < K$.

Why is it necessary to accumulate the premium when calculating option profit?

To compare the initial cost with the final payoff at the same point in time (expiration) using the risk-free rate.

Ready to test yourself?

Turn these notes into exam-style practice. Get unlimited AI questions on this topic with instant marking and explanations.

Practice This Topic

More Exam FAM – Fundamentals of Actuarial Mathematics chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practice This Topic Now