Properties of Options

FRM Part I · FRM · 6 min read

Welcome to the World of Option Properties!

Hello! Today we are diving into the "Properties of Options." This chapter is a cornerstone of the Financial Markets and Products section. Think of this as learning the "laws of physics" for options. Once you understand these rules—like why a call option price can't go below a certain level or how stock prices and strike prices interact—you’ll find the more complex formulas much easier to grasp later on. Don't worry if it seems like a lot of variables at first; we will break them down one by one.

Did you know? Most of the "rules" we’ll discuss today are based on arbitrage. This is just a fancy way of saying that if these rules were broken, traders could make "free money" without any risk. The market moves quickly to prevent that!

1. The Six Factors That Affect Option Prices

Before we look at formulas, let’s look at the "ingredients" that determine how much you pay for an option. There are six main factors:

1. Stock Price ($S_0$): As the stock price goes up, a Call (the right to buy) becomes more valuable, and a Put (the right to sell) becomes less valuable.
2. Strike Price ($K$): The higher the price you have to pay to buy the stock (Call), the less the option is worth. Conversely, the higher the price you can sell it for (Put), the more the option is worth.
3. Time to Expiration ($T$): Generally, more time is better. It gives the stock more "room to move." This usually increases the price of both Calls and Puts.
4. Volatility ($\sigma$): This is the "secret sauce." Options love volatility! Because your downside is limited to the premium you paid, but your upside is unlimited, more volatility means a higher chance of a big win. Higher volatility increases both Call and Put prices.
5. Risk-Free Rate ($r$): As interest rates rise, Call prices tend to increase and Put prices decrease. Think of it this way: buying a call is like buying a stock on "layaway"—you pay later, so you save on interest cost.
6. Dividends ($D$): When a company pays a dividend, its stock price drops. This is bad for Calls and good for Puts.

Key Takeaway: Volatility is the only factor that moves the price of both Calls and Puts in the same direction (up!).

2. Upper and Lower Bounds for Option Prices

Options have "boundaries." If the price goes outside these boundaries, an arbitrageur will step in.

Upper Bounds

- Calls: A call option can never be worth more than the stock itself. Why? Because it’s just the right to buy the stock. You wouldn't pay $ \$110 $ for the right to buy something that only costs $ \$100 $.
Formula: $ c \leq S_0 $ and $ C \leq S_0 $

- Puts: A put option can never be worth more than the Strike Price ($K$). The best a Put owner can hope for is that the company goes to zero, allowing them to sell a worthless stock for $K$.
Formula: $ p \leq Ke^{-rT} $ (European) and $ P \leq K $ (American)

Lower Bounds (European Options)

A European call cannot be worth less than the stock price minus the present value of the strike price (or zero, whichever is higher).
- Call Lower Bound: $ c \geq \max(S_0 - Ke^{-rT}, 0) $
- Put Lower Bound: $ p \geq \max(Ke^{-rT} - S_0, 0) $

Common Mistake: Forgetting to discount the strike price ($K$) for European options. Remember, European options can only be exercised at the very end, so we must use the Present Value ($e^{-rT}$).

Key Takeaway: If an option price drops below these lower bounds, it’s "too cheap," and you could make a risk-free profit by buying the option and shorting/buying the underlying stock.

3. Put-Call Parity (For European Options)

This is one of the most important concepts in the FRM! It shows a fixed relationship between the price of a European Call and a European Put with the same strike and expiration.

The Formula: $ c + Ke^{-rT} = p + S_0 $

Wait, what does this mean?
Think of two portfolios:
- Portfolio A (Fiduciary Call): A call option plus enough cash to pay the strike price later.
- Portfolio B (Protective Put): A put option plus one share of the stock.

At expiration, both portfolios will be worth exactly the same: either the stock price (if it's high) or the strike price (if the stock is low). Since they end up with the same value, they must cost the same today!

Analogy: If two different gift boxes contain the exact same $ \$100 $ toy, both boxes must be sold for the same price, regardless of how they are wrapped.

Step-by-Step Trick: If a question asks you to find the "Synthetic Call," just rearrange the formula: $ c = p + S_0 - Ke^{-rT} $.

Key Takeaway: Put-Call Parity only applies to European options. For American options, we use an inequality because of the early exercise feature.

4. Early Exercise: American Options

American options give you the right to exercise at any time. But should you? Most of the time, the answer is "No."

American Calls on Non-Dividend Paying Stocks

You should never exercise an American call early if the stock doesn't pay dividends. Why? Because by exercising, you throw away the "time value" of the option and you pay out cash sooner than you have to. It's always better to sell the option to someone else than to exercise it early.
Rule: $ C = c $ (American Call value equals European Call value if no dividends).

American Puts

Unlike calls, it can be optimal to exercise an American put early. If a company is nearly bankrupt and the stock price is close to zero, there is no room left for the stock to fall further. By exercising now, you get your cash ($K$) immediately and start earning interest on it.
Rule: $ P > p $ (American Puts are usually worth more than European Puts).

Don't worry if this seems tricky: Just remember: Calls like to wait (keep the cash in the bank); Puts might want to exercise early (get the cash into the bank).

Key Takeaway: Dividends are the only reason you might exercise an American Call early (to capture the dividend before the stock price drops).

5. The Impact of Dividends

Dividends are like a "leak" in the stock price. When a dividend is paid, the stock price drops by roughly that amount. This affects our bounds and Put-Call parity.

We represent dividends as the Present Value of Dividends ($D$) during the life of the option. We simply subtract this from the stock price in our formulas:

- Adjusted Call Bound: $ c \geq S_0 - D - Ke^{-rT} $
- Adjusted Put-Call Parity: $ c + D + Ke^{-rT} = p + S_0 $

Quick Review Box:
1. Volatility increases all option prices.
2. Call Upper Bound: Stock price.
3. Put Upper Bound: Strike price.
4. Put-Call Parity: $ c + Ke^{-rT} = p + S_0 $.
5. Early Exercise: Never for American calls (no dividends), but possible for American puts.

Congratulations! You've just mastered the fundamental properties of options. These concepts are the "logic checks" you'll use throughout your FRM journey. Keep practicing the Put-Call Parity rearrangements, and you'll be in great shape!

Quick check

Can you answer these now?

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

How do the stock price ($S_0$) and strike price ($K$) affect the value of call and put options?

For call options, value increases with stock price and decreases with strike price; for put options, value decreases with stock price and increases with strike price.

What is the effect of volatility ($\sigma$) on the price of call and put options?

An increase in volatility increases the price of both call and put options.

How do time to expiration ($T$) and the risk-free rate ($r$) typically affect option prices?

Generally, more time ($T$) increases both call and put prices; a higher risk-free rate ($r$) increases call prices and decreases put prices.

What are the maximum theoretical values (upper bounds) for call and put options?

A call can never be worth more than the stock ($ C \leq S_0 $); an American put cannot exceed the strike ($ P \leq K $); and a European put cannot exceed the present value of the strike ($ p \leq Ke^{-rT} $).

What is the lower bound for the price of a European call option ($c$) on a non-dividend-paying stock?

$ c \geq \max(S_0 - Ke^{-rT}, 0) $

What is the lower bound for the price of a European put option ($p$) on a non-dividend-paying stock?

$ p \geq \max(Ke^{-rT} - S_0, 0) $

What is the formula for Put-Call Parity for European options?

$ c + Ke^{-rT} = p + S_0 $

Is it ever optimal to exercise an American call option on a non-dividend-paying stock before expiration?

No, it is never optimal to exercise such a call early; it is always better to sell the option or hold it.

Under what conditions is it optimal to exercise an American put option early?

It is optimal when the stock price is sufficiently low, as the benefit of receiving the strike price immediately (to earn interest) outweighs the remaining time value of the option.

How do dividends affect Put-Call Parity and option bounds?

Dividends ($D$) reduce the effective stock price, leading to: $ c + D + Ke^{-rT} = p + S_0 $.

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 FRM Part I 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