Understand the principles of option pricing and valuations

Welcome to Option Pricing!

Hello there! Welcome to one of the most exciting parts of the CM2 syllabus. If you've ever wondered how financial experts decide exactly how much an option is worth, you’re in the right place. Pricing options isn't just about guessing; it's about logic, no-arbitrage, and understanding risk. Don't worry if the math looks a bit scary at first—we'll break it down step-by-step so that it makes perfect sense. Let's dive in!

1. The Basics: What Determines an Option's Price?

Before we look at formulas, let's think about common sense. What makes an option more or less expensive? There are five main "ingredients" that cook up an option's price:

• Share Price \( (S_t) \): For a Call option (the right to buy), a higher share price is good news, so the price goes up. For a Put option (the right to sell), it's the opposite.
• Strike Price \( (K) \): For a Call, a higher strike price is bad (you have to pay more to buy the stock), so the option is cheaper. For a Put, a higher strike is great (you can sell for more), so the option is pricier.
• Time to Maturity \( (T) \): Generally, more time means more opportunity for the price to move in your favor. Usually, more time = higher price.
• Volatility \( (\sigma) \): This is the "swinginess" of the stock. Options love volatility! The more the stock moves, the higher the chance it hits a big payoff.
• Risk-free Interest Rate \( (r) \): This is a bit more subtle, but generally, higher rates increase Call prices and decrease Put prices.

Quick Review: The Relationship Table

Call Price increases when: \( S_t \uparrow, \sigma \uparrow, T \uparrow, r \uparrow \) and \( K \downarrow \)
Put Price increases when: \( K \uparrow, \sigma \uparrow, T \uparrow \) and \( S_t \downarrow, r \downarrow \)

2. The Golden Rule: The Principle of No-Arbitrage

In economic modelling, we assume there are "no free lunches." This is called the Principle of No-Arbitrage. It means you cannot make a guaranteed profit without taking any risk or investing any of your own money.

Real-World Analogy: Imagine two shops on the same street. One sells gold for \$100 and the other buys it for \$110. You could buy from one and sell to the other all day for a free profit. In an efficient market, this "gap" disappears instantly. We use this logic to price options: if two different portfolios give the same payoff in the future, they must cost the same today.

3. Understanding Payoffs vs. Profits

It is vital to distinguish between what an option pays out at the end and how much money you actually made.

Payoff: The amount you receive at maturity \( T \).
For a Call: \( \max(S_T - K, 0) \)
For a Put: \( \max(K - S_T, 0) \)

Profit: The Payoff minus the accumulated value of the premium you paid at the start.

Example: You buy a Call for \$5 with a Strike of \$100. At maturity, the stock is \$110. Your payoff is \$10. However, your profit is \$10 minus the \$5 you paid (plus interest).

4. Put-Call Parity: The Most Important Equation

If you remember only one thing from this chapter, let it be Put-Call Parity. This formula links the price of a European Call and a European Put for the same stock, strike, and maturity.

The formula is: \( c_t + Ke^{-r(T-t)} = p_t + S_t \)

Why does this work? Imagine two portfolios:
Portfolio A: A European Call \( (c_t) \) and cash equal to the present value of the strike price \( (Ke^{-r(T-t)}) \).
Portfolio B: A European Put \( (p_t) \) and one share of the stock \( (S_t) \).

At time \( T \), both portfolios will be worth exactly \( \max(S_T, K) \). Since they end up with the same value regardless of what happens to the stock, they must cost the same today!

Memory Aid: "Cats Eat Kibble; People Sell Stocks." (C + K = P + S). It’s a silly way to remember \( c + K = p + S \), but it works!

Key Takeaway:

If you know the price of a Call, you can use Put-Call Parity to find the price of the Put (and vice versa), provided you have the stock price, strike, and interest rate.

5. Upper and Lower Bounds

Options can't just be any price; they have logical "fences" they must stay within. If they cross these fences, an arbitrage opportunity exists.

Upper Bounds

• A Call can never be worth more than the stock itself: \( c \leq S_t \). (Why pay more for the right to buy the stock than the actual stock?)
• A Put can never be worth more than the strike price: \( p \leq Ke^{-r(T-t)} \). (The most you can ever get is \( K \), so the price shouldn't exceed its present value.)

Lower Bounds (for European Options)

• European Call: \( c_t \geq \max(S_t - Ke^{-r(T-t)}, 0) \)
• European Put: \( p_t \geq \max(Ke^{-r(T-t)} - S_t, 0) \)

Common Mistake: Students often forget the present value factor \( e^{-r(T-t)} \). Remember, we are comparing prices today, so future money (the strike price) must be discounted!

6. American vs. European Options

European options can only be exercised at the very end (at time \( T \)).
American options can be exercised at any time up to and including time \( T \).

Did you know? Because an American option gives you more flexibility (the "right" to exercise early), it must be worth at least as much as a European option. It can never be cheaper.

Early Exercise of American Calls

On a non-dividend paying stock, it is never optimal to exercise an American Call option early. Why? Because you lose the "insurance" value of the option and the interest you could earn on the cash you're holding. It is always better to sell the option to someone else than to exercise it early. Therefore, for non-dividend stocks: American Call Value = European Call Value.

Early Exercise of American Puts

Unlike Calls, it can be optimal to exercise an American Put early, especially if the stock price drops to near zero. If the stock is bankrupt, you want your money \( K \) now to start earning interest, rather than waiting until maturity.

7. Summary Checklist

Before moving on to the complex Black-Scholes formulas in the next chapter, make sure you can:

• State the 5 factors affecting option prices and whether they increase or decrease the value.
• Explain the concept of No-Arbitrage.
• Write out the Put-Call Parity formula from memory.
• Explain why an American Call on a non-dividend stock shouldn't be exercised early.
• Identify the upper and lower bounds for Call and Put prices.

Don't worry if this seems tricky at first! Option pricing is a different way of thinking. Once you realize it's all about comparing "bundles" of assets to ensure no one gets a free lunch, the pieces will start to fall into place. You've got this!