Valuing a Derivative Using a One-Period Binomial Model

Welcome to the World of Binomial Pricing!

Hello there! Today, we are going to dive into one of the most elegant and useful tools in the CFA curriculum: the One-Period Binomial Model. If you have ever wondered how experts figure out exactly what a call or put option is worth, you are in the right place. Don't worry if math isn't your favorite subject—we are going to break this down into simple, logical steps that anyone can follow. By the end of this, you’ll see that valuing a derivative is really just a game of "filling in the blanks" to ensure no one gets "free money" (arbitrage) in the market.

1. The Core Idea: What is a Binomial Model?

The word "Binomial" sounds fancy, but it just means "two names" or "two outcomes." In this model, we simplify the world. We assume that over one period (like a month or a year), the price of a stock can only do two things: go up or go down.

The Components:
• S: The current price of the stock.
• u: The "up factor." If the stock goes up, the new price is \(S \times u\).
• d: The "down factor." If the stock goes down, the new price is \(S \times d\).
• r: The risk-free rate (the interest you’d earn on a safe bond).

Analogy: Imagine a fork in the road. You start at $100. The path to the left leads to $110 (up), and the path to the right leads to $90 (down). The Binomial Model helps us figure out what an option on that road trip is worth today.

Key Takeaway: The binomial model assumes the future has only two possible states, allowing us to calculate a precise value for a derivative today based on those future outcomes.

2. The No-Arbitrage Principle

Before we look at the formulas, we need to understand the "Golden Rule" of CFA Derivatives: No Arbitrage. This is the idea that two investments with the exact same future payoffs must have the exact same price today. If they didn't, traders would buy the cheap one and sell the expensive one to make a "risk-free profit." We use this rule to build a "synthetic" version of an option using stocks and bonds to find the option's fair price.

3. Step-by-Step: Risk-Neutral Pricing

The most common way to use the binomial model is called Risk-Neutral Pricing. This is a clever mathematical trick. We don't need to know how likely it is that the stock will go up. Instead, we calculate a "pretend" probability that makes everyone indifferent to risk.

Step 1: Calculate the Risk-Neutral Probability (\(\pi\))

This is the "probability" of an upward move in a risk-neutral world. The formula is:
\n\(\pi = \frac{(1 + r) - d}{u - d}\)

Step 2: Determine the Payoffs at Maturity

You need to know what the option is worth at the end of the period in both scenarios:
\n• \(f_u\): The value of the derivative if the stock goes up.
\n• \(f_d\): The value of the derivative if the stock goes down.

Quick Review: For a Call option with exercise price \(X\), the payoff is \(Max(0, S - X)\). For a Put option, it is \(Max(0, X - S)\).

Step 3: Calculate the Expected Value and Discount It

Now, we find the weighted average of those two payoffs and pull them back to "today's dollars" using the risk-free rate:
\n\(f = \frac{(\pi \times f_u) + ((1 - \pi) \times f_d)}{1 + r}\)

Key Takeaway: The value of a derivative today is the discounted expected value of its future payoffs, using risk-neutral probabilities.

4. A Real-World Example

Let’s put this into practice! Imagine a stock is currently trading at $100 (\(S\)). In one year, it will either go up to $120 (\(u = 1.2\)) or down to $80 (\(d = 0.8\)). The risk-free rate is 5% (\(r = 0.05\)). What is the value of a Call option with an exercise price (\(X\)) of $100?

1. Find the payoffs:
\nIf the stock goes to $120, the call is worth $20 (\(120 - 100\)). So, \(f_u = 20\).
\nIf the stock goes to $80, the call is worth $0 (you wouldn't exercise it). So, \(f_d = 0\).

2. Find the risk-neutral probability (\(\pi\)):
\n\(\pi = \frac{1.05 - 0.8}{1.2 - 0.8} = \frac{0.25}{0.40} = 0.625\)

3. Value the Call:
\n\(f = \frac{(0.625 \times 20) + (0.375 \times 0)}{1.05}\)
\n\(f = \frac{12.5}{1.05} = 11.90\)
\nThe Call option is worth $11.90 today.

Did you know? Notice we never asked how likely the stock actually is to go up. The math works regardless of the real-world probability!

5. The Hedge Ratio (Delta)

The Hedge Ratio (often called Delta) tells us how many shares of stock we need to buy to perfectly mimic the payoff of the option. This is the "replication" part of the model.

The Formula:
\(\Delta = \frac{f_u - f_d}{S_u - S_d}\)

In our example above:
\(\Delta = \frac{20 - 0}{120 - 80} = \frac{20}{40} = 0.5\)

This means that buying 0.5 shares of stock will move in price similarly to 1 call option for small changes in the stock price. If you are a dealer selling a call option, you would buy 0.5 shares to "hedge" your risk.

Key Takeaway: Delta represents the change in the derivative's price for a $1 change in the underlying stock's price.

6. Common Pitfalls to Avoid

• Mixing up u and d: Always remember that \(u\) must be greater than \((1 + r)\) and \(d\) must be less than \((1 + r)\). If they aren't, the math breaks because there would be an arbitrage opportunity.
• Forgetting to discount: Students often calculate the expected payoff but forget to divide by \((1 + r)\) at the very end. Don't leave money in the future—bring it back to today!
• Probability confusion: Remember that \(\pi\) is the probability of the "up" move. The probability of the "down" move is always \(1 - \pi\).
• Put vs. Call payoffs: Double-check if the question asks for a Put or a Call. For a Put, \(f_d\) is often the one with the value, while \(f_u\) might be zero.

7. Summary Checklist

Before you move on to the next chapter, make sure you can:
1. Identify \(S\), \(u\), \(d\), and \(r\) from a word problem.
2. Calculate the "up" and "down" stock prices.
3. Calculate the "up" and "down" derivative payoffs (\(f_u\) and \(f_d\)).
4. Calculate the risk-neutral probability (\(\pi\)).
5. Solve for the current value of the derivative (\(f\)).
6. Calculate the Hedge Ratio (\(\Delta\)).

Don't worry if this seems tricky at first! The binomial model is like a puzzle. Once you place the first piece (calculating \(\pi\)), the rest of the picture starts to fill itself in. Keep practicing these steps, and you'll be a pricing pro in no time!