The binomial option pricing model

Q: What is the fundamental assumption of the binomial option pricing model regarding stock price movements?

It assumes that over a discrete time step, the stock price can only move in two directions: Up (to $S_u = S \times u$) or Down (to $S_d = S \times d$).

Q: What condition must be met by the factors $u$, $d$, and the interest rate to avoid arbitrage?

The risk-free growth must sit between the down move and the up move: \(d

Q: What is the core principle behind the Replicating Portfolio method for pricing options?

It is based on the Law of One Price: If a portfolio of stock ($\Delta$) and bonds ($B$) has the exact same payoff as an option in every state, the option's price today must equal the cost of that portfolio ($\Delta S + B$).

Q: In the replicating portfolio method, what is the formula to find the number of shares ($\Delta$)?

$\Delta = e^{-\delta h} \frac{C_u - C_d}{S_u - S_d}$

Q: In the replicating portfolio method, what is the formula to find the bond component ($B$)?

$B = e^{-rh} \frac{u C_d - d C_u}{u - d}$

Q: How is the risk-neutral probability of an upward move ($p^*$) calculated?

$p^* = \frac{e^{(r-\delta)h} - d}{u - d}$

Q: What is the formula for calculating the option price today ($C$) using risk-neutral pricing?

$C = e^{-rh} [p^* C_u + (1 - p^*) C_d]$

Q: What is a "recombining tree" in the context of multi-period binomial models?

It is a tree where an "Up-Down" move results in the same stock price as a "Down-Up" move ($S_{ud} = S_{du}$).

Q: What is the general process for valuing an option in a multi-period tree?

Calculate stock prices at every node, find option payoffs at the final nodes (expiration), and then work backward step-by-step to Time 0 using risk-neutral pricing.

Q: How is the value at a node determined for an American option in a binomial tree?

$\text{Value at Node} = Max(\text{Exercise Value}, \text{Value of Waiting})$

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

Welcome to the World of Binomial Trees!

Hello there! If you are preparing for Exam FAM, you have likely realized that valuing options can feel a bit like predicting the future. While we don't have a crystal ball, we have something even better: The Binomial Option Pricing Model. In this chapter, we are going to learn how to price options by assuming the world is a bit simpler than it actually is. Instead of infinite possibilities, we assume a stock price can only go in two directions: Up or Down. Don't worry if this seems a bit abstract at first; once you see the patterns, it’s as simple as following a map!

1. The Core Idea: One Step at a Time

The binomial model is a "discrete-time" model. This means we look at the life of an option in specific chunks of time (steps). In the simplest version, the Single-Period Model, we look at the price today and the price at expiration.

Imagine a stock price $S$. After one period, only two things can happen:
1. The price goes Up to $S_u$ (where $S_u = S \times u$)
2. The price goes Down to $S_d$ (where $S_d = S \times d$)

Key Terms to Know:
• $u$: The "up factor" (usually greater than 1).
• $d$: The "down factor" (usually less than 1).
• $r$: The continuously compounded risk-free interest rate.
• $\delta$: The continuously compounded dividend yield (if any).
• $h$: The length of the time step (e.g., if a year is 1 and we have 2 steps, $h = 0.5$).

Did you know? For the model to make sense and avoid "free money" (arbitrage) opportunities, the risk-free growth must sit between the down move and the up move. Mathematically: $d < e^{(r-\delta)h} < u$.

Summary of the Setup

The binomial model simplifies price movements into a "fork in the road." By knowing where the stock could end up, we can work backward to find out what the option should cost today.

2. Method 1: The Replicating Portfolio ($\Delta$ and $B$)

This is often the trickiest part for students, but here is the secret: If two things have the exact same payoff in the future, they must have the same price today.

We create a "fake" option by buying some shares of stock (called $\Delta$) and borrowing or lending some money (called $B$). We want this portfolio ($\Delta S + B$) to equal the option's value whether the stock goes up or down.

The Formulas:
To find the number of shares ($\Delta$):
$\Delta = e^{-\delta h} \frac{C_u - C_d}{S_u - S_d}$

To find the amount in the bond ($B$):
$B = e^{-rh} \frac{u C_d - d C_u}{u - d}$

The Option Price: $C = \Delta S + B$

Analogy: Think of $\Delta$ as a "shock absorber." It tells you how much the option price changes relative to the stock price. If the stock goes up by $1 and the option goes up by $0.50, your $\Delta$ is 0.50.

3. Method 2: Risk-Neutral Pricing (The "Expectation" Way)

Most students prefer this method because it feels more like a standard probability calculation. We pretend that investors don't care about risk. In this "Risk-Neutral" world, the expected return on a stock is just the risk-free rate.

Step 1: Calculate the Risk-Neutral Probability ($p^*$)
This isn't the real probability that the stock goes up; it's a mathematical tool.
$p^* = \frac{e^{(r-\delta)h} - d}{u - d}$

Step 2: Calculate the Option Value
Now, we find the "average" payoff and discount it back to today:
$C = e^{-rh} [p^* C_u + (1 - p^*) C_d]$

Quick Review:
• $C_u$ is the payoff if the stock goes up (e.g., $Max(0, S_u - K)$ for a call).
• $C_d$ is the payoff if the stock goes down.
• Don't forget the discount factor $e^{-rh}$! Leaving this out is a very common mistake.

4. Multi-Period Trees: Thinking Ahead

What if the option lasts for two months and we have two one-month steps? We just extend the tree! A two-period tree looks like a web. The stock can go Up-Up, Up-Down, Down-Up, or Down-Down.

Important Note: In the standard models used in FAM, an "Up-Down" move results in the same price as a "Down-Up" move. This is called a recombining tree. It makes our lives easier because we have fewer nodes to calculate!

The Working Backward Process:
1. Start at the very end (the final nodes) and calculate the option payoffs.
2. Move back one step. Use the Risk-Neutral formula to find the values at the middle nodes.
3. Move back again until you reach "Time 0" (today).

Key Takeaway

No matter how many steps are in the tree, the process is the same: calculate the ends and "pull" the values back to the start using $p^*$.

5. American Options vs. European Options

Everything we discussed above works perfectly for European Options (which can only be exercised at the very end). However, American Options can be exercised at any time.

When you are working backward through a tree for an American option, at every single node, you must ask yourself:
"Is it better to exercise this option right now, or hold onto it?"

The Decision Rule:
Value at Node = $Max(\text{Exercise Value}, \text{Value of Waiting})$

The "Value of Waiting" (also called the continuation value) is just the discounted expected value we calculated using $p^*$ in the previous section.

Common Pitfall: Students often forget to check for early exercise on American Puts. While it is rare to exercise an American Call early (if there are no dividends), American Puts are frequently exercised early if the stock price drops significantly!

6. Summary and Tips for Success

Memory Aid for $p^*$:
Think "Forward minus Down over Up minus Down." The numerator is the forward price growth $e^{(r-\delta)h}$ minus the down factor $d$. The denominator is just the spread between the two factors $u - d$.

Quick Checklist for Problems:
• Is the interest rate $r$ annual? If so, did you multiply it by $h$ in the exponent?
• Are there dividends ($\delta$)? Remember they reduce the growth of the stock price in the $p^*$ formula.
• Is it an American option? If yes, check every node for early exercise!
• Did you use the correct strike price $K$?

Final Encouragement: The binomial model is just a logic puzzle. Once you get comfortable drawing the tree and moving backward, you'll find these are some of the most "gettable" points on Exam FAM. Keep practicing those trees!

Quick check

Can you answer these now?

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

What is the fundamental assumption of the binomial option pricing model regarding stock price movements?

It assumes that over a discrete time step, the stock price can only move in two directions: Up (to $S_u = S \times u$) or Down (to $S_d = S \times d$).

What condition must be met by the factors $u$, $d$, and the interest rate to avoid arbitrage?

The risk-free growth must sit between the down move and the up move: $d < e^{(r-\delta)h} < u$.

What is the core principle behind the Replicating Portfolio method for pricing options?

It is based on the Law of One Price: If a portfolio of stock ($\Delta$) and bonds ($B$) has the exact same payoff as an option in every state, the option's price today must equal the cost of that portfolio ($\Delta S + B$).

In the replicating portfolio method, what is the formula to find the number of shares ($\Delta$)?

$\Delta = e^{-\delta h} \frac{C_u - C_d}{S_u - S_d}$

In the replicating portfolio method, what is the formula to find the bond component ($B$)?

$B = e^{-rh} \frac{u C_d - d C_u}{u - d}$

How is the risk-neutral probability of an upward move ($p^*$) calculated?

$p^* = \frac{e^{(r-\delta)h} - d}{u - d}$

What is the formula for calculating the option price today ($C$) using risk-neutral pricing?

$C = e^{-rh} [p^* C_u + (1 - p^*) C_d]$

What is a "recombining tree" in the context of multi-period binomial models?

It is a tree where an "Up-Down" move results in the same stock price as a "Down-Up" move ($S_{ud} = S_{du}$).

What is the general process for valuing an option in a multi-period tree?

Calculate stock prices at every node, find option payoffs at the final nodes (expiration), and then work backward step-by-step to Time 0 using risk-neutral pricing.

How is the value at a node determined for an American option in a binomial tree?

$\text{Value at Node} = Max(\text{Exercise Value}, \text{Value of Waiting})$

When is early exercise most commonly considered for American Puts and Calls?

American Puts are often exercised early if the stock price drops significantly; American Calls are typically only exercised early if there are dividends paid on the underlying stock.

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