The binomial option pricing model

Q: In the replicating portfolio method, what formula determines the number of shares ($\Delta$) needed to replicate an option's payoff?

$\Delta = \frac{V_u - V_d}{S_u - S_d} = \frac{V_u - V_d}{S(u - d)}$

Q: How is the bond investment $B$ calculated in a one-period replicating portfolio for a non-dividend paying stock?

$B = e^{-rh} \frac{u V_d - d V_u}{u - d}$

Q: What is the general one-period pricing formula for a derivative $V$ under the risk-neutral probability approach?

$V = e^{-rh} [p^* V_u + (1 - p^*) V_d]$

Q: How is backward induction applied in a multi-period recombining binomial tree?

Payoffs are calculated at the terminal nodes at expiration and then discounted step-by-step back to time 0 using $p^*$ and the discount factor $e^{-rh}$.

Q: How do the replicating portfolio components $\Delta$ and $B$ differ between a European call and a European put?

A call has $\Delta \ge 0$ (long stock) and $B \le 0$ (borrowing), whereas a put has $\Delta \le 0$ (short stock) and $B \ge 0$ (lending).

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 6 min read · Updated: 8 Nov 2026

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 European-style options on a single non-dividend paying asset by assuming the world is a bit simpler than it actually is. Instead of infinite possibilities over a period, we assume the asset 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 is 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 of length $h$, 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" (with $u > 1$).
• $d$: The "down factor" (with $0 < d < 1$).
• $r$: The continuously compounded risk-free interest rate.
• $h$: The length of the time step in years (e.g., if a year is 1 and we have 2 steps, $h = 0.5$).

No-Arbitrage Condition: For the model to make sense and avoid riskless arbitrage ("free money"), the risk-free return must sit strictly between the down move and the up move. For a non-dividend paying asset, this means: $d < e^{rh} < u$.

Summary of the Setup

The binomial model simplifies asset price movements into a "fork in the road." By knowing where the underlying asset can end up and calculating the option's payoff at each outcome, we can work backward to determine the fair price of the option today.

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

Here is the foundational principle of derivative pricing: If two portfolios produce the exact same payoff at a future date across all possible states, they must have the exact same price today.

We replicate the option's payoff by creating a portfolio consisting of $\Delta$ shares of the underlying stock and an investment of $B$ in a risk-free zero-coupon bond. We set this portfolio value ($\Delta S + B$) to equal the option's value whether the stock goes up or down.

The Formulas (Non-Dividend Paying Asset):
To find the number of shares ($\Delta$):
$\Delta = \frac{V_u - V_d}{S_u - S_d} = \frac{V_u - V_d}{S(u - d)}$

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

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

Analogy: Think of $\Delta$ as a "sensitivity factor." It tells you how much the option payoff changes relative to a change in the stock price. For example, if the stock spread $S_u - S_d$ is \$1 and the option spread $C_u - C_d$ is \$0.50, your $\Delta$ is 0.50.

3. Method 2: Risk-Neutral Pricing (The Expected Present Value Way)

On Exam FAM, the risk-neutral expected present value approach is the most common method. We price derivatives as if investors are risk-neutral, meaning all assets grow at the risk-free rate $r$.

Step 1: Calculate the Risk-Neutral Probability ($p^*$)
Under risk neutrality, the expected return on the stock equals the risk-free rate: $S e^{rh} = p^* S_u + (1 - p^*) S_d$. Solving for $p^*$ gives:
$p^* = \frac{e^{rh} - d}{u - d}$

Step 2: Calculate the Option Value
Now, calculate the expected payoff under $p^*$ and discount it back to the present at the risk-free rate:
$V = e^{-rh} [p^* V_u + (1 - p^*) V_d]$

Quick Review:
• $V_u$ is the option payoff if the stock goes up (e.g., $\max(0, S_u - K)$ for a call, or $\max(0, K - S_u)$ for a put).
• $V_d$ is the option payoff if the stock goes down.
• Always include the discount factor $e^{-rh}$! Leaving this out is a very common exam mistake.

4. Multi-Period Trees: Moving Backward Step-by-Step

What if the option expires in one year and we model it with two 6-month steps ($h = 0.5$)? We extend the tree into a two-period model. In the standard recombining binomial tree, an up move followed by a down move gives the same stock price as a down move followed by an up move ($S_{ud} = S_{du} = S \cdot u \cdot d$).

The Working Backward Process:
1. Terminal Nodes: Calculate the stock prices at expiration (time $2h$): $S_{uu} = S u^2$, $S_{ud} = S u d$, and $S_{dd} = S d^2$. Determine the payoff of the derivative at each final node.
2. Step Back to Time $h$: Use the risk-neutral formula to discount each pair of terminal values back to the intermediate nodes ($V_u$ and $V_d$):
$V_u = e^{-rh} [p^* V_{uu} + (1 - p^*) V_{ud}]$
$V_d = e^{-rh} [p^* V_{ud} + (1 - p^*) V_{dd}]$
3. Step Back to Time 0: Discount $V_u$ and $V_d$ back to today to find the initial option price $V$:
$V = e^{-rh} [p^* V_u + (1 - p^*) V_d]$

Key Takeaway

No matter how many steps are in the tree, the process is always identical: calculate payoffs at the final leaves and "pull" the values back to node 0 using $p^*$ and the discount factor $e^{-rh}$.

5. Valuing Puts and Simple European-Style Derivatives

The binomial tree is universal: it works for any European-style derivative whose payoffs depend on the underlying stock price at expiration. On Exam FAM, all derivatives in scope are European-style on a single non-dividend paying asset.

European Call Option: Payoff is $C_T = \max(0, S_T - K)$. Note that $\Delta = \frac{C_u - C_d}{S_u - S_d} \ge 0$, meaning a call requires a long position in shares ($\Delta > 0$) and borrowing money ($B < 0$).

European Put Option: Payoff is $P_T = \max(0, K - S_T)$. For a put, $P_u \le P_d$ because the put is worth more when the stock price falls. Consequently:
$\Delta = \frac{P_u - P_d}{S_u - S_d} \le 0$
This means replicating a put option requires shorting shares ($\Delta < 0$) and lending money in bonds ($B > 0$).

Put-Call Parity Check: For European options with strike $K$ and time to expiry $T$, the prices obtained from the binomial model must satisfy put-call parity at every node:
$C - P = S - K e^{-rT}$

6. Summary and Tips for Success

Memory Aid for $p^*$:
Think "Growth factor minus Down over Up minus Down":
$p^* = \frac{e^{rh} - d}{u - d}$
The numerator is the growth factor $e^{rh}$ minus the down factor $d$; the denominator is the spread $u - d$.

Quick Checklist for Problems:
• Did you calculate the correct time step $h$? If expiry is $T$ and there are $n$ steps, $h = T / n$.
• Did you multiply the annual rate $r$ by $h$ in $e^{rh}$ and $e^{-rh}$?
• Are you calculating a call ($\max(0, S - K)$) or a put ($\max(0, K - S)$)?
• Did you discount each expected payoff back one period using $e^{-rh}$?

Final Encouragement: The binomial model is structured and consistent. Once you are comfortable setting up the nodes and rolling values backward with $p^*$, you will find these are reliable points on Exam FAM. Keep practicing those trees!

References

Fundamentals of Actuarial Mathematics Exam—October 2026 — Society of Actuaries
Exam FAM Details — Society of Actuaries

Quick check

Can you answer these now?

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

What is the no-arbitrage condition for the single-period binomial model for a non-dividend paying stock?

The no-arbitrage condition is $d < e^{rh} < u$.

In the replicating portfolio method, what formula determines the number of shares ($\Delta$) needed to replicate an option's payoff?

$\Delta = \frac{V_u - V_d}{S_u - S_d} = \frac{V_u - V_d}{S(u - d)}$

How is the bond investment $B$ calculated in a one-period replicating portfolio for a non-dividend paying stock?

$B = e^{-rh} \frac{u V_d - d V_u}{u - d}$

What is the formula for the risk-neutral probability of an upward move ($p^*$) in a single-period tree for a non-dividend paying stock?

$p^* = \frac{e^{rh} - d}{u - d}$

What is the general one-period pricing formula for a derivative $V$ under the risk-neutral probability approach?

$V = e^{-rh} [p^* V_u + (1 - p^*) V_d]$

How is backward induction applied in a multi-period recombining binomial tree?

Payoffs are calculated at the terminal nodes at expiration and then discounted step-by-step back to time 0 using $p^*$ and the discount factor $e^{-rh}$.

How do the replicating portfolio components $\Delta$ and $B$ differ between a European call and a European put?

A call has $\Delta \ge 0$ (long stock) and $B \le 0$ (borrowing), whereas a put has $\Delta \le 0$ (short stock) and $B \ge 0$ (lending).

What parity relationship must hold at any node for European call and put options with the same strike and time to expiration?

$C - P = S - K e^{-rT}$

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