Multi-Period Binomial Trees for American Options

Introduction

Have you ever wondered how to handle an option that you can exercise at any point up until expiration? That’s precisely the challenge with American-style options. While European-style options permit exercise only at expiration, American-style options add a little twist (and a bit of complexity) because early exercise might be optimal under certain circumstances. If you’ve ever thought, “Well, how do we account for that added flexibility?” you’re in the right place.

Let’s start by recalling that earlier in this chapter, you learned how to price options using a simple one-period binomial model (refer back to Section 10.3 for a refresher). The multi-period binomial approach just extends that idea over multiple steps. But with American-style options, we have to incorporate the possibility of early exercise at each node in that tree. In other words, at each step, we check whether the intrinsic value of immediately exercising actually beats the “hold and continue” value. That’s the gist of it, but let’s dig in deeper.

Core Concepts and Recap

American-Style Versus European-Style

• European-Style: Exercise allowed only at maturity.
• American-Style: Exercise allowed any time up to (and including) the expiration date.

For American options, you can exercise when it’s profitable to do so, perhaps if there’s a large dividend that’s about to be paid on the underlying security and you hold a call, or if you have a put and the underlying’s price takes a steep nosedive earlier than your expiration date. The presence of that early exercise right creates an “early exercise premium,” which is extra value beyond what a purely European option might have under the same parameters.

Multi-Period Binomial Trees

You can think of a multi-period binomial tree as an iterative extension of the single up/down step approach from the one-period binomial model. Over multiple steps:

1. The underlying asset price can go up or down at each discrete interval.
2. These movements typically “recombine” so that an up-move followed by a down-move ends up at the same price node as a down-move followed by an up-move.
3. At each node (date), the option value is the discounted expected value under the risk-neutral measure—or the intrinsic value if early exercise is more advantageous.

By the final step, we arrive at a tree of possible asset prices as well as a corresponding tree of option payoffs. Then, we use backward induction to discount these payoffs and check for early exercise possibilities.

Building the Multi-Period Binomial Tree

Let’s get a little more practical. Suppose you want to model a three-period binomial tree, each period being, say, one month long (so total life is three months). You’ll specify:

• The current price of the underlying (S₀).
• The up factor (u) and down factor (d).
• The risk-free rate (r) for each period.
• Possibly a dividend yield (q) if the underlying is a dividend-paying stock.

Example Setup

Imagine the stock is at $100 today. For each one-month step:

• We let u = 1.05 (meaning a 5% up-move),
• and we let d = 0.95 (meaning a 5% down-move),
• The monthly risk-free rate might be 1%.

So, if the stock goes up during the first period, at the next node it’s at $105. If it goes down, it’s at $95. Then from $105, it can go up to $110.25 or down to $99.75, and so on.

Let’s illustrate a two-period version of this tree for simplicity. A three-period tree is similar—just one more step.

Below is a sample Mermaid diagram showing a two-period binomial price tree for the underlying:

    graph LR
	    A["S₀ = 100"] --> B["S₀u = 105"]
	    A["S₀ = 100"] --> C["S₀d = 95"]
	    B["S₀u = 105"] --> D["S₀uu = 110.25"]
	    B["S₀u = 105"] --> E["S₀ud = 99.75"]
	    C["S₀d = 95"] --> F["S₀du = 99.75"]
	    C["S₀d = 95"] --> G["S₀dd = 90.25"]

Notice how the middle nodes E and F both end up at the same price 99.75. That’s the “recombining” part.

Discounting and Risk-Neutral Valuation

Within each step, we typically assume risk-neutral probabilities because that approach (developed by Cox, Ross, and Rubinstein in their 1979 paper) simplifies pricing. The idea is that, under the risk-neutral measure, the expected growth rate of the asset is the risk-free rate.

If we let Δt be the length of each period, the risk-neutral probability p is usually:

where:

• \( r \) is the risk-free rate,
• \( q \) is the continuous dividend yield (if any),
• \( u \) is the up-factor,
• \( d \) is the down-factor,
• \( \Delta t \) is the length of each period in years.

If you’re ignoring dividends, you’d just set \( q = 0 \). Then, the expected (risk-neutral) value of the option at each node is:

where \( V_\text{up} \) is the option payoff in the up state at the next step, and \( V_\text{down} \) is the payoff in the down state.

American Option Payoff with Early Exercise Check

Here’s the fundamental difference for an American-style option: After you compute that “hold” value from the risk-neutral approach, you must compare it to the immediate exercise payoff (a.k.a. intrinsic value). The node’s final value is:

For a call, the intrinsic value at any point is \(\max(S - K, 0)\), where S is the underlying price at that node, and K is the strike price. For a put, it’s \(\max(K - S, 0)\).

Step-by-Step

1. Construct the Tree for the Underlying
   Start with the current price \( S_0 \). Multiply by \( u \) or \( d \) to get the next-step prices. Continue outward until you reach the final step (expiration).

2. Compute Payoffs at Expiration
   For a call, payoff at final nodes = \(\max(S_T - K, 0)\). For a put, payoff at final nodes = \(\max(K - S_T, 0)\).

3. Work Backwards
   For each node in the penultimate step (and all prior steps), compute the “hold” value by discounting the next-step’s up/down payoffs with the risk-neutral probabilities.

4. Compare With Intrinsic Value
   At each node, set node value = \(\max(\text{Hold}, \text{Intrinsic})\).

5. Arrive at the Present
   The value at \( S_0 \) (the root node) is the price of the American-style option.

Practical Illustrations

Example: American Put on a Dividend-Paying Stock

Imagine you hold an American put on a stock that pays a significant dividend in the middle of the option’s life. The presence of that dividend can cause the stock price to drop ex-dividend, which might trigger an earlier optimal exercise for the put. Let’s do a quick numeric illustration—just at a high level.

• Current stock price: $50
• Strike: $52
• Dividend: $1 (expected halfway to expiration)
• Up-factor: u = 1.05, down-factor: d = 0.95, risk-free rate is 2% per period.

In the node just before the dividend is paid, if you anticipate the stock’s price will drop by $1 on the ex-dividend date, you might find that the immediate exercise (i.e., collect $52 - stock price right now) is more profitable than waiting. This is especially true if the probability-weighted payoff after the price drop is smaller. The multi-period binomial tree helps you see precisely at which node early exercise becomes optimal.

Case Study: Calls on High Dividend-Paying Stocks

If you have a call on a high dividend-paying stock, typically you might consider early exercise just before a big dividend if the call’s time value is less than the dividend you would gain by owning the stock. Using the binomial tree, at the node immediately preceding the ex-dividend date, you compare the “hold” value to the value if you exercise right away (which means you’d own the stock and capture the dividend). If that dividend is large enough, the early exercise might be worthwhile.

Diagram: Two-Period American Put Valuation

Let’s illustrate a simple two-period American put tree with some actual numbers. Assume:

• \( S_0 = 100 \)
• \( K = 100 \)
• \( u = 1.10 \), \( d = 0.90 \)
• \( r = 5% \) per period (assume discrete compounding for simplicity)

We’ll ignore dividends here, but the process is the same if a dividend is present (just adjust your risk-neutral probability and be extra vigilant about the ex-dividend node).

1. Compute risk-neutral probability:
   

   $$ p = \frac{(1+r) - d}{u - d} = \frac{1.05 - 0.90}{1.10 - 0.90} = \frac{0.15}{0.20} = 0.75 $$

2. Stock price evolution over two periods:

    graph LR
	    A["100"] --> B["110 (up)"]
	    A["100"] --> C["90 (down)"]
	    B["110"] --> D["121 (uu)"]
	    B["110"] --> E["99 (ud)"]
	    C["90"] --> F["99 (du)"]
	    C["90"] --> G["81 (dd)"]

3. At expiration (the second period), your put payoff is \(\max(K - S_2, 0)\):

   • Node D (S=121): \(\max(100 - 121,0) = 0\)
   • Node E (S=99): \(\max(100 - 99,0) = 1\)
   • Node F (S=99): \(\max(100 - 99,0) = 1\)
   • Node G (S=81): \(\max(100 - 81,0) = 19\)

4. Now step back to the first period:

   • At B (S=110), the “hold” value is the discounted expected payoff:
     
     $$ V_B = \frac{1}{1+r} \Bigl[p \cdot V_D + (1-p) \cdot V_E \Bigr] = \frac{1}{1.05} \Bigl[0.75 \times 0 + 0.25 \times 1 \Bigr] = \frac{0.25}{1.05} \approx 0.238 $$
     
     The intrinsic value at B is \(\max(K - S, 0) = \max(100 - 110,0) = 0\).
     So, \(\max(0.238, 0) = 0.238\). So no early exercise (it’s worthless to exercise the put at 110).
   • At C (S=90), the “hold” value is:
     
     $$ V_C = \frac{1}{1.05} \Bigl[0.75 \times V_F + 0.25 \times V_G \Bigr] = \frac{1}{1.05} \Bigl[0.75 \times 1 + 0.25 \times 19 \Bigr] = \frac{1}{1.05} \Bigl[0.75 + 4.75 \Bigr] = \frac{5.50}{1.05} \approx 5.238 $$
     
     The intrinsic value at C is \(\max(100 - 90, 0) = 10\).
     Compare 10 with 5.238 → The maximum is 10. That means it’s better to exercise at node C immediately rather than hold.

5. Finally, compute at node A (the root, S=100).

   • The “hold” value if you do not exercise:
     
     $$ V_A = \frac{1}{1.05} \Bigl[0.75 \times 0.238 + 0.25 \times 5.238 \Bigr] = \frac{1}{1.05} \Bigl[0.1785 + 1.3095 \Bigr] = \frac{1.488}{1.05} \approx 1.417 $$
     
     The intrinsic value at A is \(\max(100 - 100, 0) = 0\).
     So the option is worth \(\max(1.417,0)=1.417\) at the root.

Hence, the two-period American put is valued at around $1.417. Notice how we had an early exercise at node C, because the intrinsic value (10) was greater than the hold value (5.238). That’s the key difference from a European option.

Implementation Note (Optional)

If you’re comfortable with Python, here’s a small snippet that outlines how you might compute a multi-period binomial tree for an American put:

 1import math
 2
 3def american_put_binomial(S0, K, r, u, d, periods):
 4    # Risk-neutral probability
 5    p = (1 + r - d) / (u - d)
 6    discount = 1 / (1 + r)
 7
 8    # Initialize asset prices at maturity
 9    stock_prices = [S0 * (u**(periods - i)) * (d**i) for i in range(periods + 1)]
10    
11    # Option payoffs at maturity
12    option_values = [max(K - price, 0) for price in stock_prices]
13
14    # Step backward
15    for step in range(periods - 1, -1, -1):
16        for i in range(step + 1):
17            hold = discount * (p * option_values[i] + (1 - p) * option_values[i + 1])
18            current_price = S0 * (u**(step - i)) * (d**i)
19            exercise = max(K - current_price, 0)
20            option_values[i] = max(hold, exercise)
21    
22    return option_values[0]
23
24value = american_put_binomial(S0=100, K=100, r=0.05, u=1.10, d=0.90, periods=2)
25print("Two-period American put value:", value)

This snippet is just a demonstration; you can adapt it for continuous compounding, incorporate dividends, or refine the numeric approach.

Common Pitfalls and Best Practices

• Overlooking Dividends: If your underlying pays dividends, be sure to reflect them either by adjusting the risk-neutral probability or by explicitly adjusting the stock price after ex-dividend.
• Ignoring Early Exercise Potential: Some folks forget to compare intrinsic value vs. hold value at each node. That might significantly misprice an American put or call.
• Mis-Specifying u and d: Make sure your up and down factors align with the volatility you expect for the underlying and the time step length.
• Too Few Steps: A binomial approach gets more accurate as the number of steps grows, but more steps also increase computational complexity. There’s a trade-off.

Relationship to Other Topics

Recall back in Section 10.1, we discussed the main determinants of option value (underlying price, strike, volatility, etc.). In multi-period binomial models, those same factors come into play but in a more granular way. Also, connecting to “4.5 Option Strategies and Payoff Diagrams,” you could use the binomial approach to track the payoff paths of advanced strategies (like straddles or collars) if you break them down into calls and puts.

Exam Tips for CFA Candidates

1. Show Your Work: In a constructed-response question, lay out the binomial tree, each node’s payoff, discounting, and your final comparison (intrinsic vs. hold).
2. Understand Early Exercise Logic: This is a high-yield exam point—particularly for puts on dividend-paying stocks. They love to ask: “Why might an early exercise be optimal here?”
3. Time Management: Binomial trees can be time-consuming in an exam setting. Practice sketching tree steps quickly and systematically.
4. Be Comfortable with Simplifications: The exam might give you easy up/down factors (like ±10%) and a nice round risk-free rate. That’s to ensure your arithmetic is straightforward—focus on concept mastery.
5. Tie to Risk Management: If you recall from Chapter 6’s discussion on derivatives usage, being able to precisely price American options helps in hedging strategies.

References

• Cox, J., Ross, S., & Rubinstein, M. (1979). “Option Pricing: A Simplified Approach.” Journal of Financial Economics.
• Hull, J. C. (various editions). “Options, Futures, and Other Derivatives.”
• CFA Institute Level I Curriculum, “Binomial Trees and American-Style Options” sections.

Test Your Knowledge: Multi-Period Binomial Trees for American Options