## **Learning Outcomes**

This article explains how to use binomial trees and risk-neutral pricing to value options in a CFA Level 2 context. It focuses on translating theoretical pricing ideas into the exact calculations and interpretations commonly tested in item sets. It shows how to construct single- and multi-period binomial trees, specify up and down factors, compute risk-neutral probabilities from the risk-free rate, and roll back discounted expected payoffs to obtain arbitrage-free prices. The article distinguishes clearly between valuing European and American options, including how to test for optimal early exercise of American puts at each node. It also explains how delta arises naturally from the binomial hedge ratio and how to compute and interpret delta, gamma, vega, theta, and rho for exam-style problems. Special attention is given to how the Greeks guide hedging decisions and scenario analysis for option portfolios. Finally, the article highlights typical CFA exam pitfalls—such as using real-world instead of risk-neutral probabilities or ignoring path-dependence—so candidates can better diagnose, structure, and check their own calculations under time pressure.

## **CFA Level 2 Syllabus**

For the CFA Level 2 exam, you are expected to understand the quantitative and strategic aspects of option pricing models, with a focus on the following syllabus points:

- Applying the binomial tree to value European and American options
- Explaining the concept and calculation of risk-neutral probabilities
- Interpreting the role of risk-neutral pricing in the valuation of contingent claims
- Calculating and interpreting key Greeks (delta, gamma, vega, theta, rho)
- Understanding hedging implications of option Greeks
- Differentiating between path-dependent and path-independent options within the binomial tree framework

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What is the primary principle behind risk-neutral valuation for options?
2. In a binomial tree, which Greek is directly related to the hedge ratio at each node?
3. For a one-period binomial, if the up-move factor is 1.2 and down-move factor is 0.8, what is the risk-neutral probability if the risk-free rate per period is 5%?
4. Why can American put options be more valuable than European puts?

## **Introduction**

Options require a systematic approach for valuation and risk assessment. The binomial tree method and risk-neutral pricing underpin both understanding and application for CFA Level 2 exam questions. This article explains step-wise how to construct and use binomial trees, how risk-neutral probabilities relate to arbitrage-free pricing, and how to interpret the primary Greeks for portfolio and exam success.

> **Key Term: Binomial tree**
> A stepwise model for valuing options where, in each period, the price of the reference asset moves to one of two possible values—up or down.

## Risk-Neutral Valuation and the Binomial Tree

Options differ from standard securities because their value depends on various possible future paths of the reference asset. The binomial tree provides a transparent approach by discretizing time into steps, allowing the user to calculate possible outcomes and option values at each node.

> **Key Term: Risk-neutral probability**
> The probability, used in valuation, at which the expected discounted payoff (assuming all assets earn the risk-free rate) equals the option price if markets are arbitrage-free.

The binomial tree process involves:

1. Dividing the option life into periods.
2. Calculating possible up and down moves for the reference asset.
3. Computing the risk-neutral probabilities based on the risk-free rate.
4. Rolling back the tree using discounted expected values to determine the option's present value.

### **Worked Example 1.1**

A stock is priced at \$100. It can go up 20% or down 10% in one period. The risk-free rate is 5%. What is the value of a one-period call option with a strike of \$100?

> **Answer:**
>
> - Up move: \$100 × 1.20 = \$120, Call payoff = \$20
> - Down move: \$100 × 0.90 = \$90, Call payoff = \$0
> - Risk-neutral probability: $p = (1.05 - 0.90)/(1.20 - 0.90) = 0.5$
> - Expected payoff: $[0.5 \times 20 + 0.5 \times 0] = 10$
> - Discounted to present: $\$10 / 1.05 = \$9.52$

### Binomial Tree for Multiple Periods and American Options

With more periods, the tree branches similarly at each node. For American options, early exercise is evaluated at every step by comparing the immediate exercise value to the value from holding until the next period.

> **Key Term: American option**
> An option contract that can be exercised at any time up to and including its expiration date.

### **Worked Example 1.2**

Consider a two-period American put option: same parameters as above, strike \$100.

> **Answer:**
>
> - At expiry: Up-Up node: \$144, payoff \$0; Up-Down or Down-Up: \$108 and 97.2, payoff \$0 and \$2.8; Down-Down: \$81, payoff \$19
> - Step back and compare value from holding vs. immediate exercise at each node.
> - Calculate at each node which is higher: value by holding or exercising. American put value may be higher due to early exercise, especially for deep-in-the-money puts.

### **Exam Warning**

> A frequent mistake is to use real-world probabilities rather than risk-neutral probabilities for option valuation in binomial trees. The CFA exam expects all discounting and expectation steps to be risk-neutral.

## Greeks in the Binomial Framework

Option sensitivities, called "Greeks", quantify risk exposures to changes in model parameters.

> **Key Term: Delta**
> The rate of change in an option's value with respect to changes in the price of the reference asset. For a binomial model, it is also the hedge ratio at each node.
>
> **Key Term: Gamma**
> The rate of change of delta itself as the reference price changes; measures convexity of option value.
>
> **Key Term: Vega**
> The sensitivity of the option price to changes in the volatility of the reference asset.
>
> **Key Term: Theta**
> The rate of change of the option's value as time passes—time decay.
>
> **Key Term: Rho**
> The rate of change in an option's value with respect to changes in the risk-free interest rate.

### **Worked Example 1.3**

Suppose in a one-step binomial model, an option's price increases by \$5 when the stock increases by \$10 and decreases by \$2 when the stock falls by \$10. What is delta?

> **Answer:**
> Delta = (Option up value - Option down value) / (Stock up - Stock down)
> = (\$5 - (-\$2)) / (\$10 - (-\$10)) = \$7/\$20 = 0.35

This means the option price moves \$0.35 for every \$1 change in the stock price, at this node.

## Path-Dependence and Hedging

Options whose payoffs depend on the entire path (e.g., Asian or lookback options) require special treatment as standard binomial tree rollbacks only work for path-independent options. For hedging, delta and gamma are most relevant in managing short-term exposures, while vega is key when the asset's volatility is uncertain.

### **Revision Tip**

> For CFA calculation questions, always specify whether the option is European or American, as this affects early exercise and rollback procedures in the tree.

## **Summary**

The binomial tree and risk-neutral pricing are cornerstones of option valuation for CFA Level 2. Correct construction of the tree, understanding of risk-neutral probabilities, and careful rollback using discounted expectations allow accurate valuation of both European and American options. The Greeks—delta, gamma, vega, theta, and rho—provide a toolkit for analyzing option risks for hedging and scenario analysis.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Construction of binomial trees for option valuation
- Use of risk-neutral probabilities for arbitrage-free pricing
- Distinction between European and American options in the tree
- Calculation and interpretation of delta, gamma, vega, theta, and rho
- Hedge ratio (delta) as a guide to portfolio risk management with options

## **Key Terms and Concepts**

- Binomial tree
- Risk-neutral probability
- American option
- Delta
- Gamma
- Vega
- Theta
- Rho

Option valuation and greeks - Binomial trees and risk-neutral pricing