## **Learning Outcomes**

After reading this article, you will be able to explain the rationale behind binomial lattice models for real option valuation, construct a simple binomial tree for project appraisal, apply risk-neutral valuation to determine expected values, and critically evaluate the advantages and limitations of these methods compared to the Black-Scholes model. You will gain the skills to identify suitable application scenarios and common errors, ensuring you are prepared for ACCA AFM questions on this topic.

## **ACCA Advanced Financial Management (AFM) Syllabus**

For ACCA Advanced Financial Management (AFM), you are required to understand how real option valuation with binomial lattices and risk-neutral techniques extends traditional investment appraisal and supports superior project decision-making. In particular, revision for AFM should cover:

- The principles and construction of binomial option pricing models for financial and real options
- Application of binomial lattices to evaluate options to delay, expand, abandon, or modify projects
- The concept and calculation of risk-neutral probabilities in option valuation
- Comparison of the binomial approach and analytical models such as Black-Scholes
- Appraisal of the benefits and limitations of real option analysis in strategic project evaluation
- Identification of pitfalls and common errors in the practical application of binomial models

## **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 problem in traditional NPV appraisal does real option valuation seek to address?
2. Which of the following is a feature of the binomial lattice method of option valuation?
   a) Only suitable for European options  
   b) Assumes asset prices move in continuous time  
   c) Uses discrete time steps to model up and down movements  
   d) Requires only historical probabilities for valuation
3. Explain the meaning of risk-neutral probabilities and how they are used in binomial option valuation.
4. Briefly outline how an option to delay an investment could be valued using a binomial tree.

## **Introduction**

Traditional investment appraisal methods such as NPV do not adequately capture the managerial flexibility available in many capital projects, particularly under uncertainty. Real options analysis treats key project choices, such as whether to delay, expand, or abandon, as options. The binomial lattice model, combined with risk-neutral valuation, provides an intuitive approach to valuing these embedded flexibilities over multiple periods and project stages. This article explains how binomial trees are constructed, how risk-neutral probabilities are derived and applied, and highlights the critical steps for successfully valuing real options for the ACCA AFM exam.

## **REAL OPTION VALUATION WITH BINOMIAL LATTICES**

### The Rationale for Binomial Lattices

Conventional investment appraisal techniques assume passive management and ignore future choices. However, project managers often make decisions dynamically as new information emerges. A binomial lattice (or tree) is a flexible model that simulates the possible future values of a project (or its reference asset) at discrete intervals, allowing for “real time” evaluation of managerial options at each node.

> **Key Term: binomial lattice**
> A multi-period option pricing model that charts all possible up and down movements in the project or asset value over discrete time steps, allowing option values to be determined recursively from the end of the tree back to the present.

### Structure of a Binomial Tree

At the start of the tree, the asset or project has a known current value. Over each time step, it may move up by a factor (u) or down by a factor (d), creating a lattice of possible future values. At each “node,” a decision can be made: exercise the option (e.g., invest immediately), postpone, expand, abandon, or allow the option to lapse.

### Application to Real Options

Real project options commonly valued with binomial lattices include:

- **Option to delay:** Wait for more information before committing capital.
- **Option to expand:** Invest further if market conditions are favourable.
- **Option to abandon:** Exit the project to limit losses.
- **Option to redeploy:** Switch the use of assets depending on outcomes.

These options are analogous to European or American options on financial assets, but the exercise decision is often project-specific.

## **RISK-NEUTRAL VALUATION**

### Risk-Neutral Probabilities

Option pricing using binomial lattices does not require subjective estimates of future probabilities or risk premiums. Instead, it relies on the “risk-neutral” probability, reflecting a hypothetical world where all assets grow at the risk-free rate and all investors are indifferent to risk.

> **Key Term: risk-neutral probability**
> The probability, derived from arbitrage pricing, that makes the expected return of a risky asset equal to the risk-free rate; used to weight possible payoffs in risk-neutral valuation.
>
> **Key Term: risk-neutral valuation**
> A valuation technique where future cash flows are discounted at the risk-free rate, and weighted using risk-neutral probabilities, reflecting the no-arbitrage condition in efficient markets.

### Calculating Risk-Neutral Probabilities

The risk-neutral probability (p) is calculated as:

$$
p = \frac{e^{r \Delta t} - d}{u - d}
$$

where:

- $e^{r \Delta t}$ = exponential of the risk-free rate over the time step
- u = “up” factor
- d = “down” factor

### Valuing the Option

1. **Calculate project or asset values at each node** for all end periods, considering possible up and down moves.
2. **Compute the option payoff** at terminal nodes (e.g., NPV if exercised, zero if not).
3. **At earlier nodes, use risk-neutral valuation**:
   - Option value at each node = [p × value if up + (1 – p) × value if down] discounted at the risk-free rate.
   - For American-style (real option) valuation, check whether immediate exercise or continuation gives a higher value at each node.

Option values roll back through the tree until the present value is found at the root node.

### **Worked Example 1.1**

A company has the right (but not obligation) to invest \$200,000 in a project at any time in the next year. The present value of expected project cash inflows is \$220,000, but is uncertain. Over each half-year, the value may increase by 20% (u = 1.2) or decrease by 16.7% (d = 0.833). The risk-free rate is 5% per annum, continuously compounded. Construct a 2-step binomial tree to value the option to invest.

**Step-by-step Solution:**

> **Answer:**
>
> 1. **Set parameters:**
>    - S = \$220,000, K = \$200,000, u = 1.2, d = 0.833, r = 5%, Δt = 0.5
> 2. **Calculate risk-neutral probability:**  
>    $$ p = \frac{e^{0.05 \times 0.5} - 0.833}{1.2 - 0.833} \approx \frac{1.0253 - 0.833}{0.367} = 0.524 $$
> 3. **Find asset values at the end nodes (after two steps):**
>
> - Up-Up: \$220,000 × 1.2 × 1.2 = \$316,800
> - Up-Down = Down-Up: \$220,000 × 1.2 × 0.833 = \$220,000
> - Down-Down: \$220,000 × 0.833 × 0.833 = \$152,941
>
> 1. **Calculate option value at maturity (t = 1):**
>    - Up-Up payoff: max(316,800–200,000, 0) = \$116,800
>    - Up-Down/Down-Up: max(220,000–200,000, 0) = \$20,000
>    - Down-Down: max(152,941–200,000, 0) = \$0
> 2. **Roll back to t = 0.5 for each node:**
>    - Node value = $ [p × (option up) + (1 – p) × (option down)] × e^{-0.05 × 0.5} $
>    - For “Up” node: [0.524 × \$116,800 + 0.476 × \$20,000] × 0.9753 = \$70,651
>    - For “Down” node: [0.524 × \$20,000 + 0.476 × \$0] × 0.9753 = \$10,213
>    - Check for early exercise: If the asset at each node exceeds K, immediate exercise may be optimal.
> 3. **Value at t = 0:**
>    - Root node = [0.524 × \$70,651 + 0.476 × \$10,213] × 0.9753 = \$41,743
> 4. **Interpretation:** The real option to invest is worth approximately \$41,700—higher than the standard NPV of \$20,000. Managerial flexibility and risk-adjusted timing give added value.

## Comparison to Black-Scholes and Application Considerations

The binomial method can handle American-style (early exercise) features and multi-stage decisions, making it especially suitable for real projects where decisions are not limited to one fixed expiry date. Black-Scholes relies on several restrictive assumptions, including continuous trading, constant volatility, and lognormally distributed outcomes, limiting its application for real projects with discrete decision points.

### **Exam Warning**

> The most common error in binomial questions occurs in the application of risk-neutral probabilities. Do **not** use subjective or historical success/failure probabilities. Always apply the risk-neutral formula and be clear on the role of the risk-free rate in the model.

## Advantages and Limitations

### Advantages

- Can model complex, multi-stage project options.
- Handles American-style (early exercise) and path-dependent options.
- Intuitive for scenario-based project evaluations.

### Limitations

- Requires more computational steps than analytic models for many periods or small time steps.
- Binomial trees can become large and unwieldy for long-term projects.
- Relies on reliable inputs for volatility and project value.

### **Revision Tip**

> In Excel exam questions, clearly label binomial tree nodes, and always show your calculation and rationale for p, up, and down values. Practice simple trees by hand to ensure you understand the process before using spreadsheet functions.

## **Summary**

Binomial lattice and risk-neutral valuation methods enable ACCA candidates to value real options in investment projects by modeling all possible project value paths and optimally timing managerial decisions. Proficiency with these models allows more accurate appraisal of abandonment, delay, expansion, or redeployment features—addressing limitations of static DCF techniques.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Define and explain binomial lattice models and their application to real option valuation
- Calculate and interpret risk-neutral probabilities in the binomial framework
- Construct a simple binomial tree for project and option values over multiple time steps
- Apply risk-neutral valuation to derive present values of real options
- Contrast binomial and Black-Scholes models, identifying use case strengths and constraints
- Recognise and avoid common exam errors in binomial model application

## **Key Terms and Concepts**

- binomial lattice
- risk-neutral probability
- risk-neutral valuation

Real option valuation methods - Binomial lattices and risk-neutral valuation