## **Learning Outcomes**

After reading this article, you should be able to identify the five key Black–Scholes model inputs, explain their roles in real option valuation, and apply the model to value project flexibility in capital investment appraisal. You will also be able to assess the major limitations of the Black–Scholes model and discuss their relevance to real options analysis in ACCA Advanced Financial Management.

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

For ACCA Advanced Financial Management (AFM), you are required to understand real option valuation methods and the practical application and limitations of the Black–Scholes model. In particular, focus your revision on:

- Applying the Black–Scholes Option Pricing Model (BSOP) to value financial and real options in investment appraisal
- Identifying and explaining the five principal drivers of option value: value of the base asset, exercise price, time to expiry, volatility, and risk-free rate
- Discussing the structure, assumptions, and limitations of the Black–Scholes model
- Evaluating how real options (delay, expand, abandon) can be valued using the model

## **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. Which input in the Black–Scholes model represents the project's risk or uncertainty of cash flows?
2. Match each Black–Scholes input below with its interpretation in real option appraisal:
   a) Exercise Price  
   b) Pa (price of the base asset)  
   c) s (volatility)  
   d) t (time to expiry)  
   e) r (risk-free rate)
3. True or false? The Black–Scholes model can be applied directly without adjustment to value options in all project scenarios.
4. Briefly explain why the assumption of continuous trading in the Black–Scholes model may not hold in real investment projects.
5. List two major limitations of the Black–Scholes model for real option analysis.

## **Introduction**

Options theory is used in ACCA Advanced Financial Management to value flexibility embedded in capital investments—so-called "real options." Applying option pricing brings extra analytical power where traditional net present value (NPV) techniques fail to capture the value of management’s future decisions. The Black–Scholes model, originally developed for financial options, is adapted to estimate the value of real options such as waiting, expanding, or abandoning projects. However, to use this model correctly—and understand its limitations—you need to know the required inputs, their interpretation for real assets, and the key drawbacks when applied to non-traded projects.

## **BLACK–SCHOLES MODEL INPUTS FOR REAL OPTIONS**

The Black–Scholes Option Pricing Model is defined by five critical input variables. For real options valuation in investment appraisal, each must be carefully identified and interpreted for the scenario.

> **Key Term: real option**  
> The right, but not obligation, for managers to make certain future decisions regarding a real investment project, such as delaying, abandoning, expanding, or contracting.
>
> **Key Term: Black–Scholes model**  
> A mathematical formula originally designed to estimate the fair value of European call and put options, using variables including asset price, exercise price, volatility, risk-free rate, and time to expiry.
>
> **Key Term: volatility (as used in Black–Scholes)**  
> A measure of the expected standard deviation of returns (or value) of the asset—here, a proxy for project risk or uncertainty about future cash flows.

### The Five Key Inputs

1. **Value of the base asset (Pa):** In real option contexts, this is the present value of the project’s expected future cash flows, excluding the initial investment if the option is to delay or expand.

2. **Exercise price (Pe):** Represents the planned investment cost required to exercise the option or execute the future decision (e.g., capital outlay required to expand, redeploy, or abandon).

3. **Time to expiry (t):** The period during which the decision can be made; for example, the window allowed before the delay or investment option lapses.

4. **Volatility (s):** Estimated standard deviation (annualized) for the operating cash flows or asset value. High uncertainty increases the option’s value.

5. **Risk-free rate (r):** The theoretical rate of return on a risk-free investment (typically a government bond) matching the option term.

> **Key Term: exercise price**  
> The fixed cost or required payment to be made if the real option is exercised, such as the investment required to expand or acquire an asset.

### Applying Black–Scholes to Real Options

For real options, the Black–Scholes formula is reused but with reinterpreted parameters. The formula for a call (right to invest/expand) option is:

$$
c = P_a N(d_1) - P_e N(d_2) e^{-rt}
$$

where:

- $P_a$ = present value of project cash flows (asset value)
- $P_e$ = exercise price (cost to implement future opportunity)
- $t$ = time to expiry
- $s$ = volatility of expected base cash flows
- $r$ = risk-free rate
- $N(d)$ = cumulative normal distribution
- $d_1$ and $d_2$ are calculated using the standard formulae.

> **Key Term: N(d)**  
> The probability, under a standard normal distribution, that the outcome will be less than a given value d. Used in the Black–Scholes formula to weight future payoffs probabilistically.

## **VALUING REAL OPTIONS IN INVESTMENT APPRAISAL**

Real options typically take the form of:

- **Option to delay** (wait before committing capital)
- **Option to expand** (invest further if circumstances are favourable)
- **Option to abandon** (exit project to recover residual value)

The Black–Scholes model offers an estimate of the value managers should be willing to pay for such flexibility.

### **Worked Example 1.1**

**A technology company may build a small prototype factory for \$2 million, giving the option to commit \$10 million more within 2 years if demand justifies. If the PV of potential future cash flows from expansion is \$13 million, annual volatility in these estimates is 30%, and the risk-free rate is 4%, value the option to expand.**

> **Answer:**  
> Here, $P_a = \$13m (PV of expected future cash flows from expansion), $P_e = \$10m (further investment), $t = 2$, $s = 0.3$, $r = 0.04$. Plug these into the Black–Scholes formula. Calculate $d_1$, $d_2$, $N(d_1)$, $N(d_2)$, then find $c = Pa \times N(d_1) - Pe \times N(d_2) \times e^{-rt}$. The resulting value represents the current value of the flexibility to expand after the prototype stage.

## **LIMITATIONS OF THE BLACK–SCHOLES MODEL FOR REAL OPTIONS**

Although widely used in exam scenarios, applying Black–Scholes to real options comes with important drawbacks:

> **Key Term: European option**  
> An option exercisable only at its expiry date, not before. The Black–Scholes model prices European-style options.
>
> **Key Term: normal distribution**  
> A statistical distribution where values are symmetrically distributed around the mean, forming a bell-shaped curve; assumed in the Black–Scholes model.

### Model Assumptions and Real Option Limitations

- **European-style exercise only:** The model is meant for options exercisable only at expiry, but real decisions may be taken at various times.
- **Base asset traded and liquid:** The formula assumes the asset is traded in efficient markets, whereas projects are unique and not freely traded.
- **Constant volatility and risk-free rate:** Volatility and risk-free rate are assumed constant, but both can change over a project’s lifetime.
- **No dividends or cash flows before expiry:** Ignores interim payoffs or costs that may arise in practice.
- **Continuous trading and hedging:** The model presumes managers can hedge project risk continuously, which is impossible for real assets.

### **Worked Example 1.2**

**A pharmaceutical company can abandon a research project in 3 years, recovering \$20 million of assets. If the present value of uncertain future cash flows forgone is \$18 million, volatility is estimated at 40%, and risk-free rate is 3.5%, value the put option to abandon.**

> **Answer:**  
> This is an option to abandon (a put option). The Black–Scholes formula for a put can be used, with the asset ($P_a$) as PV of forgone cash flows, exercise price ($P_e$) as abandonment recovery, and appropriate t, s, r. The calculation shows the value of being able to walk away, but note the assumption that abandonment is allowed only at the 3-year mark.

### **Exam Warning**

> The Black–Scholes model’s output is only as reliable as its input estimates—especially volatility of base cash flows. Unrealistic inputs can lead to misleading option values.

### **Revision Tip**

_If you cannot obtain quality estimates for volatility, explain this limitation clearly in your answer and discuss the qualitative value of flexibility as part of your recommendation._

## **Summary**

Real option valuation using Black–Scholes can add significant understanding where project flexibility has value, but you must clearly state and evaluate the model’s assumptions and practical weaknesses in real-world investment appraisals. Always clarify how each input is interpreted and assess the reliability of your estimates, especially for volatility.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- The five required inputs for the Black–Scholes model and their real option interpretations
- The method for valuing delay, expand, or abandon options in project appraisal
- Key assumptions of the Black–Scholes model and their suitability for real assets
- Major limitations of the Black–Scholes model in real option contexts
- The importance of volatility estimates and the impact of model imprecision

## **Key Terms and Concepts**

- real option
- Black–Scholes model
- volatility (as used in Black–Scholes)
- exercise price
- N(d)
- European option
- normal distribution

Real option valuation methods - Black–Scholes inputs and limitations