Advanced Financial Management · Application of option pricing theory in investment decisions

Black-Scholes Option Pricing Model for ACCA AFM

Updated 11 October 2026 · Fact-checked

The Black-Scholes model values a European option from five inputs: share price, exercise price, risk-free rate, time to expiry and volatility. You calculate d1 and d2, look up N(d1) and N(d2) in the tables, then apply c = Pa×N(d1) − Pe×e^(−rt)×N(d2). Get the put from put-call parity.

Understand Black-Scholes Option Pricing Model

An option gives its holder a right, not an obligation. A European call lets you buy a share at the exercise price on the expiry date only. A European put lets you sell. The model gives a fair price today for that right.

The idea behind the formula is simple. A call is like owning the share, partly funded by borrowing the present value of the exercise price. N(d1) is how much of the share you effectively hold. N(d2) is roughly the risk-neutral probability that the option finishes in the money. So the call value is the share you hold minus the discounted exercise price you expect to pay.

Five inputs drive the value. A higher share price, a longer time to expiry, higher volatility and a higher interest rate all raise a call value. A higher exercise price lowers it. Volatility matters because the holder gains from big upward moves but loses only the premium on downward moves.

The model rests on assumptions. The option is European and can only be exercised at expiry. The share pays no dividends during the option's life. Share returns are lognormally distributed, with constant volatility. The risk-free rate is constant, and you can borrow and lend at it. There are no taxes or transaction costs. Markets are efficient and trading is continuous.

These assumptions are also its limitations. Real volatility changes and is hard to estimate. Dividends exist. Many real options are American or have no market price. In exams, state the limitation and say what it does to the answer.

Key rules to remember

Call option value: c = Pa × N(d1) − Pe × e^(−rt) × N(d2); Pa = current share price, Pe = exercise price, r = continuously compounded risk-free rate, t = years to expiry. The AFM formula sheet gives this.
d1: d1 = [ln(Pa ÷ Pe) + (r + 0.5 × s²) × t] ÷ (s × √t); s = annual volatility (standard deviation) as a decimal. Use the natural log.
d2: d2 = d1 − s × √t; Always derived from d1. Do not recalculate from scratch.
Put value via put-call parity: p = c − Pa + Pe × e^(−rt); Valid for European options on a non-dividend-paying share with the same exercise price and expiry.
Negative d values: N(−d) = 1 − N(d); Tables usually show positive values only. Use this for negative d1 or d2.

How to solve Black-Scholes Option Pricing Model questions

Use this order for any Black-Scholes question. It keeps the arithmetic tidy and shows the marker each stage.

1List the inputs: Pa, Pe, r, t (in years) and s. Convert months to years and percentages to decimals. If you are given variance, take the square root for s.
2Calculate s × √t first. You need it for both d1 and d2.
3Calculate d1 using ln(Pa ÷ Pe) + (r + 0.5s²)t, divided by s√t. Show the numerator and denominator separately.
4Calculate d2 = d1 − s√t.
5Look up N(d1) and N(d2) in the normal distribution table. For negative values use 1 − N(positive). Interpolate if the question needs it.
6Calculate Pe × e^(−rt), then apply c = Pa×N(d1) − Pe×e^(−rt)×N(d2).
7If a put is asked for, use put-call parity: p = c − Pa + Pe×e^(−rt).
8State the answer with currency and a brief comment, for example on assumptions or what the value means for the decision.

Quickest way: Fast route under time pressure

When to use it: Use when the question asks for a single call or put value and gives clean inputs.

Write s√t, then d1 and d2 on three lines. Round d values to two decimals so you can use the table.
Find N(d1) and N(d2). Check that N(d1) > N(d2) when d1 > d2.
Compute the discounted exercise price once and reuse it for both the call and the put.
Sense-check: a call cannot exceed Pa, and it cannot be below Pa − Pe×e^(−rt). A put cannot exceed Pe×e^(−rt).
Keep the working in the answer booklet or spreadsheet in a clear layout, so method marks are easy to award even if a table lookup is wrong.

Common mistakes in Black-Scholes Option Pricing Model

Using t in months or r as a whole number, such as 5 instead of 0.05.
Question data is given as 6 months and 5%, and students type it straight into the formula.
Fix: Convert first. Write t = 0.5 and r = 0.05 at the top of your working, before any calculation.
Using log base 10 instead of the natural log for ln(Pa ÷ Pe).
The calculator has both log and ln keys.
Fix: Always press ln. As a check, ln of a number slightly above 1 is close to that number minus 1.
Forgetting to take the square root of t, or using variance as s.
s√t and s² both appear in the formulas, and they get mixed up.
Fix: Compute s√t separately. If the question gives variance, square-root it to get s, then use s² only inside the 0.5s² term.
Reading N(d) wrongly for negative d values.
Tables list only positive d. Students read the table as if d were positive.
Fix: Use N(−d) = 1 − N(d). A negative d must give N below 0.5.
Forgetting to discount the exercise price.
Students write Pe×N(d2) rather than Pe×e^(−rt)×N(d2).
Fix: Write the discounted exercise price as a separate line before the final formula.
Using the model result without commenting on assumptions, or ignoring dividends.
Students treat the model as exact.
Fix: Add a short comment: the model assumes European exercise, no dividends and constant volatility, so the value is an estimate.

Worked examples

Example 1

A share trades at $100. A European call and a European put each have an exercise price of $100 and expire in 1 year. The risk-free rate is 5% and share volatility is 20% a year. No dividends are expected. Calculate the value of the call and the put. Use N(0.35) = 0.6368 and N(0.15) = 0.5596.

Show the solution

Inputs: Pa = 100, Pe = 100, r = 0.05, t = 1, s = 0.20.
s√t = 0.20 × 1 = 0.20.
d1 = [ln(100 ÷ 100) + (0.05 + 0.5 × 0.04) × 1] ÷ 0.20 = (0 + 0.07) ÷ 0.20 = 0.35.
d2 = 0.35 − 0.20 = 0.15.
N(d1) = 0.6368 and N(d2) = 0.5596.
Discounted exercise price = 100 × e^(−0.05) = 100 × 0.9512 = $95.12.
Call: c = 100 × 0.6368 − 95.12 × 0.5596 = 63.68 − 53.23 = $10.45.
Put by parity: p = 10.45 − 100 + 95.12 = $5.57.

Answer: Call value ≈ $10.45. Put value ≈ $5.57.

Example 2

A share price is $45. A European call has an exercise price of $50 and expires in 6 months. The risk-free rate is 3% and volatility is 40%. Calculate the call value, and the put value using put-call parity. Use N(0.18) = 0.5714 and N(0.46) = 0.6772.

Show the solution

Inputs: Pa = 45, Pe = 50, r = 0.03, t = 0.5, s = 0.40.
s√t = 0.40 × 0.7071 = 0.2828.
ln(45 ÷ 50) = ln(0.9) = −0.1054.
(r + 0.5s²)t = (0.03 + 0.5 × 0.16) × 0.5 = 0.11 × 0.5 = 0.055.
d1 = (−0.1054 + 0.055) ÷ 0.2828 = −0.0504 ÷ 0.2828 = −0.18 (rounded).
d2 = −0.18 − 0.2828 = −0.46 (rounded).
N(−0.18) = 1 − 0.5714 = 0.4286. N(−0.46) = 1 − 0.6772 = 0.3228.
Discounted exercise price = 50 × e^(−0.015) = 50 × 0.9851 = $49.26.
Call: c = 45 × 0.4286 − 49.26 × 0.3228 = 19.29 − 15.90 = $3.39.
Put: p = 3.39 − 45 + 49.26 = $7.65.

Answer: Call value ≈ $3.39. Put value ≈ $7.65.

Exam tips

Tables and the formula are given in the exam, but you must know what each symbol means. Write inputs down first so the marker can follow your logic.
Round d1 and d2 to two decimals so you can use the table. Show N values clearly. Small rounding differences are normally accepted if the method is right.
Questions often ask you to comment, not just calculate. Prepare two or three points on assumptions, volatility estimation and dividends.
If the question asks for a put, do not recalculate d values. Use put-call parity if the options are European and share dividends are not expected.
In a case study, link the value to a decision, such as whether a premium is good value or whether a real option adds to project NPV. This earns professional skills marks for commercial acumen.

Practice questions from Application of option pricing theory in investment decisions

Black-Scholes Option Pricing Model in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Black-Scholes Option Pricing Model: frequently asked questions

What do N(d1) and N(d2) mean in the Black-Scholes model?

They are cumulative probabilities from the standard normal distribution. N(d2) is roughly the risk-neutral probability that a call finishes in the money. N(d1) acts as the share-equivalent amount the call holder effectively owns.

Does the Black-Scholes model work for American options?

The basic model values European options, which can be exercised only at expiry. For a call on a non-dividend-paying share, early exercise is not normally worthwhile, so the value is a fair guide. For other American options it is only an approximation.

How do I get the put value in Black-Scholes?

Calculate the call first, then use put-call parity: p = c − Pa + Pe × e^(−rt). This applies to European options with the same exercise price and expiry on a share with no dividends.

What are the main limitations of Black-Scholes?

It assumes constant volatility, a constant risk-free rate, no dividends, no transaction costs and European exercise. Volatility is not directly observable and must be estimated, which makes the result uncertain. Real options also may not be traded.