Strategic Financial Management · Options

Black-Scholes Option Pricing Model for CMA Final

Updated 11 October 2026 · Fact-checked

The Black-Scholes model prices a European option from five inputs: spot price, exercise price, risk-free rate, volatility and time to expiry. Calculate d1, then d2 = d1 − σ√T, read N(d1) and N(d2) from the normal table, and compute the call as S·N(d1) − X·e^(−rT)·N(d2). Get the put by the put formula or put-call parity.

Understand Black-Scholes Option Pricing Model

An option's value today depends on how likely it is to finish in the money and by how much. The Black-Scholes model turns that idea into a formula for a European option, which can be exercised only on the expiry date.

The call value has two parts. S·N(d1) is the present value of the share you expect to receive if the option is exercised. X·e^(−rT)·N(d2) is the present value of the exercise price you expect to pay. The call is the first part minus the second.

N(d) is the cumulative standard normal probability, that is, the area under the bell curve to the left of d. N(d2) is, in the model's risk-neutral world, the probability that the call finishes in the money. N(d1) is the delta of the call, that is, its sensitivity to the share price. A higher volatility, a longer time to expiry or a higher spot price raises the call value. A higher exercise price lowers it.

The model assumes: European exercise; no dividends during the option's life (in the basic form); constant risk-free rate and constant volatility; share prices that follow a lognormal path with continuous trading; no transaction costs or taxes; and no arbitrage. These assumptions are also its limits. Real volatility changes, markets jump, options in some markets, such as US-listed stock options, are American style and Black-Scholes does not directly price early exercise, and dividends matter. So the output is a model value, not an exact market price.

Key rules to remember

d1: d1 = [ln(S ÷ X) + (r + σ² ÷ 2) × T] ÷ (σ × √T); S = spot price, X = exercise price, r = continuously compounded risk-free rate, σ = annual volatility (standard deviation), T = time in years. ln is the natural log.
d2: d2 = d1 − σ × √T; Always compute d1 first. Use the same σ√T you used in the d1 denominator.
Call value: C = S × N(d1) − X × e^(−rT) × N(d2); e^(−rT) discounts the exercise price at the risk-free rate.
Put value: P = X × e^(−rT) × N(−d2) − S × N(−d1); Use this directly, or find the put from put-call parity.
Negative d values: N(−d) = 1 − N(d); Tables usually show only positive d. This symmetry gives you the rest.
Put-call parity (no dividends, European): C + X × e^(−rT) = P + S; Use it to find the second option value, or to check your answer.

How to solve Black-Scholes Option Pricing Model questions

Follow the same order every time. It keeps the working short and lets the examiner award method marks even if a table value is slightly off.

1List the inputs: S, X, r, σ, T. Convert T to years (3 months = 0.25) and r and σ to decimals. If σ is given as variance, take the square root first.
2Compute σ√T. You need it for both d1 and d2.
3Compute d1 = [ln(S ÷ X) + (r + σ² ÷ 2)T] ÷ σ√T. Write the numerator parts separately: ln(S ÷ X), then (r + σ²÷2)T.
4Compute d2 = d1 − σ√T.
5Read N(d1) and N(d2) from the normal table. For a negative d, find N of the positive value and subtract from 1. Interpolate only if the question gives a table with finer values and asks for precision.
6Compute e^(−rT) and the present value of the exercise price, X × e^(−rT).
7Substitute into the call formula. For a put, use the put formula or parity: P = C + X e^(−rT) − S.
8State the value in rupees and, if asked, compare with the market price and recommend: buy if the market price is below the model value, sell or write if it is above.

Quickest way: Call first, then put by parity

When to use it: Use this when both the call and the put values are asked, or when you want a built-in check on the answer.

Compute d1, d2, N(d1), N(d2) once and find the call value.
Compute PV of X = X × e^(−rT) once and keep it.
Get the put as P = C + PV of X − S. This avoids two more table lookups.
Cross-check with the direct put formula only if time is left. Both should match, apart from rounding.
Round d1 and d2 to two decimals to match the table, and carry at least two decimals in the final rupee figure.

Common mistakes in Black-Scholes Option Pricing Model

Using T in months or percentages as whole numbers (r = 8 instead of 0.08).
Question data is given as '3 months' and '8%', and students plug it in unchanged.
Fix: Write T in years and r and σ as decimals before any calculation.
Forgetting to subtract σ√T when finding d2, or using σ instead of σ√T.
The two d formulas look alike and students rush.
Fix: Compute σ√T as a separate line and use it in both d1's denominator and d2 = d1 − σ√T.
Reading N(d) for a negative d directly from a table that shows only positive values.
Students look for a minus sign in the table and pick the wrong row.
Fix: Use N(−d) = 1 − N(d). For example, if N(0.20) = 0.5793, then N(−0.20) = 0.4207.
Not discounting the exercise price, that is, using X instead of X × e^(−rT).
Students copy the S·N(d1) − X·N(d2) pattern from memory.
Fix: Always compute e^(−rT) and the PV of X as a separate step.
Using the call formula's signs for the put, for example S·N(−d1) − X e^(−rT)·N(−d2).
The put looks like a mirror of the call and the order gets reversed.
Fix: Remember the put as 'PV of X times N(−d2), minus S times N(−d1)', or use parity from the call value.
Ignoring dividends or applying the basic formula to an American option without comment.
The question states a dividend or an exercise style and students overlook it.
Fix: Read the data for dividends. If a known dividend is given, the standard exam treatment is to reduce S by the present value of the dividend before computing d1. State this as your assumption.

Worked examples

Example 1

A share of an Indian company trades at ₹1,000. A 6-month European call and put both have an exercise price of ₹1,000. The risk-free rate is 10% per annum (continuously compounded) and volatility is 20% per annum. Use N(0.42) = 0.6628 and N(0.28) = 0.6103, and e^(−0.05) = 0.9512. Find the call and put values.

Show the solution

Inputs: S = 1,000, X = 1,000, r = 0.10, σ = 0.20, T = 0.5.
σ√T = 0.20 × √0.5 = 0.20 × 0.7071 = 0.1414.
ln(S ÷ X) = ln(1) = 0. (r + σ²÷2)T = (0.10 + 0.02) × 0.5 = 0.06.
d1 = 0.06 ÷ 0.1414 = 0.4243, taken as 0.42.
d2 = 0.4243 − 0.1414 = 0.2829, taken as 0.28.
N(d1) = 0.6628 and N(d2) = 0.6103.
PV of X = 1,000 × 0.9512 = ₹951.20.
Call = 1,000 × 0.6628 − 951.20 × 0.6103 = 662.80 − 580.52 = ₹82.28.
For the put: N(−d2) = 1 − 0.6103 = 0.3897 and N(−d1) = 1 − 0.6628 = 0.3372.
Put = 951.20 × 0.3897 − 1,000 × 0.3372 = 370.68 − 337.20 = ₹33.48.
Check by parity: C + PV of X = 82.28 + 951.20 = 1,033.48, and P + S = 33.48 + 1,000 = 1,033.48. It matches.

Answer: Call value ≈ ₹82.28 and put value ≈ ₹33.48.

Example 2

A share trades at ₹250. A 3-month European put has an exercise price of ₹260. The risk-free rate is 8% per annum (continuous) and the volatility is 30% per annum. Use e^(−0.02) = 0.9802, N(0.05) = 0.5199 and N(0.20) = 0.5793. Find the put value and the call value for the same exercise price and expiry.

Show the solution

Inputs: S = 250, X = 260, r = 0.08, σ = 0.30, T = 0.25.
σ√T = 0.30 × 0.5 = 0.15.
ln(250 ÷ 260) = ln(0.96154) = −0.0392. (r + σ²÷2)T = (0.08 + 0.045) × 0.25 = 0.03125.
d1 = (−0.0392 + 0.03125) ÷ 0.15 = −0.0531, taken as −0.05.
d2 = −0.05 − 0.15 = −0.20.
For the put we need N(−d2) = N(0.20) = 0.5793 and N(−d1) = N(0.05) = 0.5199.
PV of X = 260 × 0.9802 = ₹254.85.
Put = 254.85 × 0.5793 − 250 × 0.5199 = 147.64 − 129.98 = ₹17.66.
Call by parity: C = P + S − PV of X = 17.66 + 250 − 254.85 = ₹12.81.

Answer: Put value ≈ ₹17.66 and call value ≈ ₹12.81.

Exam tips

Section A may give you d1, d2 and table values and ask only for N(d), the call value or the effect of a change in one input. Know the direction of each effect: a higher σ, T or S raises the call, and a higher X lowers it.
In written answers, show d1, d2, N(d1), N(d2) and PV of X as separate lines. If a table value is misread, you still earn the method marks.
Use the values given in the question for e^(−rT) and N(d). Do not recompute them from a calculator if the paper supplies them, so that your answer matches the expected figure.
Close with a decision when the market price is given: compare it with the model value and say buy, sell or hold the option.
Write your assumptions in one line, for example 'European option, no dividends, constant σ and r'. If a dividend is given, adjust S and say so.

Practice questions from Options

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

How do I calculate d1 and d2 in Black-Scholes?

First compute σ√T. Then d1 = [ln(S ÷ X) + (r + σ²÷2)T] ÷ σ√T. Finally d2 = d1 − σ√T. Keep T in years and r and σ as decimals.

How do I use the normal distribution table for N(d1) and N(d2)?

Round d to two decimals. Find the row for the first decimal and the column for the second to read N(d), the area to the left of d. For a negative d, look up the positive value and subtract from 1.

What is the Black-Scholes put option formula?

P = X × e^(−rT) × N(−d2) − S × N(−d1). You can also get it from the call using put-call parity: P = C + X e^(−rT) − S, for a European option with no dividends.

What are the main assumptions and limitations of the Black-Scholes model?

It assumes a European option, no dividends, constant risk-free rate and volatility, lognormal prices, no costs and no arbitrage. In practice volatility changes, prices can jump, and many options can be exercised early, so the model value is an estimate.

What does N(d2) mean in the formula?

N(d2) is the risk-neutral probability that the call finishes in the money at expiry. N(d1) is the call's delta, which tells you how much the option value moves for a small change in the share price.