Economic Modelling · Principles of option pricing

Black-Scholes Model: Formula, Assumptions and Greeks

Updated 11 October 2026 · Fact-checked

The Black-Scholes model gives the price of a European option on a share whose price follows geometric Brownian motion. You find d1 and d2 from S, K, r, σ and T, then use the normal distribution function Φ. The Greeks, such as delta and gamma, measure how that price changes.

Understand Black-Scholes Model and Assumptions

An option gives you the right, not the obligation, to buy (call) or sell (put) a share at a strike price K on a date T. Black-Scholes tells you the fair price today without needing anyone's view on where the share will go.

The idea is replication. You can build a portfolio of the share and cash that copies the option payoff, and you adjust it continuously. If the portfolio copies the option, the option must cost the same as the portfolio. Otherwise there is an arbitrage. This is why the real-world drift of the share does not appear in the formula. Only the risk-free rate r appears.

The main assumptions are:

The share price follows geometric Brownian motion with constant volatility σ, so returns over any period are lognormal.
The risk-free rate r is constant and you can borrow and lend at that rate.
There are no transaction costs or taxes, and trading is continuous.
Shares are infinitely divisible and short selling is allowed without restriction.
There is no arbitrage.
The share pays no dividends during the option's life (in the basic model).
The option is European, so it is exercised only at T.

The Greeks measure sensitivity. Delta is the change in option value for a small change in share price. Gamma is the change in delta for a small change in share price. Vega is the sensitivity to σ, theta to time, and rho to r. Delta tells you how many shares to hold to hedge. Gamma tells you how often that hedge needs resetting.

The limitations come from the assumptions failing. Real volatility is not constant, so implied volatility varies with strike and gives a smile. Share prices jump. Trading has costs and is not continuous. Dividends and early exercise matter in practice. The model is still the standard benchmark, and you should be able to say both what it does and where it breaks.

Key rules to remember

d1: d1 = [ln(S/K) + (r + σ²/2)T] ÷ (σ√T); S is the current share price, K the strike, r the continuously compounded risk-free rate, σ the volatility, T the time to expiry in years.
d2: d2 = d1 − σ√T; Compute d1 first, then subtract σ√T.
European call price: c = S Φ(d1) − K e^(−rT) Φ(d2); Φ is the standard normal distribution function. Valid for a share paying no dividends.
European put price: p = K e^(−rT) Φ(−d2) − S Φ(−d1); Use Φ(−x) = 1 − Φ(x). You can also get p from put-call parity.
Put-call parity: c − p = S − K e^(−rT); For European options on a non-dividend-paying share with the same K and T.
Delta: Call delta = Φ(d1); Put delta = Φ(d1) − 1; Call delta lies between 0 and 1. Put delta lies between −1 and 0.
Gamma: Γ = φ(d1) ÷ (S σ √T); Same for a call and a put. φ(x) = e^(−x²/2) ÷ √(2π) is the standard normal density.
Vega: ν = S φ(d1) √T; Same for a call and a put. Always positive.
Share price model: dS = μS dt + σS dZ; Geometric Brownian motion. Under risk-neutral valuation μ is replaced by r.

How to solve Black-Scholes Model and Assumptions questions

Use this order for any Black-Scholes question, whether it asks for a price, a Greek or a comment on the model.

1Write down S, K, r, σ and T. Check that r and σ are annual and continuously compounded, and that T is in years.
2Check the conditions: European option, no dividends, constant r and σ. If dividends are given, say how you adjust or state the assumption.
3Compute d1 using the formula, showing ln(S/K) and the other parts separately.
4Compute d2 = d1 − σ√T.
5Look up Φ(d1) and Φ(d2) from the tables, using Φ(−x) = 1 − Φ(x) for negative values.
6Substitute into the call or put formula. For a put, you may use put-call parity as a check.
7For Greeks, use Φ(d1) for delta and φ(d1) ÷ (Sσ√T) for gamma. Remember the sign and range of each.
8Finish with a one-line interpretation, such as the hedge needed or the effect of a share price change.

Quickest way: Compute d1 once, reuse it everywhere

When to use it: Use this in multiple-choice questions and in the early part of written answers when time is short.

Compute d1 and d2 once and keep four decimal places of working.
Get the call price first. Get the put from put-call parity: p = c − S + K e^(−rT).
Use Φ(d1) for call delta and Φ(d1) − 1 for put delta. No further calculation is needed.
For gamma, work out φ(d1) directly from the density formula. You do not need the tables for this.
For conceptual questions, match the answer to a failed assumption: constant volatility, no jumps, no costs, continuous trading.

Common mistakes in Black-Scholes Model and Assumptions

Using the real-world expected return μ in the formula.
Students think the option value should depend on how fast the share is expected to grow.
Fix: Only r appears. Replication removes μ. Say this explicitly if the question asks why.
Using a percentage like 5 instead of 0.05, or a time in months.
Inputs are given in different units in the question.
Fix: Convert r and σ to decimals and T to years before you start.
Getting the sign wrong in Φ(−d2) for puts.
Tables show only positive values, so students misread the symmetry.
Fix: Write Φ(−x) = 1 − Φ(x) beside your working and apply it step by step.
Saying call delta can be negative or put delta can exceed zero.
Students memorise the formulas but not the ranges.
Fix: Call delta is in (0, 1). Put delta is Φ(d1) − 1, so it is in (−1, 0). Use this as a quick check.
Saying Black-Scholes assumes returns are normal.
Normal and lognormal get mixed up.
Fix: Log returns are normal. The share price is lognormal. Write it this way.
Listing limitations without explaining the effect.
Students recall a list of words but not the consequence.
Fix: For each limitation, state what happens. For example, jumps and fat tails mean deep out-of-the-money options are underpriced by the model.

Worked examples

Example 1

A share has price ₹100 and pays no dividends. A European call has strike ₹100 and expires in 1 year. The risk-free rate is 5% per year continuously compounded and σ = 20%. Calculate d1, d2 and the call delta. Use Φ(0.35) = 0.6368 and Φ(0.15) = 0.5596.

Show the solution

S = 100, K = 100, r = 0.05, σ = 0.20, T = 1.
ln(S/K) = ln(1) = 0.
(r + σ²/2)T = 0.05 + 0.02 = 0.07.
σ√T = 0.20.
d1 = 0.07 ÷ 0.20 = 0.35.
d2 = 0.35 − 0.20 = 0.15.
Call delta = Φ(d1) = Φ(0.35) = 0.6368.

Answer: d1 = 0.35, d2 = 0.15 and call delta = 0.6368. The hedge for one short call is to hold about 0.64 shares.

Example 2

Using the same data as above, find the call price and then the put price. Use e^(−0.05) = 0.951229, Φ(0.35) = 0.6368, Φ(0.15) = 0.5596.

Show the solution

Call: c = S Φ(d1) − K e^(−rT) Φ(d2).
S Φ(d1) = 100 × 0.6368 = 63.68.
K e^(−rT) = 100 × 0.951229 = 95.1229.
K e^(−rT) Φ(d2) = 95.1229 × 0.5596 = 53.2314.
c = 63.68 − 53.2314 = 10.4486, about ₹10.45.
Put by parity: p = c − S + K e^(−rT).
p = 10.4486 − 100 + 95.1229 = 5.5715, about ₹5.57.

Answer: The call price is about ₹10.45 and the put price is about ₹5.57.

Exam tips

Show d1 and d2 as separate lines. Marks are given for method even when a table value is slightly off.
Always state the assumptions you use, especially no dividends and a European option.
For discussion questions, link each limitation to a specific effect on the price or the hedge.
Use put-call parity to check a put price. It takes seconds and catches errors.
In computer-based paper work, set the inputs in clearly labelled cells or variables, and check the call price against the bounds S − K e^(−rT) ≤ c ≤ S.

Practice questions from Principles of option pricing

Black-Scholes Model and Assumptions 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 Model and Assumptions: frequently asked questions

Why does the expected return of the share not appear in Black-Scholes?

The option can be replicated by trading the share and cash, so its price is fixed by no-arbitrage. Any growth rate of the share is already reflected in the share price S. Only r and σ matter.

What is the difference between delta and gamma?

Delta is the rate of change of the option price with the share price. Gamma is the rate of change of delta with the share price. A high gamma means your delta hedge goes out of date quickly.

Can I use Black-Scholes for American options?

Not directly. For a call on a non-dividend-paying share, early exercise is never optimal, so the European value applies. For puts, or when dividends are paid, early exercise can matter and you need other methods such as binomial trees.

What are the main limitations of Black-Scholes?

Volatility is not constant in practice, which produces a volatility smile. Prices can jump, trading has costs and is not continuous, and interest rates and dividends vary. The model is a benchmark, not an exact description of markets.