IAI Actuarial Core Principles · Economic Modelling

Black-Scholes Derivative-Pricing Model for CM2

The Black-Scholes model prices European options by assuming the share price follows geometric Brownian motion and that you can hedge continuously at the risk-free rate. You solve it by finding d1 and d2, then applying the call or put formula. The same price comes from discounting expected payoff under risk-neutral probabilities.

What this chapter covers

This chapter is about pricing options in continuous time. You start with the model's assumptions, then the share price process: geometric Brownian motion, which gives lognormal prices. From there you build the formulas for European calls and puts, and see why the price does not depend on the share's real-world expected return.

The second half of the chapter is about using the model. You derive the pricing result through the Black-Scholes PDE and risk-neutral valuation, measure sensitivities with the Greeks, and study volatility, which is the one input you cannot observe directly. You finish with extensions to dividends, currencies and other underlyings.

In the CM2 syllabus this chapter sits within option theory, which carries 30% of the 2026 topic weighting. It links back to rational economic theory and asset valuation, where arbitrage-free pricing is set up. It links forward to hedging, risk management and stochastic models of asset prices. Option theory also appears in the computer-based Paper B, so you must be able to code or compute the formula, not just recite it.

Option theory carries 30% of the 2026 CM2 topic weighting, and Black-Scholes is the core of it. It is tested in both the multiple-choice section and the written questions of Paper A, and it suits computation in Paper B. Written questions reward the full chain: state the assumption, give the formula in standard notation, show the working and interpret the result. A student who learns the method once can score on pricing, Greeks, hedging and volatility questions. A student who only memorises the formula loses marks on the 'explain' and 'comment' parts.

Black-Scholes derivative-pricing model: topics in the order to study them

1Black-Scholes Assumptions and LimitationsEvery later result depends on these assumptions, and examiners often ask which one fails in practice.
2Geometric Brownian Motion and Lognormal Share PricesThe share price model is the input to the formula, so you need its distribution and moments first.
3Black-Scholes Formula for European Calls and PutsWith the price model in place, you can now apply the formula and check it with put-call parity.
4Risk-Neutral Valuation and the Black-Scholes PDEThis explains why the formula works, and why the real-world drift drops out of the price.
5The Greeks and HedgingThe Greeks are derivatives of the price, so you need the price formula and the hedging idea first.
6Volatility: Historical, Implied and SmileVolatility is the key input. You study it after you know how sensitive the price is to it (vega).
7Extensions: Dividends, Currencies and Other UnderlyingsThese are small changes to the base formula, so learn them last, once the base case is secure.

How to prepare Black-Scholes derivative-pricing model

Treat this chapter as one connected argument, not seven separate topics. Work in short sessions that suit a phone or a commute, and practise computation on paper or in your Paper B tool.

List the assumptions from memory, and for each one write what happens to the price or the hedge if it fails.
Write the lognormal result for the share price in your own words, with its parameters, and practise finding probabilities and expected values from it.
Learn the call and put formulas with d1 and d2 in standard notation. Do three numerical prices by hand, and check each with put-call parity.
Practise the risk-neutral argument: replace the drift with the risk-free rate, discount the expected payoff, and explain in two lines why the real-world drift does not matter.
Calculate delta, gamma, vega, theta and rho for a simple option, then explain a delta hedge and why it needs rebalancing.
Practise volatility questions: estimate historical volatility from price data, back out implied volatility by trial, and describe the smile and what it says about the model.
Finish with past-style questions that mix topics, and replicate the main calculations in R or Excel for Paper B.

Common mistakes in Black-Scholes derivative-pricing model

Using the real-world drift μ in the option price instead of the risk-free rate r.
Fix: Remember that the price depends on hedging, not on forecasts. In pricing, always use r. Use μ only for real-world probabilities.
Mixing up d1 and d2, or using N(d1) where N(d2) belongs.
Fix: Remember that d2 = d1 − σ√T. Check your answer against put-call parity and bounds, such as a call price never exceeding S₀.
Entering volatility as a percentage, such as 20, instead of 0.20, or mixing time units.
Fix: Convert σ to a decimal and T to years before you start, and write the converted values at the top of your working.
Treating the formula as valid for American options or for dividend-paying shares without adjustment.
Fix: Check the option type and the dividend assumption first. Adjust the share price or yield as the extension requires.
Giving only a number in written questions and leaving out the assumptions, notation and comment.
Fix: Use a fixed layout: state assumptions, write the formula in standard notation, show the working, give the result, then add one line of interpretation.
Confusing historical and implied volatility, or claiming the smile supports the model.
Fix: Historical comes from past prices. Implied comes from the current option price. The smile shows that Black-Scholes assumptions do not fully hold.

Last-day revision: Black-Scholes derivative-pricing model

Black-Scholes assumes continuous trading, no transaction costs, constant volatility and a constant risk-free rate.
Share price follows geometric Brownian motion: dS = μS dt + σS dW.
Under GBM, ln(S_t) is normal, so S_t is lognormal.
Call price: c = S₀ N(d1) − K e^(−rT) N(d2).
Put price: p = K e^(−rT) N(−d2) − S₀ N(−d1).
d1 = [ln(S₀ ÷ K) + (r + σ²/2)T] ÷ (σ√T), and d2 = d1 − σ√T.
Put-call parity for European options on a non-dividend share: c + K e^(−rT) = p + S₀.
The real-world drift μ does not appear in the price. Only r and σ do.
Delta of a call is N(d1). Delta of a put is N(d1) − 1.
Gamma and vega are the same for a call and a put with the same strike and expiry.
Implied volatility is the σ that makes the model price match the market price.
A volatility smile shows that constant volatility does not hold in the market.

Black-Scholes derivative-pricing model practice questions

Black-Scholes derivative-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 derivative-pricing model: frequently asked questions

Do I need to derive the Black-Scholes formula for the exam?

Learn the logic of the derivation rather than every algebra step. You should be able to explain the hedging argument and the risk-neutral result, and to apply the formula accurately. Check the IAI syllabus for the depth required.

How should I prepare for the Paper B questions on this chapter?

Practise building the price, d1, d2 and the Greeks in R or Excel from the input values. Check your output against a hand calculation and against put-call parity. Show the method clearly in the answer.

Why does the real-world expected return not affect the option price?

The option can be hedged with the share and cash, so its price is fixed by that hedge. The hedge removes the dependence on expected return. That is why you can value the option as if the share grew at the risk-free rate.

What is the quickest way to check my call or put price?

Use put-call parity and simple bounds. A European call on a non-dividend share must be at least S₀ − K e^(−rT) and at most S₀. If your answer breaks these, recheck d1 and d2.