IAI Actuarial Core Principles · Economic Modelling

Principles of Option Pricing for CM2 Economic Modelling

Option pricing finds the fair price of an option by using no-arbitrage. You build a portfolio of the underlying share and cash that copies the option's payoff, then the option must cost the same as that portfolio. Binomial trees do this step by step. Black-Scholes does it in continuous time.

What this chapter covers

This chapter explains how options are valued without guessing what the share price will do. You start with calls, puts and their payoffs. Then you learn the single idea behind everything else: no arbitrage. If two positions give the same payoff in every state, they must have the same price today.

From that idea you get price bounds, put-call parity, and the reasons why early exercise of American options may or may not pay. You then move to the binomial tree, where replication and risk-neutral valuation give a price. Black-Scholes is the continuous-time limit of the same logic, with a stated set of assumptions.

In CM2 this chapter sits beside asset valuations, liability valuations and measures of investment risk. Option theory carries 30% of the 2026 CM2 syllabus weighting. The same tools help you value guarantees and options embedded in insurance and pension contracts, and they link to the stochastic models and risk-neutral ideas used elsewhere in the paper. Expect both Paper A written questions and Paper B computer work to use them.

Option theory is 30% of the 2026 CM2 syllabus weighting, the joint-largest topic with asset valuations, so this chapter deserves real effort. It is also very rewarding. Most questions follow a small set of methods: payoff diagrams, parity, a one- or two-period tree, and Black-Scholes inputs. If you practise these, you can earn full method marks even when arithmetic slips. MCQs often test conceptual points such as how a parameter moves a price, so clear understanding also wins quick marks. In Paper B, you may be asked to build a tree or compute a Black-Scholes price in code or a spreadsheet, so the methods must be automatic.

Principles of option pricing: topics in the order to study them

1Options Basics and Payoff DiagramsYou need the payoff of every call, put, long and short position before any pricing makes sense.
2No-Arbitrage Principle and Option BoundsThis is the core idea behind every later result, and it gives you simple price limits to test answers against.
3Put-Call ParityIt is the first real use of no-arbitrage and links call and put prices directly.
4Factors Affecting Option PricesOnce you know bounds and parity, you can reason about how price, strike, time, volatility, rates and dividends move values.
5Early Exercise of American OptionsIt builds on bounds and the effect of interest and dividends to decide when exercising early can be sensible.
6Binomial Tree Option PricingIt turns no-arbitrage into a working method with replication and risk-neutral probabilities, and prepares you for the continuous model.
7Black-Scholes Model and AssumptionsIt comes last because it is the continuous-time version of the tree, and you must know its assumptions and limits.

How to prepare Principles of option pricing

Treat this chapter as one idea applied seven ways. Learn the logic first, then drill the calculations until they are quick.

Draw payoff and profit diagrams for long and short calls and puts from memory until you can do it without notes.
Write out the no-arbitrage argument in your own words, then prove parity and one or two bounds by comparing two portfolios that give the same payoff.
Make a one-page table of how each factor affects calls and puts, and practise giving a short reason for each direction.
Practise one-period trees: find the replicating portfolio, then check your price with risk-neutral probabilities. Both methods must agree.
Extend to two-period trees, including American options where you compare exercise value with continuation value at each node.
List the Black-Scholes assumptions and practise the price calculation using the standard normal distribution, noting where each input enters.
Finish with timed past-style questions, and repeat key calculations in R or Excel for Paper B, showing method, formula and result.

Common mistakes in Principles of option pricing

Using real-world probabilities to discount tree payoffs.
Fix: Always find the risk-neutral probability from u, d and r, and say clearly that you are valuing under that measure.
Applying put-call parity without checking the conditions.
Fix: State the assumptions first. If dividends are paid, adjust the share price by the present value of the dividends.
Saying an American call should never be exercised early in every case.
Fix: Quote the result only for a non-dividend-paying underlying. With dividends, early exercise just before a dividend can be optimal.
Forgetting the exercise-value check at each node of an American tree.
Fix: At each node take the larger of continuation value and immediate exercise value, and write both numbers down.
Giving the direction of a factor's effect without a reason.
Fix: Give a short reason such as 'higher strike lowers a call's payoff' or 'more time gives more chance of a favourable move'. Note that time can lower a European put's value.
Mixing up the inputs in Black-Scholes, such as using percent for r or volatility, or the wrong time unit.
Fix: Convert everything to decimals and years before substituting, and check that your answer lies within the no-arbitrage bounds.

Last-day revision: Principles of option pricing

Call payoff at expiry is max(S − K, 0); put payoff is max(K − S, 0).
No arbitrage means identical payoffs must have identical prices.
European put-call parity for a non-dividend share: C − P = S₀ − K e^(−rT).
A call price cannot be negative and cannot exceed the share price.
A higher volatility raises the value of both calls and puts.
An American option is worth at least as much as the matching European option.
For a non-dividend share, early exercise of an American call is not optimal.
Risk-neutral probability in a binomial tree: q = (e^(rΔt) − d) ÷ (u − d).
Option price at a node = e^(−rΔt) × [q × up value + (1 − q) × down value].
For American options, compare exercise value with continuation value at every node.
Black-Scholes assumes constant volatility and interest rate, lognormal share prices, no transaction costs and continuous trading.
Risk-neutral valuation is a pricing tool; it does not mean investors are risk neutral.

Principles of option pricing practice questions

Principles of option pricing in other exams

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

Principles of option pricing: frequently asked questions

How much of CM2 is option pricing?

The 2026 syllabus weighting for option theory in CM2 is 30%. It is one of the largest topics, so it is worth thorough preparation.

Do I need to memorise the Black-Scholes formula?

You should know it well enough to use it and explain each input. Spend equal time on its assumptions and on why the model is only an approximation in practice.

Is the binomial tree or Black-Scholes more important?

Both matter, but learn the tree first. It shows the replication logic that Black-Scholes relies on, and tree questions are easy to set and mark by method.

What should I practise for Paper B?

Practise building a binomial tree and computing a Black-Scholes price in R or Excel. Show the method, formula, inputs and result, and sanity-check against no-arbitrage bounds.