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IAI Actuarial Core Principles · Economic Modelling

Stochastic Models for Security Prices: Brownian Motion to Black-Scholes

Stochastic models for security prices describe how a price moves randomly over time. You start with random walks and Brownian motion, use Ito's lemma to handle stochastic differential equations, apply geometric Brownian motion for lognormal prices, then price options with Black-Scholes. Solve questions by stating assumptions, then working step by step.

What this chapter covers

This chapter in CM2 Economic Modelling builds a mathematical description of how security prices change. You begin with the random walk, move to its continuous-time limit, Brownian motion, and then write price dynamics as stochastic differential equations. Ito's lemma is the main tool for finding the dynamics of a function of a random process.

The central model is geometric Brownian motion, where the log of the price is normally distributed, so the price is lognormal. The chapter then uses this model to derive the Black-Scholes framework for option pricing. It ends with alternatives: discrete-time models, jump models and mean-reverting models. These exist because the basic model does not fit all observed behaviour.

The chapter links directly to the rest of the paper. Measures of investment risk use return distributions that come from these models. Asset valuations rely on assumptions about price behaviour. Option theory uses Black-Scholes and its assumptions. If you understand this chapter well, those later areas become much easier. For Paper B, you may also be asked to simulate or compute values in R or Excel, so practise the formulas numerically too.

Option theory and asset valuations together carry a large share of the 2026 CM2 syllabus weighting, and both depend on the models in this chapter. Questions here reward method: stating assumptions, applying Ito's lemma correctly, and using the lognormal distribution to get probabilities and expected values. These are learnable skills that give reliable marks in both written questions and computer-based work. A weak base here also costs you in later chapters, so the effort pays back more than once.

Stochastic models for security prices: topics in the order to study them

  1. 1Random Walks and Brownian MotionEverything else is built on these; learn the properties of increments first.
  2. 2Stochastic Differential Equations and Ito's LemmaYou need Ito's lemma before you can derive or manipulate any price model.
  3. 3Geometric Brownian Motion and Lognormal ModelThis applies Ito's lemma to the standard price model and gives you the lognormal distribution.
  4. 4Black-Scholes Model and Option PricingIt uses GBM and its assumptions, so it comes after you are comfortable with both.
  5. 5Other Models: Discrete Time, Jump and Mean-RevertingThese are best understood as fixes to the limits of GBM, so study them last.

How to prepare Stochastic models for security prices

Treat this chapter as a chain. Each topic uses the one before, so do not skip ahead. Mix theory with calculation from the start.

  1. Learn the defining properties of Brownian motion: starts at zero, independent increments, normally distributed increments with variance equal to the time step, continuous paths. Write them from memory.
  2. Practise Ito's lemma on simple functions until the steps are automatic. Always write the second-order term, and remember dt² = 0, dt·dB = 0 and (dB)² = dt.
  3. Derive the GBM solution yourself, then compute the mean, variance and probabilities for the lognormal price. Do this with numbers, not just symbols.
  4. Write out the Black-Scholes assumptions as a list and state what each one means. Then practise pricing calls and puts with the formula, including put-call parity.
  5. Make a comparison table on paper for the other models: what each captures, what it fixes in GBM, and where it fails.
  6. Do past-paper style questions under time. For Paper B practice, reproduce your calculations in R or Excel, and check the results against your hand working.
  7. In the last week, redo only the questions you got wrong and recite the key formulas and assumptions.

Common mistakes in Stochastic models for security prices

  • Forgetting the second-order term in Ito's lemma.

    Fix: Always write the full expansion first, then substitute (dB)² = dt before simplifying.

  • Confusing the mean of ln S with the log of the mean of S.

    Fix: Write down the distribution of ln S(t) first, then convert. Remember that the mean of ln S involves μ − σ²/2.

  • Using the real-world drift μ when pricing options.

    Fix: For option pricing, replace μ by the risk-free rate r. Say that you are working under the risk-neutral measure.

  • Mixing up volatility and variance, or using the wrong time unit.

    Fix: Convert σ to decimal and time to years before substituting. Check whether the question gives σ or σ².

  • Quoting Black-Scholes without stating its assumptions.

    Fix: Learn the assumptions as a short list and link each model in the last topic to the assumption it relaxes.

  • Writing only a final number in calculation questions.

    Fix: Show the formula, the inputs, the intermediate values such as d1 and d2, and then the result. Marks are given for method.

Last-day revision: Stochastic models for security prices

  • Brownian motion has independent, normally distributed increments: B(t) − B(s) ~ N(0, t − s) for t > s.
  • Brownian motion paths are continuous but nowhere differentiable.
  • Ito's lemma has an extra second-order term compared with ordinary calculus.
  • Multiplication rules: (dB)² = dt, dt·dB = 0, (dt)² = 0.
  • GBM: dS = μS dt + σS dB.
  • Under GBM, ln S(t) is normal, so S(t) is lognormal.
  • Under GBM, E[S(t)] = S(0) e^(μt).
  • Black-Scholes assumes constant volatility and risk-free rate, no dividends in the basic form, no transaction costs and continuous trading.
  • Call price: C = S N(d1) − K e^(−rT) N(d2).
  • Put-call parity for European options on a non-dividend-paying share: C − P = S − K e^(−rT).
  • Option prices use the risk-neutral drift r, not the real-world drift μ.
  • Jump models add sudden price changes, and mean-reverting models pull the value back towards a long-run level.

Stochastic models for security prices practice questions

Stochastic models for security prices in other exams

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

Stochastic models for security prices: frequently asked questions

Is this chapter more theory or calculation?

Both. You need to understand the properties and derivations to answer written questions, and you need to compute values such as lognormal probabilities and option prices. Practise both together so that each supports the other.

Do I need to memorise the Black-Scholes formula?

Yes, learn the call and put formulas, d1 and d2, and put-call parity. Also know the assumptions and what each input represents. Understanding the logic makes the formula easier to remember.

How should I study the other models at the end of the chapter?

Study each one by asking what feature of real prices it captures that GBM does not. Note the basic form of each model and its main weakness. Questions often ask you to compare or comment, not to derive.

How does this chapter help in the computer-based paper?

The same formulas are used in R or Excel to simulate paths, compute probabilities and price options. If you can do a calculation by hand, you can set it up in software. Check your software output against a hand calculation.