FRM Exam Part I · The Black-Scholes-Merton Model

Black-Scholes-Merton Assumptions and the Stock Price Process

Updated 11 October 2026 · Fact-checked

The Black-Scholes-Merton model assumes a stock follows geometric Brownian motion: dS = μS dt + σS dz. Prices are therefore lognormal at any future date. A portfolio of the option and the stock is riskless over a short interval, so it must earn r. That no-arbitrage argument gives the BSM differential equation.

Understand Black-Scholes-Merton Assumptions and Stock Price Process

Start with the stock price. A stock earns an expected return and also moves randomly. The BSM model captures both in one equation: dS = μS dt + σS dz. Here μ is the expected return per year, σ is the volatility per year, and dz is a tiny random shock, normally distributed with mean 0 and variance dt. This is geometric Brownian motion. The change in price is proportional to the current price, so percentage returns, not dollar changes, are what behave randomly.

Because returns are normal, the log of the price is normal. Applying Ito's lemma gives d(ln S) = (μ − σ²/2) dt + σ dz. So ln S_T is normal with mean ln S₀ + (μ − σ²/2)T and standard deviation σ√T. That makes S_T lognormal: it can never go below zero, and it has a right-skewed distribution. The term σ²/2 is a drift correction. It is why the expected log return is lower than μ, while the expected price is still E(S_T) = S₀ e^(μT).

The key assumptions are: the stock follows GBM with constant μ and σ; short selling is allowed with full use of proceeds; there are no transaction costs or taxes; securities are perfectly divisible; the stock pays no dividends during the option's life (in the basic model); there are no riskless arbitrage opportunities; trading is continuous; and the riskless rate r is constant and the same for all maturities. Real markets break several of these, which is why you see volatility smiles and jumps.

The differential equation comes from a no-arbitrage argument. Build a portfolio that is short one derivative and long ∂f/∂S shares. Over a short interval the random terms (dz) cancel, so the portfolio is riskless. A riskless portfolio must earn r, otherwise there is arbitrage. This gives: ∂f/∂t + r S ∂f/∂S + ½σ²S² ∂²f/∂S² = r f.

Look at what is missing from this equation: μ. The expected return of the stock does not appear. This leads to risk-neutral valuation. You can value the derivative as if investors were risk neutral, so the stock drifts at r instead of μ. Then you discount the expected payoff at r. The result is the same as in the real world because the equation never depended on risk preferences.

Key formulas to remember

Stock price process (GBM): dS = μS dt + σS dz; μ is expected return, σ is volatility, dz ~ Normal(0, dt). Percentage change dS/S is normal.
Log price process: d(ln S) = (μ − σ²/2) dt + σ dz; Follows from Ito's lemma. The drift of ln S is μ − σ²/2, not μ.
Distribution of ln S_T: ln S_T ~ Normal( ln S₀ + (μ − σ²/2)T , σ²T ); So S_T is lognormal. Standard deviation of ln S_T is σ√T.
Expected future price: E(S_T) = S₀ e^(μT); Mean of the lognormal variable. Under risk-neutral valuation use r in place of μ.
Variance of S_T: Var(S_T) = S₀² e^(2μT) (e^(σ²T) − 1); Standard lognormal result.
BSM differential equation: ∂f/∂t + r S ∂f/∂S + ½σ²S² ∂²f/∂S² = r f; Holds for any derivative on a non-dividend-paying stock. μ does not appear.
Riskless hedge portfolio: Π = −f + (∂f/∂S) S; Short one derivative, long ∂f/∂S shares. The portfolio is riskless only for a short interval and must be rebalanced.
Risk-neutral valuation: f = e^(−rT) E*[payoff]; E* assumes the stock drifts at r, so E*(S_T) = S₀ e^(rT).

How to solve Black-Scholes-Merton Assumptions and Stock Price Process questions

Use this method for any question on BSM assumptions, the lognormal price process or the derivation of the differential equation.

1Identify what is asked: an assumption, a distribution property, a probability or expected value, or a step in the derivation.
2If it is about the process, write dS = μS dt + σS dz and note that the price change is proportional to S.
3For distribution questions, convert to ln S_T. Mean = ln S₀ + (μ − σ²/2)T and standard deviation = σ√T. Use these to get a normal z-value.
4For expected prices, use E(S_T) = S₀ e^(μT). Do not use the mean of the log.
5For derivation questions, recall the chain: build a riskless portfolio, so the dz terms cancel, so it must earn r, which gives the PDE with no μ.
6For valuation questions, decide if risk-neutral valuation applies. Replace μ with r, take the expected payoff, then discount at r.
7Check units: μ, σ and r annual, T in years. Check that the answer is sensible (a price must be positive).

Quickest way: Three-check shortcut

When to use it: Use it when a multiple-choice question gives four plausible statements about the model and time is short.

Check whether the statement is about S or ln S. S is lognormal and ln S is normal.
Check for the σ²/2 trap. Log drift is μ − σ²/2. Price drift is μ.
Check whether μ appears in anything about option pricing. If an option value depends on μ, it is wrong.
Eliminate any option that says the stock can be negative, volatility is stochastic, or the hedge is permanent in the basic model.

Common mistakes in Black-Scholes-Merton Assumptions and Stock Price Process

Using μ as the mean of ln S_T.
Students forget the Ito correction and assume the log return has the same drift as the price.
Fix: Always subtract σ²/2: the mean of ln S_T minus ln S₀ is (μ − σ²/2)T.
Saying the stock price is normally distributed.
The percentage return and the log price are normal, so the terms get mixed up.
Fix: The log price is normal, so the price is lognormal. It is skewed and always positive.
Putting μ in the option pricing equation or the risk-neutral valuation.
Students think a riskier or higher-return stock must make the option worth more directly.
Fix: The hedge removes the randomness, so only r and σ matter. Under risk-neutral valuation the drift is r.
Thinking risk-neutral valuation says investors are actually risk neutral.
The name is misleading.
Fix: It is a pricing device. The equation is free of risk preferences, so any consistent world gives the same price, and the risk-neutral world is the easiest.
Treating the delta hedge as permanent.
The riskless portfolio is described as fixed in many summaries.
Fix: ∂f/∂S changes with S and time, so the portfolio is riskless only over an instant and needs continuous rebalancing.
Mixing up the dz variance, using dz variance as dt standard deviation.
Both dt and √dt appear in the formulas.
Fix: dz has variance dt and standard deviation √dt. Over T years the standard deviation of the shock is σ√T.

Worked examples

Example 1

A non-dividend-paying stock has S₀ = $50, expected return μ = 12% per year and volatility σ = 20% per year. Under the BSM model, what are the expected value of S_T and the expected value of ln S_T for T = 2 years? (Use e^0.24 = 1.2712.)

Show the solution

Expected price: E(S_T) = S₀ e^(μT) = 50 × e^(0.12 × 2) = 50 × e^0.24.
50 × 1.2712 = 63.56.
Mean of ln S_T = ln S₀ + (μ − σ²/2)T.
σ²/2 = 0.04 ÷ 2 = 0.02, so μ − σ²/2 = 0.12 − 0.02 = 0.10.
Drift over 2 years = 0.10 × 2 = 0.20.
ln 50 = 3.9120, so mean of ln S_T = 3.9120 + 0.20 = 4.1120.

Answer: E(S_T) ≈ $63.56 and E[ln S_T] ≈ 4.112 (the stock price at the mean of the log would be about $61.15, below the expected price).

Example 2

Using the same stock (S₀ = $50, μ = 12%, σ = 20%), what is the standard deviation of ln S_T for T = 4 years, and which of these best describes the BSM treatment of the stock's expected return when valuing a call on it? A) It enters the option equation through the drift. B) It is replaced by the risk-free rate under risk-neutral valuation. C) It is replaced by the dividend yield. D) It must be estimated using CAPM beta.

Show the solution

Standard deviation of ln S_T = σ√T.
√4 = 2, so σ√T = 0.20 × 2 = 0.40, or 40%.
The BSM differential equation contains S, t, r and σ, but not μ.
So in risk-neutral valuation the stock is assumed to grow at r, and the payoff is discounted at r.
A is wrong because μ does not appear. C is wrong because the basic model has no dividends. D is wrong because the equation does not depend on risk preferences or beta.

Answer: Standard deviation of ln S_T = 40%. The correct statement is B.

Exam tips

Expect a numeric question on the mean and standard deviation of ln S_T, then a normal probability. Remember μ − σ²/2 and σ√T.
Watch the wording: 'expected price' uses μ, 'expected log price' uses μ − σ²/2, and 'risk-neutral' uses r.
Conceptual questions often ask what is absent from the BSM equation. The answer is μ, and any risk preference.
Know the list of assumptions and which real-world features violate them: jumps, stochastic volatility, dividends, transaction costs, discrete trading.
Use a financial calculator or scientific mode for e^x and ln. Write each intermediate value to avoid rounding drift.

Practice questions from The Black-Scholes-Merton Model

Black-Scholes-Merton Assumptions and Stock Price Process 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-Merton Assumptions and Stock Price Process: frequently asked questions

Why is the stock price lognormal in the BSM model?

Returns over small intervals are normal, so the log of the price changes by normal increments. The sum of normal increments is normal, so ln S_T is normal. Exponentiating gives a lognormal price that cannot be negative.

Why does the BSM equation not contain the stock's expected return?

The hedge portfolio of the option and delta shares has no random term over a short interval. It must earn the risk-free rate, so the stock's expected return drops out of the argument. Only r and σ remain.

What is risk-neutral valuation?

It is a method that prices a derivative as the discounted expected payoff when the stock drifts at the risk-free rate. It works because the pricing equation does not depend on risk preferences. It does not mean real investors are risk neutral.

What are the main BSM assumptions I should memorise for FRM Part I?

The stock follows GBM with constant μ and σ, there are no dividends in the basic model, no transaction costs or taxes, short selling is allowed, trading is continuous and the risk-free rate is constant. There are also no riskless arbitrage opportunities.