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FRM Part I · FRM Exam Part I

The Black-Scholes-Merton Model for FRM Part I

The Black-Scholes-Merton model prices European options by assuming a stock follows geometric Brownian motion with constant volatility and a constant risk-free rate. You solve it by finding d1 and d2, then applying the call or put formula. Adjust the inputs for dividends, and treat American options separately.

What this chapter covers

This chapter covers the model that underlies most option pricing in practice. It starts with the assumptions: the stock price follows a lognormal process, volatility and the risk-free rate are constant, there are no arbitrage opportunities, and trading is frictionless. It then moves to the pricing formulas for European calls and puts, and to the two ways of getting volatility: estimated from past returns, or backed out of market option prices.

The second half extends the model. You adjust it for dividends and for other underlyings such as indices, currencies and futures. You then look at warrants and employee stock options, where exercise creates new shares and dilutes existing holders. The chapter ends with American options, where early exercise is possible and the closed-form formula no longer applies in general.

The chapter connects to several other parts of the paper. It builds on probability and the lognormal distribution from Quantitative Analysis, and on option payoffs and put-call parity from Financial Markets and Products. It feeds directly into the Greeks, hedging and value-at-risk work in Valuation and Risk Models. If you are solid here, those later topics become much easier.

Option valuation questions are numerical, so they reward candidates who can run a formula cleanly under time pressure. The 100 questions in 4 hours leave about 2.4 minutes each, so you need to find d1 and d2 and read the normal values quickly. The ideas also recur elsewhere: implied volatility, the lognormal assumption and dividend adjustments show up in risk-model questions. Mastering this chapter earns direct marks and makes neighbouring chapters easier. GARP does not publish a pass mark, so you should aim for a secure grasp rather than gambling on skipping it.

The Black-Scholes-Merton Model: topics in the order to study them

  1. 1Black-Scholes-Merton Assumptions and Stock Price ProcessEverything else rests on the lognormal price process and the assumptions, so learn them first.
  2. 2Black-Scholes-Merton Formula for European OptionsOnce the process is clear, the pricing formula and its inputs make sense and you can start practising calculations.
  3. 3Volatility: Historical and ImpliedVolatility is the one input you cannot observe directly, so you need to know how it is estimated and what implied volatility means.
  4. 4Options on Dividend-Paying Stocks and Other UnderlyingsThis is the formula with small adjustments, which is easier once the base version is automatic.
  5. 5Warrants, Employee Stock Options and DilutionThese apply the pricing ideas to instruments that create new shares, so they need the basic model first.
  6. 6American Options and Early ExerciseThis closes the chapter by showing where the closed-form model stops working and why early exercise matters.

How to prepare The Black-Scholes-Merton Model

Treat this as a calculation chapter with a conceptual layer. Build the concepts first, then drill the arithmetic until it is quick.

  1. Write out the assumptions from memory and note which ones are unrealistic in practice, such as constant volatility.
  2. Learn the call and put formulas, d1 and d2, and what each symbol means. Check you can derive the put from the call using put-call parity.
  3. Practise computing d1 and d2 with a calculator, then reading N(d) from a normal table or given values. Do at least ten full examples.
  4. Study how volatility is estimated from historical returns and how implied volatility is found from a market price. Be clear on what each tells you.
  5. Work through the adjustments for dividends and for other underlyings, and note which input changes in each case.
  6. Compare European and American options and learn when early exercise can be optimal. Finish with mixed practice questions under timed conditions.

Common mistakes in The Black-Scholes-Merton Model

  • Using the wrong sign or the wrong N(·) in the put formula.

    Fix: Learn the put as K e^(−rT) N(−d2) − S0 N(−d1), or compute the call and use put-call parity to get the put.

  • Mixing up volatility and variance, or forgetting the √T term.

    Fix: Write σ, T, σ√T and σ²÷2 as separate steps before substituting into d1.

  • Using time in months or days instead of years.

    Fix: Convert T to years first, for example six months is 0.5, and use the same units for r and σ.

  • Forgetting to adjust for dividends or yields.

    Fix: Check the underlying and any dividend information first. Deduct the present value of known dividends from the stock price, or use the yield-adjusted version.

  • Treating implied volatility as a forecast from historical data.

    Fix: Remember that historical volatility comes from past returns, while implied volatility comes from today's option prices through the model.

  • Assuming early exercise is always sensible for American options.

    Fix: Learn the conditions: early exercise of a call is mainly relevant before a dividend, and a deep in-the-money put may be exercised early.

Last-day revision: The Black-Scholes-Merton Model

  • The model assumes the stock price follows geometric Brownian motion, so returns are normal and prices are lognormal.
  • Volatility and the risk-free rate are assumed constant, and the model ignores transaction costs.
  • d1 = [ln(S0 ÷ K) + (r + σ²÷2)T] ÷ (σ√T), and d2 = d1 − σ√T.
  • European call: c = S0 N(d1) − K e^(−rT) N(d2).
  • European put: p = K e^(−rT) N(−d2) − S0 N(−d1).
  • Put-call parity for European options: c + K e^(−rT) = p + S0 for a non-dividend stock.
  • Implied volatility is the volatility that makes the model price equal the market price, and it is found by iteration.
  • Historical volatility is estimated from past returns, and it assumes the past is a guide to the future.
  • For a stock with known dividends, subtract the present value of the dividends from S0 before using the formula.
  • For options on indices and currencies, replace the dividend adjustment with a continuous yield or the foreign interest rate.
  • Warrants and employee options create new shares when exercised, which dilutes existing shareholders.
  • An American call on a non-dividend stock is never optimally exercised early, but an American put may be.

The Black-Scholes-Merton Model practice questions

The Black-Scholes-Merton Model in other exams

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

The Black-Scholes-Merton Model: frequently asked questions

Do I need to memorise the Black-Scholes-Merton formula for FRM Part I?

Yes, you should know the formulas for d1, d2 and the call and put prices. Questions are numerical and time is limited, so you cannot afford to rebuild the formula during the exam.

Can I use a financial calculator for this chapter?

A calculator helps with the logarithm, exponential and square root steps. You still need to read the normal values N(d1) and N(d2), which questions often give or which you work out from the table or function you are allowed to use. Check GARP's current calculator rules before the exam.

What is the difference between historical and implied volatility?

Historical volatility is estimated from past price returns. Implied volatility is the figure that, when put into the model, reproduces the current market price of the option. One looks backward and the other reflects what the market is pricing now.

Does the model work for American options?

Not directly in general. The closed-form formula is for European options. For American options you may need numerical methods such as binomial trees, although for a call on a non-dividend stock the European price is a valid answer because early exercise is not optimal.