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Set out what the Black-Scholes-Merton option pricing model assumes.

FRM® Part I · Valuation and Risk Models · The Black-Scholes-Merton Model · FRM1.VR.15.c

The idea

The Black-Scholes-Merton price comes from a riskless hedge of the option with the stock and a risk-free bond. This works under a set of assumptions. The stock price follows geometric Brownian motion with constant volatility, so log returns are normal. Short selling is allowed with full use of the proceeds. There are no transaction costs or taxes and securities are perfectly divisible. The stock pays no dividends during the option's life. There is no riskless arbitrage, and trading is continuous. The risk-free rate is constant for all maturities. Because the hedge removes the stock's risk, the option price depends neither on the stock's expected return nor on investors' risk preferences, which is why risk-neutral valuation gives the right answer. Exam questions ask which assumption fails in practice: fat tails, jumps, volatility smiles and dividends.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which of the following is an assumption of the Black-Scholes-Merton model?

  1. The volatility of the stock rises when its price falls.Why not: Describes the leverage effect seen in equity markets; BSM holds volatility constant.
  2. Trading takes place only at discrete dates, such as the close of each day.Why not: BSM assumes continuous trading so the hedge can be rebalanced at every instant.
  3. The volatility of the stock price is constant over the life of the option. - correct
  4. The stock price can jump by a random amount at any time.Why not: Jumps belong to jump-diffusion models; BSM assumes continuous price paths.

BSM assumes geometric Brownian motion with constant volatility, continuous trading and continuous price paths. The other statements describe empirical features that the model leaves out.

A harder one

Empirical equity returns have heavier tails than a lognormal model implies, and deep out-of-the-money index puts trade at much higher implied volatilities than at-the-money options. Which BSM assumption does this pattern most directly contradict?

  1. Log returns are normally distributed with constant volatility. - correct
  2. There are no transaction costs or taxes.Why not: Frictions affect the cost of hedging but do not explain a strike-dependent pattern in implied volatility.
  3. The risk-free rate is constant for all maturities.Why not: The rate term structure changes discounting, not the relative pricing of different strikes.
  4. Short selling is allowed with full use of the proceeds.Why not: Short-sale limits matter for hedging puts, but the pattern reflects the return distribution, not borrowing.

Higher implied volatility for deep out-of-the-money puts means the market assigns more probability to large falls than a lognormal distribution with constant volatility does.

The hardest tier

A two-year European option is priced when the volatility of the stock is known to be 20% per year in year one and 30% per year in year two, with a constant risk-free rate. A risk manager wants one volatility to use in the BSM formula. The appropriate volatility is closest to:

  1. 25.5% - correct
  2. 25.0%Why not: Averages the two volatilities directly instead of averaging the variances.
  3. 36.1%Why not: Adds the two annual variances without dividing by the number of years.
  4. 30.0%Why not: Uses the volatility prevailing at the end of the option's life rather than a time-weighted average.

With known time-varying volatility, use the average variance: sigma² = (0.20² + 0.30²)/2 = 0.065, so sigma = 25.5%.

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