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IAI Actuarial Core Principles · Economic Modelling · Black-Scholes derivative-pricing model

Under the Black-Scholes model, which statement about the volatility parameter is correct?

Black-Scholes assumes volatility is constant over the option's life. Therefore, if the model held exactly, implied volatility would be the same for all strikes and maturities, giving a flat surface. Observed smiles and skews show that this assumption is not consistent with market prices.

  1. AIt is assumed constant over the life of the option, so the model implies a flat implied volatility across strikesCorrect
  2. BIt is assumed to increase with the strike price of the option
  3. CIt is assumed to follow its own stochastic process correlated with the share price
  4. DIt is assumed to equal the risk-free rate of interest
  5. It is assumed to decline as the option approaches expiry

Explanation

Black-Scholes assumes the share price follows geometric Brownian motion with constant volatility. If the model were true, implied volatilities from options at different strikes and maturities would all be equal, i.e. a flat surface. Stochastic volatility is an extension, not part of the basic model.

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