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

Market prices of European options on the same share and expiry show implied volatility that is higher for deep out-of-the-money puts than for at-the-money options. Which Black-Scholes assumption does this pattern most directly contradict?

The pattern contradicts the assumption of constant volatility. Under Black-Scholes every strike should give the same implied volatility, so a skew or smile across strikes shows the market does not price the share as having a single constant volatility.

  1. AConstant volatility of the share price across strikesCorrect
  2. BNo arbitrage between the share and the option
  3. CContinuous trading of the share
  4. DThe existence of a risk-free borrowing rate
  5. Short selling of the share is permitted

Explanation

If volatility were constant, inverting Black-Scholes would give the same implied volatility for every strike. A volatility skew or smile shows the market prices different volatilities by strike, and this contradicts constant volatility. The other assumptions are not directly tested by the skew.

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