Implied Volatility

The Black-Merton-Scholes formula gives the price of a European option as a function \fs4 C_{BMS} = C_{BMS}(S(0),K,T,r,\sigma), where

  • \fs4 S(0) is the current spot price of the underlying asset,
  • \fs4 K is the strike price,
  • \fs4 T is the time to maturity,
  • \fs4 r is the (constant) risk free rate of return, and
  • \fs4 \sigma is the volatility of the asset returns

Out of these five determinants of the option price, only the first four are directly observed (or they are part of the contract specification, e.g. the time to maturity). The asset volatility is not.

One the other hand, many European options are traded in exchanges, which makes the option prices for these contracts also observed. The implied volatility is the value of this parameter that makes the theoretical BMS option prices equal to the observed ones. We denote the implied volatility as \fs4 \hat{\sigma}. Mathematically, the implied volatility will satisfy

\fs4 C_{obs} = C_{BMS}(S(0),K,T,r,\hat{\sigma})

Assuming that the market prices all contracts correctly, the implied volatility answers the question: How volatile should be the asset price, on average (from now to maturity), in order for the BMS formula to give the correct price? One would expect implied volatilities for options with different strike prices to be approximately the same, but this is not the case. Implied volatilities exhibit distinct patterns, which are collectively known as the implied volatility skew.

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