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Define implied volatilities and outline how the Black-Scholes-Merton model is used to back them out of options prices in the market.

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

The idea

Implied volatility is the volatility that, when put into the BSM formula, makes the model price equal the option's observed market price. Volatility is the only BSM input that cannot be observed directly, and the formula cannot be inverted algebraically, so it is found by trial and error. Guess a volatility, compare the model price with the market price, and refine the guess until the two match, using bisection, interpolation or Newton-Raphson, which uses vega to size each step. Because a European option price rises steadily with volatility, there is a single solution whenever the market price lies inside the no-arbitrage bounds. Implied volatility is forward-looking, unlike historical volatility, and it differs across strikes and maturities, which is the smile or skew. The VIX is built from index option implied volatilities, and a higher option price means a higher implied volatility.

Formulas

Newton-Raphson step for implied volatility

σ₁ = σ₀ + (cM − cB) ÷ vega

σ₀
the current trial volatility
cM
the market price of the option
cB
the BSM price at σ₀
vega
the option's vega at σ₀, in price per unit of volatility

Repeat until the model price matches. If vega is quoted per percentage point, the step comes out in points; moving the volatility the wrong way when the model price is too low is the trap.

A practice question

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

Two European calls on the same stock have the same strike and expiry, and the same risk-free rate applies. Dealer A quotes 4.20 and dealer B quotes 3.90. Which statement about their implied volatilities is correct?

  1. A's implied volatility is lower, because a higher price implies lower risk.Why not: Reverses the relationship; vega is positive, so a higher price needs a higher volatility.
  2. They are equal, because both options have the same underlying stock and therefore the same volatility.Why not: Implied volatility is backed out from each price; different prices give different implied volatilities.
  3. They cannot be compared without knowing the stock's expected return.Why not: The expected return does not enter BSM, so the comparison needs only the prices and inputs.
  4. A's implied volatility is higher, because the BSM call price increases with volatility. - correct

A European call's BSM price rises steadily with volatility (vega > 0), so the higher quote corresponds to the higher implied volatility.

A harder one

At a volatility of 20% the BSM value of a call is 4.10, and at 30% it is 5.60. The market price is 5.00. Using linear interpolation between these two points, the implied volatility is closest to:

  1. 25.0%Why not: Takes the midpoint of 20% and 30% without weighting by the position of 5.00 between 4.10 and 5.60.
  2. 26.0% - correct
  3. 24.0%Why not: Reverses the interpolation weights, using (5.60 - 5.00)/(5.60 - 4.10) = 0.4 in place of 0.6.
  4. 30.0%Why not: Takes the upper bracket without interpolating.

Fraction = (5.00 - 4.10)/(5.60 - 4.10) = 0.60. Implied volatility = 20% + 0.60 x 10% = 26.0%.

The hardest tier

A call has a market price of 7.00. At a trial volatility of 25% the BSM value is 6.40, and the option's vega is USD 0.15 per one percentage point of volatility. After one Newton-Raphson step, the new estimate of the implied volatility is closest to:

  1. 21.0%Why not: Subtracts the price gap divided by vega, moving volatility down when the model price is too low.
  2. 25.6%Why not: Adds the 0.60 price gap to the volatility directly as 0.6 percentage points, ignoring vega.
  3. 27.3%Why not: Scales volatility in proportion to the price ratio, 25% x 7.00/6.40, instead of using vega.
  4. 29.0% - correct

Newton step: new sigma = old sigma + (market price - model price)/vega = 25% + (7.00 - 6.40)/0.15 per point = 25% + 4 points = 29.0%.

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