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Show what put-call parity implies for the implied volatilities of calls and puts.

FRM® Part II · Market Risk Measurement and Management · Volatility Smiles and Volatility Surfaces · FRM2.MR.17.b

The idea

Put-call parity for European options on a non-dividend-paying asset says c + K × exp(−rT) = p + S0, with a dividend or foreign-rate term added where relevant. It is an arbitrage relationship, so it holds for market prices and also for Black-Scholes-Merton prices computed with any single volatility. Subtract the two versions and the call's gap between market price and model price equals the put's gap between market price and model price. Because both prices rise with volatility, a European call and a European put with the same strike and expiry must have the same implied volatility. That is why smiles are built from the more liquid out-of-the-money options, and why a difference between call and put implied volatilities signals an arbitrage. Parity is exact only for European options, so American options and dividend estimates can create small differences.

Formulas

Equal pricing gaps implied by put-call parity

cmarket − cmodel = pmarket − pmodel

cmarket
the market price of the European call
cmodel
the Black-Scholes-Merton call price at some volatility
pmarket
the market price of the European put
pmodel
the Black-Scholes-Merton put price at the same volatility

Same strike and expiry, European options. Because both prices rise with volatility, the call and put share one implied volatility.

A practice question

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

Why must the implied volatility of a European call equal that of a European put with the same strike and maturity?

  1. Because Black-Scholes-Merton values only calls, and a put is then priced by borrowing the call's volatility as an input.Why not: The model values puts directly; the equality comes from parity, not from a modelling convention.
  2. Parity is an arbitrage relation that model prices also obey, so both options show the same gap to the model. - correct
  3. Because a call and a put with the same strike have the same delta, so they react identically to volatility.Why not: Their deltas differ (call delta minus put delta equals 1), and equal deltas would not explain equal implied volatility.
  4. Because parity follows from the lognormal assumption and fails once the smile departs from lognormality.Why not: Parity is model-free: it depends on a replicating portfolio, not on the distribution of the underlying.

Parity is derived by arbitrage, so it holds at any price level. If c_market - c_model = p_market - p_model, and both prices rise with volatility, the call and put share one implied volatility.

A harder one

A 1-year European call and put (strike 105, spot 100, risk-free rate 3%, no dividends) are valued by Black-Scholes-Merton at 20% volatility at 7.13 and 9.02. The market price of the call is 7.93. Assuming parity holds in the market, the market price of the put is closest to:

  1. 9.02Why not: Leaves the put at its 20% model value; parity forces the put to carry the same pricing gap as the call.
  2. 8.22Why not: Subtracts the call's pricing gap; the call and put gaps have the same sign.
  3. 12.93Why not: Applies parity without discounting the strike: p = c + K - S. The strike must be discounted at 3% for a year.
  4. 9.82 - correct

Call gap = 7.93 - 7.13 = 0.80. Parity gives the same gap for the put, so the market put is 9.02 + 0.80 = 9.82.

The hardest tier

On a dividend-paying stock, a dealer finds that the implied volatility of an American call is 1 volatility point below that of an American put with the same strike and expiry. Which statement is most accurate?

  1. Parity is model-free, so any gap in implied volatility is an arbitrage whatever the exercise style or dividends.Why not: Parity becomes an inequality for American options because of the early exercise premium.
  2. The call's lower volatility shows the Black-Scholes-Merton model is invalid, so the put volatility should be used for both.Why not: A modelling difference is not proof of an invalid model, and no rule says one should override the other.
  3. Parity is exact only for European options, so a small gap need not be an arbitrage here. - correct
  4. Parity applies to implied volatilities only when the interest rate is zero, so a gap is normal otherwise.Why not: Parity holds with the discounted strike at any interest rate; the exercise style and dividends are the issue.

For European options on a non-dividend stock, parity forces equal implied volatilities. For American options parity only gives bounds, and dividend estimates enter the inversion, so small differences are expected.

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