# LO 15.2: Explain the im plications o f put-call parity on the im plied volatility o f call

LO 15.2: Explain the im plications o f put-call parity on the im plied volatility o f call and put options.
Recall that put-call parity is a no-arbitrage equilibrium relationship that relates European call and put option prices to the underlying assets price and the present value of the options strike price. In its simplest form, put-call parity can be represented by the following relationship:
c – p = S – PV(X)
where: c p S PV(X) = present value of the strike = price of a call = price of a put = price of a call = price of a put = price of the underlying security
PV(X) can be represented in continuous time by:
PV(X) = Xe-rT
where: r = risk-free rate T =
time left to expiration expressed in years
Since put-call parity is a no-arbitrage relationship, it will hold whether or not the underlying asset price distribution is lognormal, as required by the Black-Scholes-Merton (BSM) option pricing model.
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Topic 15 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
If we simply rearrange put-call parity and denote subscripts for the option prices to indicate whether they are market or Black-Scholes-Merton option prices, the following two equations are generated:
Pmkt + S0e-V = cmkt + PV(X)
P b SM + S 0e qt = CBSM + ^ ( X )
Subtracting the second equation from the first gives us:
Pmkt PBSM = Cmkt CBSM
This relationship shows that, given the same strike price and time to expiration, option market prices that deviate from those dictated by the Black-Scholes-Merton model are going to deviate in the same amount whether they are for calls or puts. Since any deviation in prices will be the same, the implication is that the implied volatility of a call and put will be equal for the same strike price and time to expiration.
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