put‐call parity

[ckyeh]

Dear David;

2010-5-a-Market-Risk page 7:
Explain how put‐call parity indicates that the implied volatility used to price call options is the same used to price put options.

Could you explain it more?

Shouldn't Call Black-Scholes = Call market, since we use Call market to get implied volatility.
and put this implied volatility to Black-Scholes equation, we get Call Black-Scholes.
Then Call Black-Scholes should equals to Call market, right?
Same as put option, Put Black-Scholes = Put market.

But how to get that put‐call parity indicates that the implied volatility used to price call options is the same used to price put options?

I couldn't understand it clearly!

very thanks!

 

[archlight]

what it says is you should use same volatility (implied) to price call and put. otherwise you will violate call-put parity which is derived by non-arbitrage approach.

 

[David Harper CFA FRM]

chyeh,

Hull (Chapter 18) derives it with TWO put-call parities:
p(BSM) + S = c(BSM)+discounted strike
p(market price) + S = c(market price)+discounted strike

Now subtract the second from the first:
p(BSM) + S = c(BSM)+discounted strike
- [p(market price) + S = c(market price)+discounted strike]
= p(BSM) - p(market price) = c(BSM) - c(market price)

IMO, this is the hard formula to grasp, it is more general that the AIM, it says the pricing error must be the same for the call and the put (if strike and maturity are the same). Here, the p(BSM) and c(BSM) have many values, one for each input volatility. Keep in mind, BSM = model price; e.g.,

p(BSM if input vol = 20%) - p(market price) = c(BSM if input vol = 20%) - c(market price), or
p(BSM if input vol = 25%) - p(market price) = c(BSM if input vol = 25%) - c(market price), or
Etc etc,

Once this is understand, which IMO is far from easy to grasp, then Hull gets to the AIM:
The implied volatility, by definition, is the volatility input into the BSM such that BSM model price = market price:

p(BSM if input vol = X) = p(market price); i.e., solving for X to make them equal.

For this solution, p(BSM if input implied vol) - p(market price) = 0, then
0 = c(BSM if input vol = X) - c(market price), such that
c(BSM if input vol = X) = c(market price); i.e., X must also be the implied volatility for the call (!), a sort of amazing.

Of course, archlight is right, too, and it can I supposed by detailed from a no-arbitrage perspective. Neither mine nor his, to me personally, is easily grasped intuitively FWIW.

Hope that helps, David