put call parity

@Stuj79  

In the example that you just presented, can you explain what you would do if the call is underpriced and how it relates to the formula?

Reason I ask is because I know that in the equation a “+ sign” = BUY while a “- sign”= sell. This logic worked perfectly in your example but I’m just wondering how you would incorporate this into the equation if the call is underpriced since the c= S + p – X/(1+r)t would have the same signs as an ‘overpriced call’ situation…

Obviously, we want to “buy low, sell high” so we would want to buy an underpriced option but I want to understand how it works in relation to the other factors in the equation.  

Remember the put/call parity formula is made up of 4 components…

The put, the call, the underlying AND the risk free bond. So saying just buy or sell underlying if put or call are overpriced or underpriced etc isn’t correct

the formula is:

S + p = c + X/(1 + r)t

So a call should be equal to c = S + p – X/(1+r)t

If the call is overpriced then you sell the call and buy the underlying, buy a put and sell a risk free bond.

I won’t go through every possible combination of call and put over and under priced but you can work them all out from the formula outlined above.

Thank you for the clarification!  @Stuj79  and @googs1484 

I neglected to mention arbitrage, which as u said is really CFAIs main driver of that whole put call parity relationship. 

@hardj if the call outlined in the previous example was underpriced as opposed to over priced, then you would just do the opposite.

Taking the formula c = S + p – X/(1+r)t:

If we are buying the left hand side as it is underpriced, then we just buy the call – and if we are SELLING the right hand side (as it is relatively overvalued) then we just do the “opposite of the sign”.

So we would sell the underlying, sell the put and buy the risk free bond.

So yeah, just remember when we are “buying” a side of the equation, do things in the same direction as the signs – if we are selling, then do the opposite.

Hope that makes sense.

@googs1484, I think most people look at the put-call parity formula in terms of identifying arbitrage opportunities and showing the general relationship between the 4 inputs – if one goes up/down, what should happen to the others etc etc.

I understand what you are saying, in terms of “if the market price of a call is lower than the price you can build a synthetic call with, AND all you want is exposure to a long call – then buy it in the market. If the synthetic call is cheaper, AND all you want is an exposure to a long call, then build the synthetic call.”

However I can guarantee to you, that is not really the way the CFA frame things. They frame the put-call parity issue as a means of identifying arbitrage opportunities by identifying mispriced put and call options, and trading both a real and synthetic version of the same asset in opposite directions. Not just gaining exposure to a call position (for example) through buying the cheaper version.

If the market price of a call is higher than the price of creating a synthetic call, then sell the call in the market AND buy the synthetic call – which would create a riskless arbitrage profit as the payoff of a real and synthetic option are equal at expiration if set up properly.

@stuj79 good explanation. That’s what I was trying to say, you were just more thorough. 

The rearrangement of put call parity you just presented is the theoretically justified call value as a synthetic, it’s not an actual call option. You really have two choices here. 

1) if the justified call price via put call parity is lower then the market then buy the stock, buy the put and borrow (short the bond).

2) if the markets call price is lower then dictated buy put call parity then simply buy the call in the open market.  

They should theoretically be the same price but if not your choices are buy the call via put call parity and use the (right side of the equation) making a synthetic call. Or buy the actual call option. Buy the low, sell the high. 

Whenever the option is overpriced you sell, underpriced you buy.  Transaction on the underlying is whatever it takes to neutralise the risk.