Put Call Parity

[kingstrider]

I am currently studying put call parity. I am stuck on one thing .

If in a put-call parity there is a call, but the risk free rate is zero. So is there any opportunity of arbitrage in this case?

 

[David Harper CFA FRM]

Hi strider -

I don't think so because the riskfree rate is needed to support the assertion that the future synthetic long position (c - p) is equal to the cash long position (S-K); i.e., in the future (at the expiration, hence the need for same maturities), we should hold a synthetic long given by: long call + short put
... which (lacking frictions) and assuming same strike, future payoff of the synthetic long = max(St-K,0) - max(K-St,0) = St-K.
... so, we need a future cash portfolio that will give exact payoff of (St-K); i.e., pay cash K to own future stock St
... in order to be certain we have future cash (K) we need to borrow K*exp(-rT); i.e., the riskless rate is to ensure future (K)

so, FWIW, this is my preferred way of viewing put call parity (and i think it's worth bearing in mind P/C parity is *merely* about the necessary equality of two portfolios under the conditions that K=K and T=T for the options):
synthetic long = cash long or: long call + short put = borrow PV(K) to buy stock
future c-p = future (St-K) and PV(St-K) = S0 - PV(k), where S0-PV(k) is the minimum value/lower bound which coincides with BSM if volatility = 0

David

 

[kingstrider]

Hi David,

Put Call Parity what I have studied is that PUT + Stock = Bond + Call. What I understand from what you have said is that i cannot earn anything from this situation. As to earn I must long a call and short a put, which shud be equal to the Future Cash which we need to borrow at Rfr. But Can you explain it to me with the formula Put + Stock = Bond + Call?

Strider