The most reliably asked options question, because it is model-free.
The relationship
For European options with the same strike K and expiry T on a non-dividend-paying stock:
C - P = S - K e^(-rT)
Call minus put equals stock minus the present value of the strike.
Why it must hold
Consider two portfolios:
- Portfolio A: one call, plus cash of K e^(-rT).
- Portfolio B: one put, plus one share.
At expiry, if the stock finishes above K, A is worth S (exercise the call using the cash) and B is worth S (the put expires worthless). If it finishes below K, A is worth K (call expires, cash matures) and B is worth K (exercise the put).
They pay identically in every state, so they must cost the same today. Rearranging gives parity.
Note what is absent from that argument: any assumption about volatility, drift, or the distribution of returns. This is pure replication, which is why parity survives when models do not.
What it lets you do
Price one from the other. Given a call price you can get the put immediately - a common quick interview question.
Construct synthetics. A long call plus a short put equals a forward. A long stock plus a long put equals a call. Desks use these constantly to manage risk without trading the underlying.
Check quotes. Any set of prices violating parity implies a risk-free profit, so it is a fast sanity check on your own market.
Dividends and early exercise
With dividends, the stock term becomes S minus the present value of the dividends.
For American options, parity becomes an inequality, because early exercise is possible. An American call on a non-dividend stock is never optimally exercised early, so it behaves like a European one - but an American put can be, which breaks the equality.
That distinction is a favourite follow-up.
Practise in finance and derivatives.