Put–call parity is the most useful equation in elementary option theory. For
European call and put
on a non-dividend-paying stock, sharing strike K and maturity
T, their prices today are locked together by
C - P = S_0 - K e^{-rT},
where S_0 is the spot price and r the
continuously-compounded risk-free rate. No probabilities, no volatility, no model of how the
stock moves — just no-arbitrage.
Know any three of C, P, S_0, K e^{-rT} and the fourth is forced. We
derive it by building two portfolios with the same payoff and invoking the law of one price.
Parity, derived line by line
Step 1 — assemble portfolio A: long one call, short
one put, same K and T. Its payoff at
maturity is the difference of the two hockey sticks:
V_T^A = (S_T - K)^+ - (K - S_T)^+.
Step 2 — collapse that payoff by cases. Split on the strike. If
S_T \ge K the call is in the money and the put is worthless:
V_T^A = (S_T - K) - 0 = S_T - K.
If instead S_T < K the call is worthless and we are short a
put that pays K - S_T, so we owe it:
V_T^A = 0 - (K - S_T) = S_T - K.
Step 3 — read off the punchline. Both cases give the same line, with
no positive part left:
V_T^A = S_T - K \qquad \text{in every state.}
Long-call-minus-short-put is just a forward: a call and a put at the same strike fuse into the
straight obligation S_T - K.
Step 4 — build portfolio B with the same
payoff. Hold one share and borrow the present value of K —
i.e. short K zero-coupon bonds, each worth
e^{-rT} today and 1 at maturity. At
T the share is worth S_T and you repay
K:
V_T^B = S_T - K.
Step 5 — price each portfolio today. Portfolio
A costs the call premium minus the put premium (you receive the put
premium for the short); portfolio B costs the share minus the value
of the borrowed bonds:
P_A = C - P, \qquad P_B = S_0 - K e^{-rT}.
Step 6 — apply the law of one price. The two portfolios have identical payoffs
S_T - K in every state (Steps 3 and 4), so by
no-arbitrage their prices must coincide:
C - P = S_0 - K e^{-rT}.
That is put–call parity. The whole derivation rests on one observation — a call minus a put is a
forward — plus the refusal to leave a free lunch on the table.
For European options on a non-dividend-paying stock with common strike
K, maturity T, and continuously
compounded risk-free rate r,
C - P = S_0 - K e^{-rT}.
It is model-free: it holds whatever process the stock follows, because it is a pure
no-arbitrage relation between portfolios with the identical payoff
S_T - K.
Rearranging parity lets you synthesise any one instrument from the rest — handy when
one is illiquid or mispriced. Solving for the call,
C = P + S_0 - K e^{-rT},
says a synthetic call is "long a put, long the stock, short the bonds": it
pays exactly what a real call pays, so by parity it costs exactly what a real call costs.
Likewise a synthetic forward is long call plus short put, and a
synthetic bond (a guaranteed K at
T) is "short call, long put, long stock". Every desk runs on these
identities.
If the stock pays a continuous dividend yield q, holding the share
earns dividends the option holder forgoes, so the share leg must be discounted by the dividends
paid before maturity. Replacing S_0 by
S_0 e^{-qT} in portfolio B (you only need
e^{-qT} of a share now to have one share's worth of growth by
T) gives the dividend-adjusted parity
C - P = S_0 e^{-qT} - K e^{-rT}.
Setting q = 0 recovers the plain version. The same swap reappears
throughout option pricing whenever the underlying throws off a yield — dividends on a stock, a
foreign interest rate on a currency, or a convenience yield on a commodity.