Put-Call Parity

Put-call parity is one of the most important relationships in options pricing — a simple equation that links the prices of call and put options and is enforced by the possibility of arbitrage. Understanding it is fundamental to anyone studying derivatives or quantitative finance. This guide explains what put-call parity is, the formula, why it holds, what happens if it's violated, and its uses — in clear, plain language. It's relevant to anyone studying financial markets, derivatives or risk, including ACCA and CIMA students meeting financial-management topics.

What is put-call parity?

Put-call parity is a relationship between the prices of a European call option and a European put option that share the same underlying asset, strike price and expiry date. It states that these prices cannot move independently — they're tied together by a fixed relationship. In essence, a portfolio combining one of the options with the underlying or cash can be made equivalent to a portfolio combining the other, so their prices must line up. If they don't, a risk-free profit (arbitrage) becomes available.

The formula

For European options on a non-dividend-paying asset, put-call parity is usually written as:

C + PV(K) = P + S

where C is the call price, P is the put price, S is the current price of the underlying asset, K is the strike price, and PV(K) is the present value of the strike (the strike discounted back at the risk-free rate — often written as Ke^−rT, where r is the risk-free rate and T is the time to expiry). Rearranged, this gives C − P = S − PV(K), which neatly shows how the difference between the call and put prices relates to the underlying price and the discounted strike.

Why it holds: the no-arbitrage argument

Put-call parity holds because of arbitrage. Consider two portfolios: (1) a call option plus cash equal to the present value of the strike, and (2) a put option plus the underlying asset. At expiry, both portfolios are worth exactly the same — the greater of the asset price or the strike — whatever happens to the asset. If two portfolios are guaranteed to have the same value at expiry, they must cost the same today; otherwise you could buy the cheaper, sell the dearer, and lock in a risk-free profit. That arbitrage pressure forces the prices to satisfy put-call parity.

A quick worked example

Suppose a stock trades at £100, and there are European options with a £100 strike expiring in one year, with a risk-free rate of about 5% (so the present value of the strike is roughly £95.24). If the one-year call is priced at £10, put-call parity tells us the put should be worth about P = C + PV(K) − S = 10 + 95.24 − 100 = £5.24. If the put were instead trading at, say, £7, it would be over-priced relative to the call — and a trader could sell the put, buy the call, and combine it with the underlying and borrowing to capture the mispricing risk-free. This is exactly the kind of discrepancy that arbitrage quickly eliminates, keeping real-world prices close to parity.

What happens if it's violated

If market prices stray from put-call parity, an arbitrage opportunity arises. A trader can buy the under-priced side and sell the over-priced side, constructing a position that costs nothing (or less than nothing) today but is guaranteed not to lose at expiry — a risk-free profit. In efficient markets, traders acting on these opportunities quickly push prices back into line, which is why put-call parity holds closely in practice. Deviations tend to be small and short-lived, reflecting transaction costs and market frictions rather than genuine free money.

The assumptions and uses

Put-call parity in its basic form assumes European options (exercisable only at expiry), no dividends on the underlying over the life of the option (the formula can be adjusted for dividends), the same strike and expiry for both options, and frictionless markets. Despite these simplifications, it's enormously useful. It lets you price one option from the other, spot arbitrage opportunities, and build synthetic positions — for example, replicating a call using a put, the underlying and borrowing or lending. It's a cornerstone of how derivatives markets are understood and kept consistent.

Frequently asked questions

What is put-call parity?

A relationship linking the prices of a European call and put option with the same underlying, strike and expiry, enforced by arbitrage — their prices can't move independently.

What is the put-call parity formula?

For European options on a non-dividend-paying asset: C + PV(K) = P + S, where C and P are the call and put prices, S is the underlying price, and PV(K) is the present value of the strike.

Why does put-call parity hold?

Because two portfolios — a call plus the discounted strike in cash, and a put plus the underlying — have identical values at expiry, so they must cost the same today, or arbitrage would be possible.

What if put-call parity is violated?

An arbitrage opportunity arises: traders buy the under-priced side and sell the over-priced side for a risk-free profit, which quickly pushes prices back into line.

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