What you will learn
- State the put-call parity relationship
- Use it to solve for a missing option price
- Understand the replication intuition behind it
- See why arbitrage enforces it without any model assumptions
One of the neatest relationships in derivatives is put-call parity, which shows that calls and puts are not independent but tightly linked. Unlike the Black-Scholes model, put-call parity requires no assumptions about distributions or volatility. It follows purely from the principle that risk-free profits cannot persist, which makes it one of the most reliable results in options theory.
Put-call parity (PrepNuggets)
A clean CFA-style explanation of the relationship and the replication behind it. Focus on the equation.
The relationship
Put-call parity fixes a relationship between the prices of a European call and put that share the same strike and expiration. In words, the call price minus the put price equals the current underlying price minus the present value of the strike. This ties together four quantities, so that knowing any three determines the fourth. Calls and puts on the same underlying, strike, and expiration cannot be priced independently. They are locked together.
- C = call price
- P = put price
- S = current underlying price
- PV(K) = present value of the strike
Key terms
- Put-call parity
- The fixed relationship C − P = S − PV(K) for a European call and put with the same strike and expiry.
- PV(K)
- The present value of the strike, the strike discounted back at the risk-free rate.
- Synthetic position
- A combination of instruments that replicates another, such as long call plus short put replicating the stock.
Solving for the put price
A stock trades at 100 dollars. A call has a price of 5 dollars, and the present value of the strike is 98 dollars. Using put-call parity, what is the put price?
- Write parity. C − P = S − PV(K).
- Plug in the knowns. 5 − P = 100 − 98, and 100 − 98 is 2.
- Solve for P. 5 − P = 2, so P = 5 − 2, which is 3 dollars.
Why it matters: With three of the four quantities known, parity pins down the fourth exactly. No volatility estimate or pricing model was needed, only the no-arbitrage relationship.
Find the put price
A stock trades at 50 dollars. A call is priced at 7 dollars, and the present value of the strike is 48 dollars. Using C − P = S − PV(K), what is the put price, in dollars?
The intuition through replication
The relationship arises because two different packages produce the same payoff, and anything with the same payoff must cost the same. Holding a call plus the present value of the strike in cash gives the same outcome at expiration as holding a put plus the underlying. Since both packages deliver identical results no matter where the price ends up, they must cost the same today, and that equality is put-call parity. This is the same replication idea that underlies Black-Scholes: payoffs that match must have matching prices.
Synthetic positions
A useful consequence is the ability to construct synthetic positions, replicating one instrument from a combination of others. Because parity links calls, puts, the underlying, and cash, you can rearrange it to manufacture any one from the others. For example, a long call plus a short put at the same strike and expiration replicates the underlying itself, a synthetic long stock position. This flexibility to build equivalent positions in different ways is central to how traders manage and transform exposures.
Put-call parity says calls and puts are two faces of the same coin. Break the relationship and you have minted free money, which is exactly why it holds.
Put-call parity (Bionic Turtle)
A more formal treatment showing the two replicating portfolios. Reinforces the no-arbitrage argument.
An arbitrage relationship
Put-call parity holds so reliably because any violation creates an arbitrage opportunity, a chance for risk-free profit. If the call, put, underlying, and discounted strike ever drift out of the relationship, a trader can buy the cheaper package and sell the more expensive one, locking in a guaranteed profit. Such opportunities are seized almost instantly, and the act of exploiting them pushes prices back into alignment. This is why parity is a no-arbitrage condition, enforced not by regulation but by the relentless action of traders eliminating any free profit. Because it needs no assumptions about volatility or distributions, it is more robust than model-dependent results like Black-Scholes.
Spot the free money
You find that C − P is 4 dollars while S − PV(K) is only 2 dollars, so the call-minus-put side is too expensive relative to the stock side. What can an arbitrageur do?
When C − P exceeds S − PV(K), the call-put side is overpriced. The arbitrageur sells that side (short call, long put) and buys the cheaper side (long stock, borrow the PV of the strike). The two packages have identical payoffs at expiration, so the initial price gap is captured as risk-free profit, and the trading restores parity.Why parity is so dependable
In your own words, explain why put-call parity is more reliable than a pricing model like Black-Scholes.
Write an answer before comparing it with the model response.
Model answer
Black-Scholes produces a price only by assuming things about the world, most importantly that returns are roughly normal with constant volatility, and those assumptions are known to be false, so its prices can be off, especially in the tails. Put-call parity assumes none of that. It follows purely from the fact that two packages with identical payoffs must cost the same, which is just the no-arbitrage principle that risk-free profits cannot persist. If parity is ever violated, traders immediately arbitrage the gap and force prices back into line, so the relationship is enforced by the market itself rather than by any model. Because it needs no estimate of volatility or any distribution, put-call parity is a model-free, self-enforcing law, which makes it far more dependable than a model that rests on shaky assumptions.
Put-call parity is a model-free law enforced by arbitrage. The next lesson returns to futures to examine the mechanics that make their leverage possible: margin.