what is the put-call parity formula

The formula is C + PV(K) = P + S, where C is the call price, P is the put price, S is the current stock price, K is the strike, and PV(K) is the present value of the strike discounted at the risk-free rate to expiry. Rearranged, it says the call minus the put equals the stock minus the discounted strike. This holds for European options that pay no dividend before expiry; dividends and carrying costs adjust the stock term. Any persistent violation of the equation implies a free, risk-free profit — which is why real markets keep it tight.

what is a synthetic position in options

A synthetic position recreates the payoff of one instrument using others, and put-call parity is what makes them possible. For instance, buying a call and selling a put at the same strike and expiry produces a synthetic long stock position — its payoff matches owning the shares. Likewise a stock plus a protective put behaves like a call. Traders use these equivalences to build a desired exposure through whichever instruments are cheapest or most liquid, and arbitrageurs use them to exploit any mispricing that breaks parity.

why does put-call parity matter

It matters because it enforces internal consistency across the options market: it ties calls, puts, the stock and interest rates into one equation, so none can drift far from the others without creating an arbitrage that traders immediately close. That makes parity a powerful sanity check — if quoted prices seem to violate it, the more likely explanations are dividends, hard-to-borrow stock, or stale quotes rather than genuine free money. Understanding parity also demystifies synthetic positions and explains why the same exposure can be built several different ways.

What is put-call parity and why it matters

Options can feel like a zoo of independent instruments, but beneath them sits one tidy equation that binds a call, a put, the stock and a bond together and refuses to let any of them wander off on its own. Put-call parity is that equation — the no-arbitrage law that pins option prices to each other and to the underlying.

One equation, four instruments

For a European call and put on the same stock, with the same strike and expiry, parity says: call + present value of the strike = put + stock, or C + PV(K) = P + S. The logic is airtight — both sides pay exactly the same amount at expiry for every possible stock price, so if they cost different amounts today, you could buy the cheap side, sell the dear side, and pocket a risk-free profit. That impossibility is what holds the equation together.

Worked example

Checking parity on a strike

Stock ₹100, strike ₹100

Stock price (S)Underlying now₹100

Strike (K), 3-monthPV(K) ≈ ₹98.5 at ~6%₹100

Call price (C)Observed₹6

Parity implies putThe fair putP = C + PV(K) − S = 6 + 98.5 − 100 = ₹4.5

If the put trades at ₹6An arbitrage — or a dividend/borrow costParity is broken

Given the call, the stock and rates, parity pins the fair put at ₹4.5. If the market quotes it at ₹6 with no dividend or borrow cost to explain the gap, an arbitrageur can lock in the difference risk-free. In practice a “broken” parity almost always points to a dividend, a hard-to-borrow stock, or a stale quote — not to free money lying around.

Check yourself

You buy a call and sell a put at the same strike and expiry on the same stock. What have you effectively created?

Simple bhasha mein

Call, put, stock — ek hi zanjeer

Call aur put alag-alag cheezein nahi — ek equation se bandhe hain: call + PV(strike) = put + stock. Dono taraf expiry pe payoff same, isliye daam bhi same hona chahiye — warna risk-free arbitrage. Isi se "synthetic" bante hain: call kharido + put becho = stock kharidne jaisa. Chain pe parity "toot" jaaye toh 99% baar dividend, borrow cost ya purana quote hai — free paisa nahi. Retail ke liye ye samajhne ka aur sanity-check ka tool hai, trade ka nahi.

What to remember

Put-call parity links a European call, put, the stock and a bond in one no-arbitrage equation.
C + PV(K) = P + S: fix the inputs and the call price determines the put price.
It enables synthetic positions — e.g. long call + short put = synthetic long stock.
Apparent violations are almost always dividends, borrow costs or stale quotes, not free money.
For retail traders parity is a sanity-check and a construction tool, not an arbitrage strategy.

You reached the endMark it done and keep your streak going.

Up nextBlack-Scholes: what an option is actually worth Previous: Expiry, assignment and settlement

Finished this lesson?

Mark it done to track your progress through the curriculum.

Common questions

Short, direct answers to what people ask about this topic.

what is put-call parity: Put-call parity is a no-arbitrage relationship that links the prices of a European call and put with the same strike and expiry to the underlying stock and a bond. It states that holding a call plus cash equal to the present value of the strike is equivalent to holding a put plus the stock — both combinations pay off identically at expiry, so they must cost the same today. If they do not, a risk-free arbitrage exists. Parity is the backbone of option pricing: it means calls and puts cannot be priced independently of each other or of the stock.
what is the put-call parity formula: The formula is C + PV(K) = P + S, where C is the call price, P is the put price, S is the current stock price, K is the strike, and PV(K) is the present value of the strike discounted at the risk-free rate to expiry. Rearranged, it says the call minus the put equals the stock minus the discounted strike. This holds for European options that pay no dividend before expiry; dividends and carrying costs adjust the stock term. Any persistent violation of the equation implies a free, risk-free profit — which is why real markets keep it tight.
what is a synthetic position in options: A synthetic position recreates the payoff of one instrument using others, and put-call parity is what makes them possible. For instance, buying a call and selling a put at the same strike and expiry produces a synthetic long stock position — its payoff matches owning the shares. Likewise a stock plus a protective put behaves like a call. Traders use these equivalences to build a desired exposure through whichever instruments are cheapest or most liquid, and arbitrageurs use them to exploit any mispricing that breaks parity.
why does put-call parity matter: It matters because it enforces internal consistency across the options market: it ties calls, puts, the stock and interest rates into one equation, so none can drift far from the others without creating an arbitrage that traders immediately close. That makes parity a powerful sanity check — if quoted prices seem to violate it, the more likely explanations are dividends, hard-to-borrow stock, or stale quotes rather than genuine free money. Understanding parity also demystifies synthetic positions and explains why the same exposure can be built several different ways.

Put-call parity: the equation that links every option

One equation, four instruments

Common questions