Put-Call Parity

Q: What is the put-call parity formula?

For European options on a non-dividend-paying underlying: C − P = S − K·e^(−rT), where C and P are the call and put prices at the same strike K and expiry T, S is the spot price and r the risk-free rate. With a futures price F for the same expiry it simplifies to C − P = (F − K)·e^(−rT), which is the form most useful on NSE.

Q: Does put-call parity apply to Nifty options?

Yes. Nifty, Bank Nifty, FinNifty and Sensex options are European and cash-settled, which is exactly the case parity is derived for. In practice the relationship holds to within bid-ask spreads and transaction costs; the Nifty futures price captures the dividend and interest effects, so the futures form of parity is the one to check.

Q: What is a synthetic long futures position?

Buying a call and selling a put at the same strike and expiry. By parity its payoff equals a long futures position entered at the strike plus the net premium. Synthetics are used when the options are cheaper to carry than the future, or to convert an existing position without closing it.

Q: Why does put-call parity sometimes look violated?

Almost always because of stale or wide quotes, especially on far strikes or illiquid stocks, or because the quotes used were not taken at the same instant. Real violations large enough to profit from after brokerage, STT and slippage are rare on NSE index options and are closed by arbitrageurs within seconds.

The one relationship every option price must obey. It is how calls and puts police each other's prices, and how traders build one position out of another.

In short: put-call parity says that a European call and a European put with the same strike and expiry are locked together by the underlying's price and the cost of money: call price minus put price equals spot minus the present value of the strike. If the relationship breaks, a riskless profit exists, so in a liquid market it does not break for long.

The intuition: two portfolios, one payoff

Build two portfolios today. Portfolio A: buy one call and set aside enough cash to grow into the strike price K by expiry. Portfolio B: buy one put and buy the underlying. At expiry, if the market is above K, A exercises the call and ends up holding the underlying; B's put is worthless and it also holds the underlying. If the market is below K, A keeps its cash worth K; B exercises the put and sells the underlying for K. Either way both portfolios are worth the same. Two things that are always worth the same at expiry must cost the same today, and that equality is parity.

The practical consequence is that a call is never priced independently of its put. Trade the call, and you have taken a view that could have been expressed with the put plus the underlying. Parity is the exchange rate between the two.

Formula

Spot form (no dividends): C − P = S − K·e^−rT

With a continuous dividend yield q: C − P = S·e^−qT − K·e^−rT

Futures form: C − P = (F − K)·e^−rT

C and P are the call and put premiums at strike K, S is spot, F is the futures price for the same expiry, r the risk-free rate, T the time to expiry in years. The futures form is the one that matters in India: the index future already embeds interest and expected dividends, so you do not need to estimate either.

Worked example (illustrative numbers)

Suppose NIFTY spot is at 24,000, the Tuesday-expiry future is at 24,060, and with seven days to go the discount factor e^−rT is close enough to 1 to ignore for a quick check. Parity says the 24,000 call minus the 24,000 put should be about 24,060 − 24,000 = 60 points.

If the 24,000 call is quoted at ₹190 and the put at ₹130, the difference is exactly 60: the chain is consistent. If instead the call were ₹190 and the put ₹115, the pair would be 15 points "too cheap" on the put side. A trader could buy the put, sell the call and buy the future at 24,060, locking in a payoff of 15 points per unit, ₹975 per 65-unit lot before costs, whatever Nifty does by Tuesday. On NSE a gap that size would be arbitraged away in seconds; in practice the observed difference stays inside the bid-ask spread.

The same arithmetic runs the other way. Given a liquid call and the future, you can derive what the put must be worth, which is how market makers quote strikes where only one side trades.

Parity is a statement about two portfolios, not a pricing model. It holds whatever model you use to price the options.

Why it matters to an options trader

Synthetic positions. Long call + short put at the same strike = long future. Short call + long put = short future. Long put + long future = long call. These let you change a position's shape without closing it, or carry it where margin or liquidity is kinder.
A sanity check on the chain. If the call-minus-put difference across strikes does not track F − K, one leg is stale. Market makers use this to spot bad quotes; you can too.
Implied futures price. Rearranged, F = K + (C − P)·e^rT. This is exactly how NSE's own India VIX methodology finds the forward level from option quotes.
Understanding why IV is the same for a call and its put. If parity holds, the same volatility must price both; a chain that shows very different IVs at one strike is reporting a quote problem, not a view.

Common mistakes

Using spot instead of the future. Spot ignores the cost of carry and dividends; the index future includes both. Checks against spot will look "violated" by the basis.
Mixing timestamps. Parity needs all three quotes at the same instant. A call last traded at 10:02 and a put at 10:40 will not agree.
Forgetting costs. Brokerage, STT at 0.15% of premium on the sell side, exchange fees and slippage can exceed the apparent edge on small discrepancies.
Applying it to American options. Early exercise breaks strict equality; American parity is a pair of inequalities. On NSE this does not arise, since all exchange-traded options are European.

Indian-market specifics

Every NSE and BSE option is European and index options are cash-settled, so parity applies in its clean form.
Nifty has weekly and monthly Tuesday expiries; the futures contract is monthly. For a weekly check, use the near-month future and remember the small carry between the weekly expiry and the futures expiry.
Index dividends are paid through the constituents; the futures basis already reflects their expected value, which is why the futures form is preferred over estimating q.
Stock options are physically settled, so a synthetic position held to expiry on a stock results in delivery; close or roll before the last day if that is not intended.

FAQ

What is the put-call parity formula?

C − P = S − K·e^−rT for European options on a non-dividend-paying underlying; with futures, C − P = (F − K)·e^−rT.

Does put-call parity apply to Nifty options?

Yes. They are European and cash-settled. Use the futures form, which already includes interest and dividends.

What is a synthetic long futures position?

Long call plus short put at the same strike and expiry. Its payoff equals a long future entered at the strike, adjusted for the net premium.

Why does put-call parity sometimes look violated?

Stale or wide quotes, mismatched timestamps, or costs. Genuine, tradable violations on liquid NSE index options are closed within seconds.

Check call-put differences across the chain

The Nifty option chain shows call and put premiums side by side at every strike, with the futures price above, so the parity check is one subtraction away. In the app, the Positions page totals the net delta and theta of the legs you record.

Nifty option chain Futures & basis

Related terms

Call Option and Put Option
Black-Scholes Model, which satisfies parity by construction
Synthetic Position
Futures Basis, the carry that the futures form absorbs