Put Call Parity Explained: Formula and Nifty Example
Put call parity is the fixed relationship between the price of a call and a put that share the same underlying, strike and expiry. In plain form it says C minus P equals S minus K times e raised to the power of minus rt, where C is the call price, P the put price, S the spot price, K the strike, r the risk free rate and t the time to expiry in years.
The relationship holds for European style options, which is what NSE lists. If it breaks by more than transaction costs, an arbitrage exists, and traders closing that gap push prices back into line.
Reading the Equation in Plain Words
The left side is the price difference between a call and a put at the same strike. The right side is the spot price minus the present value of the strike.
Think of it as two routes to owning the underlying at the strike price. You can buy a call and sell a put, or buy the asset and borrow the discounted strike amount. Both give the same payoff at expiry, so both must cost the same today. The discounting term exists because the strike is paid later, not now.
A Worked Nifty Example
Take Nifty spot at 24,000, the 24,000 strike, 30 days to expiry, and a risk free rate of 6.5 percent. These are illustrative numbers, not live quotes.
- t equals 30 divided by 365, which is about 0.082 years
- Present value of the strike equals 24,000 times e to the power of minus 0.0053, roughly 23,872
- So C minus P equals 24,000 minus 23,872, which is about 128
If the 24,000 put trades at Rs 250, the call should trade near Rs 378. Suppose it is quoted at Rs 410 instead. That 32 point gap is the mispricing, and after costs a trader could sell the call, buy the put and buy a Nifty future to lock it in. Such a trade is called a conversion.
| Component | Illustrative value |
|---|---|
| Nifty spot (S) | 24,000 |
| Strike (K) | 24,000 |
| Days to expiry | 30 |
| Risk free rate | 6.5 percent |
| Present value of strike | 23,872 |
| Fair call minus put | 128 |
Synthetic Positions Fall Out of the Formula
Rearrange the equation and every synthetic position appears. This is the practical payoff of learning parity.
- Long call plus short put equals a long forward or futures position
- Long put plus long stock equals a long call, which is the protective put
- Short call plus long stock equals a short put, which is the covered call
- Long stock equals long call plus short put plus a lent amount equal to the discounted strike
Traders use these equivalences to pick the cheaper route to the same exposure. If a synthetic long costs less than the future after brokerage and STT, it wins on execution even though the risk profile is identical.
Why Arbitrage Keeps Parity Roughly True
Nobody enforces parity by rule. It survives because a violation is free money for whoever spots it first, and Indian index options are liquid enough that institutional desks and algos monitor the gap continuously.
A conversion sells the expensive call, buys the cheap put and buys the underlying. A reversal does the opposite. Both lock a spread at entry, so a deviation gets traded away within seconds on Nifty and Bank Nifty. On illiquid single stock strikes the gap can sit open longer, because nobody can execute all three legs at the quoted prices.
Small Deviations You Will See in India
Parity is a clean equation on paper. Real Indian markets bend it slightly, and knowing why saves you from chasing a phantom arbitrage on your screen.
Dividends and Cost of Carry
Single stock options are priced off a forward that already subtracts expected dividends. For index options, dividends of Nifty constituents reduce the forward price, which is why index futures sometimes trade at a thinner premium than a pure interest calculation suggests. Ignore dividends in your parity check and you will see a gap that is not there.
Costs and Frictions
STT, exchange charges, brokerage, GST and the bid ask spread all eat into an arbitrage, so a 5 point gap on Nifty may not survive round trip costs. Margin set by NSE Clearing also ties up capital, so the locked spread has to beat what that capital earns elsewhere.
Frequently Asked Questions
Does put call parity work for American style options?
Not as an equality. Early exercise rights make it an inequality with upper and lower bounds instead of one exact price. Indian exchange traded options are European style, so the standard equation applies to them.
Can I use futures instead of spot in the formula?
Yes, and it is often more practical. Substituting the futures price removes the need to model carry separately, because the future already embeds interest and dividend expectations for that expiry.
Why do broker quotes never match the parity price exactly?
You are comparing a theoretical mid value against tradeable bids and offers. Spreads, stale quotes at illiquid strikes and a different rate assumption explain a few points of difference.
Is parity useful if I never arbitrage?
Yes. It gives a fast fair value cross check on a strike, and it shows when a synthetic structure is the cheaper way to hold the same exposure.
Key Takeaways
- Put call parity states that C minus P equals S minus the discounted strike.
- It holds for European style options such as those listed on NSE.
- Rearranging it produces every synthetic position, including the covered call.
- Conversions and reversals keep the relationship close to true.
- Dividends, carry, STT and spreads cause small deviations.
