Put-Call Parity in Indian Markets: The Real Arbitrage Math

What Put-Call Parity Actually States

Put-call parity is a no-arbitrage relationship between a European call and a European put that share the same underlying, same strike and same expiry. The clean equation is C plus PV(K) equals P plus S. Here C is the call price, P is the put price, S is the spot price of the underlying, K is the strike, and PV(K) is the present value of the strike, which is the strike discounted back from expiry at the risk free rate. The intuition is simple. Holding a call and setting aside enough cash to pay the strike at expiry gives you the exact same payoff as holding the stock and a put that protects it. Two things with identical payoffs must cost the same, otherwise someone earns a risk free profit.

For Indian traders the more practical form is the rearranged version: C minus P equals S minus PV(K). The left side, call price minus put price, is something you can read straight off the option chain. The right side, spot minus the discounted strike, is something you can compute in seconds. When these two sides match, parity holds and the options are fairly priced relative to each other. When they drift apart by more than transaction costs, one option is rich and the other is cheap, and a hedged trade can in theory capture the gap.

One caveat before any numbers: parity in its pure form assumes no dividends on the underlying before expiry. For dividend paying stocks you subtract the present value of expected dividends from the spot, so the relationship becomes C plus PV(K) equals P plus S minus PV(dividends). For Nifty and Bank Nifty index options the index is price based and the cleaner reference is actually the index future, which already bakes in the cost of carry, so the working form there is C minus P equals F minus K, all discounted.

The Arbitrage Logic, Done Properly

The old way this concept is often taught is sloppy. People write the equation, plug in numbers, notice the two sides are not equal, and then stop with a vague claim that there is an arbitrage. That is not enough. An arbitrage is a specific set of trades, and you have to know exactly which leg to buy, which to sell, and what the locked profit is. The rule is the same as in any market: sell the expensive side, buy the cheap side, and do both at once.

Define the synthetic. A long call plus a short put at the same strike behaves exactly like a long position in the underlying, because at expiry you either exercise the call above the strike or get assigned on the put below it, so you end up buying at the strike either way. This combination is called a synthetic long. Its fair cost is the call premium minus the put premium. If that synthetic long is cheaper than buying the actual underlying net of the discounted strike, you buy the synthetic and sell the real thing. If the synthetic is more expensive, you do the reverse, called a conversion.

• Reversal, also called a reverse conversion: when C minus P is too low versus S minus PV(K), you buy the call, sell the put, and short the underlying. You are buying the cheap synthetic and selling the expensive real asset.
• Conversion: when C minus P is too high versus S minus PV(K), you sell the call, buy the put, and buy the underlying. You are selling the rich synthetic and buying the cheaper real asset.
• Either way the directional risk cancels. The position is hedged, and the only thing left at expiry is the price gap you locked in at entry, minus all costs.

Worked Example on a Real Nifty Strike

Let us use realistic, illustrative levels. Say Nifty 50 spot is 24,000 and we look at the monthly options expiring in 30 days. The Nifty lot size is 65. We pick the at the money 24,000 strike. Assume the 24,000 call trades at 320 and the 24,000 put trades at 250. The risk free rate is about 7 percent a year, so for 30 days the discount factor on the strike is roughly 30 divided by 365 times 0.07, which is about 0.00575, or 0.575 percent.

Now compute both sides of C minus P equals S minus PV(K). The left side is the call minus the put: 320 minus 250 equals 70 points. The right side is spot minus the discounted strike. PV(K) is 24,000 divided by 1.00575, which is about 23,863, so S minus PV(K) is 24,000 minus 23,863, which is about 137 points. The two sides differ: the synthetic long costs 70 points but the real-minus-discounted-strike value is 137 points. The synthetic is too cheap by about 67 points, so the trade is a reversal: buy the call, sell the put, and short Nifty (in practice short the Nifty future, since you cannot short the index directly).

Per lot, 67 points times 65 equals about Rs 4,355 of theoretical gross edge. That is the gap you would lock if you could trade all three legs at exactly these prices with no friction. The point of the example is the method, not the prize, because the next two sections show how Indian costs eat most or all of that 4,355.

Why the Indian Cost Stack Usually Kills It

The 67 point gap above is deliberately large to make the arithmetic clear. In reality, liquid Nifty and Bank Nifty strikes rarely show more than a few points of parity mismatch, because market makers arbitrage it away in milliseconds. Even when a gap appears, you are paying real costs on three legs. On the options legs, Securities Transaction Tax on selling options is 0.1 percent of premium, and STT on exercised or settled in the money options is 0.125 percent of the intrinsic settlement value. On the futures leg used to hedge, STT on the sell side is 0.02 percent of turnover.

On top of STT you pay brokerage, which for F&O is typically a flat fee such as Rs 20 per order at discount brokers, plus exchange transaction charges, GST at 18 percent on brokerage and exchange charges, SEBI turnover fees and stamp duty on the buy side. With three legs and an exit, you can easily clock six to eight orders. The bid-ask spread is its own tax: if the call and put each have a 1 to 2 point spread and the future has its own spread, you can surrender 3 to 5 points just crossing the market on entry and exit.

Add it up and a realistic friction budget on a single Nifty lot reversal can run Rs 1,500 to Rs 3,000 or more once the large futures notional drives the percentage based charges. Against the illustrative Rs 5,025 gross edge you might keep Rs 2,000 to Rs 3,500, but against a realistic 5 to 8 point screen mismatch worth Rs 375 to Rs 600 gross, you are guaranteed to lose money. This is why pure parity arbitrage is a market maker game, not a retail strategy.

Indian Settlement and Style Quirks That Matter

Put-call parity in its textbook form assumes European exercise, and Indian index options fit that perfectly. Nifty, Bank Nifty, FinNifty and Sensex options are European style and cash settled, so there is no early assignment risk and no physical delivery on the index. That removes one big complication that bites American style stock options. However, single stock options on the NSE are also European in exercise but are physically settled, meaning in the money positions held to expiry result in delivery of shares. That delivery obligation changes the cost and risk of any stock parity trade significantly and can trigger an STT charge on physical settlement.

Expiry mechanics also matter. Nifty now has a single weekly expiry plus the monthly expiry, while Bank Nifty moved to monthly only expiries under SEBI rationalisation of weekly contracts. The shorter the time to expiry, the smaller PV(K) differs from K, so the parity gap from interest alone shrinks toward zero. On a weekly contract the carry component is tiny, which means almost the entire C minus P difference should equal spot minus strike, and any mismatch is mostly noise, spread or a stale quote rather than a real edge.

Synthetic Positions: Where Parity Earns Its Keep

Forget free money for a moment. The real day to day value of put-call parity is that it lets you rebuild any position out of the cheapest available pieces. Because a call minus a put at one strike equals a long forward in the underlying, you can manufacture exposures synthetically when the direct instrument is costly, illiquid or unavailable.

• Synthetic long: buy the call, sell the put at the same strike and expiry. Behaves like long the future. Useful when the future is wide or you prefer defined option strikes.
• Synthetic short: sell the call, buy the put at the same strike and expiry. Behaves like short the future.
• Synthetic long put: buy the call and short the future. By parity this replicates owning the put, handy if the put is overpriced on screen.
• Synthetic long call: buy the put and buy the future. Replicates owning the call when the call quote is rich.

This is also the lens through which to understand protective puts and covered calls. A covered call, long the underlying plus a short call, has the same payoff shape as a short put by parity. A protective put, long the underlying plus a long put, has the same payoff shape as a long call. Seeing these equivalences stops you from overpaying. If a protective put structure costs more than simply buying the equivalent call, parity tells you to buy the call instead, and vice versa.

Reading a Parity Mismatch on the Option Chain

Pull up the NSE option chain for a near month Nifty expiry and pick the at the money strike. Note the call last traded price and the put last traded price, but be careful: last traded prices can be stale. For a real check use the mid of the bid and ask on each option and on the future. Compute C minus P from the mids, then compute F minus K using the same expiry future. If the two numbers agree within a point or two, parity holds and there is nothing to do.

If you see a larger gap, slow down and ask why before assuming free money. Common innocent explanations are a stale last price, a wide spread on a less liquid strike, a dividend or special adjustment on a stock underlying, or simply that you are comparing an option expiry against a mismatched future expiry. A genuine, tradable violation on a liquid Nifty strike is rare and short lived, which is exactly what an efficient market should produce.

Taxes on Any Profit You Do Make

If a parity or synthetic trade does produce a gain, the tax treatment in India is specific. Profits from futures and options are treated as business income, not as capital gains, and are taxed at your applicable slab rate. This matters because the favourable capital gains rates do not apply to F&O. For reference, on delivery based equity the short term capital gains rate is 20 percent and long term capital gains is 12.5 percent above the Rs 1.25 lakh annual exemption, but those numbers are for shares held as investments, not for your options book.

Because F&O is business income, you can set off losses and carry forward non speculative business losses for up to eight years, and you can deduct genuine trading expenses such as brokerage, data charges and a proportion of internet and software costs. STT paid on F&O is an allowable business expense here, unlike in the capital gains world. If your trading turnover is significant, a tax audit under the Income Tax Act may apply, so keep clean contract notes. None of this is personal tax advice; confirm your own position with a qualified chartered accountant.

Common Mistakes Traders Make With Parity

• Quoting a violation without naming the trade. Saying the two sides are unequal is not an arbitrage until you specify buy this leg, sell that leg, and the rupee profit per lot.
• Forgetting the lot size. A 10 point gap is not Rs 10. On Nifty it is 10 times 75, and the costs scale with the large notional too.
• Ignoring that you trade the future, not the cash index. For Nifty and Bank Nifty the honest parity check uses the future, because you cannot short the index.
• Using stale last traded prices. Parity must be checked on live bid-ask mids; last price gaps are usually phantom.
• Treating index options like American style. They are European, but single stock options are physically settled, which changes the trade entirely.
• Forgetting dividends on stock underlyings. Skip the dividend adjustment and your parity check will look broken when it is fine.

Sources and Further Reading

For authoritative data and contract specifications, refer to the NSE Option Chain, Zerodha Varsity and NSE India. Always confirm current lot sizes, STT rates, expiry schedules and contract specifications on the official source before you trade, since SEBI and the exchanges revise these periodically. Related glossary terms include arbitrage, volatility and liquidity.

Sources and Further Reading

For authoritative data and further reading on this topic, refer to NSE Option Chain, Zerodha Varsity and NSE India. Always confirm current rules, rates and contract specifications on the official source before you trade.