Have you ever wondered why option prices can't just be anything the market wants? Why a NIFTY call at a given strike always relates in a specific way to the corresponding put? The answer is a 400-year-old mathematical relationship called Put-Call Parity — and it is the invisible rule that keeps the entire options market honest.
This guide explains the formula, shows how arbitrageurs exploit tiny violations, and teaches you how to build synthetic positions that behave like stocks.
What You Will Learn
1. The Rule — Calls and Puts Are Linked
Consider two portfolios at expiry:
- Portfolio A: Long 1 call at strike K + Cash equal to strike K
- Portfolio B: Long 1 put at strike K + Long stock at spot S
At expiry, both portfolios are worth max(S, K). Since they have the same payoff in every possible future state, they must cost the same today. This is put-call parity.
The equivalence
Call + PV(Strike) = Put + Spot
Rearranging: Call - Put = Spot - PV(Strike) = Spot - Strike × e^(-r × T)
2. The Formula for Indian Options
Indian index options (NIFTY, BANKNIFTY) are European-style (exercise only at expiry), which means put-call parity holds strictly. Single-stock options in India are also European post the 2020 transition.
Parity check on 7 DTE
In practice, parity is rarely exact due to dividends, transaction costs, and bid-ask spreads. But the gap is always small (₹5-30 for index options). If you see ₹100+ gap, something is wrong with your data.
3. How Arbitrageurs Exploit Violations
If Call - Put > Spot - PV(Strike), the call is "too expensive." Arbitrageur does:
- Sell the overpriced call
- Buy the underpriced put
- Buy the spot (long stock or long futures)
- Lock in risk-free profit = the gap
4. Synthetic Long and Short Positions
By rearranging put-call parity, we can build synthetic positions:
| Real Position | Synthetic Equivalent |
|---|---|
| Long Stock | Long Call + Short Put (same strike) |
| Short Stock | Short Call + Long Put (same strike) |
| Long Call | Long Stock + Long Put |
| Long Put | Short Stock + Long Call |
| Short Call | Short Stock + Short Put |
| Short Put | Long Stock + Short Call (covered call!) |
The last row is especially relevant: a covered call equals a short put. Both have the same P&L at expiry. This is why covered call income = cash-secured put income.
5. Conversion and Reversal Arbitrage
When put-call parity deviates slightly (typically 5-10 paise per index point), prop desks execute conversion or reversal trades:
Long stock + Short call + Long put (same strike)
At expiry, the call-put combo locks in the strike. So you effectively hold a position that pays strike at expiry regardless of spot movement. Profit = (Strike - initial cost of the combo), which is the parity gap.
Scaled to 1000 lots of 25 = 25,000 units = ₹1,25,000 risk-free. After brokerage and slippage, real desks capture 30-50% of theoretical edge.
6. How Dividends Affect Parity
For stock options with expected dividends before expiry:
Dividend-adjusted parity
Call - Put = Spot - PV(Dividends) - PV(Strike)
The present value of dividends reduces the "spot" side because the stockholder gets them but the call holder doesn't.
Near dividend ex-dates, you'll see calls become slightly cheaper and puts slightly richer — exactly the parity adjustment for expected dividend.
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