Introductory Context
"Put-Call Parity: Call − Put = Spot − PV(Strike), where PV(Strike) is the present value of the strike price discounted at the risk-free rate. For European options, this relationship must hold to prevent arbitrage. It explains the relationship between put and call prices at any strike and enables the construction of synthetic positions. "
The Put-Call Parity Relationship
For European options on the same underlying, at the same strike, and with the same expiry:
C − P = S − PV(K)
Where: C = call premium, P = put premium, S = current spot price, PV(K) = present value of strike price = K × e^(−rT) ≈ K ÷ (1 + r × T) for short durations.
For short-dated Indian options where the interest rate effect is small, this simplifies approximately to:
C − P ≈ S − K
Or equivalently: Call Price − Put Price ≈ Spot Price − Strike Price
This relationship says: the difference between the call and put premiums at any strike equals approximately the intrinsic value of the call at that strike (spot minus strike). If the call at strike 24,000 costs ₹200 more than the put at the same strike, Nifty should be approximately ₹200 above 24,000 — at approximately 24,200.
Put-Call Parity in Indian Markets
For Nifty options with the same strike and expiry: if the 24,200 CE is priced at ₹150 and Nifty spot is at 24,350, put-call parity implies the 24,200 PE should be priced at approximately ₹150 − (24,350 − 24,200) = ₹150 − ₹150 = ₹0 plus carrying cost adjustment. In practice, the 24,200 PE would be slightly above ₹0 because of the cost-of-carry term — perhaps ₹5–₹15 depending on time to expiry and interest rates.
Why Put-Call Parity Must Hold — The Arbitrage Proof
If put-call parity is violated, an arbitrage profit is available. Suppose the call is too expensive relative to the put:
• Sell the overpriced call (collect premium)
• Buy the underpriced put (pay premium)
• Buy the underlying at spot (cash outflow)
• Borrow the present value of the strike at the risk-free rate (cash inflow)
This portfolio — short call, long put, long underlying, borrow PV(strike) — has zero cost to set up (the cash flows net to zero by the parity condition) and is guaranteed to have zero P&L at expiry regardless of where the underlying ends up. If the initial cost is negative (you receive cash), you have locked in a riskless profit — the definition of arbitrage.
In efficient markets like NSE, sophisticated participants monitor put-call parity continuously and trade violations immediately. The speed of modern electronic trading means violations that arise (from large order imbalances or technical glitches) are corrected within milliseconds. Put-call parity holds closely for actively traded Indian index options.
Synthetic Positions — Building Options From Components
Put-call parity enables the construction of 'synthetic' positions — positions that replicate the payoff of one instrument using combinations of others. These synthetics are used by sophisticated traders to exploit relative mispricings and by market makers to manage risk:
Synthetic Long Call
Long put + long underlying = synthetic long call. If you own a Nifty put and simultaneously go long Nifty futures, the combined position behaves identically to owning a Nifty call at the put's strike. This is used when calls are expensive relative to puts + futures, allowing the equivalent exposure at lower cost.
Synthetic Long Put
Long call + short underlying (short futures) = synthetic long put. The combined position behaves like owning a Nifty put. Used when puts are expensive relative to calls + short futures.
Why Retail Traders Should Know Synthetics
Understanding synthetics reveals why option prices must be consistent with each other — a call cannot be significantly more expensive than an equivalent synthetic position built from puts and futures, because the arbitrage would immediately be exploited to close the gap. This is why when you see a Nifty call and the equivalent put, their prices are always 'anchored' to each other through put-call parity.
Book Reference
The complete treatment of put-call parity, synthetic positions, and conversion/reversal arbitrage strategies — including how institutional traders exploit violations in real time — is covered in Book 3 of the myfinversity Options Trading Series: How Options Are Priced: Black-Scholes, Volatility and the Greeks.