Arbitrage using options: Put-call parity

Put‑call parity is a fundamental pricing relationship that holds for European‑style equity options with the same strike price (K) and expiry (T). It states that the combined value of a call option and the present value of the strike, adjusted for any dividends, must equal the combined value of a put option and the underlying stock price. In formula form, C + PV(K) + PV(D) = P + S, where C and P are the market prices of the call and put respectively, S is the spot price of the stock, PV(K) is the discounted strike, and PV(D) is the discounted dividend cash flow.

The parity condition arises because two different portfolios – a long call + cash to pay the strike, and a long put + the stock – have identical pay‑offs at expiry. If markets are efficient, their costs today must be the same; otherwise a risk‑free profit can be earned. This concept is a cornerstone of the NISM syllabus under “Arbitrage using options”.

For the exam, remember that the relationship only holds for European options (no early exercise) and that any dividend expected before expiry must be accounted for. SEBI’s definition of “equity derivative” aligns with this, so questions will explicitly mention “European” or “no early‑exercise”.

• Call (C) – right to buy at K.
• Put (P) – right to sell at K.
• PV(K) – present value of strike, discounted at the risk‑free rate.
• PV(D) – present value of any cash dividends payable before expiry.

Consider two portfolios that expire on the same date. Portfolio A holds a long European call (C) and enough cash to pay the strike price K at expiry. Portfolio B holds a long European put (P) and the underlying stock (S). Both portfolios will deliver the same cash flow at expiry regardless of the underlying price.

If the stock price at expiry (S_T) is above K, Portfolio A exercises the call, paying K and receiving the stock, ending with S_T – K. Portfolio B lets the put expire worthless, keeping the stock, also ending with S_T – K. Conversely, if S_T is below K, Portfolio A lets the call expire and retains the cash K, while Portfolio B exercises the put, selling the stock for K, again ending with K – S_T. Because the pay‑offs match, the cost today must be identical, leading to the parity equation.

When a dividend D is expected before expiry, the stock price will drop by the dividend amount on the ex‑dividend date. To keep the pay‑offs identical, the dividend’s present value must be subtracted from the stock side, giving the full formula C + PV(K) + PV(D) = P + S. This derivation is a frequent NISM question – they may ask you to construct the synthetic positions and write the resulting equation.

When market prices deviate from the parity condition, a risk‑free arbitrage can be executed. Two common cases arise:

Case 1 – Call overpriced (C too high): Sell the call, buy the put, buy the stock, and borrow PV(K) + PV(D). At expiry the positions offset, leaving a profit equal to the initial mis‑pricing.

Case 2 – Put overpriced (P too high): Sell the put, buy the call, short the stock, and invest PV(K) + PV(D). Again, the combined payoff is constant, and the trader locks in the parity gap.

In Indian markets, SEBI requires that the arbitrage be executed within the same trading day to avoid exposure to market risk. Transaction costs, brokerage, and stamp duty must be factored in; if the net profit after costs is non‑positive, the arbitrage is not viable. Exam questions often provide the costs explicitly, testing whether you correctly net them off.

SEBI mandates that all option trades be settled in cash on the expiry date, and that dividend adjustments are reflected in the option’s settlement price. Therefore, the PV(D) term must be calculated using the actual cash dividend announced by the company, not an estimated yield.

Margin requirements for synthetic positions differ from plain options. For a synthetic long call (long put + long stock – cash), the exchange may allow a lower margin because the net market exposure is limited. However, the trader must still post the full cash amount for PV(K) + PV(D) if borrowing is involved.

Finally, the Indian market operates on a T+2 settlement for equities but options settle on the same day (T+0). This timing affects the cash flow of the arbitrage strategy, and exam questions may test your awareness of settlement cycles when you design the trade.