Put-call parity
Also written Put-call parity relationship
The arbitrage-free relationship binding a European call and put of the same strike and expiry to the spot and the discounted strike: c + X·e^(−rt) = p + S. Deviations create risk-free profit.
In plain language
Two very different-looking packages can end up in exactly the same place.
Package one: buy a call at strike X and put aside enough cash to grow to X by expiry. Package two: buy a put at strike X and buy the share.
At expiry both are worth the higher of X and the share price, whatever happens. Two things that are always worth the same must cost the same today — and that identity is put-call parity.
It is the reason a call, a put, the spot and the interest rate are not four independent prices. Fix any three and the fourth is determined. If the market quotes it at anything else, somebody can lock in a profit with no exposure at all.
How it works
The relationship works because both sides can be created from the other. Rearranged, it defines every synthetic position on the board:
Synthetic long stock = Long call + Short put + cash
Synthetic long call = Long put + Long stock → a protective put
Synthetic short put = Long stock + Short call → a covered call
That last line is why the workbook can assert that a covered-call is a short put and that a protective-put is a long call. Those are not analogies; they are put-call parity rearranged.
When the traded put sits below the parity price, the arbitrage is mechanical:
- Buy the underpriced put and buy the stock.
- Sell the call.
- Borrow the net outlay at the risk-free rate.
At expiry the position unwinds to the same profit whichever way the stock went — the difference between the fair price and the traded price.
The workbook adds two hard constraints. Parity holds only for European options, because early exercise would break the "worth the same at expiry" argument. And execution risk is real: any delay between legs leaves a naked long or short position that can produce large losses.
The formula
c + X·e^(−rt) = p + S₀
c = call premium p = put premium
X = strike price S₀ = spot price
r = risk-free rate t = time to expiry in years
Rearranged for the fair put price:
p = c + X·e^(−rt) − S₀
A worked example
A stock trades at Rs 1,251. The 1,240-strike call expiring in one month is quoted at Rs 47.50. The interest rate is 8% p.a.
What should the one-month 1,240 put cost?
X·e^(−rt) = 1,240 × e^(−0.08 × 1/12) = 1,240 × 0.993356 = Rs 1,231.76
p = 47.50 + 1,231.76 − 1,251 = Rs 28.26
The put is actually trading at Rs 23.15. It is underpriced by Rs 5.11, and that is the arbitrage.
Set up the trade:
Buy the put −Rs 23.15
Buy the stock −Rs 1,251.00
Sell the call +Rs 47.50
───────────
Net outflow Rs 1,226.65 ← borrowed at 8% p.a.
Borrowing repayable = 1,226.65 × e^(0.08/12) = Rs 1,234.86
Case 1 — the stock rises to Rs 1,275 at expiry:
Sell the shares +Rs 1,275.00
Put expires worthless 0
Short call assigned, loss −Rs 35.00
Repay borrowing −Rs 1,234.86
─────────────
Net gain +Rs 5.14
Case 2 — the stock falls to Rs 1,200 at expiry:
Sell the shares +Rs 1,200.00
Long put pays +Rs 40.00
Short call expires worthless 0
Repay borrowing −Rs 1,234.86
─────────────
Net gain +Rs 5.14
The same Rs 5.14 either way — and it is the mispricing, 28.26 − 23.15, to the paisa. The arbitrageur took no view, carried no market risk, and was paid for spotting that four numbers did not agree with each other.
On a lot of 550 shares that is Rs 2,827 of risk-free profit, before the brokerage, taxes and funding frictions the workbook insists on counting.
Why NISM asks about it
Chapter 17.3 (Arbitrage using options: Put-call parity) gives the formula, the 1,251/1,240/47.50 worked example and both expiry cases. The Chapter 17 sample questions ask directly what put-call parity relates: call and put options on the same stock with the same strike price and same maturity. Expect that, plus a fair-price computation and the European-only caveat.
Common exam traps
- Same underlying, same strike, same expiry. All three conditions, every time — the sample question exists to catch candidates who drop one.
- It applies to European options only. The workbook states the restriction explicitly, and every Indian exchange-traded index and stock option is European, so it binds here.
- The strike is discounted, the spot is not. Using X instead of X·e^(−rt) is the standard arithmetic error.
- Parity gives a fair price, not a forecast. It says nothing about where the stock goes.
- The arbitrage profit is fixed and known before expiry, and identical in every scenario — if it varies by scenario, the trade is wrong.
- Both legs must be executed simultaneously. The workbook warns that a half-built arbitrage is a naked position, not a hedge.
Where this is taught
- Series IV · Chapter 4: Exchange Traded Interest Rate Optionsintroduced here
- Series XVI · Chapter 4: Commodity Optionsintroduced here
- Series I · Chapter 4: Exchange Traded Currency Optionsintroduced here
- Series VIII · Chapter 5: Strategies using Equity Futures and Equity Optionsintroduced here
- Series V-D · Chapter 17: Strategies using Equity Futures and Equity Optionsintroduced here
- Series V-D · Chapter 21: Exchange Traded Interest Rate Options
Related terms
- Call optionA contract giving its buyer the right, but never the obligation, to buy the underlying at a fixed strike price — so the loss is capped at the premium and the gain is not.
- ArbitrageSimultaneously buying and selling an asset, or a replica of it, in different markets to profit from a price discrepancy — a trade that aims to lock in a profit rather than bet on direction.
- Protective putBuying a put option to protect a foreign currency receivable: it sets a floor rate (strike minus premium) while keeping the benefit if the currency rises. Suited to contingent cash flows.
- RhoThe option Greek that measures interest rate sensitivity — the change in an option premium for a one percentage point change in the risk-free rate. It is positive for calls and negative for puts.
- Cost of carryThe cost of holding an asset until a future date — financing interest plus storage, less any income it pays. It links spot to futures: fair futures price = spot + cost of carry − inflows.
- Covered callHolding the underlying in the cash market and writing a call against it — a way of earning premium income from a holding, at the cost of capping the gain above the strike.
- European optionAn option that can be exercised only on its expiry date. Indian equity options are European — coded CE for calls and PE for puts — unlike American options, exercisable any time up to expiry.
- Put optionA contract giving its buyer the right, but never the obligation, to sell the underlying at a fixed strike price — insurance against a fall, bought for a premium.
- Black-76 modelThe variant of the Black-Scholes model used to price Options on Futures, which substitutes the futures price F for the spot price and discounts the whole payoff at e^(-rT).
- Binomial pricing modelAn option pricing model that maps the underlying's possible prices as a tree of up and down moves at equally spaced time steps — accurate and flexible because it is iterative, but slow to compute.
- Black-Scholes modelA 1973 closed-form option pricing model using spot, strike, volatility, time to expiry and the risk-free rate; Indian exchanges use it for the theoretical price of exchange-traded currency options.
- Synthetic long callA protective put position — long the stock plus a long put at the same strike — whose payoff, limited loss with unlimited upside, mimics that of an outright long call option.