Put-call parity

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.

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:

1. Buy the underpriced put and buy the stock.
2. Sell the call.
3. 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.

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 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.

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.

• 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.