Put-Call Parityadvanced

Arbitrage when Parity is Violated

When put-call parity $C + K e^{-rT} = P + S_0$ does not hold, an arbitrageur can construct a riskless profit equal to the violation. The strategy is to buy the cheaper portfolio, sell the dearer portfolio, and pocket the difference. Two symmetric trades are called the conversion (long stock, long put, short call) and the reverse conversion (short stock, short put, long call). Transaction costs, borrowing rates, and dividends create a no-arbitrage band within which small violations are not profitable to exploit.

Why it matters

Put-call parity is a no-arbitrage condition derived from the law of one price. The portfolio $\text{Call} + \text{Bond}$ pays exactly $\max(S_T, K)$ at expiry, and so does the portfolio $\text{Put} + \text{Stock}$ . If two portfolios with identical payoffs trade at different prices, an arbitrageur sells the expensive one and buys the cheap one. The trade is funded out of the proceeds, the cash flows at expiry net to zero, and the difference banked today is risk-free profit. In practice, banks run electronic surveillance precisely to spot and close these gaps within seconds.

Formulas

If LHS is cheap (call + bond underpriced)

C + K e^{-rT} < P + S_0 \Rightarrow \text{Buy call, lend } K e^{-rT}, \text{ sell put, short stock}

Cash inflow today equals

P + S_0 - C - K e^{-rT} > 0

. At expiry all positions cancel, leaving the initial profit risk-free.

If RHS is cheap (put + stock underpriced)

C + K e^{-rT} > P + S_0 \Rightarrow \text{Sell call, borrow } K e^{-rT}, \text{ buy put, buy stock}

Cash inflow today equals

C + K e^{-rT} - P - S_0 > 0

. This trade is also called a conversion in market-maker parlance.

No-arbitrage band with frictions

|C + K e^{-rT} - P - S_0| < c_{\text{transaction}} + c_{\text{borrow}} + d \cdot e^{-rT}

Bid-ask spreads, stock borrow fees, and a dividend

d

between today and expiry must all be smaller than the apparent violation for the trade to be profitable.

Worked examples

Scenario

European options on a non-dividend-paying stock. Spot $S_0 =$ A$100, strike $K =$ A$100, risk-free rate $r = 5\%$ , time to expiry $T = 0.5$ years. Quoted prices are $C =$ A$6 and $P =$ A$4. Check parity and identify the arbitrage.

Solution

Parity check, left-hand side equals $6 + 100 e^{-0.025} = 6 + 97.531 = 103.531$. Right-hand side equals $4 + 100 = 104$. The gap is $104 - 103.531 = 0.469$. The LHS is cheap, so buy the call, lend $100 e^{-0.025} = 97.531 $, sell the put, and short the stock. Cash flow today equals$ 4 + 100 - 6 - 97.531 = 0.469$. At expiry, if $S_T > 100$ , the call delivers the stock at $K$ and the bond matures to $K$ , closing all positions. If $S_T < 100$ , the put is assigned and the stock is purchased at $K$ , again closing all positions. The result is A$0.469 of arbitrage profit per share, locked in today.

Scenario

Conversion in a market-maker book. A dealer observes ASX-listed XYZ stock at A$50, an XYZ 50-strike June call at A$2.10 bid, a 50-strike June put at A$1.50 offer, with $r = 4\%$ and 60 days to expiry.

Solution

Parity benchmark, $S_0 - K e^{-rT} = 50 - 50 e^{-0.04 \times 60/365} = 50 - 49.671 = 0.329$ . The implied $C - P = 0.329$ , but the market shows $C - P = 2.10 - 1.50 = 0.60$ . The call-minus-put is too high, indicating the call is overpriced or the put is underpriced relative to parity. The dealer puts on a reverse conversion, selling the call, buying the put, buying the stock, and borrowing the strike present value. The locked-in profit per share equals $0.60 - 0.329 = 0.271$ before transaction costs and stock borrow fees.

Common mistakes

✗Parity arbitrage needs large mispricings. Even a few cents per share is exploitable at scale. Market-makers run thousands of positions and a $0.05 violation across 10,000 shares is A$500 of riskless profit.
✗Parity applies to American options. Put-call parity in equality form holds for European options only. For American options the relationship becomes an inequality because of early exercise possibilities, which limits clean arbitrage to European-style products.
✗Transaction costs always destroy the arbitrage. Costs only matter relative to the violation. The no-arbitrage band widens with costs, but professional dealers with low marginal costs can profit from violations retail traders cannot.

Revision bullets

•Parity violation $\Rightarrow$ riskless arbitrage profit
•Conversion: long stock, long put, short call
•Reverse conversion: short stock, short put, long call
•Profit equals magnitude of the parity gap
•Transaction costs set the no-arbitrage band
•Holds for European options only

Quick check

Put-call parity holds at $C + K e^{-rT} = P + S_0$ . If the market shows $C + K e^{-rT} > P + S_0$ , the arbitrage is:

Parity violation arbitrage is most reliably available on:

Connected topics

Arb Futures PCP

In learning paths

Options Path

Sources

Hull (2022), §11.3 and §11.4
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Derives put-call parity for European options and discusses arbitrage strategies when parity is violated.
Stoll (1969), Put-Call Parity
Stoll, Hans R. The Relationship Between Put and Call Option Prices. Journal of Finance, Vol. 24, No. 5, December 1969, pp. 801-824.
Original empirical and theoretical paper that documented put-call parity violations in the US options market.
Merton (1973), Theory of Rational Option Pricing
Merton, Robert C. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, Vol. 4, No. 1, Spring 1973, pp. 141-183.
Develops the no-arbitrage bounds for option prices and the put-call parity relationship under dividends and continuous trading.
OIC, Put-Call Parity educational note
Options Industry Council. Put-Call Parity. OIC Education, accessed 2026.
Industry educational resource showing the conversion and reverse-conversion trades used by market-makers.

How to cite this page

Dr. Phil's Quant Lab. (2026). Arbitrage when Parity is Violated. Derivatives Atlas. https://phucnguyenvan.com/concept/parity-arbitrage

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