Put-Call Parity Violations and Riskless Arbitrage

Summary

The note explains how violations of European put-call parity can indicate an arbitrage opportunity. When the call is priced below the value implied by the put, stock, and discounted strike, the suggested position is to buy the call, sell the put, short one share, and borrow funds. At expiry, the option and stock cash flows offset across either price outcome, while the initial pricing discrepancy provides the profit under the stated parity relation.

For American options, the answer points to the relationship between American and European call values when there are no dividends, and to the higher value of an American put, as a way to reason about the second inequality. The explanation is brief and assumes the relevant parity conditions, financing, and ability to trade at quoted prices. It does not discuss transaction costs, early exercise details, or market frictions, so the argument is a theoretical arbitrage construction rather than a complete execution plan.

Key ideas

• A European put-call parity violation can be tested with a portfolio of options, stock, and borrowing.
• The proposed position buys the underpriced call, sells the put, shorts stock, and borrows cash.
• The expiry payoffs offset across outcomes when the parity assumptions hold.
• American option reasoning depends on dividend assumptions and the relative values of American and European options.
• Trading costs and execution constraints are outside the explanation.

Tags

# Single-period market with probability space


# Single-period market with probability space












Let $C^E$, $P^E$, $C^A$, and $P^A$ denote prices of a European call option, a European put option, an American call option and an American put option, respectively. All of them with expiry time $T$ and the same strike price $K$. we assume $r\geq 0$ to be the continuously compounded interest rate. I want to show that if it holds that,

$$C^E-P^E-S(0)+Ke^{-rT}<0,$$

then we can make a sure risk-less profit.

Furthermore, I am interested in showing that,

$$C^A-P^A-S(0)+Ke^{-rT}>0,$$

then we can also make a sure risk-less profit.

Anybody have an idea?

## Answer by Kermittfrog (score 1, accepted)

https://quant.stackexchange.com/a/58953

In both cases, you should argue along the lines of options arbitrage. If you think an asset (or a portfolio) is relatively cheap (as in: arbitrageable) then you simply buy low, sell high.

In your first case, it seems that the call is too cheap, as in:

$$ C^E<P^E+S-Ke^{-rT} $$ So let's buy the call, sell a put, short a unit of stock, and borrow some money:

At maturity $T$, either your call or the put are in the money but in any case, your future payoff is always zero. On the other hand, you paid less for the call than what you got for the put/stock/borrowing position, i.e. you have a risk-less arbitrage profit.

You should be able to follow the same line of thought for the American option situation, knowing that $C^E=C^A$ (without dividends), and $P^A>P^E$.

HTH?

Edit: Maybe slide 3ff of this source can help?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.