Extracting Arbitrage from Mispriced Stock Options

Summary

The example tests whether a stock, call, and put with the same strike are priced consistently. It proposes buying the put and the stock while selling the call. For the stated prices, the combined position costs less than its payoff at expiration, which the answer checks across stock prices below, at, and above the strike.

The payoff analysis illustrates how combining options with the underlying can expose a pricing mismatch. Its conclusion relies on the quoted prices being executable and on the options sharing the stated strike and expiration. The example does not discuss transaction costs, financing, margin, taxes, early exercise, or whether the prices are simultaneous, so it does not establish that the trade is practically risk-free in a live market.

Key ideas

• A long stock position combined with a long put and short call can be evaluated across expiration prices.
• The example identifies a positive expiration payoff relative to the initial cost for every stock price considered.
• The argument assumes the quoted prices are available and the options have matching terms.
• Transaction costs, financing, margin, and execution conditions can affect whether the apparent arbitrage is realizable.

Tags

# How do I extract the arbitrage?


# How do I extract the arbitrage?












You are looking at a particular stock ticker and its options. You can go long or short on any quantity of the following instruments:

- Each unit of stock is priced at \$10.

- A call on the stock with strike price at \$15 is priced at \$2.

- A put on the stock with strike price at \$15 is priced at \$6.

Is there an arbitrage opportunity here? If so, how do you extract the arbitrage?

## Answer by Larry Cai (score 2, accepted)

https://quant.stackexchange.com/a/71398

It may be possible with a synthetic short with a long underlying stock.

Buy 1 put and sell 1 call for a debit of $4

Buy 1 stock for a debit of $10

Net debit = $14

On expiry, if stock is:

\$0: Call and Stocks are \$0, Put is worth \$15, net \$1 gain.

\$10 (unchanged): Call is \$0, Stock is \$10, Put is \$5, net \$1 gain.

\$15 (worthless options): Call and Put are \$0, Stock is \$15, net \$1 gain.

\$X, X>\$15 : Put is \$0, Stock is \$X, Call is \$15-\$X (it's a short call position). This nets out to \$15, take away the initial debit of - \$14, and you net \$1 gain again.

There may be other opportunities, this is just one I found.

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