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Finding a Put-Call Arbitrage with a Stock, Put, and Call

Article Quant Q&A · Author: user2792941
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Summary

The document tests whether a stock, a put, and a call with different strikes create an arbitrage opportunity. The example gives a stock price of USD 30, a put struck at USD 28 priced at USD 1, a call struck at USD 29 priced at USD 8, and a one-year risk-free rate of 20%. The proposed trade buys the stock and put while selling the call, for an initial cost of USD 23.

At maturity, the position is worth at least USD 28 across all stock-price ranges: above the call strike, the shares are called away for USD 29; below the put strike, the put sells them for USD 28; between the strikes, the shares can be sold in that interval. This establishes a minimum terminal value of USD 28 against repayment of USD 27.60 for borrowing the initial cost at the stated rate, leaving a guaranteed USD 0.40. The example assumes the stated prices and financing rate can be executed, and does not include transaction costs, taxes, or other market frictions.

Key ideas

  • Buy the stock and put while selling the call to create the illustrated position.
  • The call caps the share value above its strike, while the put sets a floor below its strike.
  • Between the two strikes, the stock can be sold for a value within that range.
  • Compare the minimum maturity proceeds with the financed initial cost to assess the arbitrage.

Tags

OptionsEquitiesArbitrageDerivatives pricing
Full text
# Arbitrage problem


# Arbitrage problem












Question

A share of non-dividend paying stock is trading at USD 30. The maturity of both options is 1 year from now. A put with a strike of USD 28 is trading at USD 1 and call with a strike of USD 29 is trading at USD 8 The annual risk-free interest rate is 20%.

Is there an arbitrage opportunity? If so, demonstrate how an arbitrage profit can be calculated.

Answer

From my calculations I cannot find an arbitrage opportunity. I have tried various payoff tables,but still unsuccessful. I am preparing for my exams and am still finding these problems very confusing. Is there any tricks or hints anyone can give me.

## Answer by rajah9 (score 1, accepted)

https://quant.stackexchange.com/a/32270

You could buy one share, sell one call, and buy one put. That would cost you \$23 (= 30 - 8 + 1).

A year later, if the stock were higher than \$29, the call buyer would call away the stock for \$29. You would net \$6. (Same is true if the stock were exactly \$29.)

A year later, if the stock were lower than \$28, you would exercise the put for \$28. You would net \$5. (Same is true if the stock were exactly \$28.)

A year later, if the stock were between \$28.01 and \$28.99, both options would expire worthless, and you could sell the stock for a net between \$5.01 and \$5.99.

Of course, you'd need to pay the piper. Your net would be reduced for borrowing the \$23 for a year. At 20%, this would be \$4.60.

You would be guaranteed \$0.40 (=\$5 - \$4.60).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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