Put-Call Parity Arbitrage and the Financing of a Stock Short

Summary

The document presents a question about a European call priced below its put-call parity lower bound. In the example, an arbitrageur shorts stock, buys the underpriced call, and invests the cash proceeds. At expiry, if the stock is above the strike, exercising the call supplies shares to close the short; the question asks why the strategy appears not to charge interest on the short sale proceeds.

The central concept is the arbitrage relationship among stock, calls, and discounted strike cash flows. The example is intended to clarify how financing fits into a parity trade, but the document contains only the question and its setup, not an answer resolving the financing treatment. It therefore does not establish whether the calculation accounts for borrowing costs, short-sale proceeds, or other market frictions. Readers should treat the stated payoff as an incomplete illustration rather than a complete arbitrage analysis.

Key ideas

• Put-call parity relates a European call price to the stock price and discounted strike value.
• The example combines a stock short, a long call, and investment of the short-sale proceeds.
• The call can be exercised to obtain shares for closing the short when the stock finishes above the strike.
• The document raises but does not resolve how financing and short-sale interest affect the calculation.

Tags

# Understanding basic options arbitrage in Hull


# Understanding basic options arbitrage in Hull












I’m reading Hull’s book, Options, Futures and Other Derivatives. In Chapter 11 he discusses put-call parity and the arbitrage opportunities that can result from its violation. I’m having a basic issue, which is it seems he is shorting stock without paying interest. Here’s a concrete example from his section 11.3 concerning a European call:

Suppose that $S_0$ = \$20, K = \$18, r = 10% per annum, and T = 1 year. In this case, $$S_0 - Ke^{-rT}= 20 - 18e^{-0.1}= 3.71$$ or \$3.71. Consider the situation where the European call price is \$3.00, which is less than the theoretical minimum of \$3.71. An arbitrageur can short the stock and buy the call to provide a cash inflow of \$20.00 - \$3.00 = \$17.00. If invested for 1 year at 10% per annum, the \$17.00 grows to $17e^{0.1*1} = \\\$18.79$. At the end of the year, the option expires. If the stock price is greater than \$18.00, the arbitrageur exercises the option paying \$18.00 for the stock and uses the stock to close out the short position. This leads to a profit of \$18.79 - \$18.00 = \$0.79

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I don’t understand how in this scenario the arbitrager doesn’t need to pay interest over the 1 year on the $20 he borrowed to short the stock.

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