For a digital option, whose payoff jumps depending on whether the price is above or below the strike price at maturity, the delta becomes arbitrarily large near the strike price as maturity approaches. A hedge that holds the underlying asset in line with the delta then requires buying and selling large quantities each time the price crosses the strike price1.
Consider a hypothetical example in which a digital call with a strike price of 100 yen that pays 10 yen per unit if the price exceeds 100 yen at maturity is sold. One day (1/250 of a year) before maturity, with the price at 100 yen, an interest rate of 1%, and volatility of 20%, the delta is 3.15. The hedge holds 3.15 units of the underlying per unit sold, about 315 yen in value, which is more than 30 times the maximum payment of 10 yen. If the price exceeds 100 yen at maturity you pay 10 yen, and if it does not you pay nothing.
Notes
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Alison Etheridge, "A Course in Financial Calculus" (Cambridge University Press, 2002), Chapter 6, Section 6.1, Example 6.1.1, pages 140–141. The book states that risk from small price changes and transaction costs can outweigh the maximum liability of the digital that was sold, and calls this "pin risk". ↩