Delta Hedging Digital Options Near the Strike
Summary
The document describes why a digital option’s stock hedge changes sharply as the underlying approaches and passes its strike. A digital option pays a fixed amount when it finishes in the money and nothing otherwise, so its payoff does not rise gradually with the underlying price. Near the strike, delta becomes large and gamma changes sign; a long digital call’s delta hedge therefore calls for increasing short stock exposure as the price approaches the strike, followed by buying stock back as it moves further in the money.
A separate answer emphasizes that a stock hedge removes delta exposure but leaves gamma risk, which is especially sensitive near the strike. The idealized Black–Scholes hedging argument assumes continuous, frictionless trading and complete markets. In practice, transaction costs and the need to rebalance mean that continuous delta hedging can be costly, so a stock hedge alone does not eliminate all option risk.
Key ideas
- A digital option pays a fixed amount depending on whether it finishes in the money.
- Its delta rises sharply near the strike, while gamma changes sign there.
- A long digital call’s delta hedge changes from shorting stock near the strike to buying it back further in the money.
- Delta hedging removes delta exposure but leaves gamma risk.
- Continuous hedging assumptions can fail in practice because of trading costs and market frictions.
Tags
Full text
# Why do we only need to buy or sell stock to hedge when the underlying is close to the strike? # Why do we only need to buy or sell stock to hedge when the underlying is close to the strike? Delta mesure the slope of the digital option.It also provides hedging information. Why do we only need to buy or sell stock to hedge when the underlying is close to the strike? ## Answer by AlRacoon (score 3) https://quant.stackexchange.com/a/38421 This is due to the payoff structure of the digital option. The payoff is nothing while the option is out of the money and then instantly goes to a fixed payment amount when it is in the money. It does not gradually increase as the option becomes increasingly in the money like a plain vanilla option. What makes these options difficult to hedge is that the gamma switches sign at the strike. It is an inflection point and the delta becomes very high as the option becomes at the money, only to start declining as the option becomes in the money. So a delta hedge for a long digital call requires shorting increasing amounts of stock as the underlying approaches the strike and then buying it back as the option becomes in the money. ## Answer by David Addison (score 1) https://quant.stackexchange.com/a/38443 In the case of a digital or vanilla option, buying or selling stock to hedge an option only eliminates delta risk. Notably, when the strike is near the underlying price, gamma risk is particularly high. I.e., the change in delta is highly sensitive wrt to a change in the underlying. As AIRacoon points, gamma exhibits particularly asymptotic behavior digital as the strike nears the underlying. Black-Scholes is derived by the delta hedging argument because it eliminates risk if and only if trading is costless and frictionless, and if markets are complete (see Fundamental Theorem of Asset Pricing). Only by eliminating the drift component does the Black-Scholes differential equation (basically the Kolmogorov and/or Feynman-Kac PDE) have a risk-free probability distribution. In reality, however, none of these conditions are fulfilled. An options position which is continuously delta hedged will almost surely incur excessive trading costs. Thus, in the real world, it isn’t not necessarily true that we only need to buy or sell stock to hedge an option.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.