Delta and Gamma Close to Maturity

Take a look at the lesson on 'Sensitivity Greeks'.

Delta
Delta (Δ) measures the sensitivity of an option's price to changes in the price of the underlying asset. It ranges from 0 to 1 for call options and from -1 to 0 for put options. For a call option, delta represents the rate of change of the option's price with respect to the underlying asset's price.

Behavior Near Maturity:
As a call option approaches its maturity, its delta approaches either 0 or 1, depending on whether the option is out-of-the-money (OTM), at-the-money (ATM), or in-the-money (ITM).

• In-the-Money (ITM) Call Option: If the call option is in-the-money (the underlying asset price is above the strike price), the delta approaches 1. This is because the likelihood of the option finishing in-the-money is very high, and the option's price moves almost one-to-one with the underlying asset's price.
• Out-of-the-Money (OTM) Call Option: If the call option is out-of-the-money (the underlying asset price is below the strike price), the delta approaches 0. This reflects the low probability of the option finishing in-the-money.
• At-the-Money (ATM) Call Option: For an at-the-money call option, the delta will be around 0.5, reflecting the roughly equal probability of the option ending up in-the-money or out-of-the-money. As the option nears maturity, the delta will rapidly move towards 1 or 0 depending on small movements in the underlying asset's price.

The following two illustrations show how delta evolves as it gets closer to maturity.



Gamma
Gamma (Γ) measures the rate of change of delta with respect to changes in the underlying asset's price. It indicates how much the delta of an option will change as the price of the underlying asset changes.

Behavior Near Maturity:
As a call option approaches its maturity, its gamma increases significantly, especially for at-the-money options.

• At-the-Money (ATM) Call Option: The gamma of an ATM call option is at its highest near expiration. This is because small changes in the underlying asset's price can significantly alter the probability of the option finishing in-the-money or out-of-the-money, causing large swings in delta.
• In-the-Money (ITM) and Out-of-the-Money (OTM) Call Options: Gamma is lower for ITM and OTM options compared to ATM options, but it still increases as the option nears maturity. For ITM options, since delta is approaching 1, gamma decreases as the option becomes deep ITM. For OTM options, as delta approaches 0, gamma also decreases.

The following two illustrations show how gamma evolves as it gets closer to maturity.



Practical Implications

• Hedging
  The increasing gamma near maturity implies that delta hedging becomes more challenging and requires more frequent adjustments. For options traders and portfolio managers, this means closely monitoring positions and making quick adjustments to maintain hedged positions.
• Volatility
  High gamma near expiration means that the option's price is highly sensitive to the underlying asset's price changes, leading to increased volatility in the option's value.

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