How Drift Affects the Gamma Value of a Delta-Hedged Option

Summary

The document raises a question about how the underlying asset’s drift affects the value accumulated by a delta-hedged option. It refers to continuous delta-hedging formulas and quotes an explanation from a derivatives text: when an initially at-the-money asset moves quickly because drift is large in either direction, it may leave the region where the option has substantial gamma. The question asks why ending near the strike would be favorable for the hedged portfolio.

No answer or derivation is included, so the document does not establish a quantitative relationship between drift, gamma, and realized hedge value. It provides no assumptions about the option model, volatility, rebalancing frequency, or transaction costs. The quoted intuition is therefore a prompt for analysis rather than a complete method or result. Readers would need to examine the relevant delta-hedging framework and distinguish the effect of the assumed price path from expected option value under a pricing measure.

Key ideas

• The question concerns drift’s effect on the value of a delta-hedged option.
• It focuses on how movement away from the strike changes the option’s gamma exposure.
• The cited intuition suggests that gamma can become small after a large underlying move.
• No derivation, model assumptions, or final explanation is supplied.

Tags

# Path dependency for Delta hedge value


# Path dependency for Delta hedge value












This is actually a follow-up questions for the two threads below - value of a delta hedged option:

Delta hedge value formula

Continuous delta hedge formula

My question is that how the drift (mu) impact this hedged portfolio. In Paul Wilmott's book, he comments that “if we start off at the money ad the drift is very large (positive or negative), we will find ourselves quickly moving into territory where gamma, and hence the hedged portfolio value, is small. The best that could happen would be for the stock to end up close to the strike at expiration, this would maximize the total profit."

Can someone help to explain why? Thank you so much in advance.

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