Theoretical and Actual P&L in Delta-Hedged Calls

Summary

The document poses a question about the value change of a portfolio that holds a call option and shorts the underlying in the option’s delta. As the underlying price changes, both the option value and its delta change, so the hedge is adjusted to the new delta. The requested comparison is between the theoretical change in portfolio value over a time step and the realized change, and how the two relate.

This setup points to the core mechanics of delta hedging, including the effect of rebalancing as market conditions change. However, the document contains only the question and gives no formula or answer. It does not state assumptions about the option pricing model, interest rates, dividends, transaction costs, or discrete versus continuous hedging. Those details are needed to derive a specific theoretical or realized P&L relationship.

Key ideas

• A delta-hedged call combines a long call position with a short underlying position sized by delta.
• Changes in the underlying affect both the option value and the hedge ratio.
• The hedge is rebalanced as delta changes.
• The document asks how theoretical and realized portfolio value changes relate but provides no derivation.

Tags

# Delta hedging and PF-value


# Delta hedging and PF-value












Imagine buying a call option and shorting the delta. After some time $dt$, the stock price changes, and so does the delta and the call option value. We re-adjust our hedge using this new delta.

Question: What is the (formula for the) theoretical change in value of such a portfolio, from one period to the very next? What is the actual change in value? And how are those two related?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.