Why a Delta-Hedged Call Loses Value When Price Stalls

Summary

The document clarifies the outcome of replicating a long call with a dynamic position in the underlying and bonds. A replication portfolio is intended to match the option’s value changes, including losses from time passing; it is not designed to guarantee a profit over each interval. If the underlying stays flat, the call can lose value through theta, and the replicating position should reflect that loss.

The exchange explains that a small underlying move may also fail to offset theta, with the result depending on the position’s combined changes. This is a conceptual clarification rather than a full derivation of Black–Scholes replication. It gives no numerical example or discussion of assumptions such as continuous rebalancing, transaction costs, or model mismatch, so it should not be read as a complete account of real-world hedging performance.

Key ideas

• A replication portfolio aims to match the call’s value, not ensure that it earns money each period.
• A flat underlying can coincide with a decline in call value from theta decay.
• The hedge should reflect the option’s time-related value loss as well as price movements.
• The brief explanation does not address practical costs or the model’s replication assumptions.

Tags

# Black Scholes Replication If Underlying Does Not Move?


# Black Scholes Replication If Underlying Does Not Move?












Let's say you are long a call and want to replicate that call buy being short underlying and long bonds.

If the underlying moves up in the next period but not enough to cover theta, the option loses money. Is this loss offset by your long bond position?

Now let's say the underlying does not move in the next period. The option loses money. Does the BSM model stop working in this situation?

Thanks!

## Answer by Chris (score -1)

https://quant.stackexchange.com/a/46556

You're replicating the value of the long call. As with your small move example, other things equal, if the underlying doesn't move, your position loses money (as an actual call would) due to theta. It works exactly as it should, the position itself just loses money as a result.

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