How Option Moneyness Affects Delta-Hedge Interest Costs
Summary
The document examines how an option’s moneyness affects the interest cost of carrying a delta hedge. The accepted explanation isolates financing interest from other components of hedging, such as theta and gamma costs. Under that framing, the cash interest flow scales with the hedge’s delta, the stock price, and the interest rate. If the other inputs are held constant and the hedge requires buying shares, a deep in-the-money option has a delta closer to one and therefore entails greater financing cost than an at-the-money option.
A second response points out that the question lacks enough information to determine whether the hedge incurs a payment or earns interest: a positive stock position may require borrowing, while a short position may generate cash. The conclusion also abstracts from changing market inputs, option positions, and other hedging costs. Thus, the deep in-the-money result applies to the stated comparison of interest expense under a positive hedge, not necessarily to total hedge cost or every portfolio.
Key ideas
- The isolated interest cash flow on a delta hedge depends on delta, stock price, and the interest rate.
- For a positive stock hedge with other inputs held constant, greater delta means more financing and higher interest expense.
- Deep in-the-money calls generally have deltas closer to one than at-the-money calls.
- The sign of the hedge matters because a short stock position may earn interest instead of paying it.
- Interest expense alone does not describe total dynamic hedging costs.
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Full text
# Cost of Delta Hedging
# Cost of Delta Hedging
I am confused about the following question:
A trader holds a portfolio of short option positions. The trader limits the risk of these exposures by maintaining a delta hedging strategy. In evaluating the dynamic nature of this strategy, which of the following is correct about the interest cost of carrying the delta hedge? A. The cost will be highest when the options are deep out-of-the-money. B. The cost will be highest when the options are deep in-the-money. C. The cost will be highest when the options are at-the-money. D. The cost will be lowest when the options are at-the-money.
> I believe the answer is C, because delta is most sensitive to changes in the underlying when the option is at the money. However, the solution claims the answer is B because the deeper the options are in-the-money, the larger their deltas and therefore the more expensive to delta hedge.
Can someone please explain this?
## Answer by Newquant (score 2, accepted)
https://quant.stackexchange.com/a/76079
From the perspective of only interest payments, and not $\Theta$ or $\Gamma$ hedging costs, then the interest payment on the hedge is given by: $$ \Delta_{(S_t,r_t,T-t, \sigma_t)} * r_t * S_t * dt $$ Where the t subscript shows the value at time T. Given that for all delta values for all option strikes, r, and S are constants, all that matters is the largest value of $\Delta$, so those deep ITM options with a $\Delta$ near 1 will have the highest cost.
## Answer by Rylan (score 1)
https://quant.stackexchange.com/a/76081
I don't think there is enough information to answer the question.
The hedge involves an interest cash flow which takes the form
$$\Delta_t S_t r dt$$
Specifically, we would need to know if hedging involves borrowing to buy the stock (and thus paying interest) or shorting the stock (and thus receiving interest). In other words, $\Delta_t$ could be positive or negative.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.