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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?

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    $\begingroup$ You are right if the question is about the transactions cost, but the question is misleading you by asking about the interest cost. It is a rather strange question I must say. The stock used to hedge is purchased using borrowed money, so the bigger the delta the bigger the interest cost. But interest cost is not of much relevance to market makers, it does not enter into their decisions. $\endgroup$
    – nbbo2
    Commented Jul 10, 2023 at 5:42

2 Answers 2

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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.

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  • $\begingroup$ From the formula you gave, it makes 100% sense. On the other hand, this question comes from one of the FRM practice exams for level 1 and I don't think anyone would really deduce this from the information given in the curriculum. $\endgroup$
    – Tuo
    Commented Jul 11, 2023 at 1:43
  • $\begingroup$ Thanks TuoTuo. You can also visualise the payment here: desmos.com/calculator/oz5facscq3 $\endgroup$
    – Newquant
    Commented Jul 11, 2023 at 9:43
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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.

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    $\begingroup$ Could you please rewrite your answer so that it is self-contained? $\endgroup$
    – Bob Jansen
    Commented Jul 11, 2023 at 7:49

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