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When delta-hedging and using shares to do so, which "accounting" method should one use via their brokerage when they are executing the said delta-hedging adjustments? FIFO (first-in-first-out) or LIFO (last-in-first-out)?

Does using FIFO versus LIFO make any difference in your adjustments PnL when you finally liquidate your entire position along with any remaining hedges you have? (Say you kept your entire delta-hedged position until expiration of the Option you are long or short).

Natenberg example1

Natenberg example2

The picture above is an excerpt from Sheldon Natenberg's Option Volatility and Pricing, Chapter 5 pg. 85

He explains what to do when finding an Option different from it's theoretical value and how to delta-hedge, etc. The picture showcases the adjustments he made after buying an underpriced Call Option and dynamically hedge.

He then says:

What was the result of all the adjustments required to maintain a delta neutral position? In fact the result was a profit of $205.27. (The reader may wish to confirm this by adding up the cash flow from all the trades in the adjustment column in Figure 5-1.) This profit more than offset the losses incurred from the original hedge.

I've tried to verify for myself that the profits from the adjustments do indeed add up to 205.27, but every time I tried to do so I ended up with a wrong amount of profit. I tried using both FIFO and LIFO since it's not stated what's being used and the profits I calculated both ended up being wrong.

What further confuses me is that it's also stated (in the excerpt) that he bought back the original hedge (which was -57 contracts) for a loss of -67.38. If you look at the adjustments, you will realize that right at Week 2, whether using FIFO or LIFO, it's impossible that he was able to have contracts remaining from his original hedge (which he shorted 57 of which at 101.35). The original loss would only be the amount stated if he had actually kept the original 57 contracts and then bought them back at 102.54, but that's impossible with the adjustments observed.

And also what further confuses this is at the end of Week 10, he buys 36 contracts to eventually get back to the original hedge of -57 as he started with and then buy them back at 102.54 as he claimed above. It makes no sense really.

The only this could be possible is if somehow he had the original hedge and adjustments separated from each other by using different brokerage accounts. Because as far as I know, no brokerage allows you to have simultaneous long shares and short shares on the same stock. Which is what would have to be the case here after attempting to calculate the profits from the adjustments.

It would be great to have an answer explaining what's going on here and how it relates to FIFO/LIFO and brokerage rules, etc.

Thank you in advance.

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  • $\begingroup$ By using this equation to calculate realized PnL I was able to get to the profit number of 205.27. Sells (43 * 103.78 + 15 * 99.98 + 18 * 103.59 + 3 * 100.76 + 100.39 * 7 + 5 * 102.26) - Buys (16 * 99.07 + 29 * 99.26 + 10 * 98.28 + 36 * 102.54) = 205.27 But it still doesn't make sense per current real world brokerage rules and how in the delta-hedging example above someone can have be short shares of a stock and not have them count as a buy-to-close order when they buy more shares as an adjustment. $\endgroup$ Sep 26 '20 at 2:11
  • $\begingroup$ I do not believe institutions are subject to these rules (they mark everything to market), unlike individuals who have to be concerned about short term/long term gains for the personal income tax (so they care about LIFO vs FIFO). $\endgroup$
    – noob2
    Sep 26 '20 at 2:15
  • $\begingroup$ @noob2 I suppose that's so. But what I mean by using FIFO/LIFO an how it comes into play here is that you see in the excerpt from the book, he was short -57 contracts at first representing 5700 delta. Then he sells -5 more for a total of -62, and finally buys back 16 contracts. Which means he bought back 16 (using FIFO) or (-5+16) = 11 (LIFO) of his original hedge. But then he goes on to say at the end of Week 10 he bought back his original hedge of 57 contracts which he shorted at 101.35. How could that be possible if at Week 2 he already bought back 11 or 16 of those original 57 contracts? $\endgroup$ Sep 26 '20 at 2:49
  • $\begingroup$ You said "11 or 16". It is clear from Fig 5-1 that in week 2 he bought 16 contracts (5th column) and after doing this he owned 11 more contracts (6th column) than the original position of -57 contracts (so he had -57+11= -46 contracts). Column 5 is the trades (other than the initial one) and column 6 is the cumulative amount of those trades. In the 10th row the cumulative amount is 0 (meaning we are back to a -57 position). $\endgroup$
    – noob2
    Sep 26 '20 at 7:14
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    $\begingroup$ @noob2 I tried again & used both FIFO/LIFO methods when I re-calculated the adjustments & ended up with the correct amount of 205.27 profit Although this time I didn't include the original starting hedge & left it alone for me to get the correct amount of profit from the adjustments. Looks like the numbers & everything else makes sense when you leave the original hedge completely out of the adjustments & treat those short -57 contracts as a separate lot of some kind & then buy them back. Per the real world, brokerages usually treat adding long shares to a short position as a buy-to-close order $\endgroup$ Sep 28 '20 at 0:12

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