Managing an options portfolio at expiration – Part III

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Here is the payoff table again.

Spot priceNet deltaDelta P&LPremiumNET P&L
19443,950-$46,385-$900-$47,285
19543,950-$2,435-$900-$3,335
196-16,050$41,515-$900$40,615
197-16,050$25,465-$900$24,565
198-6,050$9,415-$900$8,515
199-6,050$3,365-$900$2,465
2008,950-$2,685-$900-$3,585
200.38,9500-$900-$900
2018,950$6,265-$900$5,365
2028,950$15,215-$900$14,315
203-21,050-$5,835-$900-$6,735
20433,115-$26,885-$900-$27,785
20533,115$6,230-$900$5,330
20633,115$39,345-$900$38,445
Remember, one way to look at this position is simply as out-of-the-money options, that will expire worthless, and deltas. Our current delta is the net amount of the underlying we are holding against the position, which is roughly 9000 lots. Our out-of-the-money options, combined, will lose about $900 in optionality (or option premium). If the spot price moves through the strike of any option before expiration, then that option effectively ‘turns into’ the underlying. If however we hedge this 100%, then it reverts again to being an out-of-the-money options (but a call rather than a put or a put rather than a call). Don’t worry if this seems like a complete jumble. It should become clearer as we walk through it.
So, if we expire right here at $200.30 we lose $900 from our options and our deltas neither make nor lose anything.  If the spot price drifts down just 0.30 cents, then our loss is c.$3500. Why? Because the options still lose their $900 of premium and our deltas also lose money (0.30 cents on circa 9000 lots). So much for that being a delta hedge! How can this be right? The point is that this delta hedge (like any delta hedge) is only temporarily right; in this case it is the ‘right’ delta hedge the day before expiration. But during expiry day itself the delta of our options changes dramatically. In fact, every single option delta goes to either zero or 100. Whereas our 9000 lots of delta hedge was arrived at when the options were thought to have different deltas.
Once we drop through the 200 strike, things change. Recall that we are (net) long 150 lots of these puts. This means that as we move through the strike, we become shorter 15,000 deltas at a price of $200. Why this quantity? Simply from the definition of these puts. Each put gives its owner the right to sell 100 lots of the underlying. Obviously if these puts are in-the-money we are going to exercise them, so we are effectively just shorter 15,000 deltas below $200. Combined with our existing deltas (long 9,000), the makes us net short around 6,000 deltas. Therefore, once the spot has dropped to $199, we are back in the black because these deltas make us $6,000. Between $200 and $199 there is a break-even point at $199.41 (the spot price where the current deltas have made back the losses we made from the move between $200.30 and $200). And another $6,000 is forthcoming if the spot drops another dollar to $198. The $198 strike puts are another long position of ours; so if the spot goes through $198, we become shorter deltas again. This time we get shorter 10,000 lots. The ‘best case scenario’ on the downside occurs at $196 where we have some $40,000 of profit. But then our short 600 lots of the 196 puts kicks in. Our short delta position switches to a long delta, as we pick up 60,000 lots of the underlying from our short put position. Why? We are short the 196 puts; if they become in-the-money, we can expect to get assigned on these puts. In other words, their owner will exercise them, giving him the right to sell the spot to us at $196. So below $196 we are long over 40,000 deltas and can therefore expect all our profit to have vanished by the time the spot reaches $195. Below that, we are losing all the way.
You can repeat this process on the upside. Up to $202, we make money because our initial deltas are being carried up and outweigh the losses in option premium. But we are short 300 lots of the 202 calls, so through that strike we become short 30,000 deltas, leaving us net short around 21,000 taking into account our initial long 9,000. The losses peak at the 204 strike. By being long 500 lots of this strike, we become long delta overall again and start to make back our losses and eventually profit for all rallies over the break-even of $204.84.
As we move through a strike, that option becomes equivalent to the underlying. At $203.99, the $204 calls expire into nothing. At $204.01, there are 100% equivalent to the underlying product. Notice how in every case, moving from a long position to a short is desirable and the reverse is undesirable. As rules of thumb go, this is not a bad one for option risk management in general, not just at expiration. In Part IV, we will consider how all this information might be used to make strategic decisions regarding risk management of the option portfolio.

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