# Managing an options portfolio at expiration – Part II

Let’s consider a portfolio of stock options that is fully delta hedged and expires 1 day from now. Assume a contract multiplier of 100 and that the stock is currently trading at \$200.30. What we are going to do in Part II is now conduct a basic analysis of the position. Although option trading software can conduct this analysis instantly, any option trader should know how to perform such an analysis with a pencil and sheet of paper (ok, we’ll let you off with a spreadsheet). You can conduct this analysis at any time, but the day or hours before expiration is typical.

First, here is the basic position.

STRIKECall/putOption delta (%)Position
196Put-10-600
198Put-29100
200Put-47-50
200Call53200
202Call30-300
204Call8500

Next, we are going to assume put-call parity holds and that any option with a strike below the spot price is a put and any option above it is a call. So, we can net the position off. Using the deltas from Table 1, we can also work out the implicit delta hedge we have against each net position. Also, we are going to make a note of the current value of the options and then work out the ‘premium’ associated with our position. In other words, we are looking at the value of each option and, depending on whether we are long or short the option, we are calculating whether we are paying to own the option or collecting money from being short the option. Remember that we are considering all these options as out-of-the-money options and regardless of whether they finish in or out-of-the-money, their optionality is going to zero. So, for example, the 200 strike we are long 200 lots of the calls and short 50 lots of the puts. But because this strike is below the current spot price (of \$200.30) we can view these options simply as puts. In other words, our net position in these puts is long 150 (200 lots of the calls – 50 lots of the puts). These puts have 60 cents of optionality left in them, so our loss from owning this strike is going to be \$9000. This comes from 150 lots * option multiplier of 100 * option premium of \$0.60.

STRIKENet positionOption value (\$)Position \$premiumDelta hedge
NET:\$9008,950
196-6000.05-\$3,000-6,000
1981000.30\$3,0002,900
2001500.60\$9,0007,050
202-3000.32-\$9,6009,000
2045000.03\$1,500-4,000

So the position as a whole is going to cost us \$900 in theta by decaying to zero. And we are long almost 9000 lots of the underlying product as a delta hedge against the whole portfolio. The next stage is to figure out the profit and loss at expiration if we do nothing at all for different spot prices. This can be seen in Table 3.

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

The way to complete Table 3 is to work out the profit and loss from the position deltas and then adjust for the option premium. What does this mean? Well, if the spot product expires at \$200.30 (ie is unchanged at expiry from its current price), then our delta hedges neither make nor lose anything. But the option portfolio as a whole decays in value to zero and we lose \$900. If however the spot market rallies, then our delta hedge will make money up to a point. There is a break-even point roughly 10 cents higher than the current spot price where our delta hedge (of long circa 9000 deltas) makes enough money to pay for the theta bill of \$900. The only complication is when we pass through a strike where we have a position. For example, if the spot price passes through the \$202 strike, then we pick up an extra short 30,000 deltas. This makes the position overall short some 21,000 deltas and will therefore lose \$210 per tick increase in the spot. This is true until the spot passes through the \$204 strike, at which point we pick up 50,000 long deltas from owning 500 of the \$204 strike. The important thing is to only p&l deltas over the range where we are carrying them. So the short deltas from the position in the \$202 strike only have an effect above \$202. Below that, they have no impact.

In Part III, we will go through this procedure in more detail before thinking about how this can help in forming risk management strategies.

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