A trading friend wanted a trade that was the equivalent of shorting 75 shares of stock. That trading friend wanted to use options to create that position. The friend had a question: would purchasing a single put with a delta of -0.75 (or -75, once the 100 multiple was applied) or three puts with deltas of -0.25 be best?

I thought this example might be good to use for an Options 101 article. My first questions would be, how long was this trade expected to last? Would any dividends be awarded while the trade was open? If the trade was expected to last only a few days up to a few months and there were no dividend payments involved, then it would probably be okay to ignore the impact of dividend payments and carrying costs on the options' values. However, if this was trade would endure a year or two, using LEAPs, then those dividend payments and carrying costs probably shouldn't be ignored.

However, for most of our short-term options trades, we can ignore those dividend payments and carrying costs, and my trading friend could, too. Even with that supposition, choosing among the two scenarios my trading friend presented is still not a simple choice. Purchasing a put position with a delta of -75 would be the equivalent of shorting 75 shares of stock only at over a short period of time and through a small price movement or change in volatility. It's a snapshot option choice. Once either price or volatility begins moving, the purchaser of that put position would have a position that was short either fewer or more deltas than the original position. The position would be the equivalent of shorting either fewer or more shares of stock than had been intended.

For example, using the Op-Eval software packaged with James Bittman's Trading Options as a Professional, I can determine that a 32-day-to-expiration, 75-strike put on a hypothetical stock with an IV of 40 percent has a delta of -0.75 when that hypothetical stock is \$68.75.

Imagine that my trading friend bought that put, theoretically priced at \$7.35. My trading friend then has a position that's short 75 deltas and is the equivalent of being short 75 shares of stock.

For about five minutes. Or, perhaps not even that long. Perhaps five seconds.

The Op-Eval software churned out some figures for various changes in price or volatility. If, within a few hours, the hypothetical stock's price drops \$3.00 and everything else remains the same, the delta for that 75-strike long put is now -0.85. For the whole position, using the 100 multiplier, my friend would now be short the equivalent of 85 shares of stock. The theoretical price of the option is now \$9.77, so there's a \$2.42 x 100 = \$242 profit to be had, excluding commissions. Because the delta of the position was getting more negative on the way down, my friend would actually have benefitted more than the expected \$225 that would have been gained if that friend had shorted 75 shares of stock.

Price seldom changes without any other changes occurring, however. If, as often happens on a strong drop, implied volatility rose, the absolute value of the delta would gravitate toward 0.50, an easy-to-remember tidbit that which Bittman revealed in his book. If, in addition to the hypothetical stock's drop of \$3.00, the volatility climbed to 45.00 percent, the rising volatility would somewhat lessen the change in deltas, working in opposition to the price change with this in-the-money 75-strike put. Let's see what I mean by that with some examples.

Bittman's software reveals that the delta for this hypothetical situation would now be -0.82 rather than the -0.85 that it would have been if price had dropped but implied volatility remained the same. If my friend had wanted to be short 75 deltas, this position was no longer the right position in this case, either. The put position would then be -82 deltas, the equivalent of shorting 82 shares of stock rather than 85 as occurred when if the hypothetical stock had dropped \$3.00 with no accompanying change in volatility. The theoretical price of the option is now, however, \$9.99, so a larger \$264 profit will be found. This effect occurs because there are still 32 days until expiration. Even though this is an in-the-money put, the put still has a lot of time value or extrinsic value. That extrinsic value is what's being plumped up by the rise in implied volatility. The intrinsic value isn't impacted by a change in volatility.

What happens if the hypothetical stock had climbed \$3.00 that afternoon? If the stock's price had climbed \$3.00 and the implied volatility had stayed at its original 40.00 percent, the Op-Eval software calculates that the theoretical delta would drop to -0.62. The position then would have been short 62 deltas when the 100 multiplier was implied. No longer would the trading friend have had the position that was equivalent to shorting 75 shares of stock. The put's theoretical price would have dropped to \$5.29, so a \$206 loss would have occurred, with commissions and fees added to that. This loss, however, is less than the \$225 loss that would have been experienced if the friend had been short 75 shares of stock that climbed \$3.00. This is because the deltas were growing less negative as the stock climbed. At the first snapshot in time, the trading friend's position was experiencing a loss that was equivalent to the loss if short 75 shares of stock, but by the time the \$3.00 price change had finished, only losing as much as would be lost if short 62 shares of stock.

If, as often happens on a sharp rally, the implied volatilities had dropped, the deltas would have been impacted, too. In the first example, the implied volatility had risen from 40 to 45 percent, a change of 5 percent. Imagine that in addition to a price rise of \$3.00, the implied volatility dropped 5 percent. (Since implied volatility is expressed in percentage terms, a clarification is necessary. I mean this drop: 40 percent - 5 percent = 35 percent. I do not mean a drop of 5 percent of 40.)

Implied volatility of that option would then measure 35 percent, and the Op-Eval software calculates that the new delta of the put option would theoretically be -0.65. Once again, for this in-the-money position, this typical direction of change in price and volatilities worked against each other. With the typical lowering of implied volatility when prices rally, the change in deltas is not as big as it would have been with the price change alone.

The 1-lot put position would now be short 65 deltas. Again, the trading friend would not be holding the expected position. The put's theoretical price would have dropped to \$4.88, so a \$247 loss would have been incurred, with commissions and fees on top of that. This loss is bigger than the \$225 the trading friend would have incurred if shorting 75 shares of stock, but that's again because that volatility changes impact that time or extrinsic value.

What about the idea of the three long puts, each with a delta of -0.25? Those puts are going to be far out of the money. They have a delta with an absolute value below 50. Remember Bittman's quick tidbit about how a rise in volatility impacts delta? That rise moves the absolute value of the delta toward 0.50. Those out-of-the-money puts with a delta of -0.25 would see their deltas moves toward (but perhaps not far toward) -0.50 with a rise in volatility. So, if prices are dropping with these far OTM options, both the drop in price and the probable attendant rise in volatility should work together this time, rather than opposite each other as they did with the ITM put with the delta of -0.75.

Perhaps it's hard to wrap your thoughts around that last supposition. Let's run through some examples. Keep in mind as we do that these options are far OTM, so they have no intrinsic value. All their value is time value or extrinsic value, that value most impacted by any change in volatility.

For the first example, taking the same hypothetical stock with price originally at \$68.75 with an implied volatility of 40 percent, Bittman's Op-Eval software predicts that a 64.00 strike put would have a delta of -0.25. Of course, a \$68.75 stock isn't going to have a 64-strike put, so we have a quandary. We're going to have to use a 65 strike if we want to be true to real-life circumstances. However, if we want to compare apples to apples in observing the effects of the price and volatility changes, we need the 64 strike that's not actually going to be available if we stay true to real trading conditions.

I decided true-to-life was better. The delta of the 65-strike put would be -0.29, so my trading friend buying three of these would have held a position with a delta of -87, already higher than the friend wanted. The theoretical cost of this put is \$1.61, so for the three puts, the total debit would be \$483, not considering commissions and fees.

Let's imagine that the same day, 32 days to expiration, the stock price dropped that same \$3.00 we used in the first example with the single in-the-money put. The delta of the 65-strike put is now -0.44, so the position, with three of these long puts, now has a delta of -132. My trading friend would now be short the equivalent of 132 shares of stock, far more than originally wanted. The put's theoretical value is now \$2.70. The total profit for all three puts would be \$327, a much bigger gain than when only one put with a delta of -0.75 was purchased. It's also much bigger than the gain my trading friend would have experienced if that friend had shorted 87 shares of stock, with the delta of the three-put position originally at 87.

If, as often happens, implied volatility rose as price dropped, in most cases, such a combination would have had a cumulative effect on the delta, both changes acting to increase the absolute value of the delta. Due to some peculiarity in this particular option or the software performing the calculation, the combination of price dropping \$3.00 and the implied volatility rising from 40 to 45 percent, the deltas would have still be -0.44, the same change seen previously. The theoretical price of the put is now \$3.09, so the profit on the three puts would now be \$444.

What if price had risen \$3.00? This time, if only price had changed, the delta of the 65-strike put would have changed from -0.29 to -0.18. My friend would have been short the equivalent of only 54 shares of stock after the \$3.00 rise in price. The three puts' theoretical price would have dropped drastically to .91 per contract. The loss on the position would be \$210.00, but that's still less than \$225 my trading friend would have lost if that friend had been short 75 shares of stock with the stock price rising \$3.00.

If, in addition, implied volatility had again dropped to 35 percent with a \$3.00 climb, the delta of the long puts would now be only -0.15. The position would have had a delta of only -45, the equivalent of being short only 45 deltas or 45 shares of stock. As with the previous far-OTM example, the rise in price and change in volatility worked in the same direction. The three puts' theoretical price would now be only .63, and the position would have suffered a \$294.00 loss.

We weren't quite comparing apples to apples because of the problems with finding the correct strikes with this setup. What should be clear is that the setup to be short (or long) a certain number of deltas is a snapshot setup only. Any change in price or implied volatility will change that snapshot. The number of short deltas will change accordingly, as would the number of long deltas if the position sought had been a long-delta position. As these cases show, that "snapshot" quality is true even if the changes occur on the day the trade was originally entered. What should be obvious is that the degree of the change will differ if the position had consisted of a 1-lot position of an in-the-money put or a 3-lot position of out-of-the-money puts, even if both positions had started out at an equivalent -75 deltas. It should be clear that my trading friend needed an idea of what would happen to price and implied volatilities before a choice was made.

Passage of time will change these relationships, too. I wanted to isolate price and implied volatility changes for this article to show how they worked opposite each other in the ITM puts and in the same direction in the OTM puts.

If my trading friend's ultimate intention had been to remain short 100 deltas instead of 75, the equivalent of being short 100 shares of stock, there would have been a way to do that and remain short that many deltas no matter what happened to price or volatility. Creating such a position is easier if the trade was a short-term one not involving dividend payments or significant carrying costs. If we can dispatch the concerns about dividends or carrying costs, we can go back to the equation defining synthetic relationships, the ultimate relationship that determines how stocks, calls and puts are related. I first wrote about that relationship several months ago.

If we assume that we can ignore those dividend payments and carrying costs, Bittman boils this relationship down to +stock = +call - put (pg. 154). Performing a little algebraic manipulation returns the relationship -stock = +put -call. If you wanted this as a written statement, the equation says that a long put and short call are the equivalent to short stock, as long as we can ignore those dividend payments and carrying costs.

Let's look at how this would have worked on our examples, starting back with the original setup, with the hypothetical stock at \$68.75 and the original implied volatility at 40 percent. The 75-strike put's delta would be -0.75, as originally determined, and the 75-strike call's would be +0.25. Because the call is being sold to establish a synthetic short-stock position, however, the +0.25 is converted to -0.25 for the whole position. Together with the put's -0.75, the position's delta is -1.00 or -100 when the 100 multiplier is applied.

What happens when the price drops \$3.00? I used the Op-Eval software to make the calculations. The put's delta is now -0.85 and the call's, +0.15. When the call is sold, its delta is subtracted or converted to -0.15, and the delta total for the short call and long put still totals -1.00. If, in addition to the price change, the implied volatility climbed to 45 percent, the call's delta is 0.18 and the put's, -0.82. When the call's delta is subtracted and the put's added, we have the same -1.00 delta again. It works the same way with a \$3.00 climb. Using this synthetic relationship, the position is always short 100 deltas.

Options are flexible enough to do almost anything that needs to be done. A trader who owns 100 shares of stock and is afraid of a decline can hedge that long position with options that create an equivalent short-stock position. However, along with the flexibility that options provide, there can be pros and cons to each choice. Looking at the differences in the way that price and implied volatility changes work in opposition or together on ITM and OTM options shows some of those pros and cons. Even if traders don't want to bother with the synthetics and want a put or call position that's a "snapshot" hedge or position, those traders should have some view on not only expected price changes but also implied volatility changes when they make the choice of which option position provides the right snapshot.

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