Naked and covered in Monte Carlo: a reappraisal of option taxation.

A possible critique of the delta standard described in Part III above is that delta itself fluctuates over the life of an option. (147) If delta changes immediately after the covered call or protective put is executed, then is it possible to apply the delta model at all? As this section will show, fluctuating deltas are not an intractable problem. Instead, this section will show that delta acts like a ratchet on constructive sales because, like real sales, constructive sales cannot ordinarily be undone. So, fluctuations in delta can only increase overall constructive sales.

Let us return to an example from the prior Part VI.A. Maya writes a covered call over 10,000 shares of ABC stock in which she has a zero basis. The current price of ABC is $30 per share, and the strike price of the call is $33 per share. Applying the other assumptions given above, we come to a Black-Scholes price on the option of $36,393 (or $3.6393 per share). (148) Under current law, the writer of a "covered call" has not triggered realization of the owned assets. Under the delta model described in Part II, however, Maya would be treated as having executed a short sale over 5658 shares of ABC stock, triggering gain under the constructive sale rules.

Let us also return to the possible walk that the stock took from $30 to $35.08 as suggested above. (149) There, we examined how the position of a call holder could be closely replicated using stock trading and borrowing. The position of a call writer is replicated in very similar, yet inverse, fashion. Here, we use short selling and investing in a risk-free asset. The replication is detailed in the following table. Note that positive numbers under "Shares Shorted" represent the execution of short sales; negative numbers represent closing of short sales.

Option Beginning Value $36,393

Shares 10,000

Invested

Proceeds

Shares from Short Week Stock Delta Shorted Selling Interest 0 $30.00 0.5658 5658 $133,347 $0 1 30.31 0.5763 105 3183 256 2 31.50 0.6239 476 14,994 263 3 32.33 0.6550 311 10,055 292 4 34.16 0.7205 655 22,375 312 5 36.07 0.7798 593 21,390 356 6 36.71 0.7973 175 6424 398 7 36.56 0.7930 (43) (1572) 411 8 35.62 0.7647 (283) (10,080) 409 9 38.09 0.8328 681 25,939 390 10 39.78 0.8702 374 14,878 441 11 36.39 0.7871 (831) (30,240) 470 12 36.92 0.8025 154 5686 413 13 33.59 0.6868 (1157) (38,864) 425 14 34.21 0.7101 233 7971 351 15 35.93 0.7715 614 22,061 367 16 37.60 0.8225 510 19,176 410 17 40.38 0.8881 656 26,489 447 18 39.50 0.8712 (169) (6676) 499 19 41.04 0.9027 315 12,928 487 20 40.33 0.8909 (118) (4759) 513 21 38.74 0.8567 (342) (13,249) 505 22 37.42 0.8209 (358) (13,396) 481 23 36.94 0.8063 (146) (5393) 456 24 36.50 0.7918 (145) (5293) 446 25 34.81 0.7242 (676) (23,532) 437 26 35.08 0.7356 114 3999 392 Total $197,840 $10,627 Ending True $(50,268)

Option

Synthetic $(49,581)

Option Beginning

Risk-Free Week Asset 0 $133,347 1 136,786 2 152,043 3 162,390 4 185,077 5 206,823 6 213,645 7 212,483 8 202,811 9 229,141 10 244,459 11 214,689 12 220,788 13 182,349 14 190,670 15 213,098 16 232,684 17 259,621 18 253,444 19 266,859 20 262,614 21 249,870 22 236,954 23 232,016 24 227,170 25 204,075 26 208,467 Total Ending

Initially, the synthetic short call is created by a short sale over 5658 shares, yielding total proceeds of $169,740. Maya can do what she pleases with $36,393 of these proceeds, which represent the premium received for the option. The remainder ($133,347) is invested in a risk-free asset. Every week, short sales and risk-free assets are rebalanced to reflect changes in delta. At the end of week twenty-six, the synthetic short call is represented by an outstanding short position over 7356 shares. Because the stock is now at $35.08, it would cost $258,048 to close this position. The risk-free asset is worth $208,467, and the overall position is a liability of $49,581, which is close to the true Black-Scholes value of $50,268.

Now, however, we must determine how to model constructive sales. The fluctuations in delta obviously trigger fluctuations in the amount of outstanding short selling. This section of the article will model these fluctuations as if short sales had a ratchet effect on constructive sales. So, when a taxpayer closes a short sale, it does not reverse any constructive sales that were triggered by it. There is a technical difficulty with this assumption--the closed-transaction exception to the constructive sale rules. Under this exception, a short sale does not trigger a constructive sale if (1) the short sale is closed within thirty days of the end of the taxable year in which it was made, (2) the taxpayer continues to hold the owned stock for at least sixty days after the short sale is closed, and (3) during those sixty days the taxpayer's risk of loss over the owned stock is not diminished by using a call, put, forward, or similar contract. (150) Arguably, the taxpayer fails (3) while continuing to engage in the delta-hedging strategy, as some short sales always remain outstanding. If not, then we have a complicated problem. (151)

Another complicated problem is the fact that the same shares could be constructively sold more than one time under section 1259. (152) In the interest of simplicity, I have assumed that constructive sales are triggered when (but only when) the total short position exceeds its prior maximum. This interpretation is consistent with the purpose of section 1259 and with the broader principle that completed sales of property cannot be reversed in order to avoid taxable gain. So, we can avoid the closed-transaction exception and the possibility of multiple constructive sales of the same shares.

Now that our assumptions are clarified, let us return to the example. The initial short sale potentially leads to a constructive sale over 5658 shares. Assuming a zero basis in ABC stock, the call would initially trigger gain of $169,740. At the end of the twenty-six week position, the short position is even greater, standing at 7356 shares, and the maximum short position over the twenty-six week position was 9027 at week nineteen. Under the ratchet theory, any increase in total short sales over the past all-time high would lead to a new constructive sale. Applying this ratchet model to the prior example leads to the following schedule of constructive sales and related gain (assuming a zero basis for ABC stock). Week Stock Delta Constructive Sales Gain Realized 0 $30.00 0.5658 5658 169,740 1 30.31 0.5763 105 3183 2 31.50 0.6239 476 14,994 3 32.33 0.6550 311 10,055 4 34.16 0.7205 655 22,375 5 36.07 0.7798 593 21,390 6 36.71 0.7973 175 6424 7 36.56 0.7930 - - 8 35.62 0.7647 - - 9 38.09 0.8328 355 13,522 10 39.78 0.8702 374 14,878 11 36.39 0.7871 - - 12 36.92 0.8025 - - 13 33.59 0.6868 - - 14 34.21 0.7101 - - 15 35.93 0.7715 - - 16 37.60 0.8225 - - 17 40.38 0.8881 179 7228 18 39.50 0.8712 - - 19 41.04 0.9027 146 5992 20 40.33 0.8909 - - 21 38.74 0.8567 - - 22 37.42 0.8209 - - 23 36.94 0.8063 - - 24 36.50 0.7918 - - 25 34.81 0.7242 - - 26 35.08 0.7356 - - Totals 9027 289,779