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

Let us assume that an investor wants to replicate this call option, but over 10,000 shares. The call option has an initial value of $36,393 and an initial delta of 56.58%. So, the investor must initially buy 5658 shares, at a total cost of $169,740. The investor pays for this purchase with $36,393, borrowing the balance of $133,347.

Hasen's example has the stock at $35.08 six months later. To demonstrate how the synthetic option should work, I generated a series of random walks that the stock could take, and captured the first that ended at $35.08. I assumed that each step was one week long. At each step, I calculated delta and rebalanced the debt/stock mixture. New purchases of stock are financed with new borrowing. Sales of stock produce cash that reduces previous borrowing. The results are summarized as follows:

Option Beginning Value $36,393

Shares 10,000

Shares Borrowed Week Stock Delta Bought Cost 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

Cumulative week Cost 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

The ending value of the synthetic option is the value of the owned stock (7356 shares at $35.08 per share, or $258,048) minus the cumulative borrowing and interest ($208,467). Thus, the synthetic option is worth $49,581, fairly close to the Black-Scholes value of $50,268. The results would be even closer using daily, rather than weekly, rebalancing.

The synthetic option produces interest expense of $10,627 in the current year, and the timing of this interest is obvious from the spreadsheet. However, there was also buying and selling of stock. The buying has no direct tax consequences, but the selling produces taxable gain or loss, the measurement of which is not obvious. Perhaps the most realistic approach to measuring the gains and losses from trading would be to assume strategic behavior by the investor. The investor would select the particular stocks to sell so as to minimize gains and maximize losses, subject to the wash-sale rules. Strategic trading is allowed by Treasury regulations, (102) subject to the wash-sale rules (discussed below).

Ultimately however I chose not to present such a simulation. One reason is complexity. Strategic trading assumes that the taxpayer maintains an inventory of stock, purchased on different dates, with each having a unique adjusted basis. (103) Modeling strategic trading leads to complex, less readable computer code. Another reason for not presenting the model with strategic trading is the lack of symmetry between short and long positions. If the investor is assumed to trade strategically, then we can expect the investor to trade differently depending on whether he is replicating for example a long call or a short call. So, the taxable gain produced by a long call may be different from the taxable loss produced by the short call.

The simulation I prepared uses a weighted-average-cost basis. At any particular time, each share held by the investor has the same adjusted basis, which equals the average cost of the prior purchases. This approach simplifies the programming code, because only one adjusted basis is needed at any time. Moreover, if the realization rule is taken as a constraint, a weighted-average-cost approach is arguably the best measure of income. (104) The stock or short sales that constitute the synthetic are fungible. Selling one versus another does not affect the pre-tax returns enjoyed by the taxpayer. Although there are other plausible methods of inventory accounting for the securities, (105) only the weighted-average-cost method is presented in this article.

The weighted-average-cost method is not allowed by current law, although it was proposed by the Clinton administration. (106) Another deviation from current law in this simulation is the absence of wash-sale rules. The wash-sale rules potentially disallow a loss on the sale of stock if either (1) the taxpayer retains other shares of the same stock purchased thirty days before the date of sale or (2) the taxpayer buys other shares of the same stock thirty days after the date of sale. (107) The purpose of the wash-sale rules is to restrain strategic trading that could realize losses and defer gains. (108) Under the weighted-average-cost simulation however there is no possibility for strategic behavior. The timing of trades is determined solely by movements in delta, and the weighted-average-cost method thwarts the taxpayer's ability to select high-basis stock to sell. In short, using a weighted-average-cost method and eliminating the wash-sale rules represent a simplifying compromise that reflects economic income while retaining the realization rule.

Let us return to the prior example, recalling that the option produced an interest expense of $10,627 in the first taxable year. Now that we have an inventory method, we can calculate the gain or loss on the sales that the movement in delta forces. That reckoning is as follows: Beginning Option $36,393

Shares 10,000

Shares Shares Realized Deferred Week Stock Needed Bought WAC G/L G/L 0 $30.00 5658 5658 $30.00 1 30.31 5763 105 30.01 2 31.50 6239 476 30.12 3 32.33 6550 311 30.22 4 34.16 7205 655 30.58 5 36.07 7798 593 31.00 6 36.71 7973 175 31.13 7 36.56 7930 (43) 31.13 234 8 35.62 7647 (283) 31.13 1272 9 38.09 8328 681 31.69 10 39.78 8702 374 32.04 11 36.39 7871 (831) 32.04 3613 12 36.92 8025 154 32.14 13 33.59 6868 (1157) 32.14 1683 14 34.21 7101 233 32.20 15 35.93 7715 614 32.50 16 37.60 8225 510 32.82 17 40.38 8881 656 33.38 18 39.50 8712 (169) 33.38 1035 19 41.04 9027 315 33.64 20 40.33 8909 (118) 33.64 789 21 38.74 8567 (342) 33.64 1743 22 37.42 8209 (358) 33.64 1352 23 36.94 8063 (146) 33.64 481 24 36.50 7918 (145) 33.64 414 25 34.81 7242 (676) 33.64 789 26 35.08 7356 114 33.66 10,409 Total $13,406 $10,409