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

Under the weighted-average-cost approach, the synthetic option produces taxable gain of $13,406. Recall that it also produces interest expense of $10,627. Therefore, the net realized income is $2779. However, the investor has not sold all the stock holdings, which still have $10,409 of unrealized appreciation. If we take the synthetic call to be our normative baseline, then the true call appears to be undertaxed. All of the gain of the true call is deferred, whereas $2779 of the gain on the synthetic call is realized currently. (109) The following table summarizes the findings of this small simulation, comparing the results of the synthetic call with those of the true call: Item Synthetic Call True Call Initial Investment $36,393 $36,393 Deferred Gain $10,409 $13,875 Realized Gain $13,406 -0- Interest Expense ($10,627) -0- Net Realized Income $2779 -0- Total Value $49,581 $50,268

This example will hopefully illustrate the potential problem of option tax under current law--deferral of taxable gain. Nevertheless, we should be cautious about inferring too much from it too quickly. First, this example is just one path the stock can take, and it happened to be a winning path. We have yet to examine what happens with other paths, on which the stock might decline. Second, even though this example showed that the synthetic-option would produce income of $2779, this amount is deferred only for a year. It is not forgiven. Third, and finally, the $2779 (or 28cents per share covered by the options) is only a portion (about 1/5) of the total economic gain on the synthetic call. By way of comparison, a comprehensive mark-to-market regime actually performs far worse than does current law in achieving the synthetic-call ideal. (110)

One asset path, over the course of six months, does not yield enough insights into the gain and loss from trading and the interest expense or deductions. Although we have a framework for examining the consequences of deferral, we now need to apply it over many different scenarios and over a greater period of time. Accomplishing this task is the goal of the next section.

V. TAXING THE NAKED OPTION: A MONTE-CARLO SIMULATION

A. Introduction

The goal of this section is to compare the consequences of taxing options under the synthetic-option ideal described in the prior section with the consequences of taxing options under the deferral method of current law. This section will focus on naked options--i.e., options that are not coupled with a position in the underlying asset. (111) The next section will focus on covered calls and protective puts--short calls and long puts combined with the underlying stock. (112) The naked options are analyzed first because they do not implicate the constructive sale rules of section 1259.

This section will ask whether current law inappropriately defers the tax consequences of options. To do so, this section will compare two types of transactions. The first is the taxation of an actual option under current law, assuming that the option is settled in cash at the expiration date. Current law defers the tax consequences of this transaction until the expiration date (assuming cash settlement). The second type is a hypothetical synthetic option created by an investor. The synthetic option is created by delta hedging with daily rebalancing. Thus, every day will potentially generate gain or loss and interest expense or income. As with the true option we will assume that the synthetic-option position is liquidated at the expiration date.

The total gain or loss will be roughly the same between the two transactions. What is different is the timing. The synthetic option will produce a series of tax items: interest expense and income and trading gains and losses. The future value of these tax items will be projected forward to the exercise date. This future value can then be compared with the tax consequences on the true option (which exist only at the exercise date under current law).

I wrote a computer simulation to compare the taxation of synthetic options with the current-law taxation of true options. One might ask why taxation of the synthetic option needs to be measured by computer simulation. After all, the price of an option can be derived directly from the Black-Scholes formula, which itself is based on a synthetic option. Unfortunately, a direct solution to the taxation of the synthetic option is unavailable because the tax consequences of a synthetic option depend upon the path the stock takes. The goal of this article however is to examine the taxation of options held or written by individual investors on the cash method of accounting. Unlike dealers, (113) investors will be subject to the realization requirement, which greatly complicates the analysis. For example, a decline in delta might prompt our investor to sell stock. The gain or loss on the sale is determined by the cost of stock previously purchased, which depends in turn on the history of stock prices. Such "path dependent" results can be estimated only by a computer simulation. (114)

The calculation will be performed using a so-called Monte-Carlo simulation that I wrote in the MATLAB computer language. The computer generated 2000 pseudo-random walks for the stock to take over the course of five years. Each pseudo-random walk is 1800 steps long, corresponding with daily price movements measured over five years. A random number generator determines the daily movement of stock, using the standard assumption of Brownian motion. (115) The simulation calculates delta and the components of the synthetic option on a daily basis. Also, the simulation records the tax items associated with each day (interest expense or income; gain or loss from trading). The future value of these tax items is taken for each pseudo-random walk. As we have 2000 pseudo-random walks, the mean of the results is reported.

The conclusion of this section is that current law may well be the best practical system for taxing naked options, although it does deviate from the synthetic-option ideal. Even though the synthetic option generates daily gains and losses from trading, they often offset each other. What current law fails to capture is the interest expense and income associated with synthetic options. Ultimately, this section concludes that ignoring this interest component is the best approach for the tax system.

B. The Hypothetical Stock and Options

This section uses one hypothetical stock on XYZ Corp. We will assume that XYZ Corp. stock pays no dividends, and that the standard deviation of its return is 25%. Its current market price is $50 per share. Let us also assume the current risk-free rate of interest is 5% for all periods.

As for the options, let us assume that the exercise price is $50 and the term of the option is five years long. We now have all of the information we need to value the options and calculate delta using the Black-Scholes pricing formula.

Current price: S = $50

Strike price: K = $50

Interest rate: r = 5%

Time to exercise: T = 5

Volatility: = 25%

The formulas return the following initial amounts:

Price of call: c = $16.25 (116)

Price of put: p = $5.19 (117)

Delta of call: [[DELTA].sub.c] = 76.63% (118)

Delta of put: [[DELTA].sub.p]= -23.37%. (119)

These results allow us to initiate the synthetic options as follows:

Synthetic Call Option:

** Buy [[DELTA].sub.c]= 0.7663 shares of XYZ stock for $38.32.

** Finance this purchase in part with an out-of-pocket contribution of c = $16.25.

** The remainder, $22.07, comes from borrowing. (120)

Synthetic Put Option:

** Sell short -[[DELTA].sub.p]= 0.2337 of XYZ stock for proceeds of $11.68.

** Invest these proceeds, plus an additional p = $5.19 out of pocket, in a debt instrument.

** The total investment in the debt instrument is thus $16.87. (121)

We can garner some immediate insights into the expected taxation of the true options based on the Black-Scholes method. For technical reasons beyond the scope of this article, cash flows are valued at risk-free rates under the Black-Scholes model. (122) As a result, we can easily arrive at expected values of the option contracts at the end of five years. The call option is expected to be worth $20.87, (123) and the put option is expected to be worth $6.67. (124) So, if the options are all settled in cash, there will be a realization event in five years. At that time, the taxpayer will have an expected on the call of $4.62 and on the put of $1.47.

The synthetic option should produce almost the same amount of total gain or loss. After all, the whole point of a synthetic option is to replicate the economic gain or loss from a true option. The key issue, which is being measured by the simulation, is whether the true option results in more or less tax deferral than a synthetic option. Finding the "typical" tax treatment of these synthetic options is impractical without a computer simulation. Even though the final value of the stock determines the option payoff, it does not determine the interim tax treatment. We must also know what path the stock took in reaching its final value. These movements in the stock will determine interim gains, losses, interest income, and interest expense.

As noted before, I estimated the timing of tax items associated with synthetic options using a Monte-Carlo simulation written in the MATLAB programming language. The simulation recalculates delta and uses the new delta to rebalance the synthetic option on a daily basis.