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

The synthetic call option has the following tax items. It will produce interest expense on a daily basis, but will never produce interest income. It will also produce gain or loss whenever stock is required to be sold (i.e., when delta declines). (125) On days when the stock is purchased (i.e., when delta rises), no gain or loss is realized.

The synthetic put option has the following tax items. It will produce interest income on a daily basis, but will never produce interest expense. It will produce gain or loss whenever short sales are closed (i.e., when the absolute value of delta declines). (126) On days when short sales are initiated (i.e., when the absolute value of delta rises), no gain or loss is realized.

The simulation will thus generate a series of daily tax items associated with the synthetic option. We can compare this series to the gain or loss on the true option by taking the future value of the series. I did not prepare express simulations for the short call and the short put. Because the simulations do not allow for any strategic trading, the results for the short positions should be the exact inverse of the simulated long positions.

C. Results of Simulation

To recap, the synthetic call option will produce interest expense plus gain or loss from stock trading, and the synthetic put will produce interest income plus gain or loss from short selling. This section reports the results of the simulation of a synthetic call and a synthetic put. In this simulation, we assume that the initial stock price is $50, the strike price is $50, the risk-free rate is 5%, the time to exercise is five years, and the volatility of the stock is 25%. The stock bears no dividends, and the options are European.

We assume that the stock moves once per day according to the standard random-walk model. Thus, 1800 steps follow the first day. At each step, the synthetic option is rebalanced to reflect the change in stock price. This process is repeated 2000 times. Thus, the computer simulation rebalances a hedging portfolio 7.2 million times (3.6 million times for each option). The following tables show how effective delta hedging is at simulating and bifurcating the true option. Bifurcation of Call Option -- Sum (Not Future Value) of Tax Items Mean Interest Expense on Synthetic Call ($6.28) Mean Net Gain from Stock Trading on Synthetic Call $10.54 Mean Difference between Synthetic and True Call ($0.00) Mean Gain on True Call (Sum of Above) $4.26 Bifurcation of Put Option -- Sum (Not Future Value) of Tax Items Mean Interest Income on Synthetic Put $4.79 Mean Net Loss from Short Selling on Synthetic Put ($3.37) Mean Difference between Synthetic and True Put ($0.01) Mean Gain on True Put (Sum of Above) $1.41

These results merely confirm that delta hedging closely replicates the returns on options and that it is possible to bifurcate the option gain into more fundamental units--interest and trading gain or loss. The ultimate goal of the simulation is to examine the timing of current-law taxation of options under a realization system. To probe this question, I calculated the future value of the trading items (gains and losses) and the future value of the interest items (income or expense). These tax items are measured at the time they accrue by the MATLAB simulation, and then projected forward to the expiration date using the assumed discount rate of 5%. The future-value calculations are listed below:

Analysis of Call Option Measured Over 2000 Simulations

Mean of

Future Value Mean of Sum Difference

(Tax-Policy (Current-Law (Current Law Tax Items from Synthetic Call Ideal) Deferral) Less Ideal) Interest Expense ($7.08) ($6.27) $0.81 Net Gain from Stock Trading $10.23 $10.54 $0.31 TOTAL $3.15 $4.27 $1.12

So, the ideal measure of expected income at the expiration of the option is about $3.15, but the expected income under current law is about $4.27. Current law thus appears to overtax the holders of call options. Most of this over-taxation comes from the failure to grant the call holder any interest deduction while the call is outstanding. We can invert the results to see that current law appears to undertax the writers of call options, as it does not impute interest income during the life of the call option.

Analysis of Put Option Measured Over 2000 Simulations

Mean of

Future Value Mean of Sum Difference

(Tax-Policy (Current-Law (Current Law Tax Items from Synthetic Put Ideal) Deferral) Less Ideal) Interest Income $5.42 $4.79 ($0.63) Net Loss from Short Selling ($3.95) ($3.37) $0.58 TOTAL $1.47 $1.42 ($0.05)

Current law taxes the put at very close to the right amount. By committing two theoretical wrongs, current law arrives at practically the right result for the put. Current law overtaxes the gains and losses from short trading, but undertaxes the interest income associated with the put. These two failures appear to cancel each other out. At least on average, the taxation of the true put and synthetic put are remarkably close.

D. Interpretation

The results given above are consistent with a more qualitative explanation of delta hedging. Under this qualitative explanation, current law is about right in its taxation of put options (both long and short). In contrast, current law overtaxes long calls and undertaxes short calls.

Consider, for example, the long call. Current law defers all gain or loss to some future realization event (e.g., exercise or expiration). Since the long call is bifurcated into stock ownership and debt, the synthetic long call produces interest expense over its life. The synthetic long call will also tend to produce more realized losses than gains before the end of the option. To see why, recall that the goal with delta hedging is always to own a number of shares equal to delta. Now, if stock prices go up, then mathematically delta will also go up. (127) The rising delta, in turn, forces the investor to buy more shares. The rising market produces gains, but they are deferred because the investor is not selling. In contrast, falling stock prices cause a falling delta, which forces the synthetic-long-call investor to sell shares. So, the losses associated with this falling market tend to be realized.

Under this qualitative theory of synthetic long calls, losses tend to be realized whereas gains tend to be deferred. Moreover, the synthetic option produces interest deductions throughout its life. In contrast, current law defers the recognition of all tax items associated with the option. Because the holder of the call must wait to take advantage of the losses and expenses produced by the synthetic counterpart, current law overtaxes the long call. The inverse will hold for the short call (interest income plus realized gains and deferred losses). Similar analysis for puts is left to a footnote, (128) and the entire results are summarized below. Type of Synthetic Trading Trading Current Law Treatment Option Gains Losses Interest of True Option Long Call Deferred Realized Expense Overtaxes Short Call Realized Deferred Income Undertaxes Long Put Deferred Realized Income Correct or ambiguous Short Put Realized Deferred Expense Correct or ambiguous

This analysis shows that the Monte-Carlo simulation of the prior Part V.C is consistent with the dynamics of synthetic options. Thus, we can be sure that current law is not a perfect representation of the ideal. Yet, as the next section will argue, it may be as close to the ideal as we can practically achieve.

E. Policy Implications

Can the tax laws improve the taxation of naked options, using synthetic options as the policy ideal? The answer is probably not, as current law achieves results close to the ideal. Synthetic options produce two types of tax items: (1) gain or loss from trading and (2) interest income or expense. Option taxation could theoretically be improved by imputing income or expense based on these items before the expiration of the option. Doing so would result in great practical difficulties and only modest improvements.

Imputing the gain or loss from trading is administratively infeasible, even if theoretically possible. In order to determine the gain or loss from trading, a taxpayer would have to create a bookkeeping account to reflect the equity and debt position associated with the synthetic option. The account would then need to be updated frequently--probably daily--to reflect the passage of time and changes in the stock price. Despite its theoretical appeal, this approach is far too cumbersome and burdensome to use for taxing real-world options.

Imputing interest may be administratively feasible. Yet, merely imputing interest would actually worsen the taxation of put options. Recall that the interest income and trading losses on the synthetic long put offset each other almost completely in the Monte-Carlo simulation. (129) By failing to recognize either, the results under current law may do a good job of reflecting the synthetic-option ideal. Imputing only the interest would destroy this balance.