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

We can replicate this sensitivity by buying 0.50 shares of ABC stock. The 0.50 shares are just as sensitive to movements in the stock price as is the option itself. Nonetheless, the 0.50 shares are only part of the replication. They would be worth $10.50 at the end of the year if the stock price fell to $21, but the option itself would be worthless. This discrepancy is easy enough to fix. We can assume that the initial purchase was made partly with borrowed funds--borrowed in an amount that require a $10.50 repayment in one year. Repayment would thus wipe out the value of the shares if the share price fell $21. Alternatively, if the share price goes up to $45, then the 0.50 shares would be worth $22.50. Paying back the $10.50 would leave $12.00--the same as the actual option. So, we have perfectly replicated the option by owning 0.50 shares subject to an obligation to repay $10.50 at the end of the one year period.

The initial cost of the option should equal the initial cost of the replicating portfolio. The 0.50 shares costs $15.00 at the start of the one year period. The $10.50 final liability brings loan proceeds of $9.50 (20) at the start of the one year period. Thus, it costs $5.50 to buy the replicating portfolio, and the market price of the actual option should also be $5.50.

In summary, the binomial approach shows that a stylized call option can be replicated with a combination of stock and debt. The key to this replication is delta, which is the sensitivity of the price of an option to changes in the price of the stock. The replication is performed as follows:

** Buy delta shares of stock. In our example, this was 0.50 shares, costing $15.00.

** Pay for part of the purchase with an out-of-pocket contribution that equals the value of the option. In our example, this was $5.50.

** Pay the remainder of the purchase with borrowed funds. In our example, this initial borrowing of $9.50, leading to repayment of $10.50 in one year.

The binomial model is obviously not the real world. Stock prices move constantly and can take a multitude of values. The next subsection will show how one can extend the basic approach just described in order to replicate real-world options. As in this subsection, the key to real-world replication is measuring the delta of an option.

D. Delta Hedging and the Black-Scholes Model

Replicating real-world options with stock and debt is critical to the approach of this article, which urges that options should be taxed according to the tax treatment of the replicating portfolio. The Black-Scholes model purports to replicate real-world options, even though stock prices are moving randomly and constantly. At its core, the Black-Scholes model is the same as the binomial model. Both hold that an option can be replicated by owning "delta" shares of stock, combined with an appropriate amount of borrowing. Recall that delta is the sensitivity of the option price to changes in the stock price. So, an investor faces the same risk by owning delta shares and owning one option. As with the binomial model, the replicating portfolio also includes an appropriate amount of borrowing.

Recall from Part II.A that there are four types of options--long calls, short calls, long puts, and short puts. The Black-Scholes formula produces a price and a delta (21) for each of these four. (22) Thus, each can be replicated with a position in equity and debt. Long calls and short puts have positive deltas, meaning they are replicated with stock ownership and borrowing. Short calls and long puts have negative deltas, meaning they are replicated with short selling and lending. Option Replicating Position in Stock Replicating Position in Debt Long Call Ownership Borrowing Short Call Short Selling Lending Long Put Short Selling Lending Short Put Ownership Borrowing

The derivation of the Black-Scholes formula is beyond the scope of this article, although the approach is similar to the binomial model. In the binomial model, we needed to know the possible values of the stock in the next period. In the Black-Scholes model, we assume that the stock price moves randomly. (23) Now, we need to know the volatility of the stock. Expanding the example from the prior subsection, suppose that ABC stock is worth $30 today and has volatility (standard deviation) of 30%. Again, we are looking for the price and delta of an option to sell ABC stock for $33, exercisable in one year. As before, let us assume that ABC stock has no dividends, and that the interest rate is 10%. (24)

The Black-Scholes formula gives a value of the option of $3.6393 (25) and a delta of 0.5658. (26) To make the numbers more meaningful, let us suppose that we are interested in replicating an option covering 10,000 shares. We can take the same approach as before in order to replicate the call option:

** Buy delta shares of stock. In our example, this was 5658 shares, costing $169,733.

** Pay for part of the purchase with an out-of-pocket contribution that equals the value of the option. In our example, this was $36,393.

** Pay the remainder of the purchase with borrowed funds. In our example, this borrowing is $133,340. (27)

The difference between the Black-Scholes model and the binomial method of the prior subsection is that the stock price--and therefore delta--can change before the expiration of the option. Therefore, we must rebalance the replicating portfolio periodically. For example, suppose that we are to rebalance the replicating portfolio weekly. At the end of the first week, the stock price has jumped from $30 to $30.31. (28) The passage of a week and the jump in the stock price causes a change in delta, which is now 0.5763. (29) The number of shares in the replicating portfolio must now be increased from 5658 to 5763. The additional 105 shares cost $3183, paid for by additional borrowing.

The process of rebalancing the replicating portfolio is known as "delta hedging." The goal of delta hedging is always to own a number of shares that equals the delta of the option that is being replicated. This way, the stock ownership and the true option have the same sensitivity to movements in the stock price. Over time, changes in the stock price will cause changes in delta. These changes will force the investor to rebalance the portfolio. This process is detailed in Appendix A.

The goal of Part V will be to implement those steps with a computer simulation and to measure the tax consequences to an investor. Implementing the actual trading model is simple, and can be done with a few lines of computer code. (30) The true difficulty comes in measuring the tax consequences that an investor would face by creating a synthetic option.

III. OPTIONS AND CHALLENGES TO THE TAX SYSTEM

A. Option Taxation and the Put-Call Parity

The tax aspects of financial innovation have spawned a rich literature in the law reviews. (31) Perhaps the seminal article is Financial Contract Innovation and Income Tax Policy, in which Professor Warren showed that the fundamental problem of current option taxation is its inconsistency with the taxation of other transactions. (32) First, let us consider the taxation of options, which Professor Warren summarizes as follows:

The purchase of an option is treated as a capital expenditure, and

there are generally no tax consequences to either party until its

exercise or disposition. If the option lapses without exercise, the

option writer is treated as if he had sold the option. If a call is

exercised, the writer includes the premium in the amount realized on

the sale of the asset, and the holder of the call includes the premium

in cost basis. If a put is exercised, the writer reduces basis by the

amount of the premium, and the holder of the put reduces amount

realized by the same amount. If an option is sold prior to exercise,

gain or loss is recognized, with the nature of the gain generally

determined by that of the underlying asset. Finally, many ... options

are written for settlement by a cash payment from one party to the

other on the date of performance, rather than by the actual delivery

of the property specified in the contract. Such payments with respect

to these cash settlement options ... are taxable events. (33)

The tax system treats options as "contingent-return instruments," waiting to apply a tax until the option has resolved itself. Similar treatment applies to corporate stock itself. Dividends on corporate stock are taxed currently, but appreciation on stock escapes taxation until the stock is sold or exchanged. (34)

Contrast this treatment with "fixed-return instruments," such as bonds. Bonds generate taxable interest income on an annual basis. Even if the actual payment of interest is deferred during the life of the bond, the Internal Revenue Code imputes annual interest under its "original issue discount" regime. (35)

Options potentially allow taxpayers to select between contingent-return and fixed-return tax treatment. (36) The put-call parity implicates the tax system because the taxation of the four elements is internally inconsistent. Recall that the put-call parity holds that stock plus a put equals a bond plus a call. (37) Algebraically--

S + p = B + c

The left side of the equation (S+p) represents contingent-return instruments (except insofar as the stock pays dividends). The right side of the equation (B+c) represents a contingent-return and a fixed-return instrument.