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

The market for equity options and related derivatives is staggering, covering trillions of dollars worth of assets. As a result, the taxation of these instruments is inherently important. Moreover, the importance is made even more acute by the use of options in creating more complex transactions and in avoiding taxes.

Consider an equity call option, which entitles, but does not obligate, its holder to buy stock at a set price at a set time in the future. Option theory gives us a way to break the option down into more fundamental units. For example, an equity call option over 10,000 shares of stock might be equivalent to buying 7500 shares of stock itself.

This financially equivalent synthetic option should serve as the model for taxing an actual option. That is not the approach of current law. Nevertheless, a Monte-Carlo simulation I wrote shows that current law does a good job of approximating the tax liability generated by the synthetic option--but only when we view the option in isolation.

The results are radically different when the investor already owns some of the stock subject to the option. If such an investor sells (rather than buys) a call option, she has effectively sold a portion of the owned stock at fair market value. For example, the issuer of a call option over 10,000 shares may have effectively sold 7500 shares that she already owns. Option theory gives us a way to measure how much stock she has effectively sold. Taxing the sale of stock implied by many option and related contracts would reflect economic reality and curtail tax-motivated investments.

I. INTRODUCTION

Finding the correct tax treatment of equity options is a crucial task for three reasons. First, the market for equity options and other equity derivatives is enormous. In June 2006, equity options and related contracts covered assets worth almost $6.8 trillion, or about one half the U.S. gross domestic product. (1) The sheer size of the market warrants attention. Second, more complex financial contracts are often based on options. (2) Finding the right tax treatment of options will thus help us find the right tax treatment of these other contracts. Third, options and related contracts are often used to avoid taxes. (3) Thus, taxing options correctly would eliminate inefficient tax arbitrage and ensure equitable treatment of taxpayers. (4)

Let us start with a call option, which entitles (but does not obligate) the holder to buy stock at a set price at a certain time in the future. For example, Maya might buy a call option over XMPL Corp. stock that entitles her to buy 10,000 shares of XMPL Corp. stock for $100 per share in five years. Suppose that XMPL Corp. stock is currently worth $100 per share. The call gives Maya a valuable right because she will enjoy any appreciation in the stock over $100 per share without any risk of decline below that price. The Black-Scholes model (5) gives Maya a concrete way of valuing her option. Using that model (and other assumptions discussed later), the option is worth $350,000. (6)

The magic of the Black-Scholes model is that it equates Maya's call option with a combination of (1) ownership of XMPL Corp. stock itself and (2) borrowed funds. We will see later how Maya's call option is equivalent to her owning about 7500 shares of XMPL Corp. stock (worth $750,000) purchased partly with borrowed funds of $400,000. (7) This combination of stock and borrowing has the same net value as the call ($350,000). And, it is the starting point in a process called delta hedging that will closely approximate the economic return on the call option itself. So, we can think of the stock and borrowing combination as a synthetic option.

In due course, this article will explain how delta hedging creates synthetic options. The important point for now is that the Black-Scholes model gives us not only a way to value the call option but also to recreate it using stock and borrowing. The Black-Scholes model (8) has been spectacularly successful, winning Nobel prizes for its inventors (9) and serving as the linchpin for the multi-trillion dollar market for derivatives. It does not, however, served as the basis for taxing options. When Maya buys her call option, she is not taxed as if she bought XMPL Corp. stock with borrowed funds. Instead, the tax treatment is held open while Maya waits to see if she will actually exercise (or perhaps sell) the option. In taxation, timing is almost everything, (10) and long-term deferral can potentially be the same as tax forgiveness.

Prior commentators have argued that tax policy should strive to tax economically equivalent transactions similarly. (11) Tax policy can achieve this goal by what has been termed "bifurcation"--if a transaction can be bifurcated or broken down into more fundamental units, then the transaction should be taxed based on the tax treatment of these units. Inconsistent treatment between the fundamental units and the transaction creates the potential for economic distortions, tax arbitrage, and inequities.

Theoretically, then, the proper way to tax Maya's call option is to tax her as if she bought 7500 shares of XMPL Corp. stock (worth $750,000) purchased partly with borrowed funds of $400,000. Doing so is theoretically possible but practically difficult. The primary difficulty is that the precise amount of stock and borrowing will need to change over time. For example, suppose Maya really did decide to replicate the option with the stock and borrowing combination. If XMPL stock goes down in value, Maya would need to sell some stock (using the proceeds to pay of some of the borrowing she incurred). If XMPL stock goes up in value, Maya would need to buy some more stock (using additional borrowing to pay for it). So, the synthetic option is a dynamic mixture of stock and borrowing, representing countless sales and purchases of the underlying stock and changes in the associated borrowing. The 7500 shares financed in part with $400,000 of borrowed funds is merely the starting point. Taxing the transactions that occur after the starting point would be administratively infeasible, even though the approach is theoretically correct.

Nonetheless, this approach is a valuable policy tool, and it is possible to examine the taxation of the synthetic option and its countless transactions using a computer simulation. The simulation tells us what the expected tax consequences will be on that dynamic combination of stock and borrowing that replicates the call option. Ideally, the tax consequences on the actual option would be the same as those produced by the simulation. In reality, we should aim for a practical system that achieves results roughly the same as the ideal results produced by the simulation. Later in this article, I report the results of a computer simulation I created using the MATLAB programming language. Surprisingly, the simulation shows that current-law treatment of options is, for the most part, the best approximation of the theoretical ideal. The current-law taxation of options survives the toughest test that tax theory can apply, but with one exception.

That exception relates to so-called covered calls, which are the sale of a call combined with ownership of the stock. (12) To illustrate, suppose we change the example so that Maya already owns 10,000 shares of XMPL Corp. stock. Her adjusted basis in the stock is zero, meaning she would realize gain of $1 million if she sold the stock today. Let us also assume that Maya sells a call option over 10,000 shares rather than buying one. So, she is now obligated to sell 10,000 shares of XMPL Corp. stock for $100 per share in five years if the buyer exercises its right to do so. (And, the buyer will exercise its right only if the stock is over $100 per share at that time.) Maya receives cash of about $350,000 for selling this option. Even though Maya receives cash of $350,000 today, she is not taxed today under current law. As before, she waits for five years to see whether she must perform on the call.

Maya's call option is the equivalent of her buying 7500 shares (worth $750,000) purchased partly with borrowed funds of $400,000. In contrast, when Maya sells the call, the equivalent combination is inverted. Now, it is as if she sells 7500 shares (obtaining funds of $750,000) and lends a portion of the proceeds she obtained (again, $400,000). Thus, Maya should be taxed as if she sold 7500 shares of XMPL stock today, recognizing immediate gain of $750,000. Instead, current law improperly allows Maya to defer the tax consequences of the implicit sale for five years while she waits to see if she is called upon to perform under the option.

The article is organized as follows. Part II is an overview of option theory, which will be used throughout the article. Part III shows how options challenge the tax system and summarizes how previous proposals would deal with this challenge. Part IV shows how the tax system can achieve the ideal by taxing true options according to financially equivalent synthetic options, created from transactions in the underlying stock and debt. Part V explores the taxation of "naked options" (i.e., option positions that are not coupled with a position in the actual stock itself) using some simple examples and a more realistic Monte-Carlo simulation. After examining how naked options would be taxed under the synthetic-option ideal, Part V concludes that current law may well be the best practical model available. Part VI analyzes covered calls and related contracts in a similar fashion. Practical steps can be taken to improve the taxation of covered calls and protective puts--namely, treating them as a partial sale of the owned asset. Part VII has some concluding thoughts.

II. AN OVERVIEW OF OPTION THEORY

A. Option Terms Defined

This article uses terms of art relating to options and short selling. For convenience, this section defines these terms of art.

1. Long Call: A call option entitles (but does not obligate) the holder to buy stock at a set price at a set time in the future. The holder must pay a premium for this right. We will call the position of a call holder the "long call."

2. Short Call: The holder of a call option has a counterparty (the call writer) who receives the premium and must sell the stock if the call is exercised. We will call the position of the call writer the "short call."

3. Covered Call: A covered call is simply a short call that is combined with the underlying asset. Without the underlying asset, the call is naked.

4. Long Put: A put option entitles (but does not obligate) the holder to sell stock at a set price at a set time in the future. The holder must pay a premium for this right. We will call the position of a put holder the "long put."

5. Protective Put: A protective put is simply a long put that is combined with the underlying asset. Without the underlying asset, the put is naked.

6. Short Put: The holder of a put option has a counterparty (the put writer) who receives the premium and must buy the stock if the put is exercised. We will call the position of the put writer the "short put."

An option is specified by the asset (e.g., 100 shares of XYZ Corp. stock), expiration date (e.g., three years from today), and the exercise price (e.g., $50 per share). (13) This article focuses on options to buy or sell zero-dividend, publicly-traded stock. In addition, the options in this article are assumed to be "European," meaning the holder can exercise the option only at the expiration date. (14)

The final concept of this section is short selling. As we will see later, the magic of the Black-Scholes method for valuing options is that it equates options with easy-to-value financial positions: debt (either borrowing or lending) and stock (either owning or selling short). Borrowing, lending, and owning stock should be familiar. Selling short may not be, but it is simply the inverse of buying stock. A leading textbook on investments summarizes short selling as follows:
 A short sale allows investors to profit from a decline in a security's
 price. An investor borrows a share of stock from a broker and sells
 it. Later, the short-seller must purchase a share of the same stock in
 the market to replace the share that was borrowed. This is called
 covering the short position....

 The short-seller anticipates the stock price will fall, so that the
 share can be purchased at a lower price than it was initially sold
 for; the short-seller will then reap a profit. Short-sellers must not
 only replace the shares but also pay the lender of the security any
 dividends paid during the short sale.

 In practice, the shares loaned out for a short sale are typically
 provided by a short-seller's brokerage firm.... The owner of the
 shares will not even know that the shares have been lent to the
 short-seller. If the owner wishes to sell the shares, the brokerage
 firm will simply borrow shares from another investor. Therefore, the
 short sale may have an indefinite term. However, if the brokerage firm
 cannot locate new shares to replace the ones sold, the short-seller
 will need to repay the loan immediately by purchasing shares in the
 market and turning them over to the brokerage firm to close out the
 loan. (15)


As we will see in Part III.B, taxpayers often combine options in order to approximate the economics of a short sale while avoiding the short sale's adverse tax treatment. Part VI will present a system for treating such combinations as short sales for purposes of taxation.

B. Put-Call Parity

This section briefly describes the put-call parity, which relates the price of stocks, bonds, put options, and call options. As Part III.A demonstrates, the put-call parity shows that the current-law taxation of options is internally inconsistent. Part III.B further reveals how the put-call parity is used to create an approximate short sale, which avoids the adverse tax consequences of short sales under current law.

Put-call parity relates the value of the stock and options given any strike price (K) and time to exercise of the option (T) as follows:

S: a share of the stock

c: a call option on the stock, exercisable at time T for strike price K

p: a put option on the stock, exercisable at time T for strike price K

B: a zero-coupon bond that will be worth the strike price K at the time of exercise T (16)

The put-call parity states:

S + p = B + c. (17)

Detailed demonstrations of the put-call parity are available in the legal literature. (18) The most intuitive way to approach the put-call parity is to note that owning a bond is equivalent to owning stock, owning a put, and writing a call. In other words,

B = S + p - c.

Suppose that the strike price of the options and the value of the bond at maturity are all equal to $100 (i.e., K=$100). We know that the left side of the equation will equal $100 (i.e., B=$100) regardless of the price of the stock. As for the right side of the equation, we consider two cases. In the first case, suppose that the price of the stock is less than $100. The value of the call is zero, and the investor will exercise the put, selling the stock for $100. So, the right side is worth $100 in this first case. In the second case, suppose that the price of the stock is greater than $100. The value of the put is zero, and the investor will be called upon to sell the stock for $100 under the call. So, again, the right side is worth $100 in this second case. Thus, the right side of the equation is always worth $100.

Part III.A will show how the put-call parity can be used for tax avoidance. Each of the four transactions listed in the put-call parity can be recreated by a combination of the other three. For example, we just saw how a bond can be recreated using a combination of stock, a put, and a short call. However, the tax treatment of the bond is different from the tax treatment of the combination. Thus, the put-call parity might allow taxpayers to choose the tax treatment they prefer.

C. Delta and the Binomial Model

Although the put-call parity demonstrates how taxpayers might use options to exploit arbitrage opportunities, it does not provide a unique method for valuing options. The value of a put is dependent on the value of a call (or vice versa) under put-call parity. Option-pricing theory supplies the unique price by showing how an option can be replicated using only stock and debt. Replicating the option using only stock and debt requires more complex analysis than does the put-call parity. Before turning to a more realistic model in the next subsection, we can see the essence of how this replication works using a simple "binomial" model.

Suppose that ABC stock is worth $30 today and we know it will be worth either $21 or $45 in one year. What, for example, is the value of a call option to sell ABC stock for $33, exercisable in one year? Let us assume that ABC stock has no dividends, and that the interest rate is 10%. (19)

We know that the option will be worth $12 if the stock goes up to $45 and will be worth $0 if the stock goes down to $21. We can view the option as the following tree, with the "?" representing the current value of the option:

[GRAPHIC OMITTED]

The key to valuing the option under the binomial model is observe how sensitive the return on the option is to changes in the price of the stock. In this example, a $24 swing in the stock price (i.e., from $21 to $45) results in a $12 swing in the return on the option (i.e., from $0 to $12). So, the sensitivity of the option to the price of the stock is 50%. This figure is known as the "delta" of the option.

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.

An investor could use the put-call parity to create a synthetic bond. Bonds are the prototype for all fixed-return transactions. However, the put-call parity allows one to receive the economic return of a bond while paying tax on a contingent-return basis. An investor could replicate a bond by buying the stock, buying a put, and selling a call. (38) Why would a taxpayer do this? A true bond generates taxable interest income on an annual basis, whether or not the bond is sold. In contrast, the stock, put, and call have no tax consequences until the sale or (in the case of the put or call) exercise or expiration. (39)

Although a taxpayer can manipulate the timing of income using put-call parity, manipulating the character is more difficult. Before the enactment of section 1258, taxpayers might be able to convert the ordinary income received from bonds into the capital gains received from stocks and options. Section 1258 would now treat the synthetic bond as a "conversion transaction," resulting in ordinary-income treatment. Section 1258 would not, however, alter the timing of income on the synthetic bond. (40) As before, the synthetic bond would likely result in contingent-return treatment.

The put-call parity might also be used to achieve an effective short sale of stock. Rearranging the equation we see--

-S = p - c - B

In other words, one can replicate the short sale of stock by borrowing cash, buying a put, and selling a call.

Suppose that Maya currently owns 100 shares of stock, which has fair market value of $30 per share and an adjusted basis of $0. If Maya sold the stock she actually owns, then she would pay tax on $3000 of gain. Before 1997, Maya could have executed a "short sale against the box." (41) Rather than selling the shares she actually owns, Maya would execute a short sale over 100 shares of stock (selling 100 shares that were borrowed from a broker). In 1997, Congress enacted the constructive-sale rules of section 1259. (42) If an investor executes a short sale and also owns appreciated shares of the same stock, then he is deemed to have sold the owned stock (rather than the borrowed stock). The constructive-sale rules apply to a short sale or any comparable transactions that "have the effect of eliminating substantially all of the taxpayer's risk of loss and opportunity for income or gain with respect to the [owned security]." (43) So, Maya would face taxable gain on the 100 shares of if she executes a short sale or a synthetic short sale, constructed with options. (44) Using put-call parity, Maya could create a synthetic short sale by buying a put, selling a call, and borrowing money. Using the notation introduced above, we describe a short sale (i.e., a negative share of stock) as follows:

-S = p - c - B

Again, suppose that the stock is worth $30 today, and Maya wants to execute a synthetic short sale. She would borrow $30, buy a put, and sell a call. The term of the put and call would have to be the same, and the exercise price of the each would have to be the future value of $30. So, if the term of the option is one year and the interest rate is 5%, the exercise price would need to be $31.54 (45) for both the call and the put. In conceptual terms, the long put and short call eliminates any risk of upward or downward movement in the stock for one year. The borrowing allows Maya to access the value of the owned stock today, rather than having to wait to sell it. The resulting synthetic short sale perfectly mimics a true short sale and would be taxed as a constructive sale under current law.

B. Equity Collars

In his 2001 article Frictions as a Constraint on Tax Planning, Dean Schizer notes how taxpayers can approximate a short sale against the box, but still avoid the constructive sale rules, with an equity collar. (46) Like the synthetic short sale, an equity collar combines a long put with a short call. The difference, however, is that the equity collar has a spread in exercise prices between the two options.

Let us return to Maya and her ABC stock currently worth $30. An equity collar might be a long put with an exercise price of $27 and a short call with an exercise price of $33. Here, there is a spread of $6 between the two exercise prices--probably enough of a spread to avoid the constructive sale rules. (47) The following illustrates the return on a short sale of ABC stock and the equity collar just described. The horizontal axis is the price of the stock in one year. The vertical axis is the gross return (above or below the current stock price of $30) on the transactions in one year.

The illustration shows how similar an equity collar is to a short sale. Despite the similarities, the short-sale triggers the constructive sale rules, whereas the equity collar does not.

[GRAPHIC OMITTED]

Economically, however, an equity collar is a partial short sale. This article will urge that the equity collar should be taxed as a constructive sale (regardless of the spread). Determining the actual extent to which an equity collar is a short sale (e.g., 50%, 75%) is no trivial matter. The put-call parity does not supply the answer to this question, because it deals only with long puts and short calls that perfectly replicate a short sale. In order to find the degree to which an equity collar replicates (however imperfectly) a short sale, one must turn to the Black-Scholes model and the model's key concept of "delta."

We can easily determine the initial short sale implied by the equity collar just described. The delta on the put is -0.2020 (48) and the delta on the call is -0.5658. (49) So, the delta on the collar is the sum of the two, or -0.7678. If the collar covered 10,000 shares, Maya has essentially executed a short sale over 7678 of those shares. Applying the constructive sale rules of section 1259 to the implicit short sale means that Maya would be treated as having sold up to 7678 shares of ABC stock. There are some serious (but surmountable) complications with this approach. One is that delta depends on the volatility of a stock, which is not readily determinable. Another is that delta is constantly fluctuating along with fluctuating stock prices. These issues are fully dealt with in Part VI.

C. Academic Proposals

Because of the size of the market for options and their use in tax avoidance, option taxation has attracted considerable attention from legal academics. This section summarizes some of the existing commentary, especially as it relates to the approach of this article.

1. Spanning Method

In his 1993 article, Taxing New Financial Products: A Conceptual Framework, Professor Strnad identifies universality and consistency as ideals that the tax system should strive to achieve in the taxation of financial transactions. (50) "Universality requires that the tax system specify a tax treatment for every possible transaction." (51) Universality gives taxpayers certainty about the tax treatment of transactions. The second goal is consistency. "A tax system is consistent if and only if every cash flow pattern has a unique tax treatment. In such a system, it is not possible to manipulate tax outcomes by repackaging cash flows into different financial vehicles." (52) The discussion of the put-call party in Part III.A showed the inconsistency of taxing options the same way as pure equity.

Professor Strnad notes that a bifurcation approach accomplishes the goals of universality and consistency. Bifurcation is accomplished as follows: First, we see if a transaction can be broken down into constituent parts. Second, we identify tax treatment of each part. Third, we aggregate the tax results on the constituent parts. This bifurcation approach is consistent and universal. Another favorable aspect of bifurcation is its continuity. A system is continuous if transactions that are nearly identical have nearly identical tax treatments. (53) Thus, "small changes in any [transaction] will not cause a 'jump' in the tax results." (54) Equity collars have a discontinuous tax treatment, because if the spread between the put and call is too narrow, they trigger the constructive sale rules. (55) As a result, small changes in the spread can cause large changes in the tax consequences.

Professor Strnad analyzes the taxation of options under a stylized model called the "spanning method," under which a stock that will take one of five known values in two years. (56) This model does not reflect the real world, and may well be incapable of capturing the effect that innumerable price fluctuations have on the performance of real-world options. Like Professor Strnad's spanning method, the delta-hedging model of this article relies on bifurcation to examine the taxation of options. The delta-hedging model improves upon the spanning method, however, by its ability to produce tax results for real-world options.

2. Quasi-Mark-to-Market Approach

Professor Hasen used delta hedging to support his proposal of what he calls a quasi-mark-to-market approach for taxing options. (57) Hasen recognizes that delta hedging produces results that are equivalent to actual options and would base the taxation of actual options on a hypothetical delta hedge. As in this article, Hasen's delta hedging model bifurcates a call option into stock and debt. Yet, Hasen's model departs from the bifurcation ideal by not taxing the stock component according to current law. Instead, Hasen would tax the stock component of the synthetic option by marking it to market.

Let us recall how delta hedging can be used to replicate a long call. (58) The Black-Scholes formula produces a number "delta," which is the sensitivity of the price of the long call to movements in the price of the stock. An investor can theoretically replicate an option by buying a number of shares equal to this delta, financing part of the purchase with borrowing. Consider the following example that Professor Hasen uses to describe his delta-hedging approach:
 ABC stock is worth $30 on Day 1 and has moderate volatility of 30%.
 On Day 1, when the risk-free rate of interest is 10%, B sells A an
 option to buy ABC stock at $33 on Day 2, one year later. The price of
 the option is $3.64. At all times from Day 1 to Day 2 B is the record
 owner of the ABC stock. ABC stock pays no dividends. (59)


Hasen reports that the delta equals 56.75%. (60) A's taxable year ends six months later, at which time Hasen assumes that the stock has increased in value to $35.08. (61) Hasen reports an option value of $5 and a delta of 73.57%. (62)

A synthetic option is initialized by the purchase of delta=0.5675 shares of ABC stock, which costs $17.03. The purchase is financed by $3.64 (the price of the call) out of pocket and $13.39 of borrowing. (63) Six months later, the synthetic option will be represented by delta=0.7357 shares (worth $25.81) and borrowing of $20.81 (i.e., the value of the shares minus the value of the option). Professor Hasen's goal is finding the appropriate tax treatment for this six-month period. The problem, however, is that a synthetic option involves daily or more frequent trading to ensure that the number of shares always equals delta. We cannot know what tax consequences these trades and the related borrowing have based solely on the value at the end of six months.

In order to approximate the actual tax consequences, Hasen posits a single adjustment to the portfolio midway between Day 1 and the end of the taxable year six months later. The interim adjustment comes from the hypothetical purchase of 0.1682 new shares, reflecting the increase in delta from 0.5675 to 0.7357. Hasen deems this purchase to have been made at a share price of $32.54 (i.e., the midway point between $35.08 and $30). (64)

Professor Hasen would tax the initial stock purchase and the interim purchase on a mark-to-market basis. This system, which he calls a "quasi-mark-to-market approach," produces gains as follows. There are gains of $2.88 on the initial purchase (65) and $0.43 on the interim purchase. (66) As a result, Hasen would subject A to short-term capital gains of $3.31. Professor Hasen appears also to allow A an interest deduction of $0.77 on the imputed debt. (67) The net income would be $2.54.

This result obviously deviates from the realization rule. Under the realization rule, A would have no gain at all for year one, because she has only bought nondividend-paying stock and borrowed money. Indeed, A might even be entitled to an interest deduction. Professor Hasen justifies deviation from the realization rule by addressing the "policy question of whether it is appropriate to tax the gain on the value of the underlying asset during the pendency of the option or to wait until some future date." (68) The realization rule gives one, rather clear, answer to this policy question, although there is no reason Hasen should not argue for a better answer. (69)

But his answer points to full (not quasi) mark-to-market treatment of the option itself. In Hasen's example, the option price has increased only $1.36 (from $3.64 to $5.00), although he would impute income of $2.54. Hasen wants to avoid marking the option to market to "avoid the difficulty of actually computing the spot prices of the option on a daily basis (or in principle even more frequently)." (70) Yet, taxing the option on a mark-to-market basis would typically require only a single year-end valuation of the option. (71) In fact, valuing the option is no more difficult than calculating delta, as the formula for both have the same dependent variables. (72) Marking the option to market is no more difficult than performing the delta calculations that Hasen proposes.

At a conceptual level, this article approaches the taxation of options in a manner similar to Hasen's. Delta hedging gives us a way to break options down into more fundamental transactions. However, this article accepts as a reality the fact that these fundamental transactions have clear tax treatments that are rather uncontroversial outside the academy. Modeling this reality--including the realization rule--is the goal of this article.

3. Professor Shuldiner's Formula Interest

The spirit of this article is most in line with the framework given by Professor Warren and the bifurcation models of Professors Strnad and Hasen. There are, however, other noteworthy proposals to reform the taxation of options. Professor Reed Shuldiner has proposed tax consequences for options based on implicit interest. Professor Shuldiner would impute interest income to the holder of puts and calls, based on the amount of premium paid. (73) Shuldiner gives the following example (subject to an interest rate of 10%):
 Diva enters into a cash-settlement call option with David to purchase
 10,000 ounces of silver in two years at $12 per ounce. Diva pays David
 $10,000 for the option....

 Diva has purchased an asset for $10,000 which she is presumed to
 expect to increase in value to $11,000 by the end of the first year
 and to $12,100 by the end of the second year. Diva should accordingly
 have income of $1000 in the first year and $1100 in the second
 year. (74)


Shuldiner's approach actually imputes interest income in the opposite direction from a delta-hedging approach. Shuldiner does not supply a current price of silver nor its volatility in his example. Nevertheless, a current price of $9 per ounce and a volatility of 26.02% are consistent with the example. (75) With these parameters, delta is 47.84%. (76) Rather than buying an option over 10,000 ounces of silver for $10,000, Diva could alternatively buy 4784 ounces of silver. The cost would be $43,056, (77) which would be financed with $10,000 out of pocket (representing the premium) and $33,056 of debt. If we simply project this debt over the next two years, Diva would have year one interest expense of $3306 (78) and year two interest expense of $3636. (79)

Recall that Professor Warren had identified two basic tax regimes: fixed return and contingent return. Current law treats options as contingent-return transactions, whereas Professor Shuldiner would treat them as fixed-return transactions. Option theory shows that an option is neither fixed- nor contingent-return in its entirety. Instead, it is a hybrid of the two. (80)

4. Proposals for Covered Calls

Professor Calvin Johnson has argued that the premium received on a short call should be taxable as ordinary income if the call writer owns the underlying asset (i.e., writes a covered call). (81) The amount of income would be equal to the lesser of (1) the premium received or (2) the unrealized appreciation in the underlying property. This approach relies on an accounting concept of income, focusing on the cash received rather than the elimination of risk in the underlying asset. Thus, this approach fails to reach a protective put, which an investor pays for, even though a protective put can eliminate risk as well as a covered call. Also, the approach would not reach an equity collar either, even though it yields a relatively certain cash return but at a future date.

Professors Cunningham and Schenk would treat the sale of a covered call as the sale of part of the underlying asset. (82) Bruce Kayle has a similar approach. He used an example in which a taxpayer owns 1000 shares of stock with fair market value of $100 and adjusted basis of $40 per share. (83) Rather than selling a covered call with a strike price of $100, the taxpayer might create an economically equivalent partnership. The partnership has two classes of ownership. Class 1--analogous to the covered call--entitles the owner to all proceeds from the pre-established sale over $100. Class 2--analogous to the retained rights--entitles the owner to all dividends until the sale, plus all sale proceeds up to $100 per share. In Mr. Kayle's example, the taxpayer sells Class 1 for $5000, retaining Class 2. Mr. Kayle concludes that the taxpayer would have gain of $3000 from the sale. Because Class 2 replicates a covered call, Mr. Kayle suggests that the covered call could have similar tax treatment.

Mr. Kayle's approach would determine the taxation of the covered call by analogy to a more complicated transaction (classes of a partnership or trust). The approach of this article, in contrast, is to determine the taxation of options by their financial equivalence to more fundamental transactions (stock ownership, short selling, borrowing, and lending). Once a consistent system for taxing options is found, we could possibly invert Mr. Kayle's approach, applying the option-tax rules to partnership interests like Class 2.

Finally, David Schizer has suggested an approach for dealing with equity collars based on the delta of a stock. Recall that an equity collar combines a long put with a short call and acts as a substitute for a short sale. (84) If the spread in an equity collar is wide enough, it will avoid the constructive sale rules. Schizer notes that one could calculate the delta of the equity collar in order to determine the extent to which any collar should trigger the constructive sale rules. (85) Schizer does not, however, develop this idea fully, stating "although the delta approach is theoretically intriguing, it is probably not practical." (86) Part III.B already gave a preliminary example of this approach. Part VI.D of this article will attempt to develop this idea more fully and will ultimately propose it as a way of dealing with covered calls, protective puts, and equity collars. (87)

IV. THE SYNTHETIC OPTION AS A POLICY IDEAL

A. Theoretical Case for Taxing True Options According to Synthetic Options

Option theory works in financial markets because it equates options with liquid, easy-to-value transactions. Owning a call option is financially equivalent to owning a certain amount of the underlying stock and borrowing a certain amount of money. (88) The difference in value between the stock ownership and the indebtedness--the equity in the position--should closely approximate the value of the option. Thus, one could say that option theory successfully bifurcates call options into stock and debt. The goal of this article is to apply this approach to the taxation of options.

Taxing financial contracts according to their constituent parts is theoretically the strongest policy response to financial contract innovation. (89) A particular strength of this bifurcation approach is its "continuity," (90) which ensures that small changes to a transaction do not result in large changes to its tax treatment. Recall the problem of equity collars, described in Part III.B. An equity collar is used as a substitute for a short sale by taxpayers. Unlike short sales, however, equity collars can be structured to avoid the constructive sale rules of section 1259. Yet at some point the spread between the call and the put becomes too narrow, and the constructive sale rules are engaged. Thus, the current-law taxation of equity collars is discontinuous.

Bifurcating the equity collar into a short sale and bond avoids this discontinuity. By definition, an equity collar is a long put and short call, both of which can be decomposed into short selling and debt investing. By determining the amount of short selling inherent in the long put and in the short call, we can determine the extent to which any equity collar should trigger the constructive sale rules. Small changes in the equity collar would thus result in small changes in the amount of constructive sale that is triggered.

A similar approach can be taken with the synthetic bond described in Part III.B. There, we saw that a bond can be created by buying stock, buying a put, and selling a call. This combination is similar to the equity collar (a long put and short call) plus stock ownership. As we just saw, a long put and short call are both combinations of short selling and lending. In this case, it is the implicit lending that is important. If the tax laws imputed interest income on this lending, then the synthetic bond would offer no tax benefits.

Prior commentators have criticized the bifurcation approach as being unsound because of the lack of unique units by which transactions can be analyzed. (91) One commentator quipped, "There are no fundamental individual particles such as quarks in the financial world." (92) Yet, breaking transactions into fundamental particles is precisely what the Black-Scholes method does. The four fundamental units are owning stock, short selling stock, borrowing money, and lending money. Setting aside short selling for a moment, we should see that the tax rules for the other three transactions are familiar and unlikely to change in the foreseeable future. (93) Stock ownership gives rise to dividend income and gain or loss upon sale. Borrowing and lending money gives rise to interest expense and income. These three transactions are not commonly considered to be "derivatives," as we do not think that the economic returns on borrowing, lending, and stock ownership are based on other financial transactions. As for short selling, it is not as familiar as the other three transactions and its tax treatment is perhaps less stable, being radically changed in 1997. (94) Yet, short selling should still be considered a fundamental transaction because it is the inverse of stock ownership.

Thus, our fundamental particles are two pairs of inverse transactions: (1) borrowing and its inverse, lending, and (2) stock ownership and its inverse, short selling. These transactions are the fundamental building blocks that option theory uses to price options. They are also the building blocks that this article uses to examine the taxation of options.

B. The Timing of Tax Items

The total gain or loss on a synthetic option will be very close to the total gain or loss on a true option. After all, the whole point of the synthetic call is to replicate the economic return from a true option. As a result, we can be sure that current law gets the amount of gain or loss on options right. The interesting issue is whether the timing of gain or loss is correct.

Under current law, an option generates only one tax item--either gain or loss at some realization event (e.g., upon exercise or expiration). Under the approach of this article, an option generates several tax items based upon the tax items that a synthetic option generates. Recall that long calls and short puts are replicated with stock ownership and borrowing. (95) These options produce gain or loss from trading in the stock and interest expense from the borrowing. Short calls and long puts are replicated with short selling and lending. These options produce gain or loss from the short selling and interest income from the lending.

Unlike current law, the delta-hedging approach of this article does not defer all tax items to some future realization event. Measuring the timing of these tax items requires some assumptions, which are summarized as follows:

1. All tax items are taken into account immediately. So, interest income that is paid on October 1 is taken into account immediately, rather than on December 31 or April 15 of the following year. This assumption simplifies the calculations in the simulation.

2. Characterization is disregarded. The focus is solely on the timing of income. This assumption may well be the most limiting, as characterization has such a dramatic effect on tax rates under current law. (96)

3. Deductions for losses and interest are fully useable. (97)

4. Interest expense is deductible immediately, even though the simulation calculates interest as being capitalized. (98)

5. All positions are liquidated at the expiration date. So, gain or loss is not deferred past the expiration date, giving us a set period during which to compare the timing of tax items from the true option and the synthetic option.

The synthetic option produces a series of tax items over its lifetime. The future value of these items can be determined as of the expiration date. We can view this future value as the ideal measure of gain or loss on the option. This future value can thus be compared with the current-law treatment of the true option, which produces gain or loss only upon the exercise date.

The ultimate goal is the accurate timing of income, subject to the realization requirement. Some might assert that an even more accurate measurement of income would come from mark-to-market taxation of the synthetic option. (99) However, mark-to-market taxation of the synthetic option is the same as mark-to-market taxation of the option itself, (100) because the economic value of the synthetic option should track the economic value of the true option. Because the realization rule is so firmly entrenched, this article does not consider a mark-to-market system for taxing options.

The simulation must measure the timing of two types of tax items: (1) interest expense or income and (2) gain or loss from trading. Measuring the timing of interest is computationally straightforward. Recall for example that a synthetic long call is created by the purchase of delta shares of stock, financed in part by borrowing. We can assume that the borrowing generates interest at the same rate used in the Black-Scholes formula. Measuring the gain or loss from trading is more difficult. As time passes and the stock price fluctuates, the investor would need to rebalance the debt/stock portfolio. The goal is always to have the number of shares owned equal delta. When delta falls, for example, the investor would need to sell some stock, generating gain or loss on the sale. Measuring this gain or loss requires us to adopt some system of inventory accounting, discussed using an example in the next section.

C. A Simple Simulation

Recall the ABC stock example used above, drawn from Professor Hasen's article:
 ABC stock is worth $30 on Day 1 and has moderate volatility of 30%. On
 Day 1, when the risk-free rate of interest is 10%, B sells A an option
 to buy ABC stock at $33 on Day 2, one year later. The price of the
 option is $3.64. At all times from Day 1 to Day 2 B is the record
 owner of the ABC stock. ABC stock pays no dividends. (101)


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


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.

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.

Imputing interest to calls may improve the performance of option taxation even without imputing trading gains or losses. Recall that that the long call produced interest expense and trading losses. (130) Recognizing only the interest expense would mitigate the shortcomings of current law.

It is not difficult to estimate the expected interest on a call option. Recall that our synthetic long call is initiated as follows: Buy [[DELTA].sub.c]= 0.7663 shares of XYZ stock for $38.32, and finance this purchase with an out-of-pocket contribution of $16.25 (which is the option value). The remainder, $22.07, comes from borrowing. (131) We could assume that the taxpayer actually does create this initial position when buying an option, but never changes it over the life of the option. Using the same 5% rate used to price the options, we see that the hypothetical interest should come to $6.27 (132)--a result that is almost exactly the same as reported for the Monte-Carlo simulation reported above. We should assume that the interest accumulates on a daily basis. Projecting that daily interest forward to future value yields about $7.08--again, very close to the same as for the Monte-Carlo simulation reported above.

This system is, however, counterintuitive. Even though the buyer of a call expects to gain from the transaction, the system imputes a deduction until a realization event occurs. The inverse is true for a call writer, who pays money for a call yet faces imputed interest income. Professor Shuldiner's system of imputing interest is more intuitive and typical, as it imputes interest income, not expense, to the call holder. Greater consistency with the synthetic-call ideal clashes with tax aesthetics.

Even if the strangeness of imputing interest according to the synthetic call does not deter us, some practical considerations may. Granting an interest deduction to a cash-method call holder may not even be consistent with the proper taxation of the synthetic call. (133) A synthetic-call holder might face serious difficulties in achieving an interest deduction before expiration under the cash method. Granting interest deductions to call holders may also open the door to tax avoidance (134) unless the deduction is subject to complex systems like the straddle and wash-sale rules. (135) Moreover, current law may approximate the overall, correct result by denying call holders any interest deductions while excusing call writers from any interest income. If call holders and writers have the same marginal tax rate on average, then current law reaches the same result as the synthetic-option approach.

Thus, taxing the interest implicit in an option may be both difficult and of limited ultimate value. More limited reforms may be feasible. Although section 1258 imposes ordinary-income treatment on a synthetic bond created by a combination of stock, a long put, and a short call, section 1258 does not alter the timing of the income from the synthetic bond. (136) Even though section 1258 treats the synthetic bond as a bond for characterization, timing of the income is still determined under the realization standard. This analysis shows why section 1258 should also impute the interest income on an annual basis.

F. Conclusion

In Part IV above, I argued that a synthetic option is the theoretical ideal for taxing equity options. This section attempts to implement this ideal, focusing on naked options. Because the taxation of synthetic options depends on the actual path that the stock price takes, synthetic-option taxation can be measured only with a computer simulation. According to the simulation, current law appears to tax put options (long and short) correctly. However, current law appears to overtax long calls and undertax short calls. These results are consistent with a qualitative account of the tax items associated with synthetic options.

Yet, achieving the perfect result for a call option is probably not feasible. Although synthetic options produce gains and losses from trading, imputing these tax items to true options is not feasible. Synthetic options also produce interest income and expense. Imputing such items to call options is feasible but ultimately unwise. Interest expense would be imputed to the long call, but allowing a deduction for this expense may present opportunities for abuse. Imputing interest income to short calls would not lead to such opportunities. Imputing interest income to options would destroy the current symmetry between short and long positions in the same option. Overall, the system for taxing options may work well, even though short calls are undertaxed and long calls are overtaxed.

The taxation of short calls and long puts will be examined again in the next section. There, the options are combined with a position in the underlying stock. The result--covered calls and protective puts--would ideally trigger the recognition of unrealized gain in the underlying stock itself.

VI. TAXING THE COVERED CALL AND PROTECTIVE PUT: A MONTECARLO SIMULATION

A. Covered Calls and Protective Puts

A covered call is a short call combined with ownership of the underlying asset. Because the writer of the call receives premium income, prior commentators and courts have struggled with the issue of whether the writer should be treated as having sold the underlying asset. Current law answers with a definitive "no." This section will show that covered calls are best analyzed as implicit short sales. It will then extend this analysis to protective puts (long puts combined with ownership of the underlying asset). Under this approach, the taxation of covered calls and protective puts depends on the delta of the position, rather than the amount of premium income received.

To illustrate the problem of covered calls, suppose that Maya owns 10,000 shares of ABC stock. She has a $0 per-share basis in the stock, which is currently worth $30 per share. Next, suppose that Maya sells call options on the stock, exercisable in one year at $33 per share. Maya will receive cash for writing the call, perhaps $36,393 ($3.6393 per share) if we use the assumptions from above. (137) The problem arises in determining whether tax law should treat Maya as having sold the stock when she writes the call on it.

Under current law, Maya would pay no tax for writing this call. (138) The explanation for this treatment is that the short call and the stock are separate transactions, although the historical basis for this treatment comes from abstruse reasoning involving the character of the premium received. (139) Although this result is settled under current law, the taxation of covered calls has attracted significant attention from commentators. (140)

Option theory bifurcates Maya's short call into a short sale and a risk-free bond. The number of shares that Maya must sell short in order to replicate the short call is given by the delta of the call. In our case, the delta is 0.5658. (141) As Maya's short call covers 10,000 shares, she has essentially made a short sale of 5658 shares. A true short sale of ABC shares would implicate the constructive sale rules of section 1259 because Maya holds an appreciated position in 10,000 shares of ABC stock. (142) If a short call was equated with short selling, however, then Maya would be deemed to have sold 5658 of her owned ABC shares short at current fair market value when she wrote the call option.

Note that Maya's gain does not directly depend on the amount of the premium she received. (143) Maya would have a gain of $169,740, (144) even though she has received a premium of only $36,393. Her gain would depend on three factors. First, the delta of the option determines the number of shares that were implicitly sold short. Second, the fair market value of the stock determines the amount realized implicitly. Third, the adjusted basis in the stock Maya actually owns is used to determine the amount of gain.

The delta standard would apply even if the taxpayer pays a premium. Let us assume that, rather than selling a call, Maya bought a put over 10,000 shares of ABC stock. If we use the same parameters as before, the put would cost her $34,989. (145) The delta for the put is -0.4342. Thus, buying this put option is the same as implicitly selling 4342 shares of ABC stock short. Current law does not tax Maya upon buying the put, even if she already owns ABC stock. Yet, the delta standard of this section would treat Maya as executing a short sale over 4342 shares, producing gain of $130,260.

Thus, option theory gives tax policy a way of dealing with the ancient problem of covered calls and the related problem of protective puts. In order to ensure consistency between the taxation of short sales and the taxation of options, these options should be treated as constructive sales (based on the Black-Scholes delta calculation). (146) The example given above shows only the initial consequences of the covered call and protective put under the delta model. It does not follow through to the end of the options. Exploring the tax consequences over the entire life of the options is the goal of the next two sections.

B. Fluctuating Deltas and Constructive Sales

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


As in the prior section, the simple example using ABC stock can demonstrate how the components of the synthetic option should be taxed. (153) Yet, as before, the ABC-stock example shows just one path that stock may take. Moreover, the twenty-six week period is not long enough to explore the full consequences from the deferral regime of current law. Measuring the likely value of this deferral requires a longer option measured over more numerous paths.

In summary, this section showed how the delta model could deal with fluctuations in delta, which in turn cause fluctuations in the implied short-sale position. Fluctuating delta creates some complexity, which can be overcome with simplifying assumptions. The approach I took treats the short sales as a ratchet that is being applied to constructive sales. Constructive sales occur only when the short-sale position reaches new highs. Measuring the consequences of these constructive sales is the goal of the next section.

C. A Monte-Carlo Simulation of Constructive Sales

Let us briefly review the problem of the prior two sections. An investor who owns appreciated stock might either write a call (a covered call) or buy a put (a protective put). Under current law, the investor has almost certainly not realized any gain on the owned stock from either the covered call or the protective put. I have argued, however, that the policy ideal is to tax options according to their constituent transactions. In the context of covered calls and protective puts, we would bifurcate the option into a combination of a short sale and investment in a risk-free asset. The short sale is what is interesting here, because it would cause the investor to realize gain under the constructive sale rules of section 1259. (154) The delta of the option, which fluctuates over time and with movements in the stock, determines the amount of short sales outstanding at any given time. Thus, the short-sale position fluctuates along with delta. In order to simplify the model, I assume that delta acts like a ratchet on short sales, which therefore trigger constructive sales only when the total amount of short sales exceeds the previous all-time high. The prior section ended with an example that illustrates the workings of this assumption. (155)

If the investor ultimately sells the stock at the end of the option, then the total gain or loss realized is no different if we measure under current law or under the delta model. (156) What is different is the timing of the gain or loss. Because constructive sales generate gain (but not losses), the policy ideal given by the delta model will tend to accelerate the realization of gain when compared to current law. The goal of this section is to measure the value of the deferral given by current law and also to examine some practical ways to curtail it according to the delta model.

Let us then return to the long-term example using XYZ stock, which was the basis for the Monte-Carlo simulation of Part V. Allow me to restate the relevant particulars of the XYZ stock, the options on the stock, and the nature of the Monte-Carlo simulation.

Initial Price: $50

Exercise Price: $50

Risk-Free Rate: 5%

Time to Exercise: 5 years

Volatility: 25%

Dividends: None

Number of Simulated Paths: 2000

Black-Scholes Call Price: $16.2520

Black-Scholes Put Price: $5.1920

Black-Scholes Call Delta: 76.63%

Black-Scholes Put Delta: -23.37%

Let us assume that an investor owns 10,000 shares of XYZ stock with a zero basis. Next, let us examine separately the proper tax treatment of a covered call and protective put, each over 10,000 shares of stock.

Under current law, the investor has no tax consequences from either buying the put or from selling the call. (If the investor did both, however, he would probably trigger the constructive sale rules. (157)) Current law defers the tax consequences with respect to the shares of XYZ stock that the taxpayer owns. The focus of this section is on when the gain from the sale of XYZ stock should be recognized. Under the simulation, the mean price of the stock is $63.9143 after five years. (158) Let us assume that the investor plans to sell the stock in five years, giving him an expected future payoff of $639,143.

The synthetic short call would be initiated by the short sale of 7663 shares, and the synthetic long put would be initiated by the short sale of 2337 shares. Initially, $38,315 of gain is triggered by selling the call, and $11,685 of gain is triggered by buying the put. Afterwards, future short sales would be executed or closed to track changes in delta, with constructive sales being triggered according to the ratchet theory (i.e., whenever total short sales exceeds the previous all-time high).

Since we are assuming the investor would sell the stock in five years, the constructive sales do not increase the sum of the gain realized. (159) The constructive sales do, however, affect the timing of gain recognition. In order to measure the benefit of deferring gain, the simulation measures the constructive sale each day. This series of sales can then be projected to future value. (160) Those results are the basis for the following table, which further breaks the future value number down between the sum of the gain (i.e., what is taxed under current law), the time-value of money on the initial constructive sale, and the time-value of money on interim constructive sales.
Means Measured Over 2000 Observations

 Short Call Long Put

Sum of Gain on Owned Stock (Current Law) 639,143 639,143
TVM of Initial Constructive Sale 108,816 33,185
TVM of Interim Constructive Sales 19,569 28,014
Future Value of Gains (Policy Ideal) 767,528 700,342


The first row of numbers (Sum of Gain) represents what current law taxes. Recall that we are concerned about the taxation of the XYZ stock itself. The gain realized on the stock is not affected by whether the investor enters into an option contract; so we see the same number in both columns. The second row is the time value of money attributable to the initial short sale implicit in the option. Under the assumptions of this section, a constructive sale is triggered immediately when the investor enters into the option. This initial constructive sale is given by the delta of the option. The second row describes the benefit taxpayers receive from deferring this initial gain. The third row represents the time value of money attributable to subsequent constructive sales generated under the ratchet theory developed in Part VI.B.

Another way of viewing the numbers is that the first row is what the law actually accomplishes, the second row is what the law could conceivably accomplish, and the third row is what the law probably cannot reach. Imputing a constructive sale on the execution of the option may be practical. But it might not be practical to impute constructive sales based on interim fluctuations in delta. Doing so would require frequent (perhaps daily) tracking of stock prices and recalculation of the short sales implied by the options. Moreover, much of the prior section dealt with the practical difficulties of modeling the constructive sales produced by interim fluctuations in delta. Nonetheless, it might be practical to impute a constructive sale based upon the initial delta of an option. Indeed, most of the time value of money that is lost under current law can be attributable to the initial constructive sale. After all, the initial constructive sale is the largest in size and the earliest in time.

In summary, this section used the same set of stock prices used in Part V.B, the set containing 2000 separate random walks, each of which is 1800 days long. This section then reported the results of a Monte-Carlo simulation of the constructive sales that would be triggered by a covered call and a protective put. The simulation shows that current law results in a significant amount of deferral when compared to the policy ideal of the delta model. Fully implementing the delta model may be impractical, however, as doing so would involve frequent recalculation of delta. That being said, a very large amount of the unwarranted deferral can be eliminated simply by determining using the initial delta of the option and ignoring interim fluctuations. The next section attempts to extend this approach to certain combinations of covered calls and protective puts known as equity collars and variable prepaid forward contracts.

D. Applying the Delta Model to Equity Collars and Variable Prepaid Forwards

Under the delta model developed by this section, a covered call or a protective put triggers a constructive sale of a portion of the stock owned by an investor. The amount of the constructive sale is given by the delta of the option, calculated at the time the option is executed. Later increases in delta would ideally trigger more constructive sales, but measuring those later constructive sales is impractical. This section extends this delta model to two close substitutes for short sales: an equity collar and a variable prepaid forward contract.

As noted in Part III.B, an equity collar combines a short call with a long put. Typically the investor also owns appreciated shares of the stock covered by the option and wishes to hedge against future movements in the stock, but without incurring the tax liability that comes with selling the stock. Even though the exercise price on the short call will be higher than on the long put, the position is economically very similar to a short sale. In fact, the goal of a tax planners is to have the collar replicate a short sale as closely as possible while safely avoiding the constructive sale rules.

Dean Schizer mentioned a delta model for equity collars previously, but ultimately dismissed it on political and technical grounds. (161) The political argument is that taxpayers and their advisors would effectively oppose a delta model. (162) Perhaps so, but perhaps tax shelters like equity collars and variable prepaid collars will become an appealing target for a Congress that wants to expand spending or cut overall tax rates. Dean Schizer's technical critique is that calculating the delta of an equity collar is impracticable or impossible. (163) Recall, however, that an equity collar is simply a short call and a long put combined. The delta for such a combination is simply the sum of the delta of the two pieces. (164)

Consider an example from Dean Schizer's article. Suppose that a share of stock is currently worth $100. The spread on the collar is from $95 to $115. (165) Suppose that the volatility is 40%, the risk-free rate is 5%, and the time to expiration is three years. (166) The collar has a value of $10.97. The put (with exercise price of $95) costs the investor $16.50. The call (with exercise price of $115) brings the investor $27.47. Thus, the investor would receive the difference ($10.97) under the Black-Scholes formula. The delta of the collar is -90%, composed of the put delta (-26%) minus the call delta (64%). Even though this transaction does not trigger the constructive sale rules, it is very close in effect to a conventional short sale. A pure short sale would have delta of -100%, compared with the -90% delta for the collar.

But what if we do not know what the volatility of the stock actually is? We do know the output of a function (the instrument price) and all inputs but one. Unfortunately, the Black-Scholes equation cannot be inverted to derive an analytic solution for the unknown input. (167) However, the solution in any particular case can be found using numerical methods. (168)

Suppose that we have a collar on a stock that lasts for three years. The stock is currently worth $100, and the collar spread is from $90 to $110. The risk-free rate is 5%. The investor receives $11 for the collar. If we thought that the $11 received was the true Black-Scholes value, then the parameters just given would imply a volatility of about 15% on the stock and a delta of the collar of about -77%. (169) Yet, the $11 may be less than the true Black-Scholes value. Dean Schizer reports that the typical fee on an equity collar is 1% of the hedged asset's value for every year the collar is open. In our example, this is $3. So, in the present example, the charge would be $3 (1% times $100 times three years). If we were sure that the investment bank was charging a commission of $3, (170) then we could calculate implied volatility by assuming the investor received $14. This implies volatility of about 32% and a delta of about -88%. A delta model for taxing equity collars might, for ease of administration, ignore the banker's fee when calculating the delta on the collar. In the example just given, assuming that the price received is the true Black-Scholes price leads to a lower delta and thus a lower constructive sale. Using this simplification, we would assume a constructive sale with respect to 77% of the shares covered by the collar, rather than 88%.

The approach just described can be extended to variable prepaid forward contracts (VPFC). VPFCs are structured as forward contracts but are, in essence, modified equity collars. Recall that an investor executes an equity collar with a protective put having a low strike price and a covered call having a higher strike price. The key modification with the VPFC is that the call covers fewer shares than the put. (171) The fraction covered by the call is usually the high strike price divided by the price of the stock at the time the VPFC is executed. For example, suppose that we have a VPFC on a stock that lasts for three years. The stock is currently worth $100, and the implied collar spread is from $100 to $125. If the put portion of the VPFC covers 10,000 shares, then the call portion would cover only 8000 shares. (172) We might also think of the VPFC payoff as combining a traditional equity collar over 8000 shares plus an additional protective put over 2000 shares. The following graph will hopefully illuminate the difference between an equity collar and a VPFC.

The VPFC can be taxed according to the delta model if we can extract an implied volatility. Once we have an implied volatility for the stock, calculating delta for the VPFC is trivially easy. Unfortunately, doing this is trickier than it was with the equity collar for two reasons. First, ignoring the banker's fee may now harm, rather than help, the taxpayer. Second, even if we can factor the banker's fee into the valuation, the VPFC might actually imply two separate volatilities, which would lead to two separate deltas.

[GRAPHIC OMITTED]

Return to the prior example and suppose the risk-free rate is 5% and the taxpayer receives a net amount of $0 for entering the VPFC. We might be ready to find an implied volatility of 54.29% and a delta of -78.65%, as they correspond with a Black-Scholes value of zero. But, we need to recall that the value observed will lower than the true Black-Scholes value because of the investment bank's fee. We could disregard this with the collar, because using the lower observed value simply lowered the implied volatility and the magnitude of delta, working to the benefit of the taxpayer. Our situation is more ambiguous with the VPFC. Assume that the true value of the position is really $1.50, which the investment bank keeps as its fees. This value actually implies two volatilities (10.20% and 24.57%) and two deltas (-47.20% and -69.90%). In this example, we might feel safe choosing the 24.57% volatility, as it is more typical for stocks. But, the choice may not always be so clear.

The failure to achieve perfection should not, however, be an argument against the delta model. Even if the taxpayer chiseled the government in the above example and claimed a 10.20% implied volatility (and a -47.20% delta), he is still being subjected to a constructive sale of 47.20% of his holdings of the stock. This is clearly an improvement over the complete deferral allowed by current law.

VII. CONCLUSION

This article set out to analyze the proper taxation of options to buy or sell stock. In the Introduction, I described a call option that Maya buys that entitles her to buy 10,000 shares of XMPL stock for $100 in five years. Current law imposes no tax consequences on Maya until she exercises the option, it lapses, or she sells it. This approach is probably the best. We can bifurcate Maya's option into a synthetic option, initially composed of 7500 shares of XMPL and $400,000 of borrowing. Theoretically, Maya should be taxed as based on the synthetic option. However, the Monte-Carlo simulation above demonstrates that doing so is not materially different from the results produced by current law. Thus, current law passes a very strong test when we examine the option in isolation (i.e., naked options).

We have a very different result if Maya sold the option and already owns 10,000 appreciated shares in XMPL. Here, the synthetic option is represented by the sale, or short sale, of 7500 shares of XMPL. If Maya sells 7500 shares of stock she already owns--or if she sells 7500 shares short--then she must pay tax on the gain. Thus, she should face a constructive sale on 7500 shares when she sells a call. A similar result applies if Maya buys a protective put or enters into a more complex contract that combines covered calls and protective puts. The two contracts discussed in this article were equity collars and variable prepaid contracts, both of which should trigger immediate taxable gain as well.

Thus, the primary shortcoming of current law is that the constructive sale rules of section 1259 do not apply to covered calls, protective puts, and related contracts. Otherwise, option taxation seems to work fine.

On a broader level, I hope that the approach of this article will reinvigorate the use of bifurcation techniques by scholars and policymakers. Financial theory supplies a rich, if arcane, set of tools that potentially allows us to break down financial contracts into fundamental units like stock and borrowing. Taxing financial contracts strictly according to their fundamental units may be impracticable. But, the approach gives us a normative baseline to use when selecting from the set of administratively and politically feasible methods of taxation.

APPENDIX A: STEPS IN DELTA HEDGING

1. Determine the initial value of the option using the Black-Scholes formula.

a. For a call, this value is "c".

b. For a put, this value is "p".

2. Determine the initial delta ([DELTA]) of the option using a formula related to Black-Scholes.

3. Note the current price of the stock (S).

4. Create a synthetic option or replicating portfolio in the following manner. (Note that in each case the initial cash flow for the synthetic option is the same as for a true option.)

a. Long call: Purchase [DELTA] shares of stock. This costs [DELTA]*S. This is financed by an out-of-pocket contribution equal to c. The remainder, [DELTA]*S-c, is obtained by borrowing.

b. Short call: Sell short [DELTA] shares of stock. This produces proceeds of [DELTA]*S. Of this, place the premium, c, in the investor's pocket. (This represents the premium payment that a call writer receives.) The remainder, [DELTA]*S-c, is used to purchase a debt instrument.

c. Long put: Sell short negative [DELTA] shares of stock. (Note that delta for a put is already negative.) Use these proceeds, plus an out-of-pocket contribution equal to p, to buy a debt instrument. (Recall that a put writer must pay a premium to buy a put.)

d. Short put: Purchase negative [DELTA] shares of stock. Borrow [DELTA]*S to do this. Borrow an additional amount, p, to place in the investor's pocket. This amount represents the premium received.

5. Every day (or week or hour or other period) rebalance the synthetic option. Passage of time and movements in the stock price cause changes in delta. The synthetic option is rebalanced when the position in the stock is adjusted to reflect the new delta.

a. Long call: Always make sure that the number of shares owned equals delta. If delta rises, buy more stock using more borrowing. If delta falls, sell stock and use the proceeds to pay of the existing borrowing.

b. Short call: Always make sure that the number of shares shorted equals delta. If delta rises, sell more stock short and use the proceeds to increase the investment in the debt instrument. If delta falls, close short sales; as the investor must buy shares to close the short sales, he will obtain the needed funds by liquidating a portion of the debt instrument.

c. Long put: Always make sure that the number of shares shorted equals negative delta. If delta rises (e.g., from -50% to -40%), close short sales, obtaining funds from the debt instrument. If delta falls (e.g., from -50% to -60%), engage in more short selling, investing the new proceeds in the debt instrument.

d. Short put: Always make sure that the number of shares owned equals negative delta. If delta rises (e.g., from -50% to -40%), sell stock, using the proceeds to pay off borrowing. If delta falls (e.g., from -50% to -60%), buy more stock, borrowing more funds to pay for the purchase.

Eric D. Chason*

* Assistant Professor of Law, College of William and Mary, Marshall-Wythe School of Law. I thank Glenn Coven, Rich Hynes, Eric Kades, Michael Knoll, John Lee, Alan Meese, and Lawrence Zelenak for comments on prior drafts of this paper. I also thank Karen Gurth, Brandon Rogers, Matt Stuart, and Will Woolston for their research assistance.

(1) See Bank for International Settlements, Semiannual OTC Derivatives Statistics at end-June 2006, 22B Equity-linked Derivatives by Instrument and Market (2006), http://www.bis.org/statistics/derstats.htm.

(2) See Mark P. Gergen & Paula Schmitz, The Influence of Tax Law on Securities Innovation in the United States: 1981-1997, 52 TAX L. REV. 119 (1997) (describing various financial instruments); cf, e.g., infra Part VI.D (discussing variable prepaid forward contracts).

(3) See David A. Schizer, Balance In The Taxation Of Derivative Securities: An Agenda for Reform, 104 COLUM. L. REV. 1886, 1886 (2004) ("It is well understood that aggressive tax planning among high-income individuals and corporations represents a threat to the U.S. tax system, and that derivatives are staples of this planning.").

(4) See Michael S. Knoll, Financial Innovation, Tax Arbitrage, and Retrospective Taxation: The Problem With Passive Government Lending, 52 TAX. L. REV. 199, 200 (1997). Professor Knoll states:
 Because the tax treatment of even very basic financial contracts is
 inconsistent, financially sophisticated parties reduced their tax
 liabilities by using innovative financial products and techniques to
 exploit these inconsistencies. In its most extreme form, parties can
 engage in tax arbitrage, the process of buying and selling the same
 cash flows to generate an after-tax cash profit from the different tax
 treatment of identical cash flows. Tax arbitrage represents a serious
 threat to the tax system because taxpayers, by merely adjusting their
 portfolios, can reduce or even eliminate their tax liabilities.


(5) See Fischer Black & Myron Scholes, The Pricing of Options and Corporate Liabilities, 81 J. POL. ECON. 637 (1973); see also Robert C. Merton, Theory of Rational Option Pricing, 4 BELL J. ECON. & MGMT. SCI. 141 (1973).

(6) See infra note 25.

(7) See infra note 27.

(8) The more common usage is to refer to the formula as "Black-Scholes." Cf., e.g., JOHN C. HULL, OPTIONS, FUTURES & OTHER DERIVATIVES 234 (5th ed. 2003) (introducing the "Black-Scholes model"). Some refer to it as the Black-Scholes-Merton model. Cf., e.g., NASSIM TALEB, DYNAMIC HEDGING 109 (1997) (referring to the "Black-Scholes-Merton" model).

(9) Purists might note that there is no "Nobel Prize" in economics as there is for peace, physics, etc. Technically, it is the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel and it was awarded to Robert Merton and Myron Scholes in 1997 for their work in option pricing theory. Fischer Black was Scholes' coauthor, but died before the award was made. See Nobel Prize.org, The Prize in Economic Sciences 1997 (1997), http://nobelprize.org/nobel_prizes/economics/laureates/1997/press.html.

(10) See, e.g., Christopher H. Hanna, Demystifying Tax Deferral, 52 SMU L. Rev. 383, 384 (1999) ("Issues relating to tax deferral and time value of money are probably the most important areas of tax study."); see also Daniel I. Halperin, Interest in Disguise: Taxing the "Time Value of Money," 95 YALE L.J. 506 (1986).

(11) See, e.g., David Weisbach, Tax Responses to Financial Contract Innovation, 50 TAX L. REV. 491, 539 (1995) ("Bifurcation provides a theoretical framework for taxing hybrids.").

(12) The same analysis will apply to protective puts, equity collars, and variable prepaid contracts. See infra Part VI.

(13) ZVI BODIE ET AL., INVESTMENTS 54 (Christina Kouvelis ed., 6th ed. 2005). The exercise price is often called the strike price. The two terms are synonymous.

(14) See HULL, supra note 8, at 705. American options can be exercised at any time before expiration. See id. at 700.

(15) BODIE ET AL., supra note 13, at 91-92.

(16) Because of the future value, we know that B=exp(-r*T).

(17) HULL, supra note 8, at 174-75. See generally infra note 18.

(18) See, e.g., Michael S. Knoll, Put-Call Parity and the Law, 24 CARDOZO L. REV. 61, 72-74 (2002).

(19) See David M. Hasen, A Realization-based Approach to the Taxation of Financial Instruments, 57 TAX L. REV. 397, 431 (2004).

(20) $9.50*exp(0.10*1)=$10.50.

(21) The so-called Greek letters describe the sensitivity of an option price to market inputs. Delta is the most significant of the Greek letters, as it measures sensitivity of the option price to changes in the stock price. Another is theta, which measures the sensitivity of an option to passage of time. See HULL, supra note 8, at 309-11. Because the passage of time is constant, there is no need to hedge for theta. See id. at 311. Another Greek letter is gamma, which measures the sensitivity of delta to changes in the stock price. See id. at 312. Although gamma is critical to real-world option traders, it is not addressed in this article. The reason is that gamma cannot be hedged with the underlying asset itself. Rather, it can be hedged only with other options. See id. at 313. The other two Greek letters are rho (which measures sensitivity to interest rate changes) and vega (which measures sensitivity to changes in volatility). See id. at 316-19. For the sake of simplicity, the model used in this article will assume that volatility and interest rates are constant.

(22) Technically, we can derive all of the deltas from the long call delta, which is produced by the Black-Scholes equation. Note that the long call delta is always positive, between 0 and 1. The short call delta is simply the inverse of the long call delta (and therefore between -1 and 0). The long put delta is the long call delta minus one (and therefore between -1 and 0). The short put delta is the inverse of the long put delta (and therefore between 0 and 1).

(23) The Black-Scholes model assumes that the return on the stock is a random variable with a lognormal distribution.

(24) Hasen, supra note 19, at 438-39.

(25) I have used the MATLAB programming language to produce the Black-Scholes calculations in this article. The function that produces the option price is blsprice, which takes as its inputs the price of the stock, the strike of the option, the interest rate, the time to expiration, and the volatility of the stock. Here, blsprice (30, 33, 0.10, 1, 0.30) returns 3.6393.

As another example, recall XMPL Corp. from Part I. The call had a strike price of $100 and a term of five years, and the current price of the stock was also $100. I used an interest rate of 4.55% and a stock volatility of 29.92%. (My goal was to create a realistic option with a round price and delta, but these numbers are typical.) In MATLAB, blsprice (100, 100, 0.0455, 5, 0.2992) returns $35.0004. After multiplying by 10,000 (the number of shares covered), I rounded down from $350,004 to $350,000 for sake of convenience.

(26) The MATLAB function that produces the delta is blsdelta, which takes the same inputs as blsprice. Here blsdelta (30, 33, 0.10, 1, 0.30) returns 0.5658.

(27) $169,733-$36,393.

As another example, recall XMPL Corp. from Part I. The call had a strike price of $100 and a term of five years, and the current price of the stock was also $100. I used an interest rate of 4.55% and a stock volatility of 29.92%. (My goal was to create a realistic option with a round price and delta, but these numbers are typical.) In MATLAB, blsdelta (100, 100, 0.0455, 5, 0.2992) returns 0.7500. After multiplying by 10,000 (the number of shares covered), we obtain 7500 shares that replicate the option. These shares cost $750,000. As the option price is $350,000, see supra note 25, the equity in the synthetic option must be $350,000. So, the XMPL option is replicated with a purchased of 7500 shares, worth $750,000, financed in part with debt of $400,000.

(28) The Black-Scholes price of the option is $37,560. By comparison, the replicating portfolio is worth $37,897. The 5658 shares of stock are worth $171,494 (i.e., 5658*$30.31). The initial debt of $133,340 has grown to $133,597 (i.e., $133,340 *exp(0.1/52)).

(29) In MATLAB, blsdelta (30.31, 33, 0.10, 51/52, 0.30) returns 0.5763.

(30) Cf., e.g., PAOLO BRANDIMARTE, NUMERICAL METHODS IN FINANCE: A MATLAB-BASED INTRODUCTION 65 (2002) (giving a MATLAB-based Monte-Carlo simulation for option valuation).

(31) For example, Tax Law Review had an entire issue devoted to the topic. See David F. Bradford, Fixing Realization Accounting: Symmetry, Consistency and Correctness in the Taxation of Financial Instruments, 50 TAX L. REV. 731 (1995); Mark P. Gergen, Afterword: Apocalypse Not?, 50 TAX L. REV. 833 (1995); Deborah H. Schenk, Taxation of Equity Derivatives: A Partial Integration Proposal, 50 TAX L. REV. 571 (1995); Daniel Shaviro, Risk-Based Rules and the Taxation of Capital Income, 50 TAX L. REV. 643 (1995); Jeff Strnad, Commentary: Taxing New Financial Products in a Second-Best World: Bifurcation and Integration, 50 TAX L. REV. 545 (1995); David A. Weisbach, Tax Responses to Financial Contract Innovation, 50 TAX L. REV. 491 (1995).

(32) Alvin C. Warren, Jr., Financial Contract Innovation and Income Tax Policy, 107 HARV. L. REV. 460 (1993) [hereinafter Warren 1993].

(33) Warren 1993, supra note 32, at 464-65 (emphasis in original) (citations omitted); accord Alvin C. Warren, Jr., U.S. Income Taxation of New Financial Products, 88 J. PUB. ECON. 899, 901-02 (2004) [hereinafter Warren 2004]. The historical development of option taxation is nicely summarized in Bruce Kayle, Realization Without Taxation? The Not-So-Clear Reflection of Income From an Option to Acquire Property, 48 TAX L. REV. 233, 237-42 (1993). As this article does not address the characterization of option gain and loss, the current-law rules dealing with character are not addressed. A summary of those rules can be found at David H. Shapiro, Taxation of Equity Derivatives, 188 TAX MGM'T PORTFOLIO (BNA) [paragraph] II.A.3 (2003).

(34) See Warren 1993, supra note 32, at 463 (citation omitted); Warren 2004, supra note 33, at 901.

(35) See Warren 1993, supra note 32, at 462-63; Warren 2004, supra note 33, at 900.

(36) See Warren 1993, supra note 32, at 470; Warren 2004, supra note 33, at 902.

(37) See supra Part II.B.

(38) Algebraically, the put-call parity can be rewritten as B=S+p-c.

(39) Most or all of the gain on the synthetic bond would be subject to ordinary income rates, rather than capital gains rates. See I.R.C. [section] 1258.

(40) I.R.C. [section] 1258.

(41) The short sale is "against the box" because Maya already holds the same shares that she is shorting.

(42) See David M. Schizer, Frictions as a Constraint on Tax Planning, 101 COLUM. L. REV. 1312, 1343 (2001).

(43) See id. at 1344 n.104 (quoting JOINT COMM. ON TAXATION, 105TH CONG., GENERAL EXPLANATION OF TAX LEGISLATION IN 1997 (Comm. Print 1997)).

(44) See I.R.C. [section] 1259.

(45) $31.54 is the future value of $30 after one year at 5% interest ($30*e^0.05).

(46) See Schizer, supra note 42, at 1345-47.

(47) Dean Schizer reports that the folk wisdom of the New York tax bar is that a spread of 10% to 20% of the value of the owned asset should avoid the constructive sale rules. See id. at 1346 n.110.

(48) In MATLAB, blsdelta (30, 27, 0.10, 1, 0.30) returns -0.2020 for the put.

(49) In MATLAB, blsdelta (30, 33, 0.10, 1, 0.30) returns 0.5658 for the call. See supra text accompanying note 26. Because we are dealing with a short call, we take the inverse of the given delta.

(50) Jeff Strnad, Taxing New Financial Products: A Conceptual Framework, 46 STAN. L. REV. 569, 572-73 (1994).

(51) Id.

(52) Id. at 573.

(53) See generally supra note 50.

(54) Strnad, supra note 50, at 598.

(55) See supra Part III.B.

(56) See Strnad, supra note 50, at 593 & n.65.

(57) See Hasen, supra note 19, at 443.

(58) See id. at 430-31.

(59) Id. at 438-39.

(60) Id. at 439. Hasen's number for delta is slightly off. The correct delta is 56.58%.

(61) Id. at 444.

(62) Id. Both of these numbers are slightly off. The correct price is $5.03, and the correct delta is 73.56%.

(63) Id. at 439.

(64) See id. at 447. This assumption seems contrary to Hasen's goal of avoiding "off-market transactions." Cf. id. at 443 ("[I]f one simplified using off-market transactions to approximate option transactions, one would substantially undercut the utility of using the dynamic hedging model in the first place, because the value of the model lies in its establishment of transactional equivalents.").

(65) 0.5675*($35.08-$30).

(66) 0.1682*(35.08-32.54).

(67) See Hasen, supra note 19, at 445 & n.146 (calculating interest of $0.12 on the borrowing associated with the interim purchase and calculating interest of $0.65 on the borrowing associated with the initial purchase).

(68) See id. at 445.

(69) It is likely that most academics would support a fuller mark-to-market regime. See Edward A. Zelinsky, For Realization: Income Taxation, Sectoral Accretionism, and the Virtue of Attainable Virtues, 19 CARDOZO L. REV. 861, 861-62 (1997). Professor Zelinsky states:
 Much contemporary scholarly literature promotes the alternative vision
 of accretionist taxation, under which the taxpayer either pays tax
 periodically on increases in his net worth, without waiting for a
 realization event, or pays upon realization an additional deferral
 charge, to compensate the fisc for the time-value of the taxes the
 taxpayer would have paid earlier under a true accretionist regime.


Id.

(70) Hasen, supra note 19, at 444.

(71) The initial value of the option would typically be set by an arm's length transaction. Receipt of a gratuitous or compensatory option would, however, require an initial valuation.

(72) Delta and the call price are both functions of the same variables: the risk-free rate, stock volatility, time to exercise, the strike price, and the market price. See HULL, supra note 8, at 246, 303.

(73) See Reed Shuldiner, A General Approach to the Taxation of Financial Instruments, 71 TEX. L. REV. 243, 308-10 (1992).

(74) See id.

(75) In MATLAB, blsprice(9*10000, 12*10000, 0.1, 2, 0.2602) = 9999.7.

(76) In MATLAB, blsdelta(9*10000, 12*10000, 0.1, 2, 0.2602) = 0.47835.

(77) 4784*9.

(78) $33,056*10%.

(79) ($33,056+$3306)*10%.

(80) Cf. Warren 2004, supra note 33, at 903 ("Although an actual call is subject to wait-and-see taxation because the return is contingent, such a synthetic call would produce current interest for the holder.").

(81) See Calvin H. Johnson, Taxing the Income from Writing Options, 73 TAX NOTES 203 (Oct. 14, 1996).

(82) Noel B. Cunningham & Deborah H. Schenk, Taxation Without Realization: A "Revolutionary" Approach to Ownership, 47 TAX L. REV. 725, 775-84 (1992).

(83) Kayle, supra note 33, at 273.

(84) See supra Part III.B.

(85) See Schizer, supra note 42, at 1364-67.

(86) See id. at 1367.

(87) Some commentary has argued that hedging transactions should not trigger realization. See, e.g., Deborah L. Paul, Another Uneasy Compromise: The Treatment of Hedging in a Realization Income Tax, 3 FLA. TAX REV. 1 (1996).

(88) See Appendix A for detailed steps.

(89) See Weisbach, supra note 11, at 539.

(90) Professor Strnad identifies another goal--consistency. Consistency ensures that unique cash flows have a unique tax result. Consistency may well be satisfied under current-law taxation of options. It would be quite difficult for most taxpayers to create a synthetic option because of the necessity of frequent trading (and the resulting trading costs). Those taxpayers capable of creating synthetic options are almost certainly dealers who are subject to market-to-market taxation whether they hold true options or synthetic options.

(91) See Randall K.C. Kau, Carving Up Assets and Liabilities--Integration or Bifurcation of Financial Products, 68 TAXES 1003, 1005-07 (Dec. 1990); Edward D. Kleinbard, Beyond Good and Evil Debt (And Debt Hedges): A Cost of Capital Allowance System, 67 TAXES 943, 947-55 (Dec. 1989); Weisbach, supra note 11, at 512 (citing David P. Hariton, New Rules Bifurcating Contingent Debt--A Mistake?, 51 TAX NOTES 235, 237-38 (Apr. 15, 1991)).

(92) See Kau, supra note 91, at 1007 (quoted in Weisbach, supra note 11, at 512).

(93) Here, I am referring to the timing of income and deductions. The actual rate that applies to many of these transactions is quite volatile because of changing capital-gains rates over the past twenty years.

(94) Cf. BORIS I. BITTKER & LAWRENCE LOKKEN, FEDERAL TAXATION OF INCOME, ESTATES, AND GIFTS 57-97, [paragraph] 57.8.2 n.4 (3d ed. 2000); David Schizer, Debt Exchangeable for Common Stock: Electivity and the Tax Treatment of Issuers and Holders, 1 DERIVATIVES REP. 10 (Mar. 2000); David M. Schizer, Hedging Under Section 1259, 80 TAX NOTES 345 (July 20, 1998); David Weisbach, Should a Short Sale Against the Box be a Realization Event?, 50 NAT'L TAX J. 495 (1997); Robert Willens, TRA '97 Closes Loopholes for Tax Deferral and Conversion of Gains Into Dividend Income, 87 J. TAX'N 197 (1997).

(95) See supra text accompanying notes 22-23.

(96) Compare I.R.C. [section] 1(h)(1)(C) (applying a 15% top rate to net capital gain), with I.R.C. [section] 1(i)(2) (applying a 35% top rate to ordinary income).

(97) But cf. I.R.C. [section] 1211(b) (limiting the current deductibility of capital losses to capital gains plus $3000).

(98) The interest is potentially deductible as investment interest because it is "paid or accrued on indebtedness properly allocable to property held for investment." I.R.C. [section] 163(d)(3)(A). Investment interest is deductible, subject to two caveats worthy of note. First, the deduction cannot exceed an individual taxpayer's "net investment income." See I.R.C. [section] 163(d)(1). We can comfortably assume that the taxpayer has sufficient investment income to allow for a full interest deduction. After all, well-to-do taxpayers are the ones most likely to buy equity options. Second, and more significantly, a cash-basis taxpayer must pay the interest in cash before he can take a deduction. See Davison v. Commissioner, 107 T.C. 35 (1996), aff'd 141 F.3d 403 (2d Cir. 1998). We could assume that our investor pays this interest out of his own funds. This assumption would give our synthetic option a different cash flow from the true option, which requires no interim payments. Or, our investor might switch his method of accounting to the accrual method. Again, however, the true option does not mandate this switch. Finally, and perhaps most consistent with the simulation, we can assume that the investor borrows money from a new lender at the end of each year to pay the year's interest expense.

(99) C.f., e.g., Michael S. Knoll, An Accretion Corporate Income Tax, 49 STAN. L. REV. 1 (1996); John Lee, President Clinton's Capital Gains Proposals, 59 TAX NOTES 1399 (June 7, 1993) (proposing mandatory passthrough of income or loss as to private C corporations and mark-to-market accrual taxation of shareholders of public C corporations).

(100) See supra notes 70-72 and accompanying text.

(101) Hasen, supra note 19, at 438-39.

(102) See Treas. Reg. [section] 1.1012-1(c) (as amended in 1996).

(103) I actually did prepare a simulation that models strategic trading, subject to the wash-sale rules, and the resulting programming code is rather lengthy and abstruse.

(104) Cf. Simon D. Ulcickas, Note, Internal Revenue Code Section 1259: A Legitimate Foundation for Taxing Short Sales Against the Box or a Mere Makeover?, 39 WM. & MARY L. REV. 1355, 1368 n.86 (1998) (citing DEPARTMENT OF THE TREASURY, 1996 GENERAL EXPLANATIONS OF THE ADMINISTRATION'S REVENUE PROPOSALS 70-71 (1996)).

(105) Perhaps the best candidate is the "first-in-first-out" method allowed by Treas. Reg. [section] 1.1012-1(c)(1). Treas. Reg. [section] 1.1012-1(c)(1) (as amended in 1996).

(106) See Ulcickas, supra note 104.

(107) See I.R.C. [section] 1091.

(108) See David Schizer, Scrubbing the Wash Sale Rules, 82 TAXES 67, 67 (Mar. 2004) ("[Without limitations on losses] the 'timing option' inherent in the realization rule would allow taxpayers to defer gains (thereby reducing the tax's present value) while accelerating losses (thereby preserving the deduction's present value).").

(109) Remember that the option covers 10,000 shares, so a synthetic option covering one share would have realized gain of about 28cents.

(110) Professor Hasen's "quasi-mark-to-market" system would yield short-term capital gain of $33,100. This is far greater than the $2779 presented in my example or even the $13,875 that a mark-to-market system would produce.

(111) Cf. Campbell Harvey, Futures and Options Glossary, http://www.duke.edu/~charvey/Classes/glossary/g_n.htm (last visited May 9, 2007). Professor Harvey's glossary of finance terms defines naked strategies as:
 An unhedged strategy making exclusive use of one of the following:
 long call strategy (buying call options), short call strategy (selling
 or writing call options), long put strategy (buying put options), and
 short put strategy (selling or writing put options). By themselves,
 these positions are called naked strategies because they do not
 involve an offsetting or risk-reducing position in another option or
 the underlying security.


Id.

(112) See id. at http://www.duke.edu/~charvey/Classes/glossary/g_c.htm, http://www.duke.edu/~charvey/Classes/glossary/g_p.htm.

(113) Dealers would be subject to mark-to-market taxation under I.R.C. [section] 475. I.R.C. [section] 475. See generally David M. Schizer, Realization as Subsidy, 73 N.Y.U. L. REV. 1549, 1586-87 (1998); Dana L. Trier, Rethinking the Taxation of Nonqualified Deferred Compensation: Code Sec. 409A, the Hedging Regulations and Code Sec. 1032., 84 TAXES 141, 168 (2006).

(114) See HULL, supra note 8, at 462. Some commentators are uncomfortable with path dependent tax results. See Herwig J. Schlunk, Little Boxes: Can Optimal Commodity Tax Methodology Save the Debt-Equity Distinction?, 80 TEX. L. REV. 859, 883-85 (2002).

(115) See BRANDIMARTE, supra note 30, at 316. Professor Strnad describes Brownian motion as follows:
 Geometric Brownian motion means that the rate of return is a constant
 plus a Brownian motion term. A constant rate of return would imply a
 smooth, geometrically increasing asset value path.... The Brownian
 motion term adds a rapidly fluctuating deviation with mean zero to the
 constant rate of return. Brownian motion is named after Robert Brown
 who observed and described the jerky and random motion of pollen
 particles suspended in liquid in 1827-28. Geometric Brownian motion is
 the usual assumption in theoretical finance models of common stock
 prices. Recent empirical evidence casts some doubt on the accuracy of
 the geometric Brownian motion assumption, but it is hard to come up
 with an obvious alternative candidate for theoretical work.


Jeff Strnad, Periodicity and Accretion Taxation: Norms and Implementation, 99 YALE L.J. 1817, 1870 n.149 (1990) (citations omitted).

(116) In MATLAB, blsprice(50, 50, 0.05, 5, 0.25) returns a value of $16.2520 for the call.

(117) In MATLAB, blsprice(50, 50, 0.05, 5, 0.25) returns a value of $5.1920 for the put.

(118) This is the call delta derived in MATLAB from blsdelta(50, 50, 0.05, 5, 0.25).

(119) This is the put delta derived in MATLAB from blsdelta(50, 50, 0.05, 5, 0.25).

(120) A short call would be initiated in inverse fashion. Sell short [[DELTA].sub.c] = 0.7663 shares of XYZ stock for proceeds of $38.32. Place c = $16.25 "in pocket" (representing the premium received by a call writer). The remainder, $22.07, is invested in a debt instrument.

(121) A short put would be initiated in inverse fashion. Buy -[[DELTA].sub.p] = 0.2337 of XYZ stock for $11.68. Borrow funds to pay for this, plus an additional p = $5.19 (representing the premium received by a put writer) for a total indebtedness of $16.87.

(122) Readers with a little finance should resist the temptation of determining expected option payoffs using expected stock values. Because of risk-free pricing, the expected return on the stock is actually irrelevant. All that matters is the volatility of the stock. See HULL, supra note 8, at 245.

(123) $16.25*e^(0.05*5).

(124) $5.19*e^(0.05*5).

(125) For a discussion of the weighted-average cost method used to calculate gain or loss, see supra notes 103-108 and accompanying text.

(126) For a discussion of the weighted-average cost method used to calculate gain or loss, see supra notes 103-108 and accompanying text.

(127) The sensitivity of delta to stock prices is given by gamma, which is always positive. See HULL, supra note 8, at 314.

(128) A synthetic long put will produce interest income over its life. When stock prices rise, delta rises as well. Since delta for a put is negative, the rising delta (e.g., from -0.4 to -0.3) means that short sales must be closed. As short sales do poorly in a rising market, this means that losses are realized. In contrast, a falling market in the stock means that delta falls as well (e.g., from -0.4 to -0.5). This means that more short sales need to be executed, triggering no gain or loss. As short sales do well in a falling market, this means that gains are deferred. In summary, a synthetic long put produces interest income, defers gains, and triggers losses. A synthetic short put is the opposite (interest expense, realized gain, and deferred losses).

(129) See supra Part V.C.

(130) See id.

(131) Put: Sell short -[[DELTA].sub.p] = 0.2337 of XYZ stock for proceeds of $11.68. Invest this amount, plus an additional p = $5.19 in a debt instrument (for a total investment of $16.87).

(132) $22.07*((e^(.05*5))-1).

(133) See supra note 98 and accompanying text.

(134) Cf. David M. Schizer, Sticks and Snakes: Derivatives and Curtailing Aggressive Tax Planning, 73 S. CAL. L. REV. 1339, 1388-89 (2000) (noting that reforms can possibly open the door to new tax planning opportunities).

(135) For example, suppose that a taxpayer buys and writes the same call option, creating a perfectly neutral position. The long call produces an interest deduction, and the short call produces interest income. Suppose that the stock price falls. The fall pushes delta down, which decreases the amount of debt implicit in either side of the contract. The taxpayer might like to sell the long call, which is now producing an interest expense greater than that implied by the current synthetic option. Contemporaneously, the taxpayer might repurchase another, identical long call. The wash-sale and straddle rules would independently deny a loss deduction here. But some similar mechanism would be needed to prevent the taxpayer from refreshing the imputed interest on only one side of the straddle.

(136) BITTKER & LOKKEN, supra note 94, [paragraph] 57.8.3 ("Section 1258 affects only characterization. Timing is not affected. Interest income is generally recognized as it accrues, but gains and losses on the positions making up a conversion transaction are recognized only when realized by a sale, exchange, or termination of those positions."); DAVID M. SCHIZER, FINANCIAL INSTRUMENTS: SPECIAL RULES [paragraph] II.A. (2005) ("Notably, though, this measure [[section] 1258] does not accelerate the timing of this income.").

(137) See supra text accompanying note 59 (assuming price of $30, strike price of $33, risk-free rate of 10%, expiration in one year, volatility of 30%, and no dividends). In MATLAB, blsprice(30, 33, 0.10, 1, 0.30) returns a price of 3.6393 for the call. In MATLAB, blsdelta(30, 33, 0.10, 1, 0.30) returns a delta of 0.5658 for the call.

(138) See BITTKER & LOKKEN, supra note 94, [paragraph] 57.3.1.

(139) See id. Bittker and Lokken state:
 The earliest expression of this rule is probably found in Virginia
 Iron Coal & Coke Co. v. CIR, 99 F2d [sic] 919, 921 (4th Cir. 1938),
 where the court noted that when the taxpayer received premiums as
 writer of a call option it was impossible to determine whether they
 were taxable or not. In the event the sale should be completed the
 payments became return of capital, taxable only if a profit should be
 realized on the sale. Should the option be surrendered it would then
 become certain, for the first time, that the payments constituted
 payments in the year in which they were made.


Id.

(140) See supra Part III.C.4.

(141) See supra note 137.

(142) See I.R.C. [section] 1259.

(143) But cf. Johnson, supra note 81 (advocating a system for taxing covered calls based on the premium received).

(144) Maya has sold 5658 shares for $30. Her gain is the entire amount realized as her adjusted basis is zero.

(145) In MATLAB, blsprice(30, 33, 0.10, 1, 0.30) returns a put price of 3.4989, and blsdelta(30, 33, 0.10, 1, 0.30) returns a put delta of -0.4342.

(146) This approach is at odds with "the popular conception that 'paper gains' do not constitute income." Deborah Schenk, A Positive Account of the Realization Rule, 57 TAX L. REV. 355, 355-56 (2004).

(147) See Schizer, supra note 42, at 1364-67.

(148) See supra note 137.

(149) See supra text accompanying notes 96-97.

(150) I.R.C. [section][section] 1259(c)(3)(A), 246(c)(4).

(151) The constructive sales for any taxable year would be based on the lesser of (1) the highest outstanding short sale for the taxable year or (2) the highest outstanding short sale for the period running from thirty to ninety days after the close of the taxable year. If prong (2) applied, we would have the additional problem of matching the short sales with actual stock prices.

(152) Suppose that delta and the stock follow a path of (1) $30, -40%; (2) $30, -50%, (3) $30, -40%, (4) $40, -50%. On day two, we have a constructive sale of 0.10 shares at $30 because delta declined by 10 basis points. Technically, we should have a new constructive sale on day four as well, because delta has again declined by ten basis points. Ignoring this constructive sale greatly simplifies the mechanics of the model. I.R.C. [section] 1259(c)(1).

(153) See supra Part V.

(154) I.R.C. [section] 1259(c).

(155) See supra Part VI.B.

(156) If the investor died owning the stock, then the basis step up of section 1014 would effectively eliminate any gain or loss that accumulated before the investor's death. I.R.C. [section] 1014(a).

(157) Since the transaction is not described in the statute, it might not actually be a constructive sale as the IRS has not yet issued regulations under section 1259 (despite the nine-year-old mandate to do so). Cf. Phillip Gall, Phantom Tax Regulations: The Curse of Spurned Delegations, 56 TAX LAW. 413 (2003).

(158) Because of the risk-free nature of Black-Scholes pricing, the model assumes that stock grows at the risk-free rate. Thus, we should expect the stock to be worth 50*e^(5%*5)=64.2013 after five years. The result given by the simulation, 63.9143, is a close approximation.

(159) The investor can adjust his basis in the stock in order to reflect constructive sales. See I.R.C. [section] 1259(a)(2)(A).

(160) Also, the entire holdings of stock are liquidated at the end of the five years.

(161) Dean Schizer is also concerned that subjecting equity collars to a wider application of the constructive sale rules would lead to "lock in" and push taxpayers into less effective hedges. See Schizer, supra note 42, at 1365. The lock-in rationale is usually invoked to support a capital-gains preference and is controversial among commentators. See generally BITTKER & LOKKEN, supra note 94, [paragraph] 3.5.7. This critique has lost any vitality that it may have once had after the capital gains rate was cut to 15% in 2001.

(162) Schizer, supra note 42, at 1366.

(163) Id. at 1365-66.

(164) Readers with calculus will easily see why. Delta is the first partial derivative of the instrument price taken with respect to the stock price. So, if an equity collar is the sum of a long put and short call, then the delta for the equity collar is the sum of the delta of the long put and the short call.

(165) See Schizer, supra note 42, at 1345-46.

(166) See id. at 1350 n.129.

(167) See HULL, supra note 8, at 250.

(168) The sought after volatility is on the stock. Theoretically, one could observe the prices of other options to derive the volatility of the stock. One problem is that most options in the United States are sold over-the-counter. Publicly traded options are of relatively short term, typically under two years. If publicly traded options were three to five years long, investors could combine them to create equity collars without the need to involve the investment bankers. The volatility observed on a short-term option may not carry over to a long-term option. This is because of the "volatility smile"--the phenomenon that implied volatility depends on the strike price and time to exercise of an instrument. Id. at 330, 334-37.

(169) Recall that this collar is created by buying a put with a strike price of $90 and writing a call with a strike price of $110. The Black-Scholes price for the put is given in MATLAB by blsprice(100, 90, 0.05, 3, 0.1505), which returns $1.98. The price for the collar is given by blsprice(100, 110, 0.05, 3, 0.1505), which returns $12.98. As we can see, the volatility of 15.05% is consistent with the $11 received for entering this collar. We can use this volatility to find the delta of the collar, which is the delta of the put minus the delta of the call. Recall that we subtract the call delta because the investor is writing, not buying, the call. The delta of the put is given by blsdelta(100, 90, 0.05, 3, 0.1505), which returns -0.1335. The delta of the call is given by blsdelta(100, 110, 0.05, 3, 0.1505), which returns 0.6331. Thus, the delta for the collar is -0.7666.

(170) 1% of $100 for each of the three years.

(171) The variable prepaid forward contracts (VPFC) has the same cash flows as the modified equity collar combined with a loan equal to the value of the shares at the time the VPFC is executed. See generally David F. Levy, Towards Equal Tax Treatment of Economically Equivalent Financial Instruments: Proposals for Taxing Prepaid Forward Contracts, Equity Swaps, and Certain Contingent Debt Instruments, 3 FLA. TAX REV. 471 (1997).

(172) 10,000 * 100/125.
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