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

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.