Chapter

4 Innovative Financial Instruments

Author(s):
Howell Zee
Published Date:
April 2004
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Author(s)
Howell H. Zee

The term “innovative financial instruments” is generally taken to mean financial derivatives. A financial derivative (henceforth, simply derivative) is a financial instrument whose price depends on, or is derived from, the price of another asset. The asset underlying a derivative could be a commodity (for example, wheat), a financial asset (for example, a stock), or another derivative. There are two basic building blocks that can be used to construct derivatives: futures contracts traded in exchanges (or forward contracts traded in the over-the-counter market) and options contacts. Derivatives can be used for either hedging or speculative purposes (or both). Some elementary concepts related to derivatives are reviewed in the appendix to this paper.1

It has generally been recognized that the rapid development and pervasive use by both businesses and individuals of derivatives pose significant challenges to the tax system. These challenges stem from the fact that, under a traditional income tax system, income could be treated differentially depending on (1) the instrument from which the income is generated (for example, debt versus equity), (2) the timing rule by which income is recognized for tax purposes (for example, accrual versus realization), and (3) the way the income is characterized (for example, ordinary income versus capital gains). To enforce such differential treatments, the tax authorities must be able to draw clear dividing lines between them. The fundamental problem with derivatives, from the perspective of tax policy, lies precisely in their ability to blur the requisite dividing lines: the more exotic a derivative, the less transparent its constituent components. If the tax system lacks clear rules for handling derivatives, a significant erosion of the tax base could result, as taxpayers engage in tax arbitrage to change instrument classification, recharacterize the nature of income, and alter the timing of income recognition for no other reason than to obtain tax benefits—that is, without substantively modifying their underlying economic positions.

Apart from issues related to instrument classification, timing of income recognition, and characterization of income, derivatives also pose difficulties for the concept of ownership as it is usually applied under tax laws and raise thorny problems in an international context, since the tax treatments of derivatives are not unified across countries, and cross-border financial flows are governed by different tax rules, depending on treaty stipulations regarding the type and character of such flows.

Nature of the Problem

An income tax system traditionally classifies financial instruments as either debt or equity and typically accords each a different tax treatment, for example: (1) the return to debt (interest) is tax deductible as a business expense to the issuer, but the return to equity (dividends) is not; (2) interest is taxed on an accrual basis, while dividends are taxed on a realization basis; and (3) unlike interest, dividends are taxed as ordinary income only when realized; they are taxed (often preferentially) as capital gains when capitalized in stock prices. Furthermore, interest and dividends paid to nonresidents may be taxed at different domestic rates and be subject to different withholding rates.

Given the above asymmetrical tax treatments between debt and equity, taxpayers could realize substantial tax benefits by transforming the instruments from one into the other—or into some new instruments—at will to suit their individual circumstances. Derivatives provide them with the means to do so.

Creating Instruments Using Derivatives

The ability of derivatives to blur the demarcations between instruments is best appreciated from the put-call parity relationship derived in the appendix. For simplicity, this relationship is reproduced here for a nondividend-paying stock (S) with the subscript on S removed:

which shows that there is a fundamental relationship among the four instruments: two derivatives (a call option (C) and a put option (P)), a stock, and a bond (B). Any one of the four instruments could be synthesized by a suitable combination of the other three. For example, a bond, in conjunction with a long call and a short put, would produce a synthetic stock. Furthermore, since a long forward contract (F) is no more than a combined right and obligation to purchase an underlying asset at a certain price at some future date, it can be replicated by a long call (representing the right to purchase the asset) combined with a short put (representing the obligation to purchase the asset) at the same price and date. Hence,

Together with equation (1), it implies that a synthetic stock can also be created by combining a bond with a long forward contract:

Likewise, a synthetic bond is created by combining a stock with a short forward contract. Though derived partly from the put-call parity relationship (1), one can, in fact, arrive at equation (3) directly from intuitive reasoning: holding a bond with the intention of using its proceeds to purchase a stock at the delivery date and price under the forward contract is equivalent to holding the stock.

Instruments created by derivatives could be grouped under two broad categories: hybrids and synthetics. Each category, either by itself or in combination with the other category, would provide taxpayers with virtually limitless tax arbitrage opportunities under traditional income tax systems.

Hybrids

A hybrid is a new instrument created by combining debt, equity, and, possibly, other derivatives. The new instrument may not be recognized by existing tax laws and, hence, its tax treatment could be uncertain. Many hybrids are available, but contingent debt (or structured notes) and swaps are probably the most popular.

Contingent Debt

Consider two portfolios: (1) a traditional bond with a principal B that matures in n years with yearly coupon payments at the rate r and a long forward contract on a stock S in the nth year for a delivery price X; and (2) an instrument requiring the same initial investment B and with the same coupon rate r and a maturity of n years, like the bond in the first portfolio, but whose payoff in the nth year is contingent on the performance of the stock—specifically, its final payoff is [B + (S - X)].

While the two portfolios would return the same stream of cash flows, their tax treatments might be very different. The first portfolio’s tax treatment is clear: the coupons from the bond would be taxed each year as ordinary income, and the capital gain/loss from the forward contract would be taxed upon the contract’s settlement. In contrast, the tax treatment of the contingent debt in the second portfolio is unclear, since the debt’s yield each period cannot be calculated until its final payoff is ascertained. If the tax point is delayed until the entire investment is closed, there would be a substantial deferral of tax liabilities, compared with the first portfolio, even though, for each of the periods before the last, the contingent debt resembles the traditional bond in all substantive aspects. Moreover, the character of the final payoff is also uncertain as to whether the difference, if any, between B and (S - X) represents capital gain/loss or ordinary income.

Swaps

Swaps are popular largely because of their inherent flexibility: they can be used to exchange (swap) almost any profile of cash flows for a different one to suit the needs of the swappers (see the appendix for a brief description of some standard swaps). Since a forward contract is equivalent to an agreed exchange of cash flows occurring at a specific date, a swap is also a convenient instrument for packaging a series of forward contracts.

The correct tax treatment of swaps is not straightforward, as the nature of the swapped profiles of cash flows could be characterized in different ways. An interest rate swap, for example, could be viewed simply as the exchange of a fixed-rate bond for a floating-rate bond, and the cash flows received by both parties of the swap could be viewed as interest payments and taxed accordingly. However, if the swap is viewed as a series of forward contracts, one party could be regarded as having bought from the other party, for the same and identical price in each contract, a distinct asset whose price is a priori uncertain. In this case, the gain and loss from each contract would have to be ascertained separately for tax purposes.

Swaps could also give rise to difficulties in determining the owners of the assets being swapped. If individual 1, who owns stock S1 and individual 2, who owns stock S2, agree to swap the dividends from, and price changes of, their stocks with each other over a fixed period of time, then, in effect, the two individuals would have swapped the economic consequences of stock ownership without changing legal ownership. Certain tax rules, however, are predicated on ownership (for example, corporate restructuring, withholding taxes). Thus, the separation of legal from economic ownership is a problematic issue for taxation.

Synthetics

A synthetic instrument is one constructed from a combination of different instruments (which could themselves include hybrids and other synthetics) in such a way as to replicate the cash flows of another instrument. As described earlier, options and forward contracts can be used to create synthetics of varying degrees of complexity.

Tax rules that are applied differentially on the basis of how instruments are classified—as is the case with traditional income tax systems—will clearly have problems dealing with synthetics, since, by implication, taxpayers would have the ability to transform one instrument into another to maximize tax benefits. For example, an individual owning a substantially appreciated stock who is interested in selling the stock for a bond could defer the realization of the capital gain by creating a synthetic bond with a short forward contract (see equation (3)). The contract’s delivery price, which would be equal to the stock’s current appreciated value compounded over the life of the contract, once set, would not change regardless of the future movements in the stock price; thus, the price of the contract takes on the character of the principal of a bond. In this way, investors are able to achieve their objectives and defer tax on both the capital gain (from selling a stock) and the interest income (from acquiring a real bond) until the forward contract matures.

Synthetics constructed from multiple components also allow taxpayers to selectively realize losses of some components without realizing offsetting gains in other components at any given point in time. Continuing with the above example, suppose the stock appreciated further after the short forward contract had been struck by the individual, in which case the value of the contract would have gone down.2 Instead of waiting for the contract to mature, the individual could now choose to close out the contract and hold on to the stock.3 In this way, a loss is realized (on the forward transaction) without realizing the accrued gain (on the stock) at the same time, which would allow the individual to reap substantial tax benefits stemming from the deferred recognition of the gain.

Addressing the Problem

The above discussions illustrate that the root cause of the problem of taxing derivatives lies primarily in the traditional income tax system’s asymmetric treatments of equivalent cash flows, both at a given point in time and over time. Investors can use derivatives to exploit such asymmetric treatments and reap artificial tax benefits. This suggests that, to be effective, any approach to tackling the problem has to address both its temporal and intertemporal dimensions. While many different approaches have been advanced, none provides a perfect solution. The arguments for and against a few of the more well-known ones are briefly assessed below. Practices in selected countries in the Organization for Economic Cooperation and Development (OECD) are summarized in Table 4.1.

Table 4.1.Taxation of Selected Derivatives in Selected OECD Countries
AustraliaCanadaFranceGermanyItalyJapanNew ZealandUnited KingdomUnited States
Interest rate swaps
General tax principles apply?YesYesYesYesYesYesNoYesYes
Flows of payment taxed?Yes;Yes;Yes;Yes;Yes;Yes;Yes;Yes;Yes;
How?ordinaryordinaryordinaryordinaryordinaryordinaryordinaryordinaryordinary
Accrual tax treatment on irregular payments?No1YesYesNoNoYesYes2Yes
Withholding tax on payment to nonresidents?No1No1NoNoNoNoNoYes3No
Payment classification applying tax treatyOrdinary and otherOrdinary and otherOrdinary and otherOrdinaryOrdinaryOtherOrdinaryOrdinary and otherOrdinary and other
Financial futures
General tax principles apply?YesYesYesYesYesNoYes4
Gains/losses taxed?Yes; capital and ordinaryYes; capital5Yes; ordinaryYes; ordinaryYes; ordinaryYes; ordinaryYes; ordinaryYes; capital6Yes; capital7
How?
Mark-to-market applied?NoYes7YesNoNoNoYes7Yes8Yes
Withholding tax on payment to nonresidents?NoNoNoNoNoNoNoNoNo
Payment classification applying tax treatyOrdinary and otherOrdinary and capitalOrdinary and otherOrdinaryOrdinaryOrdinary and otherOrdinaryOrdinary, capital, and otherOrdinary and capital
Call options on stocks
General tax principles apply?YesYesNoYesYesYes
Holders8
Premium deductible? When?Yes; at expiryYes; at expiryYes; at exerciseYes; at expiryYes; at expiryYes; at expiryYes; on acquisitionYes; at expiry
Premium part of stock acquisition cost?YesYesNoYesYesYesNoYes
Gains/losses at disposal taxed?YesYesYesYesYesNoYesYes
Mark-to-market applied?NoNoYesNoNoNoNo
Withholding tax on premium to nonresidents?NoNoNoNoNoNoNo
Premium classification when applying tax treatyOrdinary and capitalOrdinary and capitalOrdinaryOtherOrdinaryOrdinary, capital, and otherOrdinary and capital
Writers8
Premium taxable? When?Yes; at issue9Yes; at issue9Yes; at expiryYes; at issue9Yes; at issueYes; at issue9Yes; at issue9Yes; at expiry
Premium part of stock delivery price?Yes9Yes9YesYes9Yes9Yes9Yes
Gains/losses on repurchase taxed?YesYesYesYesYesNoYesYes
Mark-to-market applied?NoNoYesNoNoNoNoNo
Source: OECD, Taxation of New Financial Instruments (Paris, 1994).Note: … denotes not available.

Unless the swap is a disguised loan.

Swap payments between nontraders and nonbanks are deductible on a realization basis.

Unless payments are made to nonfinancial institutions.

Futures traded in qualified exchanges, as well as certain other types of futures, are governed by specific tax legislation.

Treated as ordinary income if realized by traders.

Treated as ordinary income for hedging transactions.

Applies to traders.

Not engaged in the business of trading options.

The taxation of the premium at issue is reversed if the option is subsequently exercised.

Source: OECD, Taxation of New Financial Instruments (Paris, 1994).Note: … denotes not available.

Unless the swap is a disguised loan.

Swap payments between nontraders and nonbanks are deductible on a realization basis.

Unless payments are made to nonfinancial institutions.

Futures traded in qualified exchanges, as well as certain other types of futures, are governed by specific tax legislation.

Treated as ordinary income if realized by traders.

Treated as ordinary income for hedging transactions.

Applies to traders.

Not engaged in the business of trading options.

The taxation of the premium at issue is reversed if the option is subsequently exercised.

Marking to Market

Marking to market for tax purposes means that, at the end of each tax period, the gain or loss on an instrument—calculated as the difference between its market value and cost basis—would be recognized by the holder as if it were sold (and repurchased). The cost basis would be adjusted upward (downward) for any cash payment paid (received) during the period. If implemented comprehensively, this approach overcomes the temporal asymmetry by applying the same rate of tax to market-determined gains and losses and overcomes the intertemporal asymmetry by eliminating the timing consideration for income recognition.

Marking broadly traded instruments to market is straightforward and, in fact, is already widely practiced (futures, for example). However, many derivatives are not traded or are traded thinly, so that a reliable market price cannot be established. For these “off-market” derivatives, marking to market would present significant valuation difficulties, even if their underlying assets are themselves broadly traded. If marking to market is applied only to some instruments but not to others because of the valuation problem, then tax arbitrage opportunities would remain and taxpayers could reengineer their portfolios so that marking to market would, in effect, be applied only at their discretion.

Another difficulty with marking to market is that taxing unrealized gains imposes a cash-flow burden on taxpayers that could become real if the gains are not eventually realized. For example, suppose an instrument is held not for speculative purposes but solely because it is a perfect hedge for an underlying asset. Then, by assumption, no net gain would materialize when the position on the underlying asset is closed out. Yet tax liabilities could be incurred if the hedging instrument is marked to market but the hedged asset is not.

Finally, marking to market is incompatible with many features of a traditional income tax system, such as the differentiation of debt from equity and of ordinary income from capital gains. To the extent that such features are maintained for other tax policy reasons, ad hoc rules would be required to resolve this incompatibility. For example, in the United States, all gains and losses from trading futures are considered capital in nature. However, for noncorporate taxpayers, they are considered 60 percent long term and 40 percent short term (the so-called “60/40 rule,”—the application of which does not depend on the length of the period the traded futures are actually held), the latter being taxed in the same way as ordinary income. For hedging transactions, gains and losses are recognized only when the associated positions on the hedged assets are closed out.

Retrospective Taxation

An alternative approach to marking to market, first proposed by Auerbach4 and later generalized by Bradford5, involves taxing gains and losses retrospectively. This approach is equally effective in neutralizing both the temporal and intertemporal dimensions of tax arbitrage engendered by the use of derivatives. The basic idea is best illustrated with a simple example. Suppose an asset bought exactly two years ago is sold today at $121, and the risk-free interest rate during this period was 10 percent per year. Under the Auerbach proposal, the asset’s acquisition price would be presumed to be $100, and the tax liability in the two years would be the applicable tax rate (say, 30 percent) on the gains calculated on the basis of the risk-free interest rate, that is, $3 in the first year and $3.3 in the second year. However, since the tax for the first year is not paid until the second year, the taxpayer would owe $0.3 in interest on the deferred tax payment, making a total gross payment in tax and interest of $6.6. If the interest payment is tax deductible, it would generate a tax deduction of $0.09. The net tax and interest payment would thus be $6.51.6

In essence, the Auerbach proposal taxes only the risk-free rate of return from holding an asset. The actual gain or loss net of the risk-free rate—which represents the risk premium—is left untaxed and is, therefore, inconsequential for tax purposes. Hence, there is no incentive for taxpayers to engage in temporal or intertemporal tax arbitrage. Implementing the Auerbach proposal requires neither information on the actual acquisition cost of the asset nor adjustments to the asset’s cost basis (as would be required under the mark-to-market approach). Of course, under this approach, a tax liability may arise even if a capital loss is incurred.

The conceptual validity of the proposal is based on the theoretical consideration that, since an income tax reduces both net-of-tax gains and net-of-tax losses (assuming losses are tax deductible), the risk-adjusted tax liability on the risk premium should be zero. The Bradford proposal generalizes the Auerbach proposal by taxing the risk premium at an artificial rate7 and becomes identical to the latter when this rate is chosen to be zero. It thus captures some income in the tax net that is left out by the Auerbach proposal but provides no opportunity—like the Auerbach proposal—for tax arbitrage.

Bifurcation

The bifurcation approach to taxing derivatives is based on the straightforward logic that, since a derivative is constructed from different components, these components should be taxed separately. Under this approach, for example, the second portfolio discussed in the contingent debt section earlier would be taxed just like the first portfolio, a fixed-interest debt and a long forward contract on the stock. In theory, bifurcation would remove the incentive to enter into derivative positions for tax reasons. Moreover, it would preserve the different tax treatments accorded to certain types of assets under traditional income tax systems.

The most serious argument against bifurcation is that there is no unique way of bifurcating a derivative, since it can be constructed from a multitude of different combinations of underlying components. Returning again to the contingent debt example above, the fixed-debt and long-forward contract components could themselves be further bifurcated into different instruments, potentially producing different tax consequences.8 Similarly, a swap between streams of fixed and variable cash flows could also result in different tax consequences, depending on whether the swap is bifurcated into fixed- and floating-rate bonds or into a series of forward contracts. Bifurcation has also been criticized on the grounds that the very act of constructing a derivative creates additional economic value (arising from synergy) that its components lack by themselves. If valid, this criticism would render bifurcation conceptually suspect.

Integration

In contrast with bifurcation, the integration approach links a derivative to other instruments with which it has some associative attributes and taxes them as a combined unit. Obviously, integration could differ in degree. Some integration rules are partial, in that the tax system identifies only specified transactions for combined taxation. A “straddle”—a combination of a long call with a long put on the same underlying asset at the same strike price and expiration date—is a classic example. While gains and losses on the call and put are always offsetting, taxpayers would benefit if they were allowed to electively realize losses. An integration rule in this case could deny such elective realization of losses until both the call and the put positions are closed out. An integration rule that specifies that gains and losses on a hedging transaction are deferred until the hedge instrument is closed out is also partial, because a fuller integration rule could, instead, defer all gains and losses on the hedge instrument until the underlying hedged asset is closed out.

Just like bifurcation, however, where the extent to which a derivative should be bifurcated for tax purposes is unclear, integration can be criticized in that the degree to which transactions should be integrated is ambiguous. Moreover, regardless of degree, integration requires the identification of linked transactions, which, in many cases, is administratively difficult to achieve, since the intent of the taxpayer would need to be ascertained. For example, the linked transactions, if any, to many hybrids (such as swaps) are almost impossible to verify.

Policy Implications

It is highly implausible that relying on a single approach would be adequate to tackle the problem of taxing derivatives. The merits and limitations of each approach suggest that the most effective policy would be to choose the best approach under different circumstances. Marking to market can easily be used on some broadly traded instruments and on certain taxpayers (such as financial intermediaries) whose accounting systems are already based on the mark-to-market principle, while bifurcation and integration can coexist and be applied selectively, depending on the nature of the instruments and transactions. Most OECD countries follow this practice.

Retrospective taxation as proposed by Auerbach and Bradford is a novel approach; it is conceptually elegant but administratively and politically untested. As noted in an earlier paper in this volume, Italy is the one country that has attempted such an approach, only to abandon it.

Appendix. Elementary Concepts Relating to Derivatives

A futures contract is an agreement to buy or sell an asset at a certain time in the future for a certain price (the delivery price). Futures are traded in exchanges in standardized terms as regards the contract size (which varies depending on the underlying asset); delivery arrangements (time, location, and method); and the convention of price quotation (which varies depending on the minimum price movement that can occur in trading). To reduce volatility, exchanges often impose limits on daily futures price movements and on the maximum number of contracts a speculator may hold with respect to a given type of futures and require investors to maintain specified margins to avoid credit risks. Futures are marked to market—with the consequent gains and losses settled in the investor’s margin account—on a daily basis. A margin call is triggered whenever the balance in the margin account falls below a specified level. An investor who cannot meet a margin call will have his position closed out (that is, sold) by the broker.

A forward contract is similar to a futures contract but is traded in the over-the-counter market in terms that best serve the purposes of the contractual parties (typically, two financial institutions or a financial institution and its client). The trading of forward contracts in foreign currencies is a common practice with banks.

There are important differences between futures and forward contracts besides their trading venues and contractual terms. Most futures are closed out by investors before the specified delivery date (closing out involves entering into a trade that is the opposite of the original one). In contrast, forward contracts are usually settled at the delivery date. Furthermore, while gains and losses in the trading of futures are recognized on a daily basis, those of forward contracts are recognized at their settlement dates.

A swap is an agreement between two parties (arranged with or without a financial intermediary) to exchange cash flows over a certain period of time. An interest rate swap involves one party agreeing to pay a fixed rate of interest on a notional principal to another party in return for a floating rate of interest on the same principal (the principal is usually not exchanged). A currency swap involves exchanging principal and interest payments in one currency for principal and interest in another currency (the principal amounts are usually specified and exchanged at the beginning and end of the specified swap period). An equity swap involves one party agreeing to pay the other the return on an equity index on a notional principal in return for a fixed or floating rate of interest on the same principal. Since swaps are used to transform one profile of cash flows into another, they can be viewed essentially as a series of forward contracts.

There are two basic types of options contracts. A call option gives the holder (the purchaser) of the option the right, but not the obligation, to buy the asset specified in the contract by a certain date (the expiration date) for a certain price (the strike price). If a call option is exercised by its holder (as when, for example, the spot price of the underlying asset exceeds its strike price by more than the transaction costs of exercising the option—typically involving paying commissions on both exercising the option and purchasing the underlying asset itself)—the writer (the seller) of the option has the obligation to make the asset available at the strike price.

In contrast, a put option gives the holder of the option the right, but not the obligation, to sell the asset specified in the contract by the specified expiration date for the specified strike price. If a put option is exercised by its holder (the incentives to do so would be just the reverse of those associated with exercising a call option), the writer of the option has the obligation to purchase the asset at the strike price.

A European option is one that can be exercised only at its expiration date, while an American option can be exercised any time during its life. Options are traded in exchanges and over-the-counter markets. Purchasers of options must pay the option price in full (“buying on the margin,” or borrowing from the broker, is not allowed). Writers of options are required to maintain specified balances in their margin accounts if such options are naked (options that are not combined with an offsetting position on the underlying assets).

For both futures/forward and call/put options contracts, the purchasing party is said to have entered into a long position, the selling party into a short position. For example, a long put refers to the holder of a put option, and a short call refers to the writer of a call option.

Both futures/forward contracts and options could be used for speculative purposes. A speculator obtains leverage from trading the former through the margin account and from trading the latter through the option price. Each instrument allows a speculator to take a large position with a relatively small up-front commitment of funds. A speculator in a long futures position hopes that the spot price of the underlying asset will rise above the delivery price, while one in a short futures position hopes for the reverse. A speculator in a long call position hopes that the spot price of the underlying asset will rise above the strike price, while one in a long put position hopes for the reverse. In contrast, a speculator in a short call position hopes the spot price of the underlying asset will fall below the strike price, while one in a short put position hopes for the reverse. Hence, the profit to a speculator in a long position is the excess, net of the option price paid, of the spot price over the strike price of the asset in the case of a call option, and of the strike price over the spot price in the case of a put option. In either case, the profit could be realized by exercising (if permitted) or selling the option. The profit to a speculator in a short position is the option price initially received if the option is not exercised (or exercised at a point where the loss from fulfilling the trading obligation of the option is less than the option price received).

Similarly, both futures/forward contracts and options can also be used for hedging purposes. Traders can take long and short positions in each instrument to reduce the risks of adverse movements in certain market variables (interest rates, exchange rates, and commodity prices) in connection with certain transactions. Hedging transactions are typically engaged in by a business whose goal is not to realize financial gains from speculative activities.

The basic hedging principles are straightforward. Using futures/forward contracts, for example, a business can lock in the price of a future purchase by going long, and it can do the same in the case of a future sale by going short. Options can also be used for hedging purposes, but they operate essentially by providing a sort of insurance against large adverse movements in the spot price of the underlying asset, rather than by neutralizing the risk of such price movements in the case of futures/forward contracts. A long put held in conjunction with a long position on the underlying asset ensures that the asset could be sold at least at the strike price of the put option, irrespective of the asset’s spot price. Likewise, a long call held in conjunction with a short position on the underlying asset ensures that the asset could be bought at no more than the strike price of the call option (to cover the underlying short asset position), irrespective of the asset’s spot price. Hedging is a fundamental tool for business risk management in today’s world.

The price of a stock option is affected by a number of variables: the current spot price of the stock (S0), the strike price (X), time to expiration (T), the volatility of the stock price, and the present value of the dividends (D) expected during the life of the option. Stock options can be priced by using the well-known Black-Scholes formula. There is also a simple but important relationship between the price of a put option (P) and the price of a call option (C), known as the put-call parity. It is easier to understand this relationship by first considering a stock that does not pay dividends. Consider two portfolios: (1) one European call option for one share of stock and a zero-coupon bond (B) with a redeemable principal equal to X at time T, and (2) one share of stock and one European put option for one share of the stock. The values of the two portfolios are the same at time T and equal to the higher of ST and X. To see this, note that if the call option in the first portfolio is exercised, the proceeds X from the bond would be used to buy one share of the stock, while if the put option in the second portfolio is exercised, the share of stock would be sold for X. If the two portfolios have the same value at time T, they must also have the same value now, since the options, being European, cannot be exercised during the interim. Thus, we have

C + B = P + S0, or C - P = S0 - B.

For a stock that pays dividends, similar reasoning would lead to the following put-call parity:

C + D + B = P + S0, or C - P = S0 - D - B.

The put-call parity holds only for European options. For American options, it can be shown that (C - P) lies within certain limits.

1Detailed treatments of derivatives at various levels of accessibility are readily available from many sources. A good balance between rigor and accessibility is provided by David Luenberger, Investment Science (New York: Oxford University Press, 1997).
2In the simple case of an existing short forward contract whose underlying asset is a nondividend-paying stock, the value of the contract at any given time is equal to the present value of the excess of its delivery price over the current forward price (the current forward price is the delivery price of a newly negotiated forward contract). Since the current forward price is always equal to the compounded value of the current stock price over the life of the contract, an appreciating stock would thus lead to a rising current forward price and a fall in the value of an existing short forward contract.
3As noted in the appendix, closing out a position involves entering into an opposite trade to the original one. Thus, closing out an existing short forward position requires going long on a new forward contract covering the remaining period in the existing contract.
4Alan J. Auerbach, “Retrospective Capital Gains Taxation,” American Economic Review, Vol. 81 (March 1991), pp. 167-78.
5David F. Bradford, “Fixing Realization Accounting: Symmetry, Consistency, and Correctness in the Taxation of Financial Instruments,” Tax Law Review, Vol. 50 (Summer 1995), pp. 731-802.
6Vickrey and Meade earlier had also proposed the idea of retrospective taxation. Their proposal entails taxing gains accrued over time at realization equal to the present value of applicable tax that could have been collected on those gains (assumed to have been accrued at the rate of interest). This proposal would not completely eliminate tax arbitrage as long as the actual path of gains accrued is not the same as the hypothetical path. See William S. Vickrey, “Averaging of Income for Income Tax Purposes,” Journal of Political Economy, Vol. 47 (June 1939), pp. 379-97; and James E. Meade, The Structure and Reform of Direct Taxation (London and Boston: Allen and Unwin, 1978).
7For this reason, the Bradford proposal would require information on the actual acquisition cost of the asset. It would also require a date—which could also be chosen arbitrarily at the time the asset is acquired—for recognizing the taxable risk premium. Details on the mechanics of the Bradford proposal are omitted for simplicity.
8For example, a fixed-rate debt can be decomposed into a series of zero-coupon bonds; a long forward can be decomposed into a long call combined with a short put.

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