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LN THE UNITED STATES COURT OF FEDERAL CLAIMS
Nos. 06-245T, 06-246T, and 06-247T (Consolidated)

MURFAM FARMS, LLC, 3 By and Through Wendell H. Murphy Jr., 5 a Partner Other Than Tax Matters Partner,
§ PSM FARMS, LLC, § By and Through Stratton K. Murphy, 5 a Partner Other Than Tax Matters Partner, § MURJ!HY PORK PARTNERS, LLC, 5 By and Through Wendell H. Murphy, Jr., a Partner Other Than Tax Matters Partner,
§

Plaintiffs,
V.

5
§ § §

UNITED STATES OF AMERICA, Defendant.

5
§

8

EXPERT REPORT (Murphy) Don M. Chance, Ph.D., CFA May 8,2007 Outline I. Expert Qualifications 1. 1 Summary of Findings III. Basic Principles of Options 111.1. Characteristics and Terminology of Options 111.2. Where Options Trade III.2.a. Exchange-Listed Markets III.2.b. Over-the-counter Markets III.2.c. Hedging Option Transactions III.3. Option Market Participants IV. How Options Work IV.1 Illustration of Whether an Option is In- or Out-of-the-Money using a Google Option Example IV.2. An Example of How Options Perform using a Google Option Example How Option Values are Determined V.

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The Effect of Volatility, Interest Rates, and Cash Flows on the Underlying V.2. The Black-Scholes-Merton Model VI. Currency Options VI.1. Currency Quotation Methods VI.2. Currency Calls and Puts VII. Digital Options VII.1. Explanation of Digital Option Payoffs using the Google Option Example VII.2. Valuation of Digital Options using the Google Option Example VIII. The Digital Currency Options in this Case VIII.1.Basic Terms of the Contracts VIII.2.The Digital Option Payoffs IX. The Probabilities of Profit M.1. How Probability of Profit is Calculated IX.2. Calculating the Probability of Profit IX.3. Probabilities of Profit of Individual Transactions M.3.a. Probabilities of Profit of Individual Transactions in this Case IX.3.b. Comparison of the Probability of Profit of Individual Transactions in this Case to the Profits from Google and S&P 500 Option Strategies IX.4. Probabilities of Profit of Combinations of Transactions Whether the Options were in- or out-of-the-money on Selected Dates X. Values of the Sold Options on the Date of Contribution XI. XII. Estimated Profits of the Transactions XIII. Summary of the Issues and My Opinions XIV. References on General Options XIV.1. Classic Articles XIV.2. BooIs XV. References on Digital Options Technical Appendix A: The Black-Scholes-Merton Model Technical Appendix B: Finding the Probability that an Exchange Rate will be above or below a Critical Level by a Certain Period of Time Technical Appendix C: Estimating Volatility Technical Appendix D: Valuation of Digital Currency Options Supporting Appendix A: Curriculum Vita Supporting Appendix B: Sworn Testimony in Previous Four Years Supporting Appendix C: Documents Reviewed Supporting Appendix D: Compensation Arrangement

V.1.

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I. Expert Qualifications

I hold the William H. Wright, Jr. Endowed Chair for Financial Services at
Louisiana State University, where I have been since 2003. From 1980-2003, I was a faculty member at Virginia Tech, where I held the First Union Professorship in Financial Risk Management when I resigned. My specialization is derivatives and risk management. I am the author of three books on this subject and numerous articles and am frequently quoted in the media, including such outlets as The Wall Street Journal,

Bloomberg, Fortline, CNBC, Forbes, Barren's, The San Francisco Chronicle, The New York Times, S t Louis Post-Dispatch, The kfontreal Gazette, and Institutioi?al Investor. I
have extensive experience on a variety of consulting projects and in conducting corporate training programs. charterholder. I have written material on digital options. See specifically pp. 499-501 of An I hold a Ph.D. from Louisiana State University and am a CFA

Introdzrction to Derivatives and Risk ilfanagement, 7"' edition (2007, Thomson
Southwestern) and pp. 175-178 of Essays in Derivatives (1998, Wiley). There is an extensive body of literature on options, much of which is highly teclmical. Sections I11 and IV, References on General Options and References on Digital Options, list the sources that I view as the most valuable contributions to the state of knowledge and also useful references for understanding the basic principles that apply in this case. Supporting Appendices A, B, C, and D contain my C,,V., previous testimony, documents reviewed, and compensation arrangement.
1 . Sumrnary of Findings 1

I have been asked to opine on the general principles of options, characteristics of
currency options and digital options, and the specific issues of this case, including (1) whether the foreign exchange digital option spread strategies include a reasonable probability of profit, (2) whether the options were in- or out-of-the-money on certain dates, and (3) the amount that the assignor of the short options would pay an assignee to assume the obligation in an arms-length transaction on the date of contribution to the partnership, and (4) the actual profits from the transactions. I found that the transactions did have a reasonable probability of profit. I drew this conclusion on the basis of a comparison of the probability of earning a profit on the

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transactions in this case to the probability of earning a profit on standard option transactions that are conducted on U. S. options exchanges. For example, I estimated the probabilities of profit of a large number of options on Google and the S&P 500 and found those probabilities to he comparable if not much lower than those of the transactions in this case. These digital option strategies involved the purchase of a call or put option with one exercise price and the sale of a call or put option with a different exercise price, with both options expiring at the same time. The purchased call has the lower exercise price and the sold call has the higher exercise price. The purchased put has the higher exercise price and the sold put has the lower exercise price. As I will explain later, there are only three outcomes, one producing a loss and two producing a profit. The loss occurs if both options expire unexercised. The second and third outcomes are both profits, one of which is quite large. I will refer to the lower of the two profits as the "low profit" and the higher of the two as the "high profit." The latter has sometimes been referred to as the "sweet spot," because it produces an extremely large profit but occurs only if the exchange rate at expiration is within a very narrow range. I calculated the probability of generating any profit, which is the probability of the low profit plus the probability of the high profit. For the latter, the probability was found to be extremely low (less than 1%). My overall probability calculations include this miniscule probability but its effect is immaterial to my conclusions regarding the probability of profitability of these transactions.

I also determined if the transactions were in- or out-of-the-money on the
transaction and transfer dates. All were out-of-the-money. All of the transactions were held to expiration and all expired out-of-the-money. These terms are explained in more detail later but basically refer to whether the underlying exchange rates were above or below the levels they needed to be for the options to generate a positive value if exercised.

I was also aslted to determine the values of the short option on the transfer date.
This amount is what an assignor would pay an assignee to assume the obligation in an arm's-length transaction

I obtained these values using the well-known Blaclc-Scholes-

Merton model for valuing currency options, adapted for the case of digital options.

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111. Basic Principles of Options In this section I will describe what options are and how we characterize them. I will then describe the markets in which they trade. 111.1. Characteristics and Terminology of Options An option is a contract between two parties that provides one party the right to either buy or sell an underlying asset from another party at a fixed price either at a specific point in time or during a defined period of time.' An option granting the right to buy is referred to as a call option or simply a call. An option granting the right to sell is referred to as a put option or simply a put. The party that has the right to buy or sell is the buyer or holder of the option. The buyer or holder is said to be long the option, and is sometimes identified as the long. The counte~party, who grants the buyer the right to buy or sell, is referred to as the seller or writer The seller or writer is said to be short the option, and is sometimes referred to as the short. To acquire the option, the buyer pays a sum of money called the price orpremiztm to the seller. Options are one type of instrument in a broader family of contracts called

derivatives, which includes futures, forwards, and swaps. A derivative is a contract that
derives its value from the value of another instrument, the underlying asset. Futures, forwards, and swaps unlike options do not provide the right to buy or sell something. They commit the parties to a buy and sell at a later date. The underlying asset of the option can be any asset but optionable assets are most commonly stocks or an index of stoclts, bonds, currencies, and commodities such as oil or gold. It has also become customary to refer to the underlying asset as simply the

underlying in spite of the grammatical impropriety.2

of his definition is a conventional and widely accepted one and provides a general description of an option. There are many options that are variations of this definition, such as the digital options in this case. I will discuss these instruments in great detail, showing how they differ from conventional options. An important point to establish, however, is that digital options and indeed all inshuments that deviate somewhat from this basic definition are huly and widely accepted as being options. Perhaps the most general defmition of an option would simply be that it is an instrument that provides a payoff depending on whether a specifically defined event happens. Such a defmition has not become commonplace, because it is so widely agreed that options can he altered in a variety of ways such that the common definition seems slightly inaccurate 'It is common to use the term "asset" to describe the underlying, but some underlyings are not technically assets. For example, there are options on weather where the underlying is a meteorological measure such as rainfall or temperature. Options also exist on interest rates, which are not technically assets. In fact interest rate options are among the most widely used options.

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As the definition states, an option gives the right to buy or sell something at a fixed price. This fixed price is referred to as the exercise price, strike price, striking

price, or strike

Options expire on a specific date, called the expiration or expiry As

noted in the above definition, some options grant the right to buy or sell only on a specific date, which is the expiration. These options are called Etiropean options or

Eziropean-style options. An option that allows the holder to buy or sell at any time prior
to and including the expiration date is said to be an American option or an American-style

option.3
When the holder of the option uses the option to buy or sell the underlying, it is said to be exerci,sing the option. There are two possible forms of exercise: physical

delivery or cash settlement. In physical delivery, the actual instrument is delivered from
either seller to buyer or buyer to seller, depending on whether the option is a call or put. In cash settlement, neither party actually exchanges the asset. Instead the seller pays a sum of money to the buyer. The exchange of this sum of money is the economic equivalent of the actual delivery of the asset, although it avoids the direct costs of delivery. Whether a contract is physical delivery or cash settlement is a point established in the contract. In other words, the parties decide on the settlement form before the contract is written.

111.2. Where Options Trade
Options trade in two primary marlcets. One is the exchange-listed market and the other is the over-the-counter market. These marlcets are both quite large, but due to the nature of the transactions and the reporting requirements, it is difficult to compare them directly by size. The over-the-counter marlcet is typically characterized by a measure called notional principal, which attempts to capture the amount of the underlying involved in the transaction. For example, an option to buy one million euros that are currently trading at $1.10 has a notional principal of $1 10 million. The actual market value of that contract is considerably less and is determined by the value of the option, a topic I will cover later The exchange-listed options marlcet is usually measured by the volume of contracts traded. A standard option contract on stock covers 100 shares. Thus, daily

he terminology "European" and "American" has nothing to do with where these contracts are traded,
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volume might be reported as 1,500 contracts, but this figure represents 150,000 individual options (that is, options on 150,000 shares). An exchange-listed contract on the euro covers 62,500 euros. If volume is 500 contracts, this figure represents 31.25 million euros. Volume figures across contracts are simply added, so in this example of options on stoclc and the euro, total volume is reported as 1,500

+ 500 = 2,000

even

though the amounts of the underlyings are quite different. This approach is simply market custom and has to be remembered when attempting to compare the sizes of these two markets. Over-the-counter marlcet size data are compiled by the Bank for Intemational Settlements (BIS) of Basel, Switzerland (www bis.org), which conducts semi-annual surveys of global financial institutions to determine their usage of options, forwards, and swaps. These surveys are essentially voluntary, but the figures are widely accepted as reasonably accurate. As of mid-.June 2006, the total worldwide notional principal of over-the-counter contracts is estimated at about 370 trillion. Of this amount, about 55 trillion represents options, of which about nine trillion are currency options.

Futures Incltlshy magazine (www.fituresindustry.org) publishes data on the
volume of exchange-listed contracts in each of its bi-monthly issues.. The March-April issue includes the final total of the previous year. For the year 2006, total volume of exchange-listed options worldwide is about 6.6 billion, of which 240 million are currency options. The next subsection describes the exchange-listed markets. The over-the-counter markets will be discussed in the following subsection of this report.

111.2.3 Exchange-Listed Marlcets
In various financial centers around the world, there are organized options exchanges. For example, in the U. S., options traded on the following exchanges: Chicago Board Options Exchange, International Securities Exchange, New York Stock Exchange Arca, Philadelphia Stock Exchange, and Boston Options Exchange. Other large and well-known options exchanges exist in major cities outside the U. S. Exchange-listed options are standardized, which means that the terms of the contract such as the underlying, the exercise price, the expiration, and the number of units of the underlying covered in a contract are defined by the exchange and cannot generally be

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altered by the parties. By standardizing the terms, the exchange creates a broad but welldefined set of options on which it will permit trading. The exchange consists of members who trade to create markets in these options. These traders are called dealers or iizarlcet makers and guarantee to trade by quoting prices at which they will either buy or sell. These traders provide liquidity, giving outside participants the assurance that the option can always be bought or sold, which leads to the fact that most options are not held to expiration. They are bought and sold, just like stocks on the stock exchange Although there are exchange-listed Europeanstyle options, American-style options are more common in the exchange-listed market, at least in the United States., Exchange-listed marlcets also serve a valuable role in managing credit risk. As described, in general, the buyer of an option pays the seller the premium and receives the right to buy or sell the underlying. Hence, the buyer assumes the risk that the seller will not hlfill its obligation upon exercise of the option. Thus, the option buyer assumes some credit risk. The option seller assumes no credit risk, because the buyer is under no obligation to do anything after having paid the premium. To eliminate the credit risk, options exchanges use an institution called a

clearinghozise, which guarantees to the buyer that if the seller does not perform its
obligation, the cleainghouse will step in and do so. To provide this guarantee, the clearinghouse sets aside the premium paid from the buyer to the seller and requires the seller to put up additional funds or securities in a margin account. The clearinghouse system has historically performed as it should with no credit losses ever experienced by the holder of an exchange-listed option. Exchange-listed options are heavily regulated by federal securities regulators. In the United States, this regulation is done by the Securities and Exchange Commission. The Commodity Futuies Trading Commission regulates only one type of option, options where the underlying is a futures contract. In most other countries, it is common to have a single regulatory body.

III.2.b.Over-the-Counter Marltets
The other type of options market is the customized or over-the-cotinter, sometimes called OTC, market. This market is comprised of a large number of dealer

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firms that create options These dealers are primarily large banks and some non-bank investment firms. They stand willing to create whatever options their customers, called
end usem, want. In contrast to exchange-listed options, OTC options are tailored to meet

the specific needs of the dealers' customers. These instruments are usually structured to be created and held until expiration. Hence, they tend to be European-style slightly more often. It is oftentimes mistakenly thought that OTC markets are illiquid. Suppose an end user purchased an option wit11 the intention of holding it to expiration, but with changing circumstances, it decides that it needs to terminate the contract before expiration. In that case, the end user would usually go baclc to the original dealer and create a transaction that is the opposite of tlle original transaction. If the original transaction could be created, it is unliltely that the opposite position could not be created later. Combining these two positions offsets or terminates the overall position. The end user is not obligated to go baclc to the original dealer, however, and could go to a different dealer. If it creates the opposite position wit11 a different dealer, however, the two transactions are not offsetting. They merely have opposite financial effects, so the risk from movements in the underlying is neutralized, but both transactions remain in place. Note that terminating an option is not the same as exercising an option. Terminating an option is economically the same as selling an option that had been previously purchased or buying back an option that had been previously sold. Exercising means to actually use the option to purchase or sell the underlying asset or engage in an equivalent cash settlement. Unlilce exchange-listed options, OTC options are subject to credit risk. The buyer of an OTC option bears the risk that the seller will default Participants in this market do, however, do an excellent job of managing the credit risk by restricting who can trade, requiring margin deposits, and settling some losses before expiration.

TII.2.c. Hedging Option Transactions
Dealers in OTC options and most dealers in exchange-listed options profit off of their ability to buy and sell options at a spread with any rislc eliminated through a hedging transaction. For example, if a company approaches a dealer and asks to purchase a call option on the Canadian dollar, the dealer will assume a short position in the option. It

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would then be exposed to increases in the Canadian dollar. The dealer will use its knowledge of option valuation and risk management models and enter into an offsetting position with a different counterparty such that the dealer benefits from an increase in the Canadian dollar. Tllis transaction could be something as simple as buying Canadian dollars or something more complex such as a swap or other currency derivative. Dealers alinost always hedge these positions. In fact, most dealers are banks and are obligated to hedge so as not to endanger their depositors' money by speculating on derivatives transactions. Even non-bank dealers almost always hedge these transactions, because it is much easier to make money by being a dealer and hedging the risk than by taking the risk that the market will move the desired direction. Also, the exposures are typically quite large and speculating would in time almost invariably lead to large losses, if not ruin. Thus, dealers typically hedge.
111.3. Option Market Participants

I have mentioned the role that dealers play in options markets. They provide liquidity to those wanting to buy and sell options. Dealers offer to take either side of a transaction and execute hedges to remove their risk. Now I will discuss the other market participants. Financial markets are normally comprised of two types of participants, hedgers and speculators. Hedgers are parties who have some exposure to the underlying. For example, a farmer would have exposure to the price of his crop. A pension fund has exposure to the values of its securities. A multinational corporation is exposed to exchange rate risk on the cash flows it generates from its foreign operations and the payments it makes in foreign currencies. These participants sometimes use derivative markets to lay off' these kinds of risks. Consider Microsoft, which has operations and generates cash flows in over 70 countries. Given that Microsoft is a U. S. corporation, the cash flows it receives in euros, for example, are subject to the risk that the euro will decline in value relative to the dollar. Microsoft states in its annual report that it uses options to hedge its foreign currency exposure, which would likely mean that it either sells calls andlor buys puts on the euro. If the euro gains (loses) in value relative to the dollar, the conversion of its euro cash flows to dollars will result in an increase (a decrease) in its dollar cash inflows. If it

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sells calls, an increase (a decrease) in the euro will increase (decrease) the value of the call. Because the call is an obligation, however, the value of this obligation increases (decreases), resulting in a partial offset of the gain (loss) on the conversion of its cash flows to dollars. If it buys puts, any increase (decrease) in the euro will decrease (increase) the value of the put, resulting in a partial offset of the gain (loss) on the conversion of its cash flows to dollars. These partially offsetting gains and losses on the currency and option positions are the factors that make the transaction a hedge. Hedging is undertaken by thousands of corporations, financial institutions, institutional investors, and some governments (though not the U. S. federal government). In a few rare cases, two hedgers with opposite exposures might engage in a transaction in such that each passes on its risk to the other, which ~esults both being hedged. But such a meeting of opposite needs is rare. More often than not, a speculator must enter the picture to talce on the risk. Speculators play a valuable role in financial markets by assuming the risk that hedgers wish to eliminate. Speculators take on this risk because they believe they have a competitive advantage in the cost of trading, the ability to spot profit opportunities more rapidly than others, and in their general ability to take and manage an acceptable amount of risk. Speculators may not always be correct, of course, and will sometimes lose money, but they believe that their gains will exceed their losses by a sufficient amount to compensate for the rislc they accept. Although speculators are sometimes viewed as the vultures of the market, without them financial markets could not function. Indeed speculators are much like insurance companies, which accept premiums from customers and agree to bear some of the customer's risk. Of course, one might wonder where the speculator comes in when a hedger engages in a transaction with a dealer for the purpose of laying off the risk. Recall that we described the dealer's role as that of an intermediary. It agrees to take the opposite side of any transaction, and it lays off its rislc by hedging. Thus, the dealer is a middleman between hedger and speculator. On occasion there are several layers of middlemen. A dealer might lay off its risk to another dealer, who lays off its risk to anotl~er dealer. Eventually, however, the risk must be accepted by either a speculator or in the rare occasion by a hedger with the exact opposite needs.

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Today's options markets are dominated by large institutions and the rapidly increasing number of hedge funds.,4 Many of these institutions are dealers but some do engage in proprietary (typically speculative) trading. Hedge k n d s are almost never dealers, but their large volume of trading does contribute to the liquidity of the market. In summarizing, the participants in these markets trade at whatever levels of risk they are comfortable with. Those noted for taking high degrees of risk are commonly referred to as speculators. Those who reduce their risk are often called hedgers. It should be noted, however, that there is no clear dividing line between these two types of participants and distinctions are largely semantic.
IV. How Options Work

To understand how options work, I will use a real example. Consider Google, the well-known Internet company. I have assembled some data on exchange-listed options on Google from the trading day of July 24, 2006. This day was selected only because it was the day following the day on which I began my work on this report. There is nothing unusual about these prices or this trading day. Google's stock closed that day at $390.90. The final option prices of the day are shown in Table 1 These options expire on August

18 and September 15
options.

An exchange-listed option contract covers 100 shares, so

technically a single option contract includes 100 options. These are American-style

Table 1. Example of Option Prices for Google on July 24,2006
Calls Exercise Price
380 390 400

puts August
$8 10 $12.10 $17 00

1
September
$12 60 $17 10 $22 10

August
$20 60 $1440

September
$27 00 $20 80 $15 75

$9 60

IV.1. Illustration of Whether an Option is In- or Out-of-the-Money using a Google

Option Example

. I

A hedge fund is a portfolio managed on behalf of a limited number of wealthy individual andlor institutional investors in which the manager has the ability to Wade across a broad range of markets using a diverse set of strategies. The risk is typically high, though some funds claim to engage in low-risk strategies (hence, the name "hedge" fund) The investors' funds are usually locked up for a defined period of time, and management is typically paid a large fee and a percentage of the profits.

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A call option in which the stock price exceeds the exercise price is said to be inthe-money. If the exercise price of a call exceeds the stock price, the call is said to be out-oj-the-money. If the stock price equals the exercise price, the option is said to be atthe-money, For puts, the terminology is just the opposite. If the exercise price exceeds

the stock price, the put is said to be in-the-money. If the stock price exceeds the exercise price, the put is said to be out-oj-the-money. If the stock price equals the exercise price, the put is said to be at-the-money. Although the Google stock price is slightly higher than 390, I will refer to the 390 options as at-the-money. The 380 calls and 400 puts are in-the-money, while the 400 calls and 380 puts are out-of-the-money. Let us first consider the calls. Fox. the at-the-money call, someone paid $14.40 for the right to buy Google at $390 any time until August 18 and $20.80 for the right to buy Google at $390 any time through September 15. For any exercise price, the option price is higher for the September calls than for the August calls, because the additional time gives the stock more time to get above the exercise price. Also note that the option price is lower the higher the exercise price. For call options, the exercise price is a hurdle. To justify exercise, the stock price must get above the exercise price. The higher the hurdle, the more difficult it is for a call option to be exercised. Now look at the puts. For any of the three exercise prices, the September puts cost more than the August puts. This result is because the additional time gives the stock more opportunity to get below the exercise price.' Also, note that puts are worth more the higher the exercise price. For puts, the stock price must get below the exercise price for the put to be exercised. Hence, the higher the exercise price, the easier it is for the stock price to get below the exercise price.

IV.2. How Options Perform using a Google Option Example
Consider an investor who purchases the Google August 390 call, paying $14.40. Let us ignore the possibility of exercising the option before e ~ p i r a t i o n . ~ Also, I will

his result is true for American puts but is not necessarily true for European puts. The exception, however, occurs only for very deep in-the-money puts None of the puts in this case became deep in-themoney, so the issue is not relevant to this case, '~arl~ exercise adds a potentially quite complex issue that is not germane to the digital option case because tile options in this case were European-style so they could not be exercised early

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assume only a single option, instead of the standard 100 units or shares that comprise each contract. Thus, the result stated here is on a per unit basis. At expiration if the stock is at $390 or below, the option expires worthless and the investor loses the entire premium of $14 40 If the stock is above $390, the investor exercises the option. Let us say that the stock is at $400 at expiration Then the investor pays $390 and receives a stoclc worth $400, thereby netting a gain of $10. The overall transaction is a loss, however, because the $10 gain does not offset the $14.40 premium. The investor profits only if the stock exceeds the exercise price by at the least the premium. Thus, the breakeven is $390 + $14.40 = $404.40. Figure 1 shows a graph of the profit compared to the price of Google stock at expiration. TI& pattern is known as a "hockey-stick" graph. It shows how the investor loses the entire premium for any stock price below the exercise price, loses a portion of the premium for any stoclc price between the exercise price and the breakeven, and makes a profit for any higher stoclc price. Note that there is no upper limit to the profit Figure 1. Profit from the Strategy of Buying the Google August 390 Call I I

I

Stock Price a t Expiration

I

To the seller of the call, the position is exactly the opposite. The buyer's gains are the seller's losses. Thus, if the buyer loses the full premium of $14 40, the seller makes the full premium of $14.40. Figure 2 shows the transaction from the point of view of the seller. Note how the unlimited gain potential of the buyer is equivalent to unlimited loss potential of the seller.

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Figure 2. Profit from tile Strategy of Selling the Google August 390 Call
$20 $10 $

0 - ~ S350 4 8 3 $370 * 3 a a $390 m 1 ~

E
a

e

-$I0 420 430 -540 -$50

Stock Price at Expiration

This example can be used to illustrate an important principle that plays a role in this case: options have a high degree of leverage. For example, suppose the stock price at expiration just barely exceeds the brealceven of $404.40. Let us say it goes to $405. The call will be worth $15 and will generate a profit of $0.60 ($15 - $14.40) to the buyer. A profit of $0.60 on an investment of $14.40 is 4.2%, which is very large considering that the position is held only one month If the stock goes to $406, the investor makes a profit of $1.60, which is 11.1% in one month. Yet if the stock does not get past $404.40, the investor sustains a 100% loss. To make a profit, the stock must rise to $404.40, which is an increase of about 3.5% in one month. Google is a very volatile stock and can malce a move that large, but the odds are high that it will not Thus, we see that this option has a low probability of being profitable, but if it is profitable, the rate of return is quite large. The August 380 call costs $20.60 and its brealceven is $380 + $20.60 = $400.60. Thus, the stock has to go up only about 2.5% to be profitable. But, the rate of retum if profitable is smaller. Suppose the stock goes to $401. Then the profit is $0.40 ($21 $20.60), a return of 1.9%. l l ~ August 400 call costs $9.60 and has a much lower chance e of profitability. The stock must get to $409.60, an increase of 4.8%. If the stock goes to $410, however, the profit is $0.40, a retum of 4.2%. The lcey point is that in-the-money options have greater chance of profit but lower leverage and generally smaller returns. Out-of-the-money options have less chance of

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profit but greater leverage and generally larger returns.

At-the-money options lie

somewhere in between. This point is important in understanding elements of this case. Similar conclusions apply to put options, though the direction is the opposite. Figure 3 shows the August 390 put, which has a premium of $12.10. To make a profit the stock must get below $390 by the amount of the premium. Thus, the breakeven is $377.90, a decrease of 3.3%. The put buyer does not have an unlimited gain, but the gain is limited by the fact that the stock can fall no lower than $0. Figure 4 shows the position of the witer, which is obviously the minor image of that of the buyer.

Figure 3. Profit from the Strategy of Buying the Google August 390 Put
1
I

$410

$430

$45 1

-$to 415

Stock Price a t Expiration

Figure 4. Profit from the Strategy of Selling the Google August 390 Put

Stock Price a t Expiration

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The strategies shown here are the basic ones. Option traders frequently combine one option with another, with the underlying, or with risk-free bonds. When two options differing only by exercise price are combined, the transaction is called a .spread,because making a profit is determined by whether the underlying ends up between a spread or range of prices.7 V. How Option Values are Determined We have seen so far that call options are worth more with lower strilce and longer time to expiration and put options are worth more with higher strike and (with only a minor exception that does not arise in this case) longer time to expiration. Thus, the price of the underlying, the exercise price, and the time to expiration are three factors that affect the value of an option. There are three more. V.1. The Effect on Option Values of Volatility, Interest Rates, and Cash Flows on the Underlying Probably the most important factor affecting the value of an option is the volatility of the underlying. Recall in our example of Google options that rather large stock price increases were required for the options to generate a profit. The reason for this result is the volatility of Google stock. Volatility is a major factor in an option's value. More volatile stocks will have more valuable options, because higher volatility increases the chance that an option will expire in-the-money. Of course, higher volatility also increases the chance that an option will expire out-of-the-money, but this increased chance of an undesirable outcome is mitigated by the fact that an option expiring out-ofthe-money results in loss of' only 100% of the premium. If the option expires deeper outof-the-money, the loss is no worse. Thus, for both calls and puts, higher volatility malces

an option more valuable.
Another factor that affects option values is the level of the risk-free interest rate.' The effect of the risk-free rate is fairly minor, however, and the explanation of this complex and unimportant to this case, so I will virtually insignificant effect is somewl~at omit it here.

'This description is one type of spread, the one used with the digital options in this case. There are other types of spreads as well. Spreads will be covered in detail later. he risk-free intere.sf rate is considered to be the lowest interest rate in the market and reflects the rate one would earn by lending money with absolute assurance that the money will be repaid.

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Another factor that affects option values is payments made on the underlying. If the underlying is a stock, it might pay dividends. If a bond, it might pay interest. If a currency, it pays interest at the foreign rate. The higher these payments are, the lower the value of a call and the higher the value of a put.
V.2 . The Black-Scholes-Merton Model

Option values are determined by an option pricing model. The most well-laown is the Black-Scholes-Merton model. The model is a formula that takes these six inputs
f (price o the underlying, exercise price, time to expiration, volatility, risk-free rate, and

payments on the underlying) and converts them into a number representing what the option is worth. Details of the Black-Scholes-Merton model are covered in Technical Appendix A that appears at the end of this report. In addition, the model produces estimates of how the option value changes with changes in the inputs. These measures are essential for dealers who need to hedge the risk as previously described. The Blaclc-Scholes-Merton model is not the only model for valuing options, but it is widely accepted and used by dealers. Other formulas, such as the binomial model, are more commonly used when the terms of the option add a complexity not captured by the Black-Scholes-Merton formula. These added complexities do not appear in this case.

VI. Currency Options
Because this case deals with currency options, I will compare currency options to options on stock. Consider the options on Google. The price of Google stock is the value of a single share of the stock. In this case, an investor pays $390.,90and receives one share of Google stock. An investor could instead purchase a foreign currency. The exchange rate is the price of the foreign cuxrency and is analogous to the price of a stock.g While holding the currency, the investor is assumed to receive interest at the foreign rate.'' This interest is analogous to a dividend that might be paid on the stock.

Vi.1. Currency Quotation Methods
There is also a special relationship that exists in foreign currency markets that is important to understand. A currency can be quoted in two ways, the direct way and the
9 ~ hterm "exchange rate" is oftentimes referred to as the "spot rate" or "spot exchange rate" to distinguish e it fiom the forward rate, which is the exchange rate set on a given day for a transaction to occur at a later date I01n other words, it is not typically assumed tliat the investor would, like a tourist, simply hold foreign currency in the form of cash

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indirect way. For example, suppose the euro is trading at $1.27. To a U. S. investor, this
is the direct way of quoting the exchange rate or price of a euro. That is, a LJ. S. investor

must give up $12 7 to acquire a euro. Alternatively, tl~e dollar could be viewed as costing 1/$1.,27= 0.7874." That is, one dollar is equivalent to 0.7874 euros. To a European, this is a natural way of quoting the exchange rate. A European would say that the exchange rate for a dollar is 0.7874 euros, because this is the price of a foreign currency (the dollar) expressed in units of his currency (the euro). It i s what the European must forgo to acquire a dollar. Likewise, $1.27 is what an American must forgo to acquire a euro. In currency markets, it is quite natural to quote currencies both ways. For example, every day The PVa12 Street Jozi~nal publishes a table showing both quotes sideby-side. In financial markets, traders must get accustomed to quoting exchange rates both ways and must know which method is being used when a quote is given. Given the predominance of the U. S. dollar as the currency of choice in international trade, most currencies are quoted in U. S dollars. It is, however, common to quote the British pound

in terms of pounds per dollar and Japanese yen in terms of yen per dollar
other currencies are quoted in this indirect manner

l2

Occasionally

These direct and indirect methods of quoting currencies are quite natural in international financial markets one dollar is worth 1/$390 90 They are quite unnatural in domestic marlcets 0.002558 shares of Google For example, we could say that one share of Google is worth $390 90, or we could say that
=

Obviously the indirect

method is legitimate but awkward for domestic transactions. In this case, some currencies are quoted in terms of dollars and others in terms of the unit of currency. Thus, it is important to understand this point.

VI.2. Currency Calls and Puts
Because currencies are so easily viewed from the opposite party's point of view, they create a slight complexity when they are used in options. For example, a call in one

"E is the symbol for the euro '?he reason why the British pound is often quoted in pounds per dollars goes back to its dominance as the preferred currency for international trade prior to World War. 1 . The Japanese yen is often quoted in yen 1 per dollar because of the magnitude of the numbers. At this time, there are about 116 yen to a dollar, which is easier to quote than the inverse, 0 00862 dollars per yen.

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currency can be viewed as a put in the other currency.

The relationship is not

straightforward, however, so I need to explain it with an example. Consider a call option on the euro with an exercise price of $125. Assume that the contract covers E100,000, meaning that the holder ofthis option has the right to buy
100,000 at $1.25 on the expiration day. Thus, he can turn over $125,000 and receive 100,000. Now consider another investor who purchases a put option on $125,000

dollars with an exexcise price of E0.80. This investor has the right to turn over $125,000 and receive $125,000 x 0.80 equivalent. Now let us move to the expiration day and assume that the exchange rate is $1.60, which is equivalent to 0.625. Thus, with an exercise rate of $1.25, the call expires inthe-money and, the holder of the call exercises, tendering $1.25 x 100,000
= =

100,000.

These two positions are economically

$125,000

and receiving 100,000 euros, which are worth 100,000 x $1.60 = $160,000 for a gain of
$35,000. Because E0.625 is less than 0.80, the put is also in-the-money and the holder

exercises, tendering $125,000 x 0.625

= 78,625

and receiving 100,000 for a net gain

of 21,875. Given the cunent exchange rate of $1.60, these euros are worth $35,000, the same as the payoff of the call. If the exchange rate at expiration is less than $1.25, which means more than 0.80, the call and put expire worthless. Thus, a call on 100,000 euros with an exercise price of $1 25 is equivalent to a put on 125,000 dollars with an exercise price of 0.80. When converted to the same currency, both options produce the same outcomes. So an investor could place an order for either option, knowing the ultimate outcomes will be the same. This point is important in this case, because even though the transactions are for

U. S. investors, some of the options are quoted in dollars and some in the foreign
currency One person might view the transaction as a call and another might view it as a put. This point can be confusing but is just a minor inconvenience. What the option pays if a given condition occurs is unambiguous in this case.

VII. Digital Options
This case deals with digital options, so I will now discuss these instmments. First, let me note that the term digital option encompasses two types of options, a cash-

or-nothing option and an asset-or-nothing option. Digitals more commonly refer to the

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cash-or-nothing option, and it is my opinion that cash-or-nothing options are used more frequently tilan asset-or-nothing options.13 Also, digitals are commonly referred to as

binmy options and sometimes all-or-nothing options.

VII.1. Explanation of Digital Options Payoffs using the Google Option Example
Let us start by examining how digitals are related to standard European options. Consider the Google August 390 call option. The option quotes discussed above are standard options. Digital options are strictly OTC instruments, so there are no prices available for trades that occurred in the past. Nonetheless, we can use these Google options to illustrate how digital options would work. The following represents the payoffs of this option on the expiration day:

lf Gaogle stock is above $390:
the option holder pays $390 and receives the stock, which is worth at least $390

IjGoogle stoclc is at or below $390:
the option holder gets nothing. Digital options break these payoffs into parts. The holder of an asset-or-nothing call receives the stock if the stock is above $390 but does not have to pay the exercise price. Thus, he basically receives the stock for free but only if the option expires in-themoney.'4 1f the option expires out-of-the-money, the option holder gets nothing. The holder of a cash-or-nothing call receives $390 if the option expires in-the-money and nothing otherwise. Consider an investor named Ms. A, who buys an asset-or-nothing call from Mr. B and sells a cash-or-nothing call to Mr. C. If the calls expire in-the-money (stock price above $390), Mr. C exercises the cash-or-nothing call, thereby forcing Ms. A to pay $390. Ms. A exercises the asset-or-nothing call, thereby forcing Mr. B to deliver the stoclc to Ms. A,. Thus, Ms. A pays $390 and received the stock. If the calls expire out-of-the-money (stock price less than $390), Mr. C does not exercise the cash-or-

There is no data published on the use of these options All digitals are OTC options, which are private transactions. I previously cited the BIS data on the size of the currency options market, but this information is not broken out into types of options " '0 not let the nation that the option holder receives the stock for Free mislead you The option holder paid the option premium when he purchased the option The reference to receiving the stock for kee simply means that the option holder does not pay the exercise price

13

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nothing call, and Ms A does not exercise the asset-or-nothing call. Thus, Ms. A neither pays nor receives anythmg. These results replicate those of a standard European call. For puts, the seller of an asset-or-nothing put delivers the stock if the put expires in-the-money and does nothing if it expires out-of-the-money. The holder of a cash-ornothing put receives the exercise price if the put expires in-the-money and nothing if the put expires out-of-the-money. Thus, a standard European put is equivalent to buying a cash-or-nothing put and selling an asset-or-nothing put. Thus, digital options can be viewed as elements of standard options In this sense, they are like atoms, complex but nonetheless fundamental components of ordinary options The options in this case are cash-or-nothing options, so henceforth I will use the term "digital options" to refer strictly to cash-or-nothing options Recalling the Google digital call, it permits the holder to pay $390 if the stock expires with a value of more than $390 In practice, most digital options are standardized to a unit payoff of $1 Thus, one digital option pays $1 if the stock price is above the exercise price at expiration A digital option paying $390 can be viewed as 390 digital options that each pay $1 The most important point, however, is that the payoff of a digital option, unlilce that of an ordinary option, is simple. It is either zero or a fixed amount. With an ordinary option, the payoff is either zero or a variable amount. As noted, the options from which this information was obtained are standard options. Digitals trade in the OTC markets, so there are no quoted prices.

VII.2. Valuation of Digital Options using the Google Option Example
Valuation of digital options is relatively simple, however, and uses a variation of the Black-Scholes-Merton model. I estimate that an August 390 digital call would cost $0.51 14 per $1 payoff and an August 390 digital put would cost $0.4886 per $1 payoff. I will use these prices to illustrate the strategies of buying digital options. Figures 5 and 6 show the profits for buyers of Google digital calls and puts. The graphs for the sellers are not shown, but they would simply be the minor image."

"III Figures 5 and 6 the nearly vertical line where the stock price at expiration is near the strike price is from the imprecision of spreadsheet graphics The line should actually be only an approximation a~ising perfectly vertical at the strike The only way to make a spreadsheet graph line look perfectly graph would be to space the stock prices at virtually infinitesimal intervals Here the spacing is 0 25

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Figure 5. Profit from the Strategy of Buying the August 390 Digital Call
I

I
$0 60 $0 40 $0 20

5

E 2

n

$000-

4

,

,

,

8

4

>

,

,

, ,

,

,

, ,

,

,

,

,

$ 80 $382 $384 $386 $388 $330 $392 $394 $396 $398 $43 1 : -$O 20 -

-$0 40 -$0 60

Stock Price at Expiration

Figure 6. Profit from the Strategy of Buying the August 390 Digital Put
I I

I

Stock Price at Expiration

I

No one knows the size of the digital options market. Although the Chicago Board of Trade began trading digital options on the federal funds rate in July 2006, the majority of digital option transactions are over-the-counter and there is no official count of OTC digital option activity. There is, however, a small body of literature on digital options, which is included in the references contained in Section XIV. The transactions in this case are digital currency options, which are discussed in two of the references (Shamah (2004) and Liu (1995)). Next I will examine the digital currency options used in these strategies.,

VIII. The Digital Currency Options in this Case

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I have been aslted to opine specifically on certain digital currency option
strategies undertaken by the clients. These transactions were all conducted with the New York branch of Deutsch Bank AG, a leading multinational bank headquartered in Germany VIII.1.Basic Terms of the Contracts Each strategy consists of two option transactions, one a purchase of a digital currency option and the other a sale of another digital currency option with the two options having the same expiration but different exercise prices. As described earlier, such a strategy is called a spread, although the tenn "spread" can be used to describe some other types of option transactions. These strategies are specifically called "money spreads," but I will just use the term "spread." Summary information on the contracts is contained in Table 2. Table 2. Summary Information for the Digital Option Strategies Murphy Porlc Partners, LLC
Bank reference number Customer Currency pair Transaction date Expiration Type of option (calllput) Strike of option purchased
24623 WHM Ventures LLC USDlCHF 4/14/00 6113/00 put CHF1.5839 24625 HDM LLC USDiCHF 4/14/00 6/13/00 put CHFl.5839 2463 1 WHM Ventures LLC USDEUR 4/14/00 6/13/00 put $0.9207

PSM Farms, LLC

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MURFAM Farms, LLC

(MURFAM Farms, LLC, continued)
Bank reference number Customer Currency pair

I

24629

WHM Ventures USDEUR

1 / 1

24630

WMC USDEUR

The first line in Table 2 is the bank reference number, which is simply a number used by Deutsch Bank to identify the transaction. I am using it because it facilitates keeping track of the transactions. The second line is the customer, which is the name to which Deutsch Bank addressed the confirmation. The third line is the currency pair, which indicates the two currencies involved, with the U.S. dollar always indicated first because it is the home currency of the cu~tomer.'~ fourth line is the transaction date, The and the fifth is the expiration date. The sixth line is the type of option, call or put. As noted in Section VI, whether a currency option is a call or a put depends on one's point of view I defined the contract

as a call or a put based on the specific language in the contract that describes the payoff.
If the language states that a payoff occurs if the exchange rate at expiration is "less than or equal to" the strike, I interpreted the contract as a put. If the language states that
%HF
= Swiss

franc, EUR = euro, USD = U. S dollar

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payoff occu~s the exchange rate at expiration is "greater than or equal to" the strilte, I if interpreted the contract as a call. Again, it is important to note that the alternative interpretation can be made but would necessitate ~estating exercise rate and contract the size in the opposing currency The interpretation deals with only the te~minologyand does not affect the substance of the transactions. In the contract language the option purchased is always designated as the "first option," so I follow this pattern by putting the purchased option first. The seventh line is the strike of the purchased option Note that in some cases the strike is in U. S. dollars; in others, it is in units of the foreign cunency This information is drawn ftom the contract language, so the parties made this designation. Below the sQike, we see the initial premium, which is the amount paid by the customer to purchase the long option. The next line contains the payment received if this option expires in-the-money. As a digital option, this payment is the same for any value of the underlying at expiration, provided the underlying value is above the exercise price for a call or below for a put. The tenth line is the strilce of the option sold, followed by the initial premium and the payoff if the option expires in-the-money. For the option sold, the initial premium is received by the customer, and the payment at expiration is made by the customer, provided of course that the option expires in-the-money.

VIII.2.The Digital Option Payoffs
The information in the previous tables summarizes the contracts and is sufficient to completely identify the possible payoffs of these options. I will now take the first transaction and explain. Consider the transaction # 24623 for WHM Ventures. At the start of the transaction, the customer pays $4,000,000 to buy the first option and receives $3,880,000 to sell the second option for a net initial premium of $120,000. Both options are puts and at expiration the customer will receive $8,000,000 if the Swiss franc is at CHF1.5839 or below and will pay $7,730,000 if the Swiss franc is at $1.583701 below. These payoff combinations produce the following outcomes:
Value of Swiss franc at Option Expiration

/

Below CHF1.5837

I

Between and inclusive of

/

Above CHF1.5839

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CHFl 5837 and

-

CHFl 5839 Customer receives on 1" option Customer pays on 2""ption Total
I

$8,000,000 $7,730,000 $270,000
I

$8,000,000 $0 $8,000,000
I

$0 $0 $0 -$120,000

Profit

after

accounting

for

$1 50,000

$7,880,000

-

premium paid

The table above lays out how the two options pay off in the three possible outcomes and how these payoffs combine to produce three possible overall results. Table

3 summarizes this information for all of the transactions in this case Table 3. Summary Information for Profits
Murphy Pork Partners

PSM Farms

MURFAM Farms, LLC
Bank reference number Initial premium paid Underlying currency Payoffs and Profits Exchange rate expiration is less than 24622 $519,000 CI-IF CHFl.5837 24624 $291,000 CHF

/ / /

24626 $90,000 CHF CHF1.5837

I

CHFl.5837

I

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Profit Exchange rate at expiration is between and inclusive of and Profit Exchange rate at expiration is greater than Profit

$648,750 $363,750 CHFl.5837 CHF1.5837 CHFl.5839 CHFl.5839 $34.08 1,000 $19,109,000 CHF1.5839 CHF1.5839 -$519,000 -$291,000

$1 12,500 CHF1.5837 CHF1.5839 $5,910,00 CHF1.5837 -$90,000

=AM

Farms, LLC, continued
Bank reference number Initial premium paid
24629 $792,000

1 1

24630 $159,000

All of the transactions are characterized by only three possible profits. One of these three profits is an outcome in which both options expire out-oflthe-money, and the amount lost is the net initial premium. Another possible outcome is a very large payout that occurs over a very narrow range. A third outcome is a smaller but positive payout. Figure 7 provides a general illustration of the two patterns of profits, with Panel A representing calls and Panel B representing puts. The horizontal axis is the exchange rate at expiration and the vertical axis is the profit. The diagrams are useful but represent the general pattern across these transactions and not any one transaction. Also, they are not drawn to scale. (Note: MURPHY did no call transactions, but the call graph is shown for comparison purposes.)

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Figure 7. General Profit Graphs for Digital Option A. Digital Call Spread

Low shike

li
4

FVill?br this range, /lze long call e.Tpirer in-/lien10ite.yand tlie slzorl CON expires out-of-/Ire-~noney Jbr 4 large overaNprojif

-Both oprions expire out-of-//re-nroneyin lltis range /or a snrall ove~all losr

4
I

-/ -

4

Both options erpire in-~l~e-mo,zey //xis br range for a sn8oN ovemllprq/it

Exchange rate at expiration

B. Digital Put Spread

-

IVillrin /his range. tlre longpi11erpires in-themoney and Ore rhorrpsr erpirer oil/-ofithe-nio,iqy ./or a large overallprofil

Low strike

.
,...
1 '

E

.

I-ligh shike

e a

$0

.
Borh optio~a erpire in-tire-nzorie.yin tliis range for a small overa/lpro/it

1

4.-

Boflr optio~rsrrpirc 0111-ofitlte-moneyin dir rangefir a rnroN overall lass

Exchange rate at expiration

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IX. The Probabilities of Profit

I have been asked to opine on the probabilities that these transactions would be
profitable. I will do this in two parts. First, I will estimate the probability of the profitability of each individual transaction. Second, I will estimate the probability that a group of transactions is profitable.

IX.l. How Probability of Profit is Calculated
Probability plays a critical role in determining the values of options. There is no perfect model for valuing any financial instrument, but option valuation is widely based in theory and in practice on the Black-Scholes-Merton model This model was published in an article in the Journal of Political Economy in 1973 by Fischer Black and Myron Scholes and in another article later that year in the Bell Jotlrnal of Economics and Management Science by Robert C. Merton. In 1997 Scholes and Merton received the Nobel Prize f o ~ contribution. Black had died in 1995 and the Prize is not given this posthumously.

I have testified in several cases in which Wall Street firms used the BlackScholes-Merton model to determine the values of options. Any option that is Europeanstyle can usually be valued reasonably well using Black-Scholes-Merton. A derivatives dealer first determines the option value using the model. It then marks up options sold and marlcs down options purchased, so as to build in a margin to cover costs and generate a profit. It then uses the model to tell it how to trade to hedge away the risk. If successful, as dealers usually are, they capture the difference between wbat they pay or receive for an option and what it costs to hedge away the risk. Technical Appendix A describes the Black-Scholes-Merton model. The Black-Scholes-Merton model can be adapted to give us the probability that the exchange rate will be above or below a certain critical level by a certain date. Two of the terns in the formula are probabilities that exist under the assumptions of the model. To adapt the problem to the digital currency options, we first need to lcnow what critical level the exchange rate must get above or below for the transaction to be profitable. For digital call spreads, the critical level is the lower of the two strikes. For digital put spreads the critical level is the higher of the two strikes. This point can be seen in Figure

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7 above. Note for calls, Panel A, that the profit is positive beyond the low strike. For
puts, Panel B, the profit is positive below the high strike.

IX.2. Calculating the Probability of Profit
To estimate the probability that the exchange rate will be above or below this critical level by a certain date, it is necessary to know the current exchange rate, the critical level, the amount of time, and the volatility. As described in Technical Appendix A, two other pieces of information used in currency option valuation are the risk-free rate of interest in both countries. Option valuation theory requires that the user treat the currency as if it is expected to increase at a rate equal to the domestic interest rate minus the foreign interest rate. It is questionable, however, to make this assumption when estimating the probability that a strategy will be profitable. There is no a priori reason why the exchange rate would be expected to increase at this net rate. For options on other instruments, we would require the expected rate of return on the underlying, which is a difficult variable to estimate. When the currency is the underlying, however, even the expected rate of return is a problematic measure. In using it, one is assuming that the currency is expected to increase at this rate for domestic investors, at the expense of the currency weakening from the perspective of foreign investors. A currency cannot consistently strengthen, to the advantage of one set of investors at the expense of the other set.I7 Hence, it seems appropriate to simply assume a zero expected return for ciurencies. Moreover, given the very short term of these instruments, a zero expected return is in my opinion quite reasonable. Under standard assumptions used in currency option valuation, the probability that an exchange rate will exceed a critical level over a period of time, given the volatility, can be easily estimated using a formula and statistical calculation related to the normal probability distribution. Details are provided in Technical Appendix B. As noted above, the lower exercise price fo