GAMES 99

Revised: 11/28/99. The Dilemma. All's Fair. A Golden Opportunity – Solution.
9/28/99. Sanity Check. A Golden Opportunity. How Many Ways Can I Appreciate Thee? – Solution. The Internal Auditor Considers a Bet (II) – Solutions. The Case of the Treacherous Trader – Solution. The Two Fund Managers – Solution.
7/28/99. How Many Ways Can I Appreciate Thee? Client Pays Ultimate Price for Bad Choice – Solution. The Internal Auditor Considers a Bet (I) – Solution. The Internal Auditor Considers a Bet (II).
6/28/99. The Ultimate Price for Choosing Bad Management. The Internal Auditor Considers a Bet. A Baffling Prediction – Solution.
5/28/99. A Baffling Prediction. Raiders of the Lost Arb – Solution.
3/28/99. Raiders of the Lost Arb. Commonality – Solution.
2/28/99. Celia in Derivativesland – Solution. The Case of the Villainous Vendor – Solution.
1/28/99. The Case of the Treacherous Trader. The Case of the Villainous Vendor.

The Dilemma (11/28/99)

1. You're a hedge fund manager with $600 million under management, who finds himself in a fix. The market's tanking and you can either sell out or hang tough. (a) If you sell out, you will save $200 million for your investors. (b) If you hang tough, the probability is 1/3 that the market will recover the entire $600 million, but the probability is 2/3 that your fund will be worth nothing. What would you do?

2. Same situation, but your possibilities are: (a) You can get flat and lose $400 million. (b) Bet the farm and face a probability of 1/3 that you won't lose anything and 2/3 that your fund will lose everything. What would you do?

All's Fair (11/28/99)

Andy, an option trader, and Barry, his quant, are sharing a cab ride from Grand Central to Lower Manhattan. Their custom is to flip a coin, loser paying for the trip. After two years of this ritual, Barry has noticed that when they use Andy's coin and Andy calls the toss, Andy always chooses heads and wins nine out of ten times. It's almost bonus time and Barry doesn't want to make any rash claims about the fairness of Andy's coin. Instead, he devises a way to get equally probable outcomes from Andy's coin. What is Barry's method?

All's Fair – Solution (4/28/00)

"Barry suggests that they both toss coins, and he or Andy has to call whether the two coins are the same, i.e., both tails or both heads; or different, i.e., one head one tail. Then the bias of Andy's coin disappears and the chance of a correct call becomes 50%.

"Demo: suppose Andy tosses 20 times. 18 times it's a head, and twice it's a tail. On average 9 times when a head lands, Barry's coin is also a head, and once when a tail lands, Barry's coin is also a tail. So 10 tosses out of 20 the coins are both the same. QED."

regardless of the bias in Andy's coin. If you replaced the 1/2s by b and (1-b), things don't work out so nicely.

All's Fair – Solution (5/28/00)

Sean O'Neil submitted the following solution: "…I assumed they would only be using Andy's coin, though in retrospect you did word the question without discounting the additional usage of Barry's fair coin.

“Before flipping Andy's coin, they each choose one of the two sequences, heads/tails or tails/heads. The coin is flipped twice. If it is either of those two sequences, the person who chose that one wins. If the coin comes up heads/heads, or tails/tails, the game is replayed. Continue until there is a winner.

“Of course, this game will probably take longer than the one using both coins, but the expected waiting time is pretty much the same as that for the less likely face to come up, which isn't too bad.”

Thanks for your alternative solution. It would work, but could be extremely slow, depending on how unfair Andy's coin is. In the extreme case, suppose Andy's coin is two-headed or two-tailed. Then Andy and Barry would have to wait infinitely long. However, before that, Andy would probably stop trying to cheat Barry this way and look for an easier mark. – Dr. Risk

Sanity Check (9/28/99)

"A securities firm has hired external consultants to weed out traders who are clinically psychotic, estimated reliably to be 1 in 1000. The test they have devised is 99% reliable for traders who aren't, and 100% reliable for traders who are. Now it is reliably estimated that 1 trader in 1000 is clinically psychotic. The traders are tested, and the chief swaps trader tests positive for clinical psychosis. What is the probability that he actually is [psychotic]?"

Sanity Check – Solution (11/28/99)

A Golden Opportunity (9/28/99)

Harry Lorenzo of Deloitte Touche's New York office challenges us with the following question:

"A consultant for a treasury division of a bank is on an engagement for 7 days. Because of recent currency fluctuations, he insists on being paid every day and in gold. You have just procured a seven inch bar of gold to pay for the 7 day engagement. His fee is one inch of gold per day. How can you pay him exactly one inch per day with only making two cuts to the bar of gold (to minimize gold dust waste)."

A Golden Opportunity – Solution (11/28/99)

How many ways can I appreciate thee? (7/28/99)

The head of global derivatives trading has $19 million available for his team's bonus pool. He has decided to award it in chunks of four, three, two, and (regrettably often) zero million dollars. How many ways can he do this? Explain.

(Inspired by the following problem, given to my daughter's second grade class: "We are going fishing on a lake. There are canoes that can hold 2 people and rowboats for 3 or 4 people. How many of each kind of boat could we use so that all 19 children in our class could go fishing?" Surely the teacher was looking for any one solution, rather than the systematic discussion of all solutions that we sent in!)

How many ways can I appreciate thee? – Solution (9/28/99)

"Ignoring who gets what, the answer is 10. You have to give out an odd number of 3s.
If you give out one 3, there is 16 left and you can give out either:
eight 2s;
six 2s and one 4;
four 2s and two 4s;
two 2s and three 4s;
four 4s.
If you give out three 3s, there is 10 left and you can give out either:
five 2s;
three 2s and one 4;
or one 2 and two 4s.
If you give out five 3s, there is 4 left and you can give out either:
two 2s;
or one 4.

Client Pays Ultimate Price for Bad Choice (6/28/99)

Ellen Dishowicz was defending O.J. Watson, a hedge fund manager, charged with murdering a dissatisfied client. The prosecution wanted to admit as evidence O.J.’s record of physically abusing clients. She argued that fewer than 0.4% of the hedge fund managers who abuse their clients kill them, hence O.J.’s record of abusing clients was irrelevant for the case. Is Ms. Dishowicz’s claim sound? Or does it take a lot of chutzpah for her to advance that argument?

Client Pays Ultimate Price for Bad Choice – Solution (7/28//99)

Reader Stephen Gould explained the weakness in Ms. Dershowitz's argument this way on 7/1/99: "(slow day at work) and say .004% of fund managers who do not abuse their clients kill them. Therefore someone who abuses his clients is 100 times more likely to kill them than someone who does not. Hence her argument is justified only if the murder rate for the two groups is the same."

This problem is based on an argument that Alan Dershowitz made in the O.J. Simpson case: fewer than 0.1% of the men in abusive relationships kill their mates, which made O.J.’s record of wife abuse irrelevant for his case. (Source: John Allen Paulus, once upon a number. New York: Basic, 1998.)

Dershowitz’s argument raised in Dr. Risk’s mind the same question that the odd claim from the occasional hedge fund manager has raised: Is this guy
(a) a "chochem", who sees things that I don’t see, and from whom I have much to learn,
(b) an "idiyot", who can’t count, or
(c) a "chutzpenik", who has the nerve to say something that he knows is wrong, and a "goniff", who is trying to get my money by deception.

In the case of Mr. Dershowitz, attorney, author of Chutzpah!, Professor at Harvard Law School, I’ll give him the benefit of the doubt and assume that he knew perfectly well that his argument made no sense, but he thought that it might mislead someone on the jury, and that nobody on the prosecution would see it for the non sequitur it was. The same is true for Ms. Dishowitz.

Simply, the 0.1% is the right answer to the wrong question. To see this, we apply Bayes Theorem to the following hypothetical data for one year on one million hedge fund investors:

	Abused by MM	Not Abused by MM	Total
Murdered by MM	4	1	5
Murdered by others	1	99	100
Not Murdered	995	998,900	999,895
Total	1,000	999,000	1,000,000

Dishowitz reports the following probability:
Pr{MM murdered investor | MM abused investor}
= Pr{MM murdered investor Ç MM abused investor } / Pr{MM abused investor } = 4/1,000 = 0.004
and says that it proves that the abuse is irrelevant, because it's a small number.

More relevant is
Pr{MM murdered investor | MM abused investor Ç someone murdered investor}
= Pr{MM murdered investor Ç MM abused investor Ç someone murdered investor} / Pr{MM abused investor Ç someone murdered investor}
= 4/5 = 0.80,
which sugggests that the hedge fund manager's abusive relationship provides circumstantial evidence of his guilt. In a civil case it might provide "preponderance of evidence", even if it is not "beyond the shadow of a doubt."

The Internal Auditor Considers a Bet (I) (6/28/99)

The head trader and the internal auditor have been playing a game of cat and mouse for some time. The auditor is certain that the trader is dirty, but can't prove it. Nevertheless, the auditor hounds the trader as relentlessly as Inspector Javert hounds Jean Valjean in Les Miserables. Finally, in what the auditor interprets as a sign of desperation, the head trader prepares a portfolio of four deals, two swaps and two options. The internal auditor would select two deals at random, without replacement, from this portfolio. An independent public accounting firm assures the auditor that the probability of randomly choosing any one of n deals remaining is 1/n. If the auditor picks two option deals, then the trader will resign. The auditor has given up hope of catching the dirty trader in a firing offense. If the auditor accepts this bet, what is the probability that he will pick two options?

The Internal Auditor Considers a Bet (I) – Solution (7/14//99)

Okay, see if the following, revised version of the problem is more challenging.

The Internal Auditor Considers a Bet (II) (6/28/99)

The head trader and the internal auditor have been playing a game of cat and mouse for some time. The auditor is certain that the trader is dirty, but can't prove it. Nevertheless, the auditor hounds the trader as relentlessly as Inspector Javert hounds Jean Valjean in Les Miserables. Finally, in what the auditor interprets as a sign of desperation, the head trader prepares a portfolio of four deals, each of which is either a swap or an option. The internal auditor would select two deals at random, without replacement, from this portfolio. An independent public accounting firm assures the auditor that the probability of randomly choosing two swaps in two choices, without replacement, is 1/2. If the auditor picks two option deals, then the trader will resign. The auditor has given up hope of catching the dirty trader in a firing offense. If the auditor accepts this bet, what is the probability that he will pick two options?

The Internal Auditor Considers a Bet (II) – Solutions (9/28//99)

A Baffling Prediction (5/28/99)

The head trader prepares a pile of confirms for forty derivative product confirms. The underlying risk factors are commodity prices, interest rates, equity prices and indexes, and currency exchange rates. None of the contracts have hybrid risk, hence, no contract has both commodity and currency risk, etc. In each category of risk, one contract has notional amount of one million dollars, one has two million dollars, etc., up to ten million dollars. Thus, exactly four contracts in the portfolio have precisely six million dollars notional, etc.

Under an internal auditor's careful supervision, the trader randomizes the order of the list of contracts, squares up the stack of paper, and places it face down on the table.

The trader then writes one contract's type of risk and size on a piece of paper, folds the paper carefully, and hands the paper to the internal auditor. The auditor places the the paper in a clean mayonaise jar, which he then sets in clear view on a nearby bookshelf.

The trader asks the auditor to remove any three numbers from one to ten, cut each of the corresponding rows from the spreadsheet and paste it in row one of a separate spreadsheet, then delete the blank rows from the first spreadsheet. The auditor should then cut the first seven rows from the first sheet, paste them on the same sheet, immediately below the rows containing the remaining contract information, and delete the seven blank rows.

For illustration, let's that the auditor has selected contracts with notional amounts of 4, 8, and 9 million dollars.

The trader has the auditor cut sufficient rows from the top of the first spreadsheet and paste them onto the other four sheets to bring the total in each sheet to ten, in the following sense. He pastes six rows below the four on sheet two, counting "5, 6, 7, 8, 9, 10." He pastes two rows below the eight on sheet three. He pastes one below the nine on sheet four. The auditor then deletes the blank rows at the top of the first sheet.

The controller now adds the values of the three contracts: 4 + 8 + 9 = 21. The trader now asks the controller to read the 21st row on the first sheet. The trader exposes the palm of his left hand.

Please help the internal auditor figure this out. Until he does, he won't be able to focus his attention on any possible shenanigans.

A Baffling Prediction Solution (6/28/99)

Stephen Gould turned in a correct answer on 6/8/98: "... [T]he trader peeked at the bottom confirm (easy to do) before he wrote down the confirm details, and the rest of the process is simply a way of turning up that bottom confirm.
Method: the 40 confirms are equivalent to a pack of cards missing royalty, i.e. A through 10. Take 10 cards, turn up 3, and put 7 at bottom. Pile is now 37 cards. Let the turned up cards have pip values x, y, z, totalling T. Number of cards dealt out on each card will be (10 - x, 10 - y, 10 - z), i.e., total 30 - T. Number of cards remaining in pile is now (37 - (30-T)), i.e., 7 + T. So counting down, the last card before the bottom 7 will be the Tth card automatically, and will therefore have been the bottom card at the beginning."

Raiders of the Lost Arb (3/28/99)

Eventually, Indiana Jones tired of archaeology and outsmarting evil characters, both civilized and uncivilized. He decided to become an options arbitrageur and settled into trading from an office in the eastern time zone. One Wednesday, precisely at noon, he called his New York dealer and found that he could buy a one-week call option (i.e., it expired the following Wednesday at noon) on Sony shares at a 75 vol. (Trivia: Sony owns TriStar, which produced "Raiders of the Lost Ark".) While he had New York on the line, he called his Chicago dealer and found that he could sell the same one-week call option on Tri-Star shares for $10,000 more. Both options had the same strike, were on the same stock, and expired at noon, dealer's local time. Quickly he sold an option to Chicago and bought the obviously corresponding option from New York, netting $10,000 on the difference in premium. Now, all he had to do was exercise his New York option optimally.

One week later, when the New York market opened, he checked his Bloomberg and saw that the options were $100,000 in the money. He kept an eye on the stock price and it didn't move until five minutes before noon, his local time, when Chicago called and notified him that the buyer had exercised his option. Immediately (at 11:56 a.m., his local time), he called New York and exercised his option. However, New York told him that he was late, and that his option had expired, already.

1. Why was the New York option cheaper than the Chicago option?
2. How did he miss the New York expiration?
3. Where was Indiana Jones's home office? (Hint: The Metropolitan Statistical Area's population is greater than 1,000,000 people (1990 US Census Data).)

Raiders of the Lost Arb – Solution (5/28/99)

Christopher Merrill supplied the first correct solution: "Neat problem– Indy's trading room in (appropriately enough) Indianapolis just became a temple of doom. Most of Indiana (including Indianapolis) does not observe daylight savings time– it's permanently on EST. (Exception: NW corner of Indiana is under Central Daylight Time, just as Chicago is.) When the option was written, just before the first Sunday of April, Indy's clock coincided with his NY dealer's clock and was one hour ahead of the Chgo dealer's clock. By expiration day, the DST time change moved both the NY clock and the Chgo clock one hour ahead. To Indy's surprise and despair, his local time was Chgo local time at expiration. So the Chgo option was one hour longer lived than the NY option, hence the extra option premium. But, of course, this would be the case regardless of Indy's local time zone– the puzzle is more interesting with the Indiana time twist.– Christopher Merrill

Richard Wendt supplied a subsequent, correct solution. He adds: "I'm in Philly now, but I spent four years at Notre Dame in the mid-60's, plus frequent trips back for football games. Indiana has always had confusing time zones and they've changed the rules several times over the years. My most confusing experience was driving from South Bend to Niles, Michigan (just over the border) and finding a different time. Forget about options expiration dates, dinner reservations were more confusing."

Harry Lorenzo of New York also solved the problem and adds: "This is a daylight savings problem. And a problem where Indy was careless about his contract specs."

The Case of the Treacherous Trader (1/28/99)

Once upon a time, the head of the internal audit department, with the power to fire a dishonest trader on the spot (This is a fairy tale.), sent his loyal lieutenant to ferret out corruption by auditing his firm's four regional departments: North America, Western Europe, Asia/Pacific, and Emerging Markets. Each region traded European call and put options, FRAs, and vanilla interest rate swaps. After completing a global investigation, the ferret had informed internal audit that one of the traders in one of the departments was deliberately using a a faulty pricing model that was off by one basis point of notional value, and had promised a complete report in the morning. However, the ferret hadn't mentioned which department was committing the fraud, which derivative pricing model was in error, or whether the error was to overprice or underprice the product. To make things worse, the ferret had gone missing and the police suspected foul play.

The internal auditor went to the risk management committee and proposed calling in an expert in model risk to review the company's systems and find the problem over the next month. The trading desks all agreed that this would hurt morale, disrupt the desks, tie up technology support, expose company secret technology, lead to bad publicity, and cost too much money, and that the firm couldn't afford to let the process drag on for a month. They argued convincingly that the expert from afar should limit his investigation to pricing only one portfolio on one day, base his conclusion on whatever that told him, file his report within 48 hours, and remove his annoying person from company property immediately after that – if not before.

The internal auditor managed to delegate to the expert the authority to specify any portfolio he wanted to price, and use any inputs he wanted.

A day later, the expert had found the problem in Western Europe, with overpriced long European call positions. How did he do it?

The Case of the Treacherous Trader – Solution (9/28/99)

Western Europe's value for this portfolio came out $100 higher than the other three values. If the value were $100 lower, then the system would underprice the long European calls. A different error in the system would have led to a similarly distinctive error in portfolio value.

The Case of the Villainous Vendor (1/28/99)

The new trader for Western European European call options was trying to find software to replace the faulty software that had cost his predecessor his job. He had received demo copies of nine systems. His chief assistant had thoroughly tested the software, but not written down a report, as was tradition at the firm. Then a competitor hired the assistant away at twice his previous salary and a handsome guaranteed bonus for the first two years. As the assistant went out the door, he shouted to his boss, "They all work fine, except for one that overprices long European call options." The trader didn't catch the vendor's name as the assistant disappeared into the elevator, just before the doors shut.

As the trader set to work to find out which system overpriced the long European calls, he got a shock. The assistant had set up the software in a crazy way, so he couldn't get out the price of a single swap with a single system. Instead, the computer would let the user compare two groups of systems. He could pick one product and two groups of systems at a time, hit the "Run" button, and the computer would return two numbers – each the sum of the prices produced by the systems in one of the groups.

After a little thought, the trader figured out how to find the bad system with just two computer runs. How did he do it?

The Case of the Villainous Vendor – Solution (2/28/99)

John Davidson of Deutsche Bank sent in a correct solution, almost by return e-mail:

"First run...
Compare the results of two groups of three systems. If the results are the same then the system that overprices is one of the three not tested. If the results are different then the system that overprices is one of the three in the group that returned the higher result.

"Second run...
Consider the three candidate systems. Compare the results of any two systems. If the results are the same then the system that overprices is the one that has not been tested. If the results are not the same then the system that overprices is the one that returned the highest result."

Harry Lorenzo, Espen Gaarder Haug, and Kishore Laud sent in essentially the same solution, a few days or weeks after Mr. Davidson. Espen Gaarder Haug wins bonus points with his key, supplementary advice:
"Remember not to tell the Software Company about the error. Try instead to get the customer list from the Software Company that sells the error system. You have just got yourself into European option arbitrage trading!"

Derivative Games^TM from 1999

The Dilemma (11/28/99)

All's Fair (11/28/99)

All's Fair – Solution (4/28/00)

All's Fair – Solution (5/28/00)

Sanity Check (9/28/99)

Sanity Check – Solution (11/28/99)

A Golden Opportunity (9/28/99)

A Golden Opportunity – Solution (11/28/99)

How many ways can I appreciate thee? (7/28/99)

How many ways can I appreciate thee? – Solution (9/28/99)

Client Pays Ultimate Price for Bad Choice (6/28/99)

Client Pays Ultimate Price for Bad Choice – Solution (7/28//99)

The Internal Auditor Considers a Bet (I) (6/28/99)

The Internal Auditor Considers a Bet (I) – Solution (7/14//99)

The Internal Auditor Considers a Bet (II) (6/28/99)

The Internal Auditor Considers a Bet (II) – Solutions (9/28//99)

A Baffling Prediction (5/28/99)

A Baffling Prediction Solution (6/28/99)

Raiders of the Lost Arb (3/28/99)

Raiders of the Lost Arb – Solution (5/28/99)

The Case of the Treacherous Trader (1/28/99)

The Case of the Treacherous Trader – Solution (9/28/99)

The Case of the Villainous Vendor (1/28/99)

The Case of the Villainous Vendor – Solution (2/28/99)

Derivative GamesTM from 1999

The Dilemma (11/28/99)

All's Fair (11/28/99)

All's Fair – Solution (4/28/00)

All's Fair – Solution (5/28/00)

Sanity Check (9/28/99)

Sanity Check – Solution (11/28/99)

A Golden Opportunity (9/28/99)

A Golden Opportunity – Solution (11/28/99)

How many ways can I appreciate thee? (7/28/99)

How many ways can I appreciate thee? – Solution (9/28/99)

Client Pays Ultimate Price for Bad Choice (6/28/99)

Client Pays Ultimate Price for Bad Choice – Solution (7/28//99)

The Internal Auditor Considers a Bet (I) (6/28/99)

The Internal Auditor Considers a Bet (I) – Solution (7/14//99)

The Internal Auditor Considers a Bet (II) (6/28/99)

The Internal Auditor Considers a Bet (II) – Solutions (9/28//99)

A Baffling Prediction (5/28/99)

A Baffling Prediction Solution (6/28/99)

Raiders of the Lost Arb (3/28/99)

Raiders of the Lost Arb – Solution (5/28/99)

The Case of the Treacherous Trader (1/28/99)

The Case of the Treacherous Trader – Solution (9/28/99)

The Case of the Villainous Vendor (1/28/99)

The Case of the Villainous Vendor – Solution (2/28/99)

Derivative Games^TM from 1999