Derivative GamesTM from 1997
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The Meaning of Arbitrage (VI) – Forward, into the Futures! (11/30/97)
Lots of people have trouble understanding the relationship between forward and futures contracts and prices, and whether they can arb differences in the two.
- If interest rates are nonstochastic, what's the simplest example that you can produce that illustrates the identity of forward and futures prices? How would you arb a difference between them?
- If interest rates are stochastic, what's the simplest example that you can produce that illustrates the difference between the two prices? What could happen if you tried to arb the difference between them?
The Meaning of Arbitrage (V) – Horsing Around (9/30/97)
Martin Baxter and Andrew Rennie (Financial Calculus, Cambridge, Cambridge University, 1996) illustrate clearly how arbitrage pricing works in the context of betting on a horse race. Here are two problems, based on their illustration.
#1
In a special race at B.T. Lemon Race Track only two horses are running: Ant Fanny and Brother Willie. The fix is in, and two coin tosses will determine the race's winner: Ant Fanny will win if both coins come up heads (p=1/4); otherwise, Brother Willie will win (p=3/4). However, betters have placed $5 million on Ant Fanny and $10 million on Brother Willie. The track has quoted odds of 13-5 against Ant Fanny and 15-4 on Brother Willie. (That means, a bet of $5 on Ant Fannie will win $18 = $13 + $5, and a bet of $15 on Brother Willie will win $19 = $15 + $4.)
- How much will the track profit if Ant Fanny wins?
- How much will the track profit if Brother Willie wins?
- What is the implied probability that Ant Fanny will win?
- What is the implied probability that Brother Willie will win?
- What do the implied probabilities add up to, and how do you interpret that?
#2
Same problem as #1, but now the track quotes odds of 9-5 against Ant Fanny and 5-2 on Brother Willie.
- How much will the track profit if Ant Fanny wins?
- How much will the track profit if Brother Willie wins?
- What is the implied probability that Ant Fanny will win?
- What is the implied probability that Brother Willie will win?
- What do the implied probabilities add up to, and how do you interpret that?
The Meaning of Arbitrage (V) – Horsing Around – Solution (10/21/97)
Congratulations to Joe Spivack, Vice President in the Corporate Risk Management Department at Prudential Insurance Co., for the correct solution to these problems.
#1
- Profit = -$3,000,000 = $15 M - $5 M x (5+13) / 5 = $15 M - $18 M
- Profit = $2,333,333 = $15 M - $10 M x (15+4) / 15 = $15 M - $12.667 M
- Prob (Ant Fanny wins) = 5 / 18 = 27.8%
- Prob (Brother Willie wins) = 15 / 19 = 78.9%
- Each implied probability exceeds the fair probability, and the implied probabilities add up to 106.7%, which one might think would indicate an edge for the bookie. Indeed, the bookie's expected gain is $1,000,000 = -3,000,000 x (1/4) + 2,333,333.33 x (3/4). However, the bookie has a 25% chance of losing three million dollars.
#2
- Profit = $1,000,000 = $15 M - $5 M x (5+9) / 5 = $15 M - $14 M
- Profit = $1,000,000 = $15 M - $10 M x (5+2) / 5 = $15 M - $14 M
- Prob (Ant Fanny wins) = 5 / 14 = 35.7%
- Prob (Brother Willie wins) = 5 / 7 = 71.4%
- The probabilities add up to 107.1%. This time, the 7.1% = 1/14 is the $1,000,000 guaranteed profit as a percentage of the guaranteed winnings of $14,000,000.
Instant Derivatives Math (8/26/97)
Estimation is important in Derivatives calculations, even if you have a desk full of workstations that give answers to ten decimal places. You don't care about the last five decimal places, if you've got a handle error. Every good trader I've worked with had amazing intuition. Maybe he'd know that his American Option pricing model was giving weird numbers. Maybe he could price some weird Derivative Product on the back of an envelope, within ten percent of my price, using some approximation that sounded intuitively sort of plausible, but also somewhat far-fetched.
See if you can solve these problems in your head and explain your thought process:
#1
Andy's IT guys just delivered a new front office system for him to test. Before he takes delivery, he says, "Let's see if it can price a simple European ATM Call on 100 shares of stock that's selling at 5/8. It's dividend yield is two percent. It's vol is 40.11. The interest rate is four percent. Make the option real short-term, 1/100 of a year. What's the price?"
Almost before he stopped speaking, the IT guy said, "The price is $1.10."
Immediately, the trader said, "Not good enough! Way off! This system's worthless!"
What are the right numbers? How did the trader know the system's outputs were wrong?
#2
Barry's IT guys just delivered a new front office system for him to test. Before he takes delivery, he says, "Let's see if it can handle a European Cross Currency Call on 375 French francs, put on 100 Deutschemarks. Dollar-mark's 1.60. Dollar-Paris is six francs. Make it real short-term, 1/100 of a year. Assume that mark interest rate is four and the franc interest rate is six percent. Let the dollar interest rate be 14%. The cross vol is 40.11. What's the kappa for the cross vol?"
Almost before he stopped speaking, the IT guy said, "The kappa is 2.25."
Even before the IT guy stopped speaking, the trader said, "Not good enough! Way off! This system's worthless!"
What is the right number? How did the trader know the system's output was wrong?
Instant Derivatives Math – Solution (9/28/98)
 |
Espen Gaarder Haug of Tempus Financial Engineering – the author of the popular The Complete Guide to Option Pricing Formulas (New York, McGraw-Hill, 1997) – submitted correct solutions to these problems. |
#1
"The value is approximately $1. This value can easily be
calculated in your head or on the back of a Playboy magazine as I
did. The value of an ATM call or put (ATM-forward) option is
approximately equal to
0.4*S*vol*sqrt(T)*number of shares.
So the value must be close to
0.4*5/8*0.4*10,
which is approximately equal to $1 .(The very short time to
expiration should make the spot approx. = to the forward price so
the approx. should work fine).
In other words I sell the option to the IT guy at $1.1 ."
#2
"This is hard for a retired bond option trader. But let
me try:
The spot cross is 6/1.6=3.75 FRF/DEM, so the option is about
at-the-money forward. So the kappa=zeta=vega can be found by the
approximation [Differentiate the approximation in #1 with
respect to vol. – Ed.]
0.4*S*sqrt(T) = 0.4*3.75*0.1.
Multiply times 100 marks = 15 francs or $2.5."
[Close enough for government work, but not for managing risk.
– Ed.]
The Twilight Zone in Space and Time (7/26/97)
#1
Bobby trades credit derivatives for a large commercial bank in New York. His girlfriend called him at his apartment on the Upper West Side. He said, "I can’t talk now. It’s Thursday night. I’m in the middle of watching Seinfeld – a new episode. Then I want to watch some other stuff. Call me Friday morning at 11:00, Sweetheart.
His girlfriend replied, "One, you're an insensitive jerk, and I wouldn't call you if I could. Two, I can’t call you then. It’s already Friday at 11:15."
Explain reason # 2. (Guys, bonus points for explaining #1. Gals, no bonus points for you. Sorry, but it's too easy for you.)
#2
One bright, sunny day, Bobby was traveling from his vacation home in the Hamptons to the local grocery store. He saw that old Officer Smith had a problem. Officer Smith has two signs for directing traffic. The red sign means stop, and the green one means go. Neither sign contains any writing. He has been using them for years and everybody understands what they mean, without words, with only colors. Suddenly, he dropped them. He is too old and disabled to pick them up himself, so he asked his son, "Will you please hand me the red one. Don’t bother with the green one, because I have a new green one." However, the son picked up both of them, because he couldn’t tell them apart. The son is not color blind.
What was the problem?
#3
Bobby decided to go sky diving and witnessed something horrifying. A man jumped out of an airplane without a parachute. He fell 10,000 feet without dying. Yet, five minutes later rescuers found him dead.
How did he survive the drop?
How did he die?
#4
Bobby read a weird story on "My Excite Channel." A man ate a hearty supper. He woke up hungry after a few hours and asked his wife to fix him something to eat. She said she’d fix him a large breakfast as soon as the sun rose. She kept her promise, but he starved before he could eat it.
What happened?
The Twilight Zone in Space and Time – Solutions (12/17/97)
Congratulations to Pedro Silveira Assis, fund manager at Caixagest S.A. in Lisbon, Portugal, for sending us his solutions – most elegantly expressed – of parts 1, 3, and 4.
#1
"Reason #2 is that Bobby's sweetheart was calling him from her luxurious apartment in Tokyo, where it was 14 hours later. The sun was high and it was the middle of Friday's morning there, while it was 45 minutes before Thursday's midnight in Montevideo, Uruguay, and 9:15 p.m. Thursday evening in New York city. Reason #1 is related to the fact that she's a woman. I'm not, so I wouldn't have a clue." [Said like a true guy. – Ed.]
#2
"Bobby was horrified as he watched the man jumping from an airplane flying at an altitude of 12,000 feet. This man wore no parachute, so he dropped like a stone. He had already dropped more than 10,000 feet and was cursing and blaming his memory, as he frequently forgot important details. Bobby couldn't watch this anymore and covered his eyes. Moments later, the unfortunate, free-falling man hit the ground and met his maker."
#3
Officer Smith, who directed traffic for the Harbor Patrol, dropped the signs into deep water. His son dove to the bottom to retrieve the red sign. However, at that depth the water above had absorbed so much of the sunlight that he couldn't distinguish between red and green.
#4
"This poor, unfortunate man was the victim of his own lack of bargaining power. He woke up in the middle of the night, in his house at the North Pole. It was a cold winter night and his wife didn't want to get up just then. In fact she was a bit fed up with him because he did the same thing day in and day out. He accepted her promise but forgot that nighttime is a long time in this situation. She knew she could wait a few months and still keep her promise and did just that." [Let's hope Santa Claus doesn't make this mistake. – Ed.]
Celia in Derivativesland (7/9/97)
Carroll Lewis, a don at Christ Church College, Oxford, runs a hedge fund on the side. He specializes in buying and selling short-dated average price currency options, for which the underlying price is the average of the daily closing exchange rates over the life of the option. For example, recently, one Friday he bought a one week, Average Price Call Option that expires on the following Friday, and that pays off the greater of zero and the average of the five daily closing prices.
Also, Lewis has an interest in child photography. Namely, he has a studio where he takes pictures of little girls. From time to time he employs one of those little girls, Celia, age 10, as a derivatives quant. Unfortunately, Celia is mathematically challenged and is still counting blocks and building (sometimes elaborate) figures with them. Fortunately, Celia is highly reliable when asked to perform tasks within her range.
Lewis has devised a childishly simple, nonalgebraic algorithm for computing a key component of the correct volatility to plug into a Black-Scholes option calculator, so he can price his Average Price Options. Celia carries out this algorithm flawlessly. Lewis takes her output, performs a single arithmetic operation on it, then takes the square route of that operation's result, and multiplies it by the volatility for the currency.
What is Carroll Lewis's algorithm for Celia? (Assume for simplicity the usual Black-Scholes framework, and that only business days matter, ignoring weekends and holidays.)
Celia in Derivativesland – Solution (2/14/99)
Espen Haug supplies a correct solution that requires nothing more from Celia than the four basic arithmetic operations, plus squaring.
"I am not sure if my algorithm is simple enough, but since Celia not longer is 10 [We posted the problem circa 7/9/97 – Ed.] in but at least 11 or probably 12 years old I believe she can handle it. I am also wondering if she is Asian or American? If she is Asian I am sure she can handle it, if she is American she could have big problems understanding it. [Warning: Although your statement may be factually correct, it is not politically correct. While it does not disturb me, it may limit your dinner invitations in some parts of Liberal America. – Ed.] Well all she need to do is taking the number of days in the period and add that number togheter twice as many times as the number of days in the period, then add the number of days in the period three times more, then add 1 to that number. Then Mr. Carroll Lewis have to divide it all by 6*days^2, then take the square root of it and multiply it by the spot volatility. More precisely :
AverageVolatility=SpotVolatility * sqrt[(2n^2+3n+1)\(6n^2)]
Where n is the number of days/fixings. The formula is based on the assumption that the next fixing is one day (fixing period) away (no weekends, including weekends wold make the algorithm a bit more "complex"). The formula works excellent for discrete geometric averages, and should also be good enough for Government work on discrete arithmetic average options. [Perhaps the BIS allows this calculation for computing capital requirements. – Ed.] For a very large n the formula converges to the volatility of a continuous time geometric average rate volatility:
SpotVolatility /sqrt(3)."
While Dr. Risk can’t really quarrel with this algorithm, it might be asking a bit much, even for most little Asian girls. Maybe not too much for many little Korean girls! Dr. Risk's algorithm is simple enough for even a seven-year-old American girl.
First, the background, which is not so simple:
A = (S1 + S2 +
… + S5) / 5
= [(S0 + DS1)
+ (S0 + DS1
+ DS2) + … + (S0
+ DS1 + … +
DS5)] / 5
= (5 S0 + 5 DS1
+ 4 DS2 + 3
DS3 + 2
DS4 + 1
DS5)
/ 5
T = 5 Dt, Var(DSi) = S2 ´ sS2 ´ T, and S ~ A
Var(A) = A2
´ sA2
T = A2
´ sA2
´ 5
´ Dt
= [25 Var(DS1)
+ 16 Var(DS2)
+ 9 Var(DS3)
+ 4 Var(DS4)
+ 1 Var(DS5)
] / 25
» (25 + 16 + 9 + 4 + 1) S2
´ sS2
´ Dt /
25
Hence,
A2 ´ sA2 ´ 5 ´ Dt » (25 + 16 + 9 + 4 + 1) S2 sS2 ´ Dt / 25
and
sA2 » (25 + 16 + 9 + 4 + 1) sS2 / (5 ´ 25) = sS2 ´ 55 / 125
Here's where Celia gets involved. Carroll asks Celia to
construct two figures from blocks and count the blocks involved:
1. Construct a big cube that has 5 little cubes along an edge. It
contains 125 little cubes.
2. Construct a big pyramid that has a base of 5´5 little cubes, a layer of
4´4 cubes on top of that, etc., with one
small cube at the peak. That has 25 + 16 + … + 1 = 55 little
cubes in it.
Even a 10-year-old American girl ought to be able to handle that. My second-grade daughter (age 7) was able to perform it this morning between finishing breakfast and leaving for school. Okay, I helped a bit.
Then, Carroll multiplies the underlying variance by the ratio of the pyramid’s volume to the large cube’s volume. In the limit, as n --> ¥ , this ratio goes to 1/3, as we can see from Espen Haug's expression and we know from high school geometry. The volatility is the square root of the variance.
Down and Out in NY (6/5/97)
Don Lufkin, Jr. is puzzled. An important customer wants him to quote a market for a four-month, Down-and-Out Call Option with a linearly declining rebate. The underlying asset is a liquid stock index, with dividends reinvested, currently at 7000. His systems are down, so he can't find the index's implied or historical volatility. However, he can recall that this index tends to move smoothly, without significant discontinuities. The option's Lower Barrier is 6300. The rebate starts at 21 and declines linearly over time to zero when the option expires. The rate of interest is one percent per annum.
Don has fifteen minutes to quote a price for this option. That's not enough time to build or modify a pricing model. If he's willing to make the usual Black-Scholes assumptions, how can he price that option and manage the risk?
The Bernoulli Option (5/14/97)
Christopher Merrill of Chicago contributed today's game.
Joe is a new trader in the S & P 500 futures options pit. It's a slow day, so he strolls back to a relatively unpopulated spot near the pit to read the WSJ. He gets a funny feeling that someone's staring at him – he looks up and sees an older, eccentric-looking trader, whose acronym is "BER". His name is Dave Bernoulli – he's quite well-known among the locals. Everyone calls him Bernie. He rarely trades and carries no sheets nor computing devices of any kind, except that one day he carried an abacus to the pit, just to be funny. The locals have learned to not be on the other side of Bernie's infrequent trades, which tend to be big winners.
"Psst. Hey buddy, wanna buy an option?" Bernie says, trying to be funny.
"Huh?" Joe says, "The pit's over there, dude."
Bernie looks disdainfully at the other traders, milling about, a few debating loudly about secondary reactions to Greenspan's latest cautionary remarks. "Dull stuff – and even duller today."
Bernie makes a point to know the background of his fellow traders, so Bernie also knows that Joe has a Harvard MBA and specialized in quantitative finance. Bernie continues: "You seem like a bright guy – if you're sporting, you should try to price my option."
Joe's eyes widen with a dawning realization ... he's looking for a side-bet!
"Uh, no dice man ... look I better get back – "
"Suit yourself," he interrupts, "but I have an unusual option you might be interested in ... here's the deal: I'll tell you what the option is, how it pays off, etc., and you think about how you'd compute its value. If you can come back to me in a couple of days with the correct fair value, delta, and how you'd go about hedging it, I'll sell you the option for one-half of that fair value. Then you can hedge away the risk and your risk free profit is as good as money in the bank."
"How 'bout just giving me the one-half fair value in cash?" Joe says, half-jokingly.
Bernie continues, unamused. "I'll pay the proper payoff according to the following rules. If you can compute the value and delta, you should be able to hedge it with SPUs (S&P futures) for a risk-less profit...at my expense."
"Okay. I'm listening," says Joe.
Bernie describes his option: "The option's underlying futures contract is the S&P futures price for the contract with delivery thirteen months from now. Let F(t) denote its price at time t. Let dx be the tick size, the minimum dollar amount by which the futures price may change. The moment after I write the option, we record the current futures price, X=F(0).
"The option pays off dx if the next tick in F is down. The payoff is 2dx if F ticks up, then down. The payoff is 4dx if F ticks up twice consecutively, then down. The payoff is 8dx if F ticks up three times consecutively, then down. In general, the payoff is (2^n)*dx dollars, where n is the number of consecutive up ticks – after we record F(0) – before the first downtick.
"The option expires in T years (the future expires in at least T years, of course) if it's still alive (no downticks). At that point it would pay (2^((F(T)-X)/dx))*dx dollars, as you'd expect. The option writer – that's me – must pay immediately after the first downtick – we don't wait until expiration to settle up.
"When you come back to me with the fair value and delta, you should be able to tell me, in words, four things:
- Which binomial model, geometric or arithmetic, you chose for the underlying and why.
- What is the riskless hedging strategy – if it exists?
- What is the option's fair value?
- What would happen to the theoretical value in the continuous time limit, as dt approaches 0?"
Pretend that you're Joe. A correct solution to this problem has five parts:
- Answer Bernie's four questions correctly.
- Explain why you would or wouldn't accept Bernie's offer to buy his option at ½ fair value.
While you work, use the following assumptions:
- a constant unconditional annual variance of s2,
- a known and constant, annual riskless rate of interest
- all of the other usual Black-Scholes assumptions, except continuous trading
- a binomial model is appropriate due to the tick size, dx.
- each period of length dt the price moves at most one tick up or down.
The Meaning of Arbitrage (IV) (4/14/97)
Tex was in Ciudad Juarez, Mexico with a 100 dollar bill in his pocket. He moseyed across the Rio Grande river bridge to El Paso, sauntered into a tavern, bellied up to the bar, and ordered a beer. The bartender poured it and said, "That'll be a dollar." Tex handed over the hundred. The bar tender asked, "Don't you have anything smaller?"
"Sorry," said Tex.
"How about if I give you this 1000 peso note in change?" asked the bar tender.
"Okay," said Tex. He took the note and drank the beer.
Tex moseyed across the Rio Grande bridge to Ciudad Juarez, sauntered into a cantina, and ordered a beer. The bar tender poured it and said, "That'll be a ten pesos." Tex handed over the 1000-peso note. The bar tender asked, "Don't you have anything smaller?"
"Sorry," said Tex.
"How about if I give you this 100 dollar note in change?" asked the bar tender.
"Okay," said Tex. He took the note, drank the beer, and returned to El Paso.
Tex made ten more identical round trips before he collapsed in Juarez in an alcoholic coma. Who paid for Tex's beer?
The Meaning of Arbitrage (IV) – Solution (7/9/97)
Congratulations to Phaedon Sinis, who graduated recently from Dartmouth and submitted even more recently the first solution to "The Meaning of Arbitrage (IV)". His solution, slightly edited, follows:
"Because Tex exchanges $99 for 1000 pesos in Texas, and 990 pesos for $100 in Mexico, he is in effect buying dollars at 9.9 p/$ and selling dollars at 10.10 p/$. Tex is buying dollars at a lower price in Mexico and selling them at a higher price in the U.S. The bartenders are on the other side of the transactions, so they are contributing toward his beer. However, Tex is performing work for that money, namely, moving currency from where it is cheap to where it is dear. Thus, Tex is paying for his beer with his labor. Tex is doing currency arbitrage.
"At a more detailed level, Tex isn't doing currency arbitrage, but only currency speculation. Note that the peso/$ rate is volatile and may change while Tex is running back and forth.. For example, the bartender in El Paso might say, 'If you give me the hundred dollar bill plus $1, I'll give you a beer plus a 1000 peso note.' The bartender in Juarez might say, 'If you give me the thousand peso note plus 10 pesos, I'll give you a beer plus a $100 bill.' In such a case, Tex would return to El Paso with two beers in his belly, but minus $1 and 10 pesos."
The Two Hedge Fund Managers (4/6/97)
Two hedge fund managers – Tsouris and Uberlaufer – bet exclusively on moves in the S&P 500 Index. They express their views entirely through cash, Spiders, and three-month, ATM Call and Put Options on the S&P 500 Index.
At the close of business on the last trading day of 1996, both managers started with all their money in cash. Assume for sake of argument that the market and index were flat the entire first trading day of 1997, and for simplicity that markets are all frictionless (with no taxes or transactions costs) . Immediately after the market opened on 1997's day one, each manager invested half his fund's money in the maximum possible option position of either Calls or Puts. (We're not saying which.) Then, in the afternoon, just before the market closed, each manager invested the other half of his fund's money in the maximum possible position of either Calls or Puts. Thus, at the end of the day, each manager could have been long Calls, long Puts, or long Straddles.
The managers made their decisions independently, in every sense of the word. Nevertheless, as luck would have it, both ended up with at least some long position in Calls. In particular, manager Uberlaufer bought Calls in the morning, and may or may not have bought more after noon. Both managers held their positions until the options expired in March. At expiration, the market and index were both up significantly.
Now, please, based on the facts and assumptions that appear above, answer some questions about P&L and performance at expiration, as a percent of initial investment. As usual, you get all your points for your explanations.
- What is the probability that Tsouris bought winners in the morning and the afternoon?
- What is the probability that Uberlaufer bought winners in the morning and the afternoon?
- What is the probability that Uberlaufer outperforms Tsouris?
- What is the probability that Tsouris outperforms Uberlaufer?
- What is the probability that they perform equally well?
The Two Hedge Fund Managers – Solution (9/28/99)
This problem was more difficult than I imagined, at first, because I made an implicit assumptions, which I realized only upon coming back to it more than two years later and trying to solve it. Most important, I assumed that the managers picked their investments randomly. I hope that is obvious from the statement about making decisions "independently." Also, I made specific assumptions about probabilities of events. Nevertheless, a general solution exists, without making any specific assumptions.
Let C1 (P1) indicate that the manager purchased calls (puts) in the morning, and let C2 (P2) indicate that the manager purchased calls (puts) in the afternoon. Then the possible events for one manager are C1ÇC2, C1ÇP2, P1ÇC2, and P1ÇP2. Independence means that Pr{E1ÇE2} = Pr{E1}´Pr{E2}, etc., E = C or E = P. The fact that a manager had at least some calls in his portfolio means that he didn't choose puts in both the morning and afternoon, which rules out P1ÇP2. The fact that Uberlaufer bought calls in the morning rules out P1ÇC2, also. Let ~E mean that event E didn't happen.
- For Tsouris, the probability of picking calls in the morning and afternoon
is
Pr{C1ÇC2|~(P1ÇP2)} = Pr{(C1ÇC2)Ç~(P1ÇP2)} / Pr{~(P1ÇP2)} = Pr{C1ÇC2} / Pr{~(P1ÇP2)}. - For Uberlaufer, the probability of picking calls in the morning and
afternoon is
Pr{C1ÇC2|C1} = Pr{(C1ÇC2)ÇC1)} / Pr{C1} = Pr{C1ÇC2} / Pr{C1}. - Here, I'll assume that making decisions "independently, in every
sense of the word" means that the basic probabilities are the same for
the two managers. Then the problem boils down to computing the probability
that Uberlaufer buys calls in the morning and afternoon, given that he buys
them in the morning, and Tsouris buys puts in either the morning or
the afternoon:
Pr{C1ÇC2} / Pr{C1} ´ [1 - Pr{C1ÇC2} / Pr{~(P1ÇP2)}] - This is the probability that Uberlaufer buys calls only the in the morning
and Tsouris buys them morning and afternoon.
[1 - Pr{C1ÇC2} / Pr{C1}] ´ Pr{C1ÇC2} / Pr{~(P1ÇP2)} - This is the probability that both managers buy calls either only in the morning (as we know Uberlaufer did) or both in morning and afternoon, which equals1 - the sum of the probabilities in parts 3 and 4.
No one submitted a solution for this challenge, perhaps, because I didn't include some assumptions that would have made the problem simpler. Well, then, that opens up the possibility of making your own, convenient assumption to permit a solution.
For example, assume that each manager based his decision on the flip of a
coin, so the probability of purchasing a call (or a put) in the morning (or the
afternoon) is ½. Then
Pr{C1} = Pr{P1} = Pr{C2} = Pr{P2} =
½.
and
Pr{E1ÇE2}
= ½ ´
½ = ¼, etc.
With these probabilities in hand, things are simpler than the general
case:
- Pr{C1ÇC2|~(P1ÇP2)} = 1/4 / 3/4 = 1/3
- Pr{C1ÇC2|C1} = 1/4 / 1/2 = 1/2
- 1/2 (1 - 1/3) = 1/2 ´ 2/3 = 1/3
- (1 - 1/2) ´ 1/3 = 1/6
- 1/2 ´ 2/3 + 1/2 ´ 1/3 = 1/2
Monte Hell (1/30/97)
Monte had made a market in exotic equity derivatives every day that the market was open for several years. His "signature product" was an option on the maximum of the thirty stock prices in the Dow-Jones Industrials. For fun, on weekends, Monte competed in sports car rallies.
During his last rally – in the Catskills – Monte got lost early on and took a few chances to get back on schedule. Unfortunately, he skidded off a mountain road and rolled down the mountainside to a fiery demise. Monte found himself in a hot and inhospitable resting place, because of a few less-than-candid comments to some elderly sellers of correlation, as well as a few credit card receipts for hotel rooms for "Mr. and Mrs. Montana", dated during his wife's annual trips to see her mother in South Philadelphia.
The Devil – a sporting man – offered Monte a chance to gamble his way out of his hole. Here are the rules:
- Immediately after the market closed on day one, Monte
would guess which stock in the Dow-Jones Industrials
would have the largest percentage price gain the next
day.
The Devil would cut Monte off from all price information for the next 24 hours.
Immediately after the market closed on day two, the Devil would – excluding Monte's pick and the top performer from consideration – identify two of the other 28 or 29 stocks.
Monte could then stick with his original guess or switch to one of the remaining 27 stocks.
If Monte guessed right, the Devil would transport him promptly to heaven, where St. Peter had agreed to admit him.
For example, suppose that Monte picked AT&T, which may or may not have been the best performer. The next day the Devil tells Monte accurately that neither General Electric nor General Motors was the best performer. Monte could keep AT&T as his pick, or switch to one of the 27 unmentioned stocks, such as IBM or Sears.
Now, here are the questions for you to answer:
- What is Monte's probability of winning if he doesn't switch?
- What is Monte's probability of winning if he does switch?
- Explain your answers in 1 and 2.
Monte Hell – Solution (3/10/97)
Congratulations to Christopher Merrill of Chicago, who sent the first solution, which follows, edited slightly.
- "Assuming that Monte has lost his touch in picking the blue chips, his probability of seeing Heaven if he stays with AT&T is 1/30 (approx. 3.33%).
- "Poor Monte doesn't have much of a chance of seeing the Pearly Gates, but if he switches, the probability of seeing Heaven is 29/30 * 1/27 (approx. 3.58%).
- "The key to this problem is recognizing that Monte has a 29/30 probability of being wrong on his first guess. In other words, the probability that the winner is in the remaining 29 is 29/30. So when the Devil reveals two definite losers from that 29, the probability that the winner is in the remaining 27 is still 29/30. The condition probability of picking the right one of those stocks, given that the winner is in the remaining 27 (i.e., that his first pick was wrong) is obviously 1/27. The joint probability of switching and winning (what we want) equals the product of those two probabilities. To see that the info is valuable, consider the situation where the Devil reveals – not two – but 28 losers, leaving just AT&T – Monte's pick – and one other stock. Clearly he'd want to switch to this very special stock the Devil didn't mention. Then, his probability of winning would be 29/30 = 29/30 * 1/1."
You may have noticed that this puzzle resembles the classic "Monte Hall" television show puzzle that has entertained so many puzzlers for too many hours. Mr. Merrill did. He adds:
"By the way, as you probably know, the Monte Hell problem is a neat variant of a problem I saw years ago in Marilyn Vos Savant's (Ms. 200-something IQ) Sunday-supplement column. In her problem, we have a prize behind one of three closed doors – I choose one door, then the game show host (Monte Hall of course!) opens a certainly-empty door and offers me a chance to switch to the remaining door.
"Marilyn argued (correctly) that switching is always optimal (probability of winning doubles 1/3 to 2/3). The funny thing about this problem is that the solution is so counter-intuitive that she had many people, even some mathematicians, denying that switching is optimal and further criticizing her 'irresponsibly incorrect' answer as somehow indicative of the sad state of mathematics education in the U.S.!"
Mr. Merrill is right about the relationship between the Monte Hell and Monte Hall problems. When someone presented the Monte Hall problem to Dr. Risk's group at Bankers Trust Co., circa 1988 or 1989, three of us worked on it for longer than I care to say, using conditional probability theory, common sense, and Monte Carlo modeling. We all got the right answer, eventually! So you can solve this problem many ways.
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