Derivative Games^TM from 1998

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Commonality (12/28/98)

A salesman named Les is the host of a special party at The Grind After the Grind for some friends from the office – Art, Lance, Megan, and Roger. Their occupations are, not in this order, compliance officer, programmer, risk manager, and trader. Cande R. is dancing in Roger's lap. Les says, "I brought us together, tonight, because I discovered that each of us has the same relationship between his or her name and his or her occupation."

After a few minutes, Lance says, "If that's the case, would you pay for Mattie Lup Roset to come over and join us for a while?"

Megan blows up. "That does it. Tomorrow morning, I file a sexual harassment suit."

What is each person's occupation?

(Based on an idea in Marilyn Vos Savant, "Ask Marilyn", Parade, 1/17/99.)

Commonality – Solution (3/28/98)

We thank Aravind Rajagopalan for the answer to "Commonality".

"Each person's name is a part of what they do:
ART                  TRAder
MEGAN            risk MANaGEr
LANCE             CompLiANcE officer
ROGER             pROGRammEr
(I cannot say what the other name is, though, not online!!!!)"

Cande R. is a Lap DANCER, from northern Finland. Mattie Lup Roset is a MALE PROSTITUTE, hired to meet an Equal Employment Opportunity Commission quota. – Ed.

A Perpetual Double Barrier Option – Revised (9/28/98)

Thanks to Peter Carr for inspiring the following Perpetual, "European", Double Barrier Call Option:

The market rate of interest is eight percent.
Let S₀=100 denote the underlying price at inception. The stock's dividend yield equals three percent. Its volatility is eleven percent.
The lower barrier is ten percent below S₀, initially, and grows at eight percent.
The upper barrier is 15% above S₀. ].
The strike price (K_t) equals the lower barrier, at all times. The payoff upon exercise equals Max[0, S_t-K_t]. No voluntary, early exercise.
If the underlying price hits either barrier, the option expires immediately. If the option is in the money, exercise is automatic. The option's rebate upon striking either barrier equals (e^d ^t-1) S_t.
For simplicity, all interest rates, growth rates, and dividend yields are quoted as continuously compounded rates.

What's the value of this option? How do you prove it?

[Thanks to Mark Lake for pointing out why the previous version of this problem (5/28/98) had no "simple" solution. Apologies to Peter Carr for making it sound as though the precise problem was his. He posed a similar problem with a "simple", but not obvious, solution. – Dr. Risk]

Smiling at Barrier Options (7/28/98)

Suppose that the underlying stock price is 100, dividend yield is zero, the market's Black-Scholes-Merton smile structure is completely flat at 20%, and the rate of interest is two percent as far as the eye can see. You need to price a six month, down-and-out call option, struck at 95. Consider two cases:

You have only a BSM pricing model for an ordinary call option. How can you price the D&O call? What principle allows you to do that?
You have a fancy model that allows you to input an entire "smile" structure of BSM implied vols. What smile do you input? How does that help you?

Links to Recreational Math Sites (6/28/98)

Centre for Innovation in Mathematics Teaching. The University of Exeter's center.
University of Toronto. Puzzles.
Stanford University. Recreational Math Problems.

Delta for a Compound Option (4/30/98)

Dear Dr. Risk – I'm trying to build a computer model to calculate risk of a compound option for my MBA thesis. I completed a program to calculate the cumulative bivariate normal distribution which is used to price a compound option (Geske, 1979). However, I'm having trouble taking partial derivative of the cumulative bivariate normal distribution to derive its delta risk. Do you know if there is a closed form solution for this? Thanks – Hpwally

Dear Hpwally – Yes, I know. – Dr. Risk

Who can help Hpwally? If there is a closed form solution, what is it? If there isn't one, why not? [Hurry, before the end of the term. – Dr. Risk]

Anniversary Celebration! (1/1/98)

We thank Pedro Silveira Assis, fund manager at Caixagest S.A. in Lisbon, Portugal, for suggesting the underlying puzzle for which the following challenge is a derivative.

To celebrate the first anniversary of the "Derivative Games" page, Dr. Risk has rented the Belasco Theater in New York and invited 1000 of the world's most elite Derivatives players to enjoy an evening of Derivative Games and magic. The evening begins with an effect by famous magicians Penn & Teller. Initially, Penn and three standard light switches (A, B, and C) are on one side of a curtain. Teller and three fancy (e.g., 12.1" screens and active matrix display) Dell Latitude LM notebook computers (1, 2, and 3) on the other side. Each light switch controls the power to exactly one computer. All computers are initially off.

Penn can operate any two – but only two – of the three switches, although he can turn those two on and off as many times as he wants over the course of his time on stage. While Penn futzes with the switches and entertains the audience with his wild patter, Teller juggles sharp knives and gargles with razor blades to keep the trick from growing tiresome. Once Penn's through with the switches he must go on to the other side of the curtain, where he must match each computer with the switch that operates it. Once he's passed to the other side of the curtain he cannot go back, and therefore cannot touch the switches again. How can he perform this task without no help from Teller or anyone else, and with no risk of being wrong?

Anniversary Celebration! – Solution (1/1/98)

We thank Stinson Gibner, a member of Enron's ECT research group in Houston, Texas, for his solution to this Derivative Game:

"All three computers are initially off. One is turned on and is allowed to finish the rebooting process while Penn and Teller captivate the audience. Then Penn turns on the other switch that he is allowed to operate and steps to the other side of the curtain. He can then identify each of the three computers with the switches. One remains off. Another is in the process of rebooting, and the third has completed its reboot."

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Derivative GamesTM from 1998