Tough Question - for Experts only

[Arber]

Hi, I have the following tough scenario:

A gold merchant craves for more gold. He decides to trade gold certificates against gold ingots. He makes a million fair-coin flips. For each coin flip, if a Heads turns up, each gold certificate is worth 10 gold ingots. If a Tails turns up instead, each gold certificate is worth 1 gold ingot.

You have decided to trade the gold certificates. The question is: at what price (in terms of gold ingots) should you trade each gold certificate for, at time t=0? (at time t=1, 1 million fair coins are flipped)
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Calculation Method A:

At time t=1, for each of the 1 million fair-coin flips,
Heads: 1 gold certificate is worth 10 gold ingots
Tails: 1 gold certificate is worth 1 gold ingot

At time t=0, with expectation of approximately 500,000 Heads and 500,000 Tails,
Total
= 500,000 gold certificates worth 5 million gold ingots + 500,000 gold certificates worth 500,000 gold ingots
= 1 million gold certificates worth 5.5 million gold ingots

Value of each gold certificate to you is 5.5 gold ingots. Therefore, you are willing to pay up to 5.5 gold ingots for each gold certificate. (Equivalently, you are willing to pay up to 1 gold ingot for 2/11 of a gold certificate)
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Calculation Method B:

At time t=1, for each of the 1 million fair-coin flips,
Heads: 1 gold ingot is worth 1/10 of a gold certificate
Tails: 1 gold ingot is worth 1 gold certificate

At time t=0, with expectation of approximately 500,000 Heads and 500,000 Tails,
Total
= 500,000 gold ingots worth 50,000 gold certificates + 500,000 gold ingots worth 500,000 gold certificates
= 1 million gold ingots worth 550,000 gold certificates

Value of each gold ingot to you is = 550/1000 = 11/20 of a gold certificate. Therefore, you are willing to pay up to 11/20 of a gold certificate for each gold ingot. (Equivalently, you are willing to pay up to 1 gold certificate for 20/11 gold ingots)
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Puzzle:
From A: You are willing to purchase each gold certificate by paying up to 5.5 gold ingots.
From B: You are willing to sell each gold certificate by receiving at least 20/11 gold ingots (because you are willing to pay up to 1 gold certificate for 20/11 gold ingots).

THAT MEANS I CAN ARBITRAGE AGAINST YOU!!!

I will sell you gold certificates, with me receiving 5.5 gold ingots for each gold certificate sold to you.
Then I will purchase gold certificates from you, buying back all the gold certificates previously sold to you, paying you 20/11 gold ingots for each gold certificate.

Therefore, for each gold certificate that I sell and subsequently buy back from you, I earn (11/2 - 20/11) = (121/22 - 40/22) = 81/22 gold ingots.
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Any thoughts?

 

[Frankie]

It's a trick question, method B doesn't actually follow the statement:

For each coin flip, if a Heads turns up, each gold certificate is worth 10 gold ingots. If a Tails turns up instead, each gold certificate is worth 1 gold ingot.

The statement says that there is a 50/50 chance that a Certificate is worth either 1 or 10 Ingots. What method B assumes is a 50/50 chance that an Ingot is worth either 1 or 1/10 of a Certificate, which is a totally different probability distribution to the original statement, hence giving a different result.

There is no arbitrage here, only 2 different pricing conditions.


I wonder whether this post is a trick question to see who's the first to state the obvious... :P