Solutions to Homework 6 FFM
Prof. Emiliano Pagnotta
Topic 7: Arbitrage
1. The stock PolarBear.com trades on both the South Pole Stock Ex-
change and the North Pole Stock Exchange.
(a) Suppose the price on the North Pole is $18. What does the No-
Arbitrage Condition say about the price on the South Pole? (As-
sume no trading costs.)
To rule out arbitrage, the price on the South Pole must be the
same as on the North Pole, $18.
(b) Suppose the price on the North Pole is $18 and the price on the
the South Pole is $17? How can you make an arbitrage profit?
(Assume no trading costs.)
You can make an arbitrage by buying on the the South Pole for
$17 and selling on the North Pole for $18. This transaction has a
riskless profit of $1 per share.
(c) Suppose that the price on the North Pole is $18, that buying or
selling on the North Pole costs $2, and that buying or selling on
the South Pole is free. What does the No-Arbitrage Condition say
about the price on the South Pole?
The price on the South Pole must be between $16 and $20. If the
price is lower than $16 then you can make an arbitrage profit, net
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, of trading costs, by buying on the South Pole and selling on the
North Pole. Similarly, you can make an arbitrage if the price is
higher than $20.
2. Suppose that there are two securities RAIN and SUN. RAIN pays $100
in there is any rain during the next world cup soccer final. SUN pays
$100 in there is no rain. Suppose that the world cup soccer final is 1
year from today (although this is not true), and suppose that RAIN is
trading at a price of $23 and SUN is trading at a price of $70.
(a) If you buy 1 share of RAIN and 1 share of SUN, what is your
payoff after 1 year, depending on the weather?
If you buy 1 share of RAIN and 1 share of SUN, then your payoff
after 1 year is $100, no matter what the weather is.
(b) What does the No-Arbitrage Condition imply about the price of
a 1-year zero-coupon bond? (Assume no trading costs.)
A portfolio of 1 share of RAIN and 1 share of SUN replicates the
payoff of a zero-coupon bond, that is, it has a sure payoff of $100
just like a zero. Therefore, the zero must have the same price as
the portfolio, namely $93.
(c) Suppose that a 1-year zero-coupon bond is trading at $90. Show
how you would set up a transaction to earn a riskless arbitrage
profit. (Assume no trading costs.)
If the 1-year zero-coupon bond is trading at $90 then you can earn
a riskless arbitrage profit as follows. You buy 1 zero-coupon bond,
short-sell 1 RAIN, and short-sell 1 SUN. This gives you a profit
of $23+$70-$90=$3 today, and your net cash flow next year is 0,
no matter what the weather is.
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Prof. Emiliano Pagnotta
Topic 7: Arbitrage
1. The stock PolarBear.com trades on both the South Pole Stock Ex-
change and the North Pole Stock Exchange.
(a) Suppose the price on the North Pole is $18. What does the No-
Arbitrage Condition say about the price on the South Pole? (As-
sume no trading costs.)
To rule out arbitrage, the price on the South Pole must be the
same as on the North Pole, $18.
(b) Suppose the price on the North Pole is $18 and the price on the
the South Pole is $17? How can you make an arbitrage profit?
(Assume no trading costs.)
You can make an arbitrage by buying on the the South Pole for
$17 and selling on the North Pole for $18. This transaction has a
riskless profit of $1 per share.
(c) Suppose that the price on the North Pole is $18, that buying or
selling on the North Pole costs $2, and that buying or selling on
the South Pole is free. What does the No-Arbitrage Condition say
about the price on the South Pole?
The price on the South Pole must be between $16 and $20. If the
price is lower than $16 then you can make an arbitrage profit, net
1
, of trading costs, by buying on the South Pole and selling on the
North Pole. Similarly, you can make an arbitrage if the price is
higher than $20.
2. Suppose that there are two securities RAIN and SUN. RAIN pays $100
in there is any rain during the next world cup soccer final. SUN pays
$100 in there is no rain. Suppose that the world cup soccer final is 1
year from today (although this is not true), and suppose that RAIN is
trading at a price of $23 and SUN is trading at a price of $70.
(a) If you buy 1 share of RAIN and 1 share of SUN, what is your
payoff after 1 year, depending on the weather?
If you buy 1 share of RAIN and 1 share of SUN, then your payoff
after 1 year is $100, no matter what the weather is.
(b) What does the No-Arbitrage Condition imply about the price of
a 1-year zero-coupon bond? (Assume no trading costs.)
A portfolio of 1 share of RAIN and 1 share of SUN replicates the
payoff of a zero-coupon bond, that is, it has a sure payoff of $100
just like a zero. Therefore, the zero must have the same price as
the portfolio, namely $93.
(c) Suppose that a 1-year zero-coupon bond is trading at $90. Show
how you would set up a transaction to earn a riskless arbitrage
profit. (Assume no trading costs.)
If the 1-year zero-coupon bond is trading at $90 then you can earn
a riskless arbitrage profit as follows. You buy 1 zero-coupon bond,
short-sell 1 RAIN, and short-sell 1 SUN. This gives you a profit
of $23+$70-$90=$3 today, and your net cash flow next year is 0,
no matter what the weather is.
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