Suppose that there are two securities RAIN and SUN. RAIN pays $100 in there is a

Question

Suppose that there are two securities RAIN and SUN. RAIN pays $100 in there is any rain during the next world cup soccer nal. SUN pays $100 in there is no rain. Suppose that the world cup soccer nal 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 payo after 1 year, depending on the weather?

(b) What does the No-Arbitrage Condition imply about the price of a 1-year zero-coupon bond? (Assume no trading costs.)

(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 prot. (Assume no trading costs.)

(d) Suppose that trading zero-coupon bonds is costless, but trading RAIN and SUN each cost $2 per $100 face value. Can you still make an arbitrage profit?

Explanation / Answer

a) If u buy 1 share of RAIN and 1 share of SUN, then your pay after 1year is $100, no matter what the weather is.

b) A portfolio of 1 share of RAIN and 1 share of SUN replicates the payo of a zero-coupon bond, that is, it has sure payo of $100 just like a zero. Therefore, the zero must have same price as the portfolio, namely $93.

c) If the 1-year zero-coupon bond is trading at $90 then you can earn a riskless arbitrage prot as follows. You buy 1 zero-coupon bond, short-sell 1 RAIN, and short-sell 1 SUN. This gives you a prot of $23 + $70 - $90 = $3 today, and your net cash ow next year is 0, no matter what the weather is.

d) If trading RAIN and SUN each cost $2 per share then the above strategy has a trading cost of $4. This is the grater than the value of the mispricing $3. Therefore, you cannot make a arbitrage prot.

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