Actuarial Outpost Which of the following creates an arbitrage oppertunity
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#1
08-11-2017, 12:46 PM
 nickeisenberg Non-Actuary Join Date: May 2017 College: UNLV Posts: 16
Which of the following creates an arbitrage oppertunity

I have a question from the second lesson from the ASM manual.

It says, The ask price for a share of ABC company is 100.5 and the bid price is 100. Suppose an investor can borrow at an annual effective of 3.05%. and lend at 3% annual effective. Assume there are no transaction costs and no dividends. Determine which of the following create an arbitrage opportunity.

1) Short one share, and enter into a long one-year forward contract on one share at price of 102.5

2)Short one share, and enter into a long one-year forward contract on one share at price of 102.75

3) Short one share, and enter into a long one-year forward contract on one share at price of 103

4)Buy a share with borrowed money, enter into a short one year forward contract on one share with forward price 103.6

5) Buy a share with borrowed money, enter into a short one year forward contract on one share with forward price 103.75

For 1, 2, and 3 im not really understanding the set up. The book says if we short a share and invest it at 3% then we have 103 at the end of the year and if the forward is less than 103 we profit. Using this logic it says 1 and 2 are arbitrages but 3 is not but in the book it defines an arbitrage to be an event when you either make a profit or loose nothing and in 3 we lost nothing so why isnt that an arbitrage.

Also I dont under stand the point of short selling a share of stock. If we short now we get 100 but we still need to pay back 100(1.03)^t when we buy back the share we shorted so its like that original 100 dollars wast even ours to begin with. In this problem we shorted and got 100 and time 0 and it turns into 103 and time 1 so we can use that, in 1 for example, to complete the forward contract and buy a share for 102.5 which gives us 103-102.5 = .5 leftover. But we still owe the 100 from the short with 3% interest.

Hopefully someone can clear this up for me. Thank you!

EDIT: I think I just answered my own question for the second part. We now own a share with .5 left over so to complete the short sell we just give back the share we own now and we are left with .5 in profit. But Why isnt 3 an arbitrage because we did not lose money?

Last edited by nickeisenberg; 08-11-2017 at 12:56 PM..
#2
08-11-2017, 01:31 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 9,192

For an arbitrage to exist there has to be a positive probability of profit and a zero probability of loss. In number 3 there is no chance of profit. When you short a share you buy it back at the prevailing market price, or forward price if you have locked in. You receive cash at the time of the transaction.

When you buy on the margin as in 4 and 5 you have to pay back the amount borrowed with interest.
#3
08-11-2017, 02:01 PM
 nickeisenberg Non-Actuary Join Date: May 2017 College: UNLV Posts: 16

Thank you, theres another short question that hopefully you can explain as well. It says determine which of the following has the same cash flows and a short stock position.

So if we short the stock we get money now but need to buy back a share later.

The answer to this was short foward and short zero-coupon bond. Is this because if we short the zero coupon bond we get money now and need to buy back a bond later but at that same time we get money for shorting the forward and then later owe money for buying back a share from the short forward. So its like at time 0 we get money, at time 1 we owe money for buying back the bond but also get money from the short forward so they cancel out, and at time 2 we owe money for buying a share from the short forward contract? So in the end it seems like we only had cash inflow at time 0 and cash out flow at time 2?
#4
08-11-2017, 02:18 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 9,192

Shorting the forward involves no cash flow at time 0. The cash flow at time one is the forward price determined at contract inception minus the market price of the stock. This can be either positive or negative. The forward price under all of the assumptions of the problem should be equal to the maturity value of the 0 coupon bond (which was designed to generate the current price of the stock at time 0).

The proceeds from the bond sale are therefore equal to the short sale proceeds at time 0, and the total cash flow from the short forward and loan payoff replicate the required repurchase of the share.
#5
08-11-2017, 02:59 PM
 nickeisenberg Non-Actuary Join Date: May 2017 College: UNLV Posts: 16

There is a cash flow at time 0 though right because we short the 0 coupon bond?
#6
08-11-2017, 04:30 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 9,192

Quote:
 Originally Posted by nickeisenberg There is a cash flow at time 0 though right because we short the 0 coupon bond?
Yes, or alternatively you could look at it as taking out a loan.

 Tags forwards, short