DSE 2010

[poonam]

suppose there is only one future period with two states s1 or s2. Return on a company share is 5 in s1 and -1 in s2. Future return on a government bond is 1 independent of state. A third assets offers 3 in s1 and 0 in s2. price of stock =3 and price of bond=1. If the price of the new asset rules out the possibility of arbitrage, what is its price?

Please let me know how to do this question. Shouldn't the price depend on the probabilities of the future states ??

[Amit Goyal]

This is how you do it. And this will also show why it does not depend on probabilities.

Return vector on a share of a given company is (5,-1)
Return vector on a bond is (1,1)
If you hold a portfolio of s shares and b bonds then your return from the portfolio is (5s+b,-s+b) where 5s+b is the return in state s1 and -s+b is the return in state s2. Solving for the portfolio of the above two assets which is equivalent to the third asset in terms of the returns it generates i.e. (3,0), we get b=s=0.5.
Since the prices of the stocks and bonds are 3 and 1 respectively, cost of this portfolio is 2. And by no arbitrage condition this must also be the price of the third asset.

[poonam]

thank you so much !!

[SHIKHA]

WHY DID WE EQUATE RETURNS OF S1 AND S2 TO 3 AND 0 RESPECTIVELY...PLZ EXPLAIN...

[Amit Goyal]

By no arbitrage condition, two assets giving the same return must cost the same. Now a portfolio of (0.5 bond, 0.5 stock) also generate a return vector (3, 0) (this is the reason why we equated). We know the cost of this portfolio is 1(0.5)+(3)(0.5) = 2  (because the cost of stock is 3 and cost of bond is 1). This third asset also generates a return vector (3, 0) so it must also cost 2 (by no arbitrage condition).