Question

1. Company A is currently suing company B for IP infringement. Their stock prices currently are pA = $10 and pB = $8. If A wins the lawsuit, its stock will go up to $15 and B’s stock will drop to $5. If A loses, however, its stock will be worth $9, and B’s stock will rise to $10. Neither stock pays dividends. Assume no arbitrage and consider whether A wins or not to represent the only two states of the world.

a. Calculate the Arrow-Debreu security prices, qA-wins and qA-loses.

b. If there is a riskless asset in the economy that pays $1 in both states, what would the price of such asset be? What would be the riskless interest rate?

c. Now suppose there actually is a “riskless bond” in the economy that offers a 5% interest rate between now and when the outcome of the lawsuit is announced. Is there an arbitrage opportunity? If so, how might you trade to exploit it? (Note the asset we priced in part b is not assumed to exist, nor are Arrow Debreu securities.)

Answer 1

(a.)

Let q₁ and q₂ are two state of prices.

q₁ = qA wins

q₂ = qA loses

The payoff structures of the two stocks as well their current prices :

15q₁ + 9q₂ = 10 -------------- equation 1

5q₁ + 10q₂ = 8 -------------- equation 2

solve both equation mathematically

multiplying equation 2 by 3 we get,

3 * ( 5q₁ + 10q₂ = 8 )

15q₁ + 30q₂ = 24   -------------- equation 3

subtract equation 1 from equation 3

15q₁ + 30q₂ = 24  - (15q1 + 9q2 = 10)

15q1 + 30q2 = 24 -15q1 - 9q2 = -10

0 + 21q₂ = 14

21q₂ = 14

q₂ = 14 / 21

q₂ = 0.67

by putting value of q₂ in equation 2 we get ,

5q₁ + 10 * 0.67 = 8

5q₁ + 6.7 = 8

5q₁ = 8 - 6.7

5q₁ = 1.3

q₁ = 1.3 / 5

q₁ = 0.26

State prices are q₁ = qA wins = 0.26 and q₂ = qA loses = 0.67

(b)

State prices as calculate above,

Price of riskless asset (P_f ) = q₁ + q₂

= 0.26 + 0.67

= 0.93

Riskless interest rate (R_f ) =

  

=  0.0753 Or 7.53%

(c.)

Yes, there is an arbitrage opportunity, because riskless bond in overpriced today.

the price of a riskless bond = 1 / (1 + 0.05) = 0.95

0.95 is higher than 0.93 (as above calculated)

For arbitrage opportunity one would like to short the riskless bond and long the stocks.

One could long 5 units of stock A and 7 units of stock B , and short 120 units of riskless bond.

According to this strategy profit of 120 * 0.95 - ( 5 * 10 + 7 * 8 ) = 114 - 106 = 8 today and nothing happen in next period.