108156.pdf

264 Mathematics for Finance

• Use dealerAto change 1 dollar into 1. 58441 ∼= 0 .6312 pounds.


• Use dealerBto change 0.6312 pounds into^00 ..^63126401 ∼= 0 .9861 euros.


• Use dealerAto change 0.9861 euros into 0. 9861 × 1. 0202 ∼= 1 .0060 dollars.
  The arbitrage gain will be about 0.0060 dollars.
  1.4We want b othxS(0) andyA(0) to be equal to a half of the initial wealth. This
  givesx80 = 5,000 andy100 = 5,000, sox=62.5andy= 50. The value of
  this portfolio at time 1 will be

V(1) = 62. 5 S(1) + 50A(1) =

{

11 ,750 if stock goes up,

9 ,250 if stock goes down,

and hence the return on this portfolio will be

KV=

{

0 .175 if stock goes up,

− 0 .075 if stock goes down.

Now we can compute the expected return

E(KV)=0. 175 × 0. 8 − 0. 075 × 0 .2=0. 125 ,

which is 12.5%, and the risk

σV=

√

(0. 175 − 0 .125)^2 × 0 .8+(− 0. 075 − 0 .125)^2 × 0 .2=0. 1 ,

that is, 10%.

1.5The following strategy will realise an arbitrage opportunity. At time 0:

• Borrow $34.


• Buy a share of stock for $34.


• Enter into a short forward contract with forward price $38.60 and delivery
  date 1.
  At time 1:


• Sell the stock for $38.60, closing the short forward position.


• Pay 34× 1 .12 = 38.08 dollars to clear the loan with interest.
  The balance of 38. 60 − 38 .08 = 0.52 dollars will be your arbitrage profit.
  1.6Suppose that a sterling bond promising to pay£100 at time 1 is selling forx
  pounds at time 0. To findxconsider the following strategy. At time 0:


• Borrow 1. 6 xdollars and change the sum intoxpounds.


• Purchase a sterling bond forxpounds.


• Take a short forward position to sell£100 for $1.50 to a pound with
  delivery date 1.
  Then, at time 1:


• Cash the bond, collecting£100.


• Close the short forward position by selling£100 for $150.


• Repay the cash loan with interest, that is, 1. 68 xdollars in total.
  The balance of all these transactions is 150− 1. 68 xdollars, which must be
  equal to zero or else an arbitrage opportunity would arise. It follows that a
  sterling bond promising to pay£100 at time 1 must sell forx= 1150. 68 ∼= 89. 29
  pounds at time 0.