108156.pdf

Solutions 301

0 .64% which allows us to findk(2,3; ud) = 0.20%.The other missing returns

can be computed in a similar manner, firstk(1,3; d),thenk(2,3; dd).

11.3The bond prices are given in Figure S.11.

Figure S.11 Bond prices in Solution 11.3

11.4The money market account is given in Figure S.12. Note that the values for

the ‘up’ movements are lower than for the ‘down’ movements. This is related

to the fact that the yield decreases as the bond price increases, and our trees

are based on bond price movements.

Figure S.12 Money market account in Solution 11.4

11.5The pricesB(1,2; u) = 0.9980 andB(1,2; d) = 0.9975 are found by discounting

the face value 1 to be received at time 2, using the short ratesr(1; u) and

r(1; d). The priceB(0,2) = 0.9944 can be found by the replication procedure.

11.6At time 2 the coupons are 0.5227 or 0.8776, depending on whether we are in

the up or down state at time 1. At time 1 the coupon is 0.9999.

11.7At time 1 we find 18.0647 = (0. 8159 ×20 + 0. 1841 ×10)/ 1 .0052 in the up

state and 1.7951 = (0. 1811 ×10 + 0. 8189 ×0)/ 1 .0088 in the down state. Next,

applying the same formula again, we obtain 7.9188 = (0. 3813 × 18 .3928 +

0. 6187 × 1 .7951)/ 1. 01.

11.8There is an arbitrage opportunity at time 1 in the up state. The price

B(1,2; u) = 0.9924 implies that the growth factor in the money market is

1 .00766, whereas the prices of the bond maturing at time 3 imply growth fac-

tors 1.01159 and 1.00783. To realise arbitrage, bonds with maturity 3 should

be bought, the purchase financed by a loan in the money market.