302 Mathematics for Finance
11.9Using formula (11.5) and the short rates given, we find the following structure
of bond prices:
B(2,3; uu) = 0. 9931
B(1,3; u) = 0. 9859 <
/B(2,3; ud) = 0. 9926
B(0,3) = 0. 9773
\ B(2,3; du) = 0. 9924
B(1,3; d) = 0. 9843 <
B(2,3; dd) = 0. 992311.10It is best to compute the risk-neutral probabilities. The probability at time 1
of the up movement based on the bond maturing at time 3 is 0.76, whereas the
probability based on the bond maturing at time 2 is 0.61. The present price of
the bond maturing at time 2 computed using the prices of the bond maturing
at time 3 and the risk-neutral probabilities computed from these prices is
0 .9867. So, shorting at time 0 the bond maturing at time 3 and buying the
bond maturing at time 2 will give an arbitrage profit.
11.11At time 2 the option is worthless. At time 1 we evaluate the bond prices by
adding the coupon to the discounted final payment of 101.00 at the appropriate
(monthly) money market rate: 0.521% in the up state and 0.874% in the down
state. The results are 101.4748 and 101.1213, respectively. The option can
be exercised at that time in the up state, so the cash flow is 0.1748 and 0,
respectively. Expectation with respect to the risk-neutral probabilities of the
discounted cash flow gives the initial value 0.06598 of the option.
11.12The coupons of the bond with the floor provision differ from the par bond at
time 2 in the up state: 0.66889 instead of 0.52272. This results in the following
bond prices at time 1: 101.14531 in the up state and 100.9999 in the down
state. (The latter is the same as for the par bond.) Expectation with respect
to the risk-neutral probability gives the initial bond price 100.05489, so the
floor is worth 0.05489.