108156.pdf

252 Mathematics for Finance

value of the following random cash flow: At time 2 we receive $20 in the

state uu, $10 in the states ud and du, and nothing in the state dd. No

payments are due at other times.

Exercise 11.8

Find an arbitrage opportunity for the bond prices in Figure 11.14.

Figure 11.14 Bond prices in Exercise 11.8

Exercise 11.9

Suppose that the risk-neutral probabilities are equal to^12 in every state.

Given the following short rates, find the prices of a bond maturing at

time 3 (with a one-month time step,τ= 121 ):

r(2; uu) = 8.3%

r(1; u) = 8.5% <

/r(2; ud) = 8.9%

r(0) = 9.5%

\ r(2; du) = 9.1%

r(1; d) = 9.8% <

r(2; dd) = 9.3%

The next proposition gives an important result, which simplifies the model
significantly.

Proposition 11.2

The lack of arbitrage implies that the risk-neutral probabilities are independent
of maturity.