B. Solutions to the problem sets
Solution 3.3 (a ) A risk-free bond costs β =
249 αs = 0.9803 = ρ −1 ; thus,
(b ) α ̃ = (1.02)α = (12.50%, 25.00%, 37.50%, 6.25%, 18.75%).
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the risk-free interest rate ρ equals 2.0%.
(c) Using the decomposition idea, we get a price of
5 × α1 + 5 × α2 + 2 × α3 + 7 × α4 + 4 × α5 = 3.7375.
Solution 3.4 (a) Yes. There are five assets, just sufficient to span the five states at least potentially. There is no obvious collinearity between different rows or columns, so the payoff matrix appears to be regular. To test this hunch we use Excel’s MDETERM function (or the corresponding function of any other spreadsheet software) and find that the determinant of the payoff matrix is −24, so different from zero, indicating that r is indeed regular; and thus the market is complete.
(b ) By reverse decomposition (Box 3 . 11), we have α = q · r −1 . Inverting a 5 × 5 matrix is not trivial without the help of a computer. With a computer, however, it’s a snap. For instance, with Excel’s functions MINVERSE and MMULT you can easily compute the vector of Arrow prices. In this specific example it is
α = (0.1197, 0.2370, 0.3573, 0.0600, 0.1790).
(c) All Arrow prices are positive; therefore there are no arbitrage oppor-
tunities.
(d) The price of a risk-free bond equals β = s αs = 0.9530. The gross risk-free rate of return ρ is β−1 = 1.0493. Thus, in this example, the risk-free rate of return is 4.93%.
(e ) The risk-neutral probabilities are simply ρ times the Arrow prices, so α ̃ = 1.0493α = (12.56%, 24.87%, 37.50%, 6.30%, 18.78%).
(f ) The payoff of a call option is max{q − e, 0}, where e is the exercise price and q is the state-dependent price of the underlying asset. Applied to this example, a call option on a share of company Y with exercise price 5 gives rise to a state-dependent cash flow of (7, 3, 0, 0, 0).
The most straightforward way to compute the price of this option is by the decomposition idea (see Box 3 . 2). The price is simply the price of seven state 1 Arrow securities plus three state 2 Arrow securities,
price = 7 × 0.1197 + 3 × 0.2370 = 1.549.
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