STOCHASTIC

(d) Show that
p(.t) = exp(-{T-1) R(t)) [ 1 + r J T exp((T-T) R(X)) rfr] .
Consider the special case in which p(t) is linearly strictly decreasing and SO in [0, T], and
suppose that i?(0) = r. Suppose also that/>(0) = 1; that is, the bonds are issued at par.
(e) Show that pit) is strictly increasing in some closed interval [0, 0, and
strictly decreasing in some closed interval [t2, T],t2< T.
(f) Show that any stationary point of p(t) must be a strict local maximum ofp(-).
(g) Show that/>(-) has one and only one stationary point in (0, T), and this point is the
unique global maximum of p(-) in [0, T\. Let this point be tm.
(h) If p"(0) g 0, show that p"(t) < 0 for all / 6 (0, T]; thus pi.-) is strictly concave in
(0,7/].
(i) If p"(Q) > 0, show that there exists /0 e (0, /„) such that p{•) is strictly increasing and
strictly convex in (0, t0), and strictly concave in (t0, T].
(j) Show that (h) holds iff rg2//j(0).
[Hint: In (e)-(j). analyze the differential equation (c) directly, rather than the solution (d).]
Referring to the Pye paper, suppose that c(t) is a nonincreasing convex function, and
c(D = l.
(k) Show that Pye's equation (2) is a sufficient as well as necessary condition for
optimality of a call at time t.
11. Referring to the article by Pye concerned with call options on bonds. Suppose
n = 5, ip1,p2,p3,pi,p!,) = (0.04,0.045,0.05,0.055,0.06), T=4, r = $50, pTJ, = $1000,
c = (d, a, c3, d) = (1080,1060,1020,1000), and
Q =
0.6
0.2
0
0
0
0.3
0.5
0.1
0
0
(a) Utilize Eq. (5) to calculate the p„.
(b) Calculate the vtl via Eq. (4).
(c) Describe and interpret the optimal policy.
(d) Referring to the Appendix, show that each row of Q dominates the preceding row.
(e) Show that Ac, S 0 but A 2 c, is not nonnegative.
(f) Suppose c = (1080,1040,1020,1000). Show that Ac, S 0 and A 2 c, a 0.
(g) For this new value of c, calculate the critical values associated with the proposition
in the Appendix.
(h) Describe and interpret the optimal policy using the results in (g).
12. Referring to Exercises ME-7 and 8 formulate the call option problem (Exercise 11):
(a) as a pure entrance-fee problem, and
(b) as a linear programming problem.
0.1
0.2
0.8
0.1
0.1
13. Referring to Pye's paper in Chapter 2 of this part, suppose that the maximum possible
asset price increases and decreases are unequal; that is, a=£ b.
(a) Show that an optimal nonsequential minimax policy for asset selling is to sell amounts
s, at t = 0,1,..., T, where
0
0.1
0.1
0.6
0.2
0
0
0
0.3
0.7
26 . . , . „ . , (a~b)T+a + b
s, = , 0SlSr-l, and sT = .
(a + b)(T+\) (a + b)(T+l)
COMPUTATIONAL AND REVIEW EXERCISES 669