STOCHASTIC

202 GORDON PYE
Given that p,- is the one-period interest
rate in t + 1, the maximum expected present
value of the option in I + 1 will be the
larger of the following two quantities: (a)
the value of exercising the option in I + 1,
which is pt+u — Ci+i; or (A) the maximum
expected present value of exercising the option
in a period subsequent to t + 1 which
is Fi+iy. If pi is the one-period rate in I, the
probability that p in I + 1 is p, is qt,: Therefore,
given that p in t is p., the expected
maximum present value of the option in
1+ 1 is
n
/, la max(/>i+i; — ct+\,Vt+u).
)-i
The maximum expected present value in £
of exercising the option subsequent to t
which is Vn must be the maximum expected
present value of the option in / + 1 discounted
for the one-period rate of interest
in t. Therefore, V n satisfies the relation
X max. (pt+u ^ ct+i ,V,+u)
(»=1, 2, ...,n).
(4)
For a bond maturing in T periods the value
of exercising the option subsequent to T is
clearly zero. Therefore, one has that Vr i =
0 for ally. Given the pn, the V,< can be calculated
by iterating backward using equation
(4) and this initial condition. 8
The pn may be calculated iteratively in
a similar manner. If p turns out to be p,- in
1 + 1, a bondholder in I will have in ( + 1
an asset worth r + pt+u where r is the cou-
• A dynamic programing approach similar to
that used here has previously been used by Karlin
(1962) in studying the optimal time to sell an asset.
However, he has not applied his model to options
nor considered uncertain discount rates which are
basic to the bond option problem.
Bachelier (1900), Kruizenga (1956), and Boness
(1964) have studied the value of a call option on
stock. However, they have not studied the multiperiod,
sequential decision case or the case of uncertain
discount rates which is relevant for bonds.
pon payment. Given that p in Ms pi, the
expected value of the bond in ( + 1 will be
r + 2 9nPi+n •
,_i
Discounting this for the interest rate between
/ and t + 1 gives the expected present
value of the bond in t or pi