STOCHASTIC

present value. Let ( be time, r the interest
rate, and V the present value of the option.
V will be given by
V=[p(t)-cU)}exp[-f'r(r)dr].(.i)
Setting the derivative of V with respect to
/ equal to zero gives as a necessary condition
for the optimal time to call the bond:
r{t)\p{i) - c(l)\ = p'(l) - c'(l) . (2)
In other words, the bond should be called
only if the relative rate of growth of the
profit from doing so is equal to the market
rate of interest. This condition should not be
surprising because formally the bond problem
is just like the classical capital theory
problem of when to cut down a tree or sell
a cask of wine.
This optimization condition can be used
to evaluate the effect of a shift in the schedule
of call prices on the optimum time for
call. In particular, assume that the schedule
of call prices is shifted up by the factor X so
that it becomes \c(t). Replacing c in the
optimization condition by \c, implicitly differentiating,
and assuming c'{t) < 0, gives
iL= c '~ rc >n (3)
d\ d{p'-\c'-r(p-\c)]/dl
K '
The denominator is negative from the second-order
condition on the maximum. Thus,
shifting up the call price schedule by a constant
factor will postpone call of the bond.
The analysis so far has been from the
viewpoint of the issuer rather than that of
the bondholder. The other side of the profit
of p — c obtained by the issuer when he
calls the bond is a loss of p — c inflicted on
the bondholder. When call occurs, the bondholder
receives only c for a future cash flow
of interest and principal which is, worth p.
The bondholder will assume that the
issue will be called at the optimal time from
the issuer's point of view. Thus, at any point
of time the present value of the interest and
principal payments to a bondholder on a
callable bond are reduced by the maximum
present value of p — c. If p* is the market
CALL OPTION ON A BOND 201
price of the callable bond and V the value
of the option, this means that p* = p — V.
The issuer of the bond will not call the
bond as long as the value of the option is
greater than the profit of exercising it immediately.
Thus, before the option is exercised,
it follows that V > p — c. Combining
this inequality with the fact that p* = p —
V gives p — p* > p — c or c > p*. Thus,
a callable bond will sell below its call price
before it is called. When it is called, its price
just becomes equal to its call price. In fact,
it follows that callable bonds will not sell
above their call price in any case. If a callable
bond were to sell above its call price,
the issuer could make a profit indefinitely by
calling the bond and reissuing an identical
callable issue. Bondholders realizing the
bond would be called at a loss to them if
its price were above the call price would
never pay more than the call price for the
bond.
A MODEL UNDER UNCERTAINTY
It will now be assumed that the future
interest rates and bond prices are uncertain.
However, their probability distribution will
be assumed to be known. Let time be divided
into periods and assume that the oneperiod
interest rate can take on a finite
number of possible values, pi, pi, ... , p„. In
particular, assume that the probability distribution
of the one-period rate in any period
depends only on the one-period rate in the
previous period. 2 The probability that the
one period interest rate is py given that it
was Pi in the previous period will be denoted
by qt,.
The value of future cash flows will be
assumed to be their expected present value.
Let pu be the price of a bond in the (th
period when p in the (th period is pi. Let Vu
be the maximum expected present value of
the call option in period / when p in J is pi
and when the option is required to be exercised
subsequent to period t.
2 Elsewhere it has been shown that this assumption
is rich enough to imply under suitable restrictions
several observable features of the term structure
of interest rates (Pye, 1966).
548 PART V. DYNAMIC MODELS