STOCHASTIC

3. MODELS OF OPTION STRATEGY
Reprinted from THE JOURNAL OF POLITICAL ECONOMY
Vol. LXXIV, No. 2, April 1966
Copyright 1966 by the University of Chicagc
Printed in U.S.A.
THE VALUE OF THE CALL OPTION ON A BOND*
GORDON PYE
University of California, Berkeley
ALMOST all corporate bonds and some gov-
/\_ ernment and municipal bonds have
call provisions. This call provision gives the
issuer the option of buying back his bonds
from whomever is holding them at a stated
time and price. The call price is typically
above par and declines as maturity approaches.
Frequently, call is not permitted
until several years after the bond has been
issued.
The call option has value to the issuer for
several reasons. In the future the borrower
may wish to remove restrictions placed on
him by the bond indenture. For a corporation
these might be restrictions on merger,
the sale of assets, or the payment of dividends.
Without the call provision, the bondholders
by hard bargaining might be able to
utilize their monopoly position to extract a
large premium from the issuer before selling
back the bonds or agreeing to change these
clauses.
A second source of value is that the borrower
may find that he wants to decrease
the amount of his borrowing before the
bonds mature. Essentially the same effect as
retiring his own bonds could be obtained by
buying on the market similar bonds issued
by someone else. However, because of the
transactions costs involved in making interest
payments, the cost of bonds to the
issuer will always be somewhat greater than
their value on the market. There will therefore
be some saving to the issuer in retiring
his own bonds rather than buying someone
else's.
The third and probably most significant
source of value of the option to the issuer is
the ability it gives him to refinance the issue
* Financial support was provided by the Institute
of Business and Economic Research of the
University of California, Berkeley.
in the future if interest rates should fall.
The analysis of this paper will be confined
to considering this aspect of the value of the
option. Two models are considered. In one
the future price of the bond is assumed to
be known and in the other the probability
distribution of the future price is assumed
to be known and to obey certain conditions.
Optimal policies for exercising the option
are given for both cases and expressions for
the value of the option are derived. 1
VALUE OF THE OPTION UNDER CERTAINTY
First, consider the case where future bond
prices and interest rates are known with
certainty. Suppose that p is the market
price of a non-callable issue with the same
coupon and maturity as a callable issue.
Suppose the call price of the latter is c.
Suppose that p is greater than c, that transactions
costs can be neglected, and that the
issuer is too small to affect the market. If
the issuer calls the bond at c and issues the
otherwise identical non-callable bond at p,
he makes a profit of p — c. His contractual
obligations with respect to the future payment
of interest and principal remain unchanged.
Therefore, the present value of the
interest and principal payments are unchanged
and independent of when the option
is exercised. The value of the option
to the issuer is, of course, dependent on
when it is exercised.
The optimal time to exercise the option
will be the time at which the present value of
p — c is at a maximum. The value of the
option to the issuer will be this maximum
1 A technical discussion of the value of the call
option on a bond has previously been given by
Crockett (1962). The approach taken in this paper
is different from hers and additional results are
obtained.
3. MODELS OF OPTION STRATEGY 547