As we saw in the previous chapter, important assumptions of the
Black and Scholes application are
- constant risk free domestic interest rate
(approximately fulfilled by stock options with life spans of
9
months)
- independence of the price of the option's underlying from
the interest rate
(empirical research shows that for stocks
this is approximately fulfilled).
Both assumptions are violated for bond options and the longer time periods which are typically found in
the analysis of these options.
A bond produces at the time of maturity
a
fixed amount
, the nominal value, and if applicable at
predetermined dates before
dividend payments (coupon).
If there are no coupons, the bond is referred to as a zero coupon
bond or zero bond for short.
When valuing a Bond Option coupons can be treated as discrete
dividend payments when valuing stock options.
To begin we calculate the bond value
at time
with a
time dependent but known interest rate
.
From the assumption of no arbitrage we conclude that a bond's
change in value over the time period
with possible
coupon payments
coincides with the change in value of a
bank account with a value
and with an interest of
Together with the boundary restrictions
it
follows that:
 |
(10.1) |
with
, the antiderivative of
For a zero bond this simplifies to:
Due to the uncertainty associated with the future development of
the interest rate,
is modelled as a random variable. In
order to have an unambiguous, fixed interest rate, one usually
considers the interest rate of an investment over the shortest
possible time period:
spot rate = Interest rate for
the shortest possible investment.
does not follow a geometric Brownian motion so that the
Black-Scholes application cannot be used. There are a number of
models for
, that are special cases of the following general
Ansatz which models the interest rate as a general
Itô Process:
 |
(10.2) |
represents as usual a standard Wiener process.
Three of the most used models are simple, special cases and indeed
are the coefficient functions of the models from
In general
and
can be conveniently
chosen, in order to replicate in the model empirically observed
phenomena. In the following we write
for
,
in order to clearly differentiate between the function
and the constant
, which appears as a
parameter in the three models mentioned above.
A stock option can be hedged with stocks, and Black and Scholes
use this in deriving the option pricing formula. Since there is no
underlying financial instrument associated with a bond, bonds with
varying life spans have to be used to mutually hedge each other,
in order to derive the equation for valuing bonds.
Consider a portfolio made up of a zero bond with a remaining life
time of
and
zero bonds (i.e.,
sold
zero bonds) with a remaining life time of
. The value of
the portfolio at time
for the current interest rate
is:
where
stands for the value function of both bonds. We
write
. Using Itôs
Lemma it follows that
By hedging the risks the random component disappears. This is
achieved by choosing
The insertion and comparison of the portfolio with a risk free
investment and taking advantage of the no arbitrage assumption,
that is the equality of the change in value of the portfolio and
investment:
produces altogether
 |
|
|
|
 |
|
|
(10.6) |
This is only correct when both sides are independent of the
remaining life times
.
therefore satisfy both of the following differential equations
for the function
which is independent of one of the
remaining life times. With the economically interpretable value
this produces with the abbreviation
the zero bond's value equation for
 |
(10.7) |
with the boundary restrictions
at the time of
maturity and with additional boundary restrictions dependent on
. It should be noted that in the equation
stand for functions of
and
.
The value
has the following interpretation.
Consider a risky portfolio, that is not hedged, consisting of a
bond with the value
For the change in value
within the time period
we obtain using Itôs Lemma and the
zero bond's value equation:
Since
the mean change in value
is
above the increase in value
of a risk free
investment.
is therefore the bonus on the
increase in value, which one receives at time
with a current
interest rate
for taking on the associated risks.
is thus interpreted as the market price of
risk.
Consider the special case:
Inserting the solution assumption
into the zero bond's value equation results in the two equations
with boundary restrictions
(since
For the time independent
there is
an explicit solution, which with a remaining life time of
has the form
with
Choice of
parameters:
- 1)
- The spot rate volatility is
With this representation
can be
estimated from historical data, in a fashion similar to the
historical volatility of stocks.
- 2)
- Taking the yield
curve (see Section 10.4.3) into consideration, the
discussion of which goes beyond the scope of this section,
estimators for
and
can be derived.
As an example consider a European Call with a strike price
and
a maturity
on a zero bond with a maturity of
, i.e., the right is given to buy the bond at time
at a price
.

Value of the bond at time

with the current interest rate

Value of the Call at time

with the current interest rate
is only dependent on the random variable
and time
and therefore itself also satisfies the zero bond's value
equation, but with the boundary restrictions
This equation, analogous to the corresponding Black-Scholes
equation can be numerically solved.