6.2
Option Valuation under Certainty
Introduction

How
would you value a call option in a world with certainty, where stock prices
exhibit constant continuous growth?
The
answer to this question provides a useful introduction to the option valuation
problem in a world with uncertainty. To
start answering the question, we will need to introduce some notation.
Notation
X
= strike price of the option;
T
= expiration date of the
option;
S
= current market price of the stock;
ST
= price of the stock when the option expires; and
r
= riskfree interest rate
Continuous Time
The
return on a stock between periods t and t+1 is
given by
This
is the "continuously compounded" return, which means that if the stock
grew continuously between periods t and t+1 at rate m,
then St would grow to St+1.
In other words,
If
m is constant
over time, then we know the value of the stock price at the time of the option's
expiration, T .
or,
equivalently, the current stock price is the present value of the stock price at
the time of the option's expiration:
Certainty Option Pricing Model
In
our world, since the stock price at time T
is known, the terminal value of the option is also known.
From the definition of a call option, the call is exercised only if it
finishes inthemoney (i.e., the stock price at time
T exceeds the strike price).
Thus, it is either worth zero (and left unexercised), or it is
inthemoney and worth the difference between the stock price at time
T and the strike price. Mathematically,
this value is expressed as:
Call
Value = max {0, ST
 X}
Since
there is no uncertainty, we can find the current value of the call option: it is
simply the terminal value discounted back to today at the riskfree interest
rate:
Current
Call Price = PV( max {0, ST  X} )
And,
with continuous compounding at r, the
present value is:
If
the call option has zero value at time T,
then the call option has zero value at any other time. If the call finishes
inthemoney, then the current value is
We
know that if the growth rate of the stock is
r, then S is the present value of
the stock price at time T.
But if there is no uncertainty, and
r is the riskfree
interest rate, then S must grow
at rate r. Otherwise,
there is an arbitrage opportunity (if, for example, S
grows at a smaller rate, you should sell the stock and invest the proceeds in
the bond).
Therefore,
if the call finishes inthemoney, its value is
This
price has an easy interpretation. It
equals the stock price minus the present value of the exercise price.
In
Chapter 5, topic 5.2, Binomial Option Pricing:
NPeriods, you saw that this equation has the same structure as the binomial
model. The major difference is that
in the binomial model, the first and second terms are weighted by probabilities,
to allow for uncertainty.
Application of the Certainty Model
So
how well does the certainty model perform?
Consider the following data for call options trading on IBM, the price
information taken from FAST trading room at Carnegie Mellon University on
Friday, October 15,1993, at 14:55:43 (i.e., 2:55).
Last
traded price for IBM: 44 1/2
Current
bid price for IBM: 44 3/8
Current
ask price for IBM: 44 5/8
Yield
to maturity for a Treasury Bill with approximately the same life as the option:
2.867%.
Exdividend
date for IBM: 11/5
Dividend
payment date: 12/10
Estimated
IBM dividend during life of option: $0.25
Continuously
compounded yield to maturity for a tbill
maturing around this dividend payment date: 2.922%.
Maturity
date for November 1993 options: November 20.
Option
information from the CBOE for the November options at this same time is
presented in Table 6.1.
Table 6.1
Chicago Board of Exchange Data

November
Options 
 

Call 
 
Strike 
Bid 
Ask 

35

9 3/8 
9 7/8 

40

4 7/8 
5


45

1 1/16 
1 3/8 

50

1/4 
3/8 

We
will now apply the certainty model to this data.
The
first problem is that IBM will pay a dividend after the option's
expiration date, but the stock goes exdividend prior to the
November expiration date. This affects the stock price during the life of the
option because the stock price will fall by the amount of the
dividend on the exdividend date.
To
accommodate this problem, we will adjust the current stock price
by subtracting the present value of the dividend payment D.
The option holder is only concerned with the stock after
the exdividend date. Therefore,
=
0.25 x exp(.02922*58/365)
=
0.2488
The
present values of the strike prices are computed in Table 6.2.
Table 6.2
Present Values of Strike Prices
Strike 
PV 

35 
34.901 

40 
39.887 

45 
44.873 

50 
49.859 

The
certainty option pricing model is applicable by
adjusting for the dividend.
That is:
Certainty
Value: Max (0, S 
PV(Dividend) 
PV(X))
Substituting
the numbers, we get the following values for the
November call options, compared with the bid/ask spread
in the options market:
Table 6.3
Call Options
Strike 
Model
Price 
Actual
Spread 
 
35 
9.350 
9.375 
9.875 

40 
4.364 
4.875 
5 

45 
0.0 
1.375 
1.4375 

50 
0.0 
.25 
.375 

You
can see that the certainty model has systematically
underpriced the options.
Why
is this the case?
There are at least two factors we have
missed. First,
these are American options, and there may be
some value to early exercise.
Second,
the stock price at the terminal date is
not known.
This uncertainty raises the
value of the option.
To
see why, consider a simple example.
Suppose the
price of a nondividendpaying stock at
time T
equals the present value of the strike price X.
In the certainty model, the call has
zero value.
Now, suppose the terminal stock price
is either increased by some small positive
number, e, or decreased by e
with equal probability.
The expected terminal stock price is
the same, but the call has positive value,
because in the "plus" event, you
get:
while
in the "minus" event, you still get
zero. In
fact, if you make e
larger (i.e., increase the volatility), the
call will increase in value.
Intuitively, what is happening is that
as volatility increases, there is a greater chance that the call
will be deeper inthemoney.
As
a result, you should be able to do better than
the certainty model in the presence of
volatility.
You can read this in the next topic Stock
Price Dynamics.