9.2
Constant Continuous Dividend Yield OPM
T 
he BlackScholes option
pricing model assumes that the underlying asset pays zero dividends over the
life of the option. In practice
this is rarely the case. A useful
extension of the BlackScholes model addresses an underlying asset that pays a
dividend continuously with a known yield. This
assumption allows us to apply the model to a wide range of options, including
currency options and options on futures. Recall that in Chapter 6, topic 6.5, Interpretation
of the BlackScholes Model, you saw that the BlackScholes option pricing
model consists of two terms. The
first term is an expected present value of the stock price, conditional upon the
terminal stock price exceeding the strike price (i.e., finishing inthemoney).
The second term is the expected present
value of the strike price, also conditional on the option being
inthemoney.
Black and Scholes
approach the problem by assuming a wellbehaved dynamic stock price process, in
which the option value is a function of the underlying stock price and time.
From this assumption, a partial differential equation (PDE) that includes
the unknown option valuation function and some of its derivatives is derived
from an instantaneously riskless portfolio.
Finally, they solve this differential equation subject to the appropriate
terminal option values to get a closedform solution for the option value.
From the binomial option
pricing model, you should have a good understanding of how the principle of
dynamic replication maintains the riskless hedge over the life of the option.
Whenever this principle is applicable, there is an easier approach for
valuing options, namely, the RiskNeutral
Valuation Principle (see Chapter 2, topic 2.7).
Application of this principle yields the expected present value with
respect to a riskneutral probability distribution.
Although the constant
continuous dividend extension to the BlackScholes pricing model can be derived
formally from partial differential equations, the riskneutral valuation
principle provides a more intuitive approach.
It lets us view the value of a continuous dividendpaying stock as
equaling the sum of two streams: the
present value of the continuous dividend plus the present value of the stock
that is stripped of its dividend.
The former has no impact
upon the value of the option. Instead
it is the terminal distribution of the latter that determines the value of the
option.
Approach
Suppose you hold one
share of security S that pays a
constant continuous dividend yield q.
Changes in the value of your holding arise from two components 
realized capital changes, and dividends. Over
a small interval of time Dt, denote the realized capital gain/loss as
DS and the dividend as qSDt.
The current stock price
for this security, S, reflects both
the present value of capital gains/losses plus the present value of dividends.
If the dividends were retained by the stock, then the "cum"
dividend stock price at any future time, T,
is larger than the actual terminal exdividend stock price, ST,
by:
As a notational note,
the present is denoted as 0. Therefore,
the interval of time from the present to time T is T  0 = T.
In terms of present values, the present value of the stock stripped of
all claims to future dividends is:
The terminal
distribution of this stripped stock value is
the underlying asset that the optionholder has a claim against at the time of
the option's maturity.
As a result, the standard BlackScholes option pricing model applies to
the present value of this stripped stock.
Option Pricing Model for the Stripped Stock
Denote S'
as:
Applying the standard
option pricing model to S' yields:
where
Substituting the
stripped stock price for S' gives the
constant dividend yield option pricing model:
and since ln(S'/X)
= ln(S/X)  qT
Similarly, the European
put option price is either obtained indirectly from putcall parity or directly
expressed as:
where d1 and
d2 are as before.
EXAMPLE
One
application of the constant dividend yield model is to value options on stock
indices. A stock index such as the
S&P500 is a valueweighted index of 500 stocks. This means that the index is calculated as follows:
Since an
index such as the S&P500 contains a large number of stocks which pay
dividends at different times, it is sometimes held that the constant dividend
yield model is a good approximation for valuing options on this index.
The S&P500 index options also have the simpler feature of being
European style options, so the option pricing formula developed in this chapter
can be applied without modification.
Consider
the following data. On September
29, 1993, the value of the S&P500 index at 1:45 pm was 460.38.
The last traded prices on the October options (maturing on October 15)
are:
Strike 
Call 
Put


455 
7 7/8 
2 1/4 

460 
4 3/8 
3 7/8 

465 
1 7/8 
6 3/8 

The
interest rate on a Treasury bill maturing on October 14 was 2.835.
Here,
S = 460.38
T = 0.0438
r = 0.02835
The
dividend yield on the S&P500 index is estimated at 2%, so q = 0.02.
We will use the volatility estimate implicit in the atthemoney
option prices. You can do
this using the Option Calculator subject of Option Tutor as
follows.
First,
select Call Option, then European.
Now, select “On” for implied volatility.
Enter 4 3/8 = 4.375 for the Option Price. Put in the numbers for the asset price (460.38), the maturity
(0.0438), the interest rate (0.02835), and the dividend yield (0.02).
Enter 460 for the strike price, and click OK.
The
implied volatility (or sigma) is 0.1067.
Now,
enter this value for the volatility, and select “Off” for implied
volatility. Now, we can use
this volatility estimate to value the other options.
This yields the following put and call prices, compared to the
actual market prices.

Actual 
Estimated 
 
Strike 
Call 
Put 
Call 
Put 

455 
7 7/8 
2 1/4 
7.430 
1.887 

460 
4 3/8 
3 7/8 
4.375 
3.827 

465 
1 7/8 
6 3/8 
2.270 
6.716 

You can see that applying this model with a constant
volatility estimate, undervalues the 455 strike options and
overvalues the 465 call.
Such a systematic effect is often referred to as the
"volatility smile."
The constant dividend
yield model provides a useful model for pricing a wide variety of
derivatives such as currency options and options on futures. It is therefore useful for a wide range of risk management
problems. We
provide the comparative statics for this model in the next
topic, Comparative
Statics.