Fischer Black
on
OPTIONS
Vol. 2, No. 1
June, 1977
THE SEARCH FOR PUT OPTION VALUES
Put values and call values are affected by a common list of variables: (1) the
stock price, (2) the strike price, (3) the option maturity, (4) the stock
volatility, (5) interest rates, and (6) future dividends on the stock.
An increase in the stock price increases the value of a call and decreases the
value of a put. This means that call deltas are positive and put deltas are
negative. An increase in the strike price decreases the value of a call and
increases the value of a put. An increase in a stock's volatility increases
the values of both its calls and its puts.
Now let us look at the relation between put values and call values in the simplest
case: when interest rates are zero and the underlying stock pays no
dividends.
No Interest and No Dividends
With no interest and no dividends, it will almost never pay to exercise either
a call or a put early. A person who holds an option will be better off selling
it than exercising it early, except in very unusual tax or trading cost sit.uations.
Copyright Fischer Black 1977
Fischer Black on OPTIONS Page 2
Either conversion or reverse conversion will be riskless. Conversion means
buying a put, selling the corresponding call, and buying the stock. Reverse
conversion means buying a call, selling the corresponding put, and selling the
stock short. This implies an exact relation between the put price and the
call price. The call price minus the put price must equal the stock price
minus the exercise price.
CALL - PUT = STOCK - STRIKE
If the put price is lower than that, conversion will be profitable. If the
put price is higher, reverse convers ion will be profi table. So converters
and reverse converters will tend to keep put and call prices in line with
this relationship.
No Early Exercis.e
Now suppose that interest rates are positive and stocks pay dividends. But
suppose that we're working with European options that cannot be exercised
before maturity. Suppose further that when a reverse converter sells stock
short, he receives interest on all margin money he puts up, including the
proceeds of the short sale.
Both converters ~nd reverse converters continup. to have riskless positions,
so making a profit means making more than the interest rate. The converter
receives dividends on his stock and the reverse converter pays dividends,
so dividends count toward the profit of a position. Thus, the call price
minus the put price must equal the stock price minus the exercise price
plus interest on the exercise price minus dividends.
Fischer ~lack on OPTIONS Page 3
CALL - PUT = STOCK - STRIKE + INTEREST - DIVIDENDS
If there is no early exercise, and if the dividends are known, this relationship
should hold independently of any formula, and independently of the volatility
of the stock. If it does not hold, there will be profit opportunities,
at least gross of the costs of puttjng on and taking off positions.
Effects of Early Exercise
An option that can be exercised early will always be worth at least as much
as an option that can only be exercised at maturity. Allowing early exercise
normally increases the values of both calls and puts. With a call, early
exercise is forced by a large dividend, and normally occurs only just before
dividend ex-dates. With a put, early exercise can occur whenever the stock
price is low enough.
A low stock price can cause early exercise on a put because waiting to exercise
a put means losing interest on the exercise price. This is offset in
part by the fact that waiting to exercise a put means gaining dividends on the
stock.
Because of early exercise, neither conversion nor reverse conversion is really
riskless. A converter may want to exercise his put early, or he may see the call
he has written exprcised early. If he closes ou_ one side of his position. ~e'll
probably want to close out the other side too, so the whole position may be
closed out early. A position that is closed out early will generally show a
profit or a loss.
The reverse converter is in a similar position. He may want to exercise his call
early, or he may see the put he has written exercised early. In either case, he
will close out his position with a profit or a loss.
Fischer Black on OPTIONS Page 4
If converters and reverse converters make early exercise decisions correctly,
and if other investors do not, then converters and reverse converters will
be more likely to end up with a profit than with a loss. It doesn't matter
whether the other investors exercise too early or too late. Either one will
give profits to the corresponding option writers.
Dividends, Interest, and Early Exercise
A higher interest rate makes calls worth more and puts worth less. Thus
it tends to increase the difference between the values of at-the-money calls
and at-the-money puts.
A higher dividend rate makes calls worth less and puts worth more. Thus
it tends to reduce the difference between the values of at-the-money calls
and at-the-money puts. If the dividends are high enough, they will outweigh
the interest effects and cause at-the-money puts to be worth more than the
corresponding at-the-money calls.
A higher dividend rate makes early exercise more likely (or equally likely)
for calls and less likely (Dr equally likely) for puts. The value of a call may
drop to parity with a dividend increase, which makes early exercise pay, and
the value of a put may rise above parity with a dividend increase. No option
worth more than parity should be exercised early.
As calendar time moves forward, two things happen to a put option with dividends.
The time to expiration goes down, whicr. ~ends to reduce the option
value. But the dividends come closer. which tends to increase the option
value. When a put option is well in the money. the second effect may dominate
and the option value may go up as calendar time goes forward. if the
stock price does not change.
Fischer Black on OPTIONS Page 5
Strategies with Puts and Calls
If puts are priced too low relative to calls, then converters will step in.
If puts are priced too high relative to calls, then reverse converters who
can get full use of the proceeds of a short sale of stock will step in.
Some brokerage firms are in a position to do that by selling short stock
in customer margin accounts.
If puts are overpriced and calls are correctly priced or overpriced, it may
pay to sell combinations of puts and calls. For a neutral position, the
right ratio will generally be other than 1:1. For many investors, though,
this kind of position will have certain disadvantages.
A position involving writing puts and writing calls will have negative
curvature: gains will occur if the stock doesn't move, and losses will
occur if it moves a lot. If the position is not adjusted, potential
losses are unlimited. And the margin rules for many i.nvestors who write
naked puts and calls will mean tying up a good deal of capital.
On the other hand, if puts and calls are both underpriced, it may pay to
buy combinations of puts and calls. This kind of position will have no
extra margin requirements, potential losses will be limited, and the curvature
will be positive: losses will occur if the stock doesn't move, and
gains will occur if it moves a lot.
If stock volatilities go up, positions with negative curvature will lose,
and positions witt. ?ositive curvature will gajn. Since there is no limit
to how much stock volatilities can go up, this is another sense in which
positions with negative curvature are more speculative, and positions
with positive curvature are more conservative.
If puts are overpriced relative to calls, than it may pay to buy a call
Fischer Black on OPTIONS Page 6
and sell a put instead of buying a stock. Except when early exercise
becomes profitable, the return from buying a call and selling a put will
be very close to the return from buying the stock. If the put is
enough overpriced, the return will almost always be greater than the return
from buying the stock.
Similarly, if puts are overpriced relative to calls, it may pay an investor
who has been buying stock and writing calls to write naked puts instead.
For this to be profitable, he must be able to earn full interest on any
margin money he puts up, and on the difference between what he puts up to buy
stock and what he puts up to write naked put options. If the stock pays
taxable dividends and the interest earned is tax exempt, there may even be tax
benefits to the strategy that sells puts rather than buying stock and selling
calls.
Figuring Put Values
The method I use for figuring put values makes the same assumptions as the
method for figuring call values: that the stock volatility is constant,
that there are no taxes or trading costs, and that there are no penalties
to short selling options or stock.
There is a formula for a European put option (where there is no early exercise),
and that formula can be adapted easily to include dividends. But
there is no formula for an American put option (where early exercise is
possible}, and there's not likely to be one that would be simple enough
to be helpful.
There are several methods for constructing tables of put values using a
computer. Each column of such a table would represent a different date,
and each row would represent a different stock price. We start by stating
the value of a put at maturity for each different stock price. From that
Fischer Black on OPTIONS Page 7
column, we can create a column of values for the put on a date just before
maturity. We continue working backward, constructing columns for earlier
and earlier dates, until we come to the present.
I chose the method developed by Michael Parkinson, because it seemed the
simplest, most efficient, and most accurate of the methods for constructing
tables on a computer. It is described in the January, 1977 issue of the
Journal of Business. His paper does not tell how to handle dividends, but
it's not hard to change the method to include dividends.
If what you want is to construct a table of values, then the Parkinson
method does the job. But if what you want is to find a single value, it's
very costly to have to construct a whole table of values just to get the
one value you want. There had to be an approximate method for finding a
single value.
Dan McFadden (of Monchik-Weber Associates) and I spent many months looking
for a good approximate method for American puts. We had help and advice
from John Cox, Steven Grossman, and Robert Merton. In the end, we came
up with a relatively simple method. In fact, it is as fast or faster to
compute put values using this method as it is to compute call values using
the call option formula.
The general method we decided on was to construct a group of tables that
can be used for all options, rather than a sep~rate table for each option.
Parkinson has a group of tables in his paper that can be used for this purpose,
but he aSSumes no dividends on the stock. So we had to construct a
group of tables like his for each of several possible dividend rates.
When we want to value a particular option, we generally find that it doesn't
appear in any of the tables. So we look for similar options that do appear
Fischer Black on OPTIONS Page 8
in the tables, and average their values, giving greater weight to the options
that are most similar. Normally, we will average the values of 16 options
that appear in the tables to get an estimate of the value of the option we
want.
Each table assumes that the stock pays dividends continuously. But real companies
don't pay small dividends every day; they pay larger dividends at
quarterly or greater intervals. The hardest problem that McFadden and I
faced was in figuring how to use tables for continuous dividends without
significant errors.
An obvious possibility is to assume that all the dividends between now and
an option's maturity are spread evenly over the life of the option, with a little
paid each day. This works well when there are several dividends, or
when there's a dividend near the end of the option's life, and when the dividends
are not too large. It does not work well when there's a large dividend
in the near future, especially when the option is well in the money.
With a big dividend coming up, it generally won't pay to exercise a put option
now, even if it is far in the money. It's better to wait till after the exdividend
date to get the benefit of the dividend. After the stock price falls
as it goes ex-dividend, it may well pay to exercise the option. Spreading
the dividend out and moving it to the future may completely eliminate the
benefit of the dividend because it may suggest immediate early exercise.
Just moving the di·,idend to the future may reduct ~he benefit of the dividend
by as much as interest on the exercise price for the time over which the dividend
is moved.
Another reason to be careful in moving dividends iR that moving a dividend
changes the effective volatility of the stock. When a dividend is paid,
the dollar volatility of a stock goes down because the value of the stock goes
down. Thus pushing a dividend to the future causes a change in effective
Fischer Black on OPTIONS Page 9
volatility that tends to increase the value of a put.
However, these two effects tend to cancel. Moving a dividend to the future
tends to decrease the value of a put because it delays the effect of the
dividend on the stock price, but it tends to increase the value of a put
because it delays the effect of the dividend on the stock's effective
volatility.
Balancing all these factors, we came up with a method of approximating the
dividends on a stock in two parts: (1) an initial dividend, which we subtract
from the stock price, and (2) a continuous dividend, which we use in
deciding which tables to choose option values from. The two parts add up
to the total dividends that we project for the stock over the life of the
option.
We find the initial dividend by adding up the amounts by which each dividend
exceeds interest on the exercise price between that dividend and the one before
it. For the first dividend, it's the amount by which the dividend
exceeds interest on the exercise price in the time till the first dividend.
One reason this works is that if the dividends are greater than interest on
the exercise price, they will largely eliminate the possibility of early
exercise. Any added dividends can be moved to the present without changing
the early exercise pattern significantly.
Overall, this approximation for the put dividends is probably better than
the approximation I have been using for the call dividends. We're working
on the possibility of using a similar method for calls.