Guaranteed Profits in Stock Market Options?
An Introduction to Linear Programming
As Applied to Stock Market Options
The table below shows the Nov. 30, 2001 closing prices for a few of the
S & P 100 Dec., 2001 Stock Index Options as given in Barron's. If
you knew ahead of time exactly what the stock market was going to do,
all you would have to do is buy or sell the appropriate option and you
could get very rich very quickly. The bad news is nobody knows what the
stock market is going to do.
An interesting secondary question can be asked. Is there
any combination of buying and/or selling a mixture of these options (at
the given prices) that can guarantee a profit no matter what the market
does? Interestingly the answer is YES.
In fact, given any set of option prices (real, arbitrary,
real time, closing values, this year, next year, etc.) there are always
buy/sell combinations that will guarantee a profit no matter what the
market does. There are a few conditions that must be met.
1) Brokerage companies will subtract commissions from the profit shown
by the calculations. (Good news - these are usually small enough that
they are not a problem.)
2) The securities laws restrict some of the combinations. (Eliminates
some of the combinations, but other combinations are open.)
3) Once you have made your commitment, the options remain unexercised
until they expire.
4) For any combination, you have to execute all of the orders at (or
near) the given price - usually this must be done simultaneously.
(There is a long frustrating story regarding this. If you are an
outside investor, your orders will probably never be executed. However,
this is an example of how brokerage companies operate in the financial
For now, assume that you can buy or sell at these actual
closing prices. We will show how to calculate a strategy that can
guarantee a profit for any arbitrary (or real) prices for any group of
options. We also assume that a malevolent manipulator can dictate any
subsequent stock market price after you have made your commitment.
Nov. 30, 2001 closing prices for a few of the S&P 100 Dec. options
as reported in Barron's.
Price Options Options
First, we construct a table showing the net change in your
finances if you bought (or sold) each of the above options vs. some of
the subsequent possible S & P 100 prices when the contract expires.
(The value of the S&P 100 when you start does not matter.)
Buy/Sell Strike Buy/Sell <----
If the S&P goes to ----->
Buy Call 580 15.00
-1,500 -1,500 -1,000 1,500 40,500
Buy Call 585 11.30
-1,130 -1,130 -1,130 1,370 40,370
Buy Call 590 10.20
-1,020 -1,020 -1,020 980
Sell Call 580 15.00
1,500 1,500 1,000 -1,500 -40,500
Sell Call 585 11.30
1,130 1,130 1,130 -1,370 -40,370
Sell Call 590 10.20
1,020 1,020 1,020 -980 -39,980
Buy Put 580 10.80
56,920 920 -1,080 -1,080
Buy Put 585 13.00
57,200 1,200 -1,300 -1,300 -1,300
Buy Put 590 15.50
57,450 1,450 -1,050 -1,550 -1,550
Sell Put 580 10.80
-56,920 -920 1,080
Sell Put 585 13.00
-57,200 -1,200 1,300 1,300
Sell Put 590 15.50
-57,450 -1,450 1,050 1,550
(In practice, you would
want to include additional options (rows) not included here (can only
improve the final results), and include additional S&P prices
If you look at the table, no single row can guarantee a
profit. In fact if you sell something, and the final S&P 100 price
goes the wrong way, the loss can be staggering. Nevertheless, a mixed
strategy can be computed that will guarantee a profit. In fact, our
calculations will find the mixed strategy that guarantees the maximum
possible profit assuming the malevolent manipulator can designate any
possible S&P 100 price when the options are exercised.
The solution method we will use is called Linear
Programming. It is used to find the Maximum or Minimum value of
something within the bounds defined by a series of constraints. In our
case, the value that we will maximize is: Maximum possible profit vs.
any worst possible final S & P 100 price. The constraints involved
state that our finances must show a profit of at least this amount no
matter what happens to the S & P 100.
First, we define the unknowns in the problem. They will indicate the
proportional amounts of each option contract to include in our mixed
strategy. These proportions will sum to 1.000. (We could alternately
solve for the actual cash amount invested in each option in our mixed
strategy, but this usually leads to annoying fractions.)
Let X1 = the proportional position in buying the 580 call at 15.00
Let X2 = the proportional position in buying the 585 call at 11.30
Let X3 = the proportional position in buying the 590 call at 10.20
Let X4 = the proportional position in selling the 580 call at 15.00
Let X5 = the proportional position in selling the 585 call at 11.30
Let X6 = the proportional position in selling the 590 call at 10.20
Let X7 = the proportional position in buying the 580 put at 10.80
Let X8 = the proportional position in buying the 585 put at 13.00
Let X9 = the proportional position in buying the 590 put at 15.50
Let X10 = the proportional position in selling the 580 put at 10.80
Let X11 = the proportional position in selling the 585 put at 13.00
Let X12 = the proportional position in selling the 590 put at 15.50
Let X13 = the guaranteed profit vs. the worst possible outcome for the
S & P
Next, we define what we want to maximize/minimize.
In this case:
Maximize Z = X13
The first five constraint equations state that our results
must be positive for any possible S & P result. Each equation uses
a column from the above table. The "-X13" term appears in each of the 5
equations as it must be maximized.
Eq. 1) - If the S & P 100 goes to 0:
-1500X1 -1130X2 -1020X3 +1500X4 +1130X5 +1020X6 +56920X7 +57200X8
+57450X9 -56920X10 -57200X11 -57450X12 -X13 >= 0
etc. through Eq.5)
If the S & P 100 goes to 1000:
40500X1 +40370X2 +39980X3 -40500X4 -40370X5 -39980X6 -1080X7 -1300X8
-1550X9 +1080X10 +1300X11 +1550X12 - X13 >= 0
and finally the sum of the proportions must equal 1.000
X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 + X12 = 1
These 6 equations form a Linear Programming system that
can be solved by various packages including spreadsheets. The solution
X2 (Buy 585 Call at 11.30) = .25
X6 (Sell 590 Call at 10.20) = .25
X9 (Buy 590 Put at 15.50) = .25
X11 (Sell 585 Put at 13.00) = .25
X13 = 35
In English this means: For each package (mixed strategy)
that includes 1/4 of an option contract as indicated above, you are
guaranteed to make a profit of at least $35.
This solution technique can be used on any family of
options as long as they are for the same security and expire at the
same time. The only restriction is they can not be exercised
prematurely. There is always a mixed strategy that will guarantee a
profit. (In theory it possible to have a worst case of "break even",
but this will probably never happen in practice.) The bad news is that
an outside investor must have some brokerage company execute the mixed
strategy on an "all or none" basis. Unfortunately, this has proven to
be virtually impossible. However, it is an example of the action in
derivatives that many brokerage companies use.
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