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 derivatives markets.)

   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.

  Strike        Call        Put
  Price      Options   Options
--------------------------------
   580        15.00      10.80
   585        11.30      13.00
   590        10.20      15.50

   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.)

                   Option
Buy/Sell   Strike  Buy/Sell    <---- If the S&P goes to  ----->
Action     Price   Price        0    560     585    610     1000
-----------------------------------------------------------------
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   39,980
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   -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  1,080    1,080
Sell Put   585     13.00  -57,200  -1,200   1,300  1,300    1,300
Sell Put   590     15.50  -57,450  -1,450   1,050  1,550    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 (columns).)

   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
and finally
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

Eq. 6)
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 gives:
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's 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.


For another example using Linear Programming, please see the author's web page at:
http://www.durangobill.com/TravelingSalesman/TravelingSalesman.html
The Traveling Salesman Problem



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