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# Game Theory - Inf Camp (Bullish), Sup Camp (Bearish) And Optimal Stock & Option Pricing On An Option Expiration Day

Summary

Maximum Gain is defined as the optimal option or its underlying stock pricing strategy using optimal control theory

Great tool to complement Maximum Pain which is known to the investment community

Why do we need another algorithm now that we have Max Pain? It is very important to understand Max Gain exists in its own rights and should gain popularity among investors/speculators alike for offering an alternative optimization solution.  The estimate from Max Gain is usually better than Max Pain where the accuracy the algorithm determines the profit or loss for investors/speculators. As better estimate will normally leads to higher profit and lower loss, it is expected Max Gain to gain more followers and win the popularity contest. Furthermore, the existence of computationally efficient algorithm makes it possible to calculate the solution by hand which is a great relief if he (or she) is in a hurry to make a trade and has no time to try it on a computer.  Thus a back of envelop approach may be all you need for a short term solution.

Max Gain is dynamic, easy to apply and a very versatile trading tool

XYZ company has been in business for a while and its stocks have been publicly traded.

A game was created to meet investor's need. If you think its price is going up above its infimum, then you join the inf camp. You become a bull.  In investment community, you will buy a "call" option.

However, if you think its price is going down below its supremum, then you join the sup camp.  You become a bear.  In investment community, you will buy a "put" option.

To play, each investor pays \$x dollars/contract to participate by choosing an interger target price Pt (called Strike Price in investment community), # of contracts v (each contracts controls 100 shares), type of options (call or put) and a date (option expiration day).

Assume futher the target price ranges from 1 to 1000.

On the option expiration day, if you are a bear (mathematically, you are in Sup Camp) and P < Pt, you win and you are rewarded by R = v * 100 * (Pt - P) = v * 100 * |P - Pt| in dollars.  But if P >= Pt, then you lose 100% of your investment  I = x * v * 100  in dollars.

Similarly, if you are a bull (mathematically, you are in Inf camp) and P > Pt, you win and you are rewarded by R = v * 100 * (P - Pt) = v * 100 * |P - Pt| in dollars.  But if P <= Pt, then you lose 100% of your investment I = x * v * 100 in dollars.

Assume volume info for each target price is available and the objective is to minimize the payouts to invesors and maximize the bets which becomes worthless. What is the optimum pricing strategy?
(1) Pairwise optimum
Consider only two prices P1 & P2. Specify the conditions to determine Popt. i.e. What is your condition c such that
If c, then Popt = P1,
Else Popt = P2
(2) Global optimum
Extend your results in (1) to global optimum.
(3) Is there a computationally efficient way to calculate Popt. If so, explain how and why.  Yes, the global optimum is called Maxim Gain (for the option sellers).  The strategy will be posted in the upcoming website called "Max Gain".

(4) Is this a linear programming problem? If so, can we use known techniques to solve this problem and what are these techniques.

If not, is this a nonlinear programming problem? Are there any tools available? Please specify.