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.