Can we make a bet on stock price ?

In http://www.wilmott.com/messageview.cfm?catid=8&threadid=48034, lord12 asked a simple question that r = 0.1 u = 1.2 d=.9. John thinks the stock will go up and makes a bet with Jerry. If S1>S0, John will $100 from Jerry. If S0 is smaller than S1, John will give $100 to Jerry. Who has the advantage?

My idea is the following:
I think they can hedge out the risk which depends on the situation of option at that time. The clue is :


1) They can not hedge out risk if only hold/sell the stock...

In the side of Jerry:

Jerry can buy X stock(Let us suppose the price is 1) at time 0, if stock is up, he will get X*1.2-100 ,if stock is down, he will get X*0.9+100. So, he need X*1.2-100>(1+r)*x=1.1X and X*0.9+100>1.1X to hedge out any risk, 0.1X>100 and 100>0.2X hold at the same time. it is impossible .

In the side of John,

he need X*1.2+100>1.1X and X*0.9-100>1.1X So X>-1000 and -100>0.2X (X<-500). We get -500<-1000! it is impossible if John buy stock.

2) But it seems promising if they can use call/put option to hedge out the risk. Eg: John can sell the call option at time 0 and the strike is K. the call option price per stock is then (1.2-K)*2/3. Suppose he sell X call option to his client, the total money at time 1 is if stock is up, he will get (1+r)*X*(1.2-K)*2/3-(1.2-K)*X+100. So if 100>(1.2-K)*0.27X, he will win 100- (1.2-K)*0.27X bucks.
Notice we need X<100/((1.2-k)*0.27) 100=""> 0.73*(1.2-K)X+KX-0.9X-100, we need 0.23K*X-0.02X-100>0 to get profit, so he needs X>100/(0.23K-0.02) .

In this way, we know if 100/((1.2-K)*0.27)>100/(0.23K-0.02)=> 0.23K-0.02>0.32-0.27K=> 0.5K>0.34 => K>0.68=> Always hold if we know u=1.2 d=0.9.

In this example, if John is a banker, he can bet with Jerry as he can hedge out the risk completely!

Research Note 4/17/07

To calculate the Tr(I-M) which is a lower bound for the block entropy in bipartite fermion systems in any kind of shape, I find that we need to modify the ordinary Fejer kernel. I will keep thinking in this problem.