# Creating a doubling and halving position

I want to create a position that either multiplies with $1+u$ (outcome $U$) or $1-d$ (outcome $D$). The probability of $U$ is denoted by $P(U) = \pi$. The initial value of the position is $V_0$. Given outcome $U$ the value of the position is $V_U = (1+u)V_0$, and given outcome $D$ it is $V_D = (1-d)V_0$.

More specifically, I am trying to create a doubling or halving position with more or less equal probability of $U$ and $D$ occurring, i.e. $\pi \approx 0.50$. I was thinking of using binary options that pay 1 when $U$ is the outcome. I was also thinking the position could be created using binaries and holding cash. So, $V_U = 2V_0 = \frac{V_0}{2} + N*P_U = \frac{V_0}{2} + N, P_U = 1$ where $N$ is the number of options in the position and $P_U$ is the option payout at success. $V_D = \frac{1}{2}V_0 = V_0 - NP_0$ where $P_0$ is the initial price of the binary option. I want to find $N$ and $P_0$ such that the value is doubled at success and halfed at non-success.

We find that $N = \frac{3}{2}V_0$ and that $P_0 = 1/3$. However, this breaches the $\pi$ condition. If I am purchasing daily binary options on whether S&P 500 closes above its last close, one would assume that the binary option would be priced near $0.50$ at the beginning of the day.

How should I construct this portfolio?

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Can you borrow at the risk free rate? Is your market free of arbitrage? If yes, then you can have doubling halving position irrespective of that physical probability $\pi$ value. –  Alexey Kalmykov Oct 27 '12 at 19:11

Now, as for creating a doubling and halving position, it is actually Kelly's criterion---with the odds stated (available to bet) in a market-neutral martingale measure but the true odds in a frequentist market measure---that determines the optimal fraction of $V_0$ to bet on the binary option each day, and whether you really want to double or halve, or if some other fraction is more appropriate.