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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
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First of all, don't forget that there are two different probability measures at play here: the frequentist market measure that reflects actual observation of the market "in the long run" and the market-neutral martingale measure which is pertinent for pricing options. More or less, we can take the frequentist measure, and "back out" the effects of market beta in order to arrive at a martingale measure that avoids counting a position as an arbitrage solely because it is long in the market and participates in the natural tendency of the market to go up over time.

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.

Also, remember at the beginning of the day, at least 17.5 hours in most cases have elapsed since the last close, the market in all likelihood has already gapped significantly up or down, and only 6.5 hours remain till close. Whatever time remains in the day, that is the arbitrage you have to take advantage of, but if the option is at or near the money, there is still time to take full advantage of it. In this case, though, it seems that the arbitrage consists of simply being long in the market, at least on an elementary theory that markets are efficient.

But then again, I'm a skeptic of "quant," and I believe that markets are generally more efficient than people give them credit for.

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