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6

At the first glance, what you are asking for is a model admitting arbitrage, so there is a zero chance of losing money and positive chance of yielding profits. Well, many equilibrium models start with assuming arbitrage is not possible (otherwise it would be trivial wouldn't it). But, in my opinion, what you actually seek is the Efficient Markets ...

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This will depend on the nature of your tree. For a re-combining binomial tree, the number of nodes, including the initial one, will be \begin{align*} \sum_{i=1}^n i = \frac{n(n+1)}{2}. \end{align*} For the paths, as at each time $j$, there are two possibilities from each node, the total path number is $2^n$.

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Let $B_t$ be the value of the risk-free asset at time $t$. Then $B_0=1$ and $B_{t+1} = (1+R) B_t$. Moreover, let $\beta_t$ be units invested in the risk-free asset at time $t$. It is clear that $\beta_0 = w_0 - \Delta_0 S_0$. Since the strategy is self-financing, \begin{align*} \Delta_{t-1} S_{t-1} + \beta_{t-1} B_{t-1} = \Delta_t S_{t-1} + \beta_t ...

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This doesn't really suffice as an existence proof, but you can start with a series of mathematical results collectively known as no free lunch theorems. The linked paper proves the average performance of any optimization algorithm over arbitrary problem domains is independent of the algorithm. That is, no single algorithm can ever be better than others on ...

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This is where one needs the concept of no free lunch with vanishing risk (NFLVR), whose proof you can find in: Delbaen & Schachermayer (1994). Though, as a warning, I should mention it is pretty involved.

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Why do you think such a theorem would exist? I will give you a counterexample: You have two assets A and B. Both are completely identical in every respect, except price: The price of A is USD 1 the price of B is USD 2. Your strategy is simple: You (short) sell B for a gain of 2 and buy A for 1. This strategy requires no capital and leaves you with an ...

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