Financial Markets

• Kelly Criteria/Merton Share for sizeing bets

\begin{equation} l = \frac{\mu}{\sigma^2} \end{equation}

• Use Markowitz to determine optimal portfolio composition.

\begin{equation} r_{(1,2)} = x_1 r_1 + x_2 r_2 \\ \text{var}(r_{(1,2)}) = \text{var}({r_1}) x_1^2 + \text{var}({r_2}) x_2^2 + 2 x_1 x_2 \text{cov}(r_1, r_2) \end{equation}

In general this works for \(n\) risky assets with respective normally distributed returns \(r_i : i \in (1..n)\):

\begin{align} r_{(1..n)} &= \sum_{i \in (1..n)}{x_i r_i} \\ \text{var}(r_{(1..n)}) &= \sum_{(i,j) \in ((1..n), (1..n))} 2x_ix_j\text{cov}(r_i, r_j) \\ \sigma(r_{(1..n)}) &= \sqrt{ \sum_{(i,j) \in ((1..n), (1..n))} 2x_ix_j\text{cov}(r_i, r_j)} \end{align}

• The above math generatlizes to log normal distributions:

\begin{align} r_{(1..n)} &= \sum_{i \in (1..n)}{x_i r_i} \\ \text{var}(r_{(1..n)}) &= \sum_{(i,j) \in ((1..n), (1..n))} 2x_ix_j\text{cov}(r_i, r_j) \\ \sigma(r_{(1..n)}) &= \sqrt{ \sum_{(i,j) \in ((1..n), (1..n))} 2x_ix_j\text{cov}(r_i, r_j)} \end{align}

• Using Target Prices and standard deviations of price movements to determine sizing

• RE-Bench: Evaluating frontier AI R&D capabilities of language model agents against human experts. Explains how training AI agents yield very interesting performance relative to time spent. In particular it looks like human participants deliver exponential increases relative to time spent, while AI agents hit a performance wall and do not gain improving results as time spent increases. https://arxiv.org/pdf/2411.15114
• Portfolio Diversification and Supporting Financial Institutions(Harry Markowitz). Yale Class on Optimizing portfolio compositions to reduce risk exposure and hence optimizing risk return. https://www.youtube.com/watch?v=_B_24GUWdSM
• The Kelly criterion: How to size bets. A really nice interactive representation of the kelly Criterie for determining optimal bet sizes when considering a bet with positive expected outcome. https://explore.paulbutler.org/bet/
• Merton's portfolio Problem. Explains how to use a statical approach similar to Kelly, but based on knowledge of the statistical distribution in a continuous form. https://en.wikipedia.org/wiki/Merton%27s_portfolio_problem
• Tactical Investment Algorithms(Marcos López de Prado). [Not read] The idea of this paper is to construct a properbalistic model of the world, and test strategies against this model. We then test the real world against the model to determine if we expect our strategies to work. https://caia.org/sites/default/files/tactical.pdf