financial traders

Abstract: We study adversarial market making with orders that remain active until execution. The maker operates on two markets: a primary market where orders are made and a secondary market where the resulting positions are hedged. In each round, the maker posts a quote on the primary market before observing both primary and secondary market prices. An order executes when the primary price reaches its quote, and the resulting position is immediately offset at the secondary price. When both price sequences are chosen by an adversary over $T$ rounds, we give a simple deterministic policy with regret of order $\log T$ against the best fixed quote (static comparator) and $\sqrt T$ against the best quote for each submission date (dynamic comparator), plus two terms measuring exposure to future primary and secondary prices. These rates remain optimal even when both exposure terms vanish and lower bounds show that the exposure terms are necessary. Our policy naturally extends to simultaneous bids and asks, where we show that static and dynamic regret are both characterized (up to constant factors) by the exposures terms (where the primary price exposure $V_T^{\mathrm{two}}$ now accounts for both bids and asks) and a term of order $\sqrt{V_T^{\mathrm{two}}\log T}$. Our policy is simple, efficient and fully parameterless. We extend this setting by demonstrating how natural assumptions on the price processes lead to improved guarantees without changing the policy.

Abstract: The limit order book (LOB) records outstanding buy and sell orders across multiple price levels. Its state evolves over time across market side, price level, and order attribute, naturally forming a multiway structure. However, existing LOB models typically flatten these dimensions into a matrix model, thus obscuring the existing structural relationships. Starting from first principles and the market-microstructure intuition of the LOB, we design TensorLOB to operate entirely in the tensor domain. TensorLOB uses mode-wise tensor contractions for feature extraction and directly contracts the learned representation with low-rank class-weight tensors for prediction, retaining the distinct roles of time, side, level, and attribute modes. Experiments on the FI-2010 benchmark show that TensorLOB achieves the highest accuracy, recall, and F1 score across all four evaluated horizons. This proof of concept highlights the potential of high-dimensional modelling of multi-way latent information intrinsic to financial markets, which has so far been largely overlooked.

Abstract: In this note we explore the following simple question: How does the classic \emph{gambler's ruin} phenomenon change if we allow the gambler \emph{financial leverage}? We show that, with a moderate amount of leverage, it is possible for an (idealized) gambler to sustain seemingly improbable profits over a remarkably long time horizon without ever going bankrupt, \emph{even} if the gambler makes only fair bets. With a leverage ratio of $L = 10$ (less than what Lehman Brothers Holdings had from 1995--2008), the probability of multiplying one's money by $1000\times$ without going bankrupt is roughly $50\%$. More generally, given $L$-fold leverage, and given the goal of multiplying one's money by a factor of $r > 1$, the optimal gambling strategy has probability roughly $r^{-1/L}$ of success. The results in this note are designed as part of the course development for the CMU randomized algorithms course.

Abstract: We study prophet inequalities for a random walk reward stopping problem with sample-based information. The goal is to stop as close as possible to the maximum of a random walk with i.i.d. increments, measuring performance by the ratio between the expected reward when stopping and the expected true maximum. We consider a sample-based model in which the increment distribution is unknown and the decision maker has access to $K$ independent sample paths of the reward process. For the infinite-horizon setting, we establish a sharp prophet inequality with constant $(K/(K+1))^{K+1}$. The guarantee is attained by a randomised stopping rule based on the ladder height decomposition of random walks. As $K\to\infty$, this recovers the classical $1/e$ prophet inequality from the full-information setting. For the finite-horizon setting, where the process terminates after $n$ steps, we first prove a tight no-information prophet inequality with constant $1/H_n$, where $H_n$ is the $n$-th harmonic number. For $K\ge1$ samples, we show that a prophet constant of $1/4$ is attainable. Finally, we prove that, with $K$ samples, the prophet constant is at most $(K/(K+1))^{K+1}+(6+6H_K)/H_n$ for $n\ge 2K^2$, implying convergence to the infinite-horizon constant as $n\to\infty$. Our approach combines random walk theory, including ladder heights and Spitzer's identity, with linear programming duality. Our results contribute to random walk stopping theory and to sample-based prophet inequalities for correlated rewards, an area that remains largely unexplored. To the best of our knowledge, our tight sample-based prophet inequalities are the first whose performance is parameterised exactly, rather than only up to constants, by the number of available samples.

Abstract: Large language models are being organized into multi-agent systems with specialized roles, but whether such specialization produces distinct forecasts and whether subsequent synthesis improves utility remains unclear. In this study, we carried out a live, prospective evaluation over the final 56 matches of the information-dense 2026 FIFA World Cup, keeping a frontier foundation model constant while assigning two primary forecasting agents contrasting specialist roles: a quantitative specialist focusing on structured performance statistics and a news specialist focusing on current injuries, tactics and information from press conferences. Their forecasts were then reviewed by a separate critic before being combined by a meta-agent, resulting in a sequential four-agent model. Forecasts from the betting market served as an external benchmark. The news specialist obtained the highest mean probability-weighted Top-3 utility and matched the betting market in Top-3 exact-score hits. Nevertheless, the two specialist forecasters agreed on at least two of the three scorelines in 50 out of 56 matches, and the meta-agent never generated more than one scoreline outside the specialists' forecast set. These findings show that rapidly changing, unstructured information can provide a valuable forecasting signal alongside structured statistics, whereas adding critic and meta-agent stages does not necessarily create complementary information or improve on the strongest specialist.