We consider the dynamic price competition between two firms operating within an opaque marketplace, where each firm lacks information about its competitor. The demand follows the multinomial logit (MNL) choice model, which depends on the consumers'

Options and structured products, as pivotal financial derivatives, provide contract holders with specific payoff structures based on the performance of underlying assets at predetermined times and conditions. They serve as effective tools for investment institutions to manage risk, hedge exposures, and optimize investment portfolios. With the continuous development of financial markets and the diversification of investor demands, financial institutions have invented a wide variety of exotic options based on the principles and experience of standard options. Exotic options can be further categorized according to their complexity: relatively simple exotic options such as Asian options, barrier options, lookback options, and ratchet options primarily add a single feature to standard options; while highly complex structured products like snowball products, phoenix notes, shark fin options, and cumulative products feature multiple path-dependent conditions and intricate payoff structures. These innovative financial instruments not only broaden investor choices but also provide powerful tools for more refined and personalized risk management and investment strategies[1]. Precisely because exotic options and structured products exhibit high levels of diversity, customization, and structural complexity, accurate pricing remains a core challenge for all market participants.

Applications range from technical analysis of a company's fundamental value, wider market sentiment, factor analysis and most tasks involving some form of natural language processing (NLP) [1, 2]. The implications to trading systems will likely be a dramatic increase in the rate and volume of market insights that can be generated to inform decisions. The overall capabilities of LLMs have dramatically increased over the last five years [3]. This has led to an increase in the number of LLMs available, both as proprietary models from frontier labs or as smaller models with open-weights which can be run locally. Given this, the influence of LLMs on trading decisions is expected to be varied and highly model specific. Early work is starting to compare and benchmark these models in tasks specific to financial applications, such as trading decisions, portfolio optimisation, and market analysis [4-10]. As the number of models increases, and their underlying strengths and weaknesses become more apparent, it is expected that different classes of pre-trained models will be more regularly deployed to achieve certain objectives [11, 12]. While these objectives are likely to be significantly linked to NLP-based tasks, such as text summarisation, analysis, and generation, recent LLM architectures give early evidence that more complex tasks can also be automated. These LLMs, such as the'o' series from OpenAI or'R1' from DeepSeek, generate'reasoning' tokens which result in the model performing more in-context analysis of the generated output and has lead to improved performance over a number of key evaluation measures [13, 14].

We consider a popular problem in finance, option pricing, through the lens of an online learning game between Nature and an Investor. In the Black-Scholes option pricing model from 1973, the Investor can continuously hedge the risk of an option by trading the underlying asset, assuming that the asset's price fluctuates according to Geometric Brownian Motion (GBM). We consider a worst-case model, in which Nature chooses a sequence of price fluctuations under a cumulative quadratic volatility constraint, and the Investor can make a sequence of hedging decisions. Our main result is to show that the value of our proposed game, which is the "regret" of hedging strategy, converges to the Black-Scholes option price. We use significantly weaker assumptions than previous work--for instance, we allow large jumps in the asset price--and show that the Black-Scholes hedging strategy is near-optimal for the Investor even in this non-stochastic framework.

Finance, especially option pricing, is a promising industrial field that might benefit from quantum computing. While quantum algorithms for option pricing have been proposed, it is desired to devise more efficient implementations of costly operations in the algorithms, one of which is preparing a quantum state that encodes a probability distribution of the underlying asset price. In particular, in pricing a path-dependent option, we need to generate a state encoding a joint distribution of the underlying asset price at multiple time points, which is more demanding. To address these issues, we propose a novel approach using Matrix Product State (MPS) as a generative model for time series generation. To validate our approach, taking the Heston model as a target, we conduct numerical experiments to generate time series in the model. Our findings demonstrate the capability of the MPS model to generate paths in the Heston model, highlighting its potential for path-dependent option pricing on quantum computers.

We consider the learning dynamics of a single reinforcement learning optimal execution trading agent when it interacts with an event driven agent-based financial market model. Trading takes place asynchronously through a matching engine in event time. The optimal execution agent is considered at different levels of initial order-sizes and differently sized state spaces. The resulting impact on the agent-based model and market are considered using a calibration approach that explores changes in the empirical stylised facts and price impact curves. Convergence, volume trajectory and action trace plots are used to visualise the learning dynamics. Here the smaller state space agents had the number of states they visited converge much faster than the larger state space agents, and they were able to start learning to trade intuitively using the spread and volume states. We find that the moments of the model are robust to the impact of the learning agents except for the Hurst exponent, which was lowered by the introduction of strategic order-splitting. The introduction of the learning agent preserves the shape of the price impact curves but can reduce the trade-sign auto-correlations when their trading volumes increase.

This note aims to provide a basic intuition on the concept of filtrations as used in the context of reinforcement learning (RL). Filtrations are often used to formally define RL problems, yet their implications might not be eminent for those without a background in measure theory. Essentially, a filtration is a construct that captures partial knowledge up to time $t$, without revealing any future information that has already been simulated, yet not revealed to the decision-maker. We illustrate this with simple examples from the finance domain on both discrete and continuous outcome spaces. Furthermore, we show that the notion of filtration is not needed, as basing decisions solely on the current problem state (which is possible due to the Markovian property) suffices to eliminate future knowledge from the decision-making process.