Non-confidential indicative sample
How stock prices fluctuate
How stock prices fluctuate asks for the mechanism—the patterns, models, and mathematical processes that describe the movement of prices over time. Why stock prices fluctuate seeks the reasons behind those movements—fundamental, macro‑economic, behavioral, and market‑structure factors that cause the price changes.
Research in Stock Price Prediction Using Financial Mathematics, Econophysics, and Machine Learning
My research in stock price prediction is driven by a curiosity for quantitative finance, creative problem solving, and self-learning. My work focuses on developing mathematical models to describe the complex dynamics of stock price movements and applying these models to build ensemble econometric models for accurate price forecasting. This research spans multiple disciplines, including financial mathematics, econophysics, and machine learning, to understand and predict market behaviour.
Mathematical Modelling of Stock Price Movements
The nature of stock price movements is a key area of study within both finance and statistical physics (also known as econophysics). Current models explore the random processes, volatility dynamics, fractals, and non-linear behaviour that characterize financial markets. Commonly, stock prices are modeled using stochastic processes such as Geometric Brownian Motion (GBM), with additional techniques like GARCH (Generalized Autoregressive Conditional Heteroskedasticity) for volatility modelling, power-law distributions for extreme events, and agent-based models for simulating market dynamics.My research aims to synthesize these existing models, integrating insights from various theories to create a unified framework for predicting future stock prices. This involves testing the scientific validity of these models by comparing them against empirical market data, refining and adjusting them to improve their accuracy.
Building Econometric Models for Stock Price Prediction
A core aspect of my work involves developing ensemble econometric models that combine multiple forecasting techniques. These models use extracted features, signals, and categories from mathematical representations of stock price movements. In the field of quantitative finance and algorithmic trading, such features are critical for making informed predictions, identifying market trends, and automating decision-making processes.To enhance prediction accuracy, I incorporate neural networks, particularly deep learning models, which excel in handling large datasets and capturing complex, non-linear relationships that traditional financial models may miss. These advanced machine learning techniques are increasingly popular in financial prediction and are widely applied in the realm of robo-advisors and investech (investment technology).
Ongoing Research and Scientific Evaluation
The validity of financial models is often debated, as their effectiveness depends on several factors, including assumptions, time frames, and market conditions. My research also involves the scientific evaluation of financial models, continuously testing and refining these models to ensure they remain relevant and accurate under varying market scenarios.
Conclusion
Therefore, in conclusion, my research and development activities are driven primarily by my curiosity, research based goal oriented self-learning and need for creative problem solving. It involves research into techniques for mathematically describing the nature of stock price movements (and by extension / generalization of such techniques to describe similar random / semi-random in nature.
Main Theoretical Frameworks for Price‑Movement Mechanisms
| Theory / Model | Core Idea | Typical Formulation | What It Captures |
|---|---|---|---|
| Random Walk | Future price changes are independent of the past. | (P_{t+1}=P_t+\epsilon_t) with (\epsilon_t\sim\mathcal{N}(0,\sigma^2)) | Lack of predictable trends; martingale property. |
| Geometric Brownian Motion (GBM) | Continuous‑time log‑normal diffusion with constant drift and volatility. | (\frac{dS_t}{S_t}= \mu\,dt + \sigma\,dW_t) | Exponential growth, proportional volatility, basis for Black‑Scholes. |
| Mean‑Reverting Processes (Ornstein‑Uhlenbeck, Cox‑Ingersoll‑Ross) | Prices tend to revert toward a long‑run equilibrium level. | (dX_t = \kappa(\theta – X_t)dt + \sigma dW_t) | Commodity prices, interest rates, assets with anchoring forces. |
| Jump‑Diffusion Models (Merton, Kou) | Diffusive motion punctuated by discrete jumps. | (\frac{dS_t}{S_t}= \mu dt + \sigma dW_t + J dN_t) where (N_t) is a Poisson process. | Sudden news, earnings surprises, market crashes. |
| Stochastic Volatility Models (Heston, SABR) | Volatility itself follows a random process. | (\begin{cases} dS_t = \mu S_t dt + \sqrt{v_t} S_t dW_t^S \ dv_t = \kappa(\theta – v_t)dt + \xi\sqrt{v_t} dW_t^v \end{cases}) | Volatility clustering, implied‑vol smile. |
| Regime‑Switching Models (Markov‑Switching) | Model parameters switch between discrete states (e.g., bull vs. bear). | (\mu_t,\sigma_t) depend on hidden state (Z_t) governed by a Markov chain. | Different dynamics in distinct market phases. |
| Agent‑Based / Microstructure Models | Prices emerge from interactions of heterogeneous traders, order flow, and limit‑order book dynamics. | Simulations of order arrivals, cancellations, and price impact functions. | Short‑term price formation, liquidity effects, bid‑ask bounce. |
| Fractal / Multifractal Models (Mandelbrot, Multifractal Random Walk) | Price changes exhibit scaling laws and long‑range dependence. | (X(t)=\sum_{i=1}^{N(t)}\epsilon_i) with time‑deformation (N(t)) following a multifractal cascade. | Heavy tails, volatility clustering beyond simple diffusion. |
| Information‑Based Models (Elliott‑Kopp, Kyle) | Prices move as new information is incorporated, with market makers adjusting quotes based on order flow. | Price impact (\Delta P = \lambda Q) where (Q) is net order flow and (\lambda) reflects information asymmetry. | Gradual incorporation of private information, price discovery. |
How the Models Are Used
- Simulation & Risk Assessment – Monte‑Carlo draws from GBM, jump‑diffusion, or stochastic‑vol models generate possible future paths.
- Option Pricing – GBM (Black‑Scholes), Heston, or SABR provide the dynamics needed for closed‑form or numerical pricing.
- Statistical Inference – Maximum‑likelihood or Bayesian methods estimate parameters ((\mu,\sigma,\kappa,\theta), etc.) from historical price series.
- High‑Frequency Analysis – Agent‑based and microstructure models explain intraday price changes, order‑book resiliency, and spread dynamics.
These theories together form the contemporary toolkit for describing how stock prices fluctuate across time scales—from tick‑by‑tick microstructure to multi‑year investment horizons.
Emerging Frameworks (2020 – 2025)
| New Approach | Core Idea & How It Models the Path | Typical Implementation | What It Adds to Classic Theory |
|---|---|---|---|
| Neural Stochastic Differential Equations (Neural‑SDEs) | Parameter‑rich drift (\mu_\theta(\cdot)) and diffusion (\sigma_\phi(\cdot)) functions are learned by neural nets, then plugged into an SDE solver. | (\displaystyle dS_t = \mu_\theta(S_t,t)\,dt + \sigma_\phi(S_t,t)\,dW_t) Training via likelihood‑based or score‑matching objectives on historical price series. | Captures highly nonlinear, state‑dependent dynamics while retaining a continuous‑time stochastic foundation. |
| Deep Generative Path Models (GAN‑/VAE‑based) | A generator network produces entire price trajectories conditioned on market features (e.g., macro data, sentiment). | Conditional GAN: (G(z,\,x_{\text{cond}}) \rightarrow {S_{t_0},\dots,S_{t_T}}). VAE‑style latent space encodes plausible path shapes. | Enables realistic multi‑modal path distributions and easy scenario generation for stress testing. |
| Transformer‑Based Time‑Series Diffusion Models | Diffusion‑probabilistic models treat a price path as a sequence gradually “noised” and then denoised by a transformer. | Forward process adds Gaussian noise step‑wise; reverse process learned by a transformer that predicts the denoised price at each step. | Handles long‑range dependencies and complex conditional information (news, order‑book snapshots) better than Markovian SDEs. |
| Reinforcement‑Learning Market Simulators | The market is modeled as an environment where multiple RL agents (liquidity providers, speculators) interact; the resulting price path emerges from their policies. | Multi‑agent RL framework (e.g., OpenAI Gym‑style) with reward functions reflecting profit, inventory risk, and market impact. | Provides a bottom‑up view of price formation, useful for testing execution algorithms and understanding microstructure feedback loops. |
| Hybrid Agent‑Based + Stochastic‑Vol Models | Combines macro‑level stochastic volatility with micro‑level order‑flow agents; the agents’ aggregate order flow drives the volatility process. | Stochastic volatility SDE for (v_t) coupled to an agent‑based order‑book simulation that supplies net order flow (Q_t). | Bridges the gap between high‑frequency microstructure effects and longer‑horizon volatility dynamics. |
| Graph Neural Network (GNN) Market‑Structure Models | Treats the market as a graph of assets linked by correlation, supply‑chain, or ownership ties; price dynamics propagate across the graph. | GNN learns edge‑wise influence functions (\phi_{ij}) that modify each asset’s drift: (\mu_i(t)=\sum_j \phi_{ij} S_j(t)). | Captures cross‑asset contagion and sector‑wide co‑movement patterns that classic single‑asset SDEs miss. |
| Fractional‑Order Stochastic Processes | Extends Brownian motion to fractional Brownian motion with Hurst exponent (H\neq 0.5), allowing long‑range dependence. | (dS_t = \mu S_t dt + \sigma S_t dB_t^{(H)}). | Models persistent ( (H>0.5) ) or anti‑persistent ( (H<0.5) ) price trends observed in high‑frequency data. |
Why These Frameworks Matter
- Non‑linearity & Regime Shifts – Neural‑SDEs and transformer diffusion models can learn abrupt changes in drift/volatility that static (\mu,\sigma) cannot capture.
- Scenario Diversity – Generative GAN/VAE approaches produce many plausible future paths, aiding stress testing and risk‑budgeting.
- Micro‑Macro Integration – RL market simulators and hybrid agent‑based/stochastic‑vol models link order‑book dynamics to longer‑term volatility, offering a unified view.
- Cross‑Asset Interaction – GNN‑based models reflect the reality that a shock in one sector quickly ripples through related stocks.
- Long‑Memory Effects – Fractional processes address empirical evidence of autocorrelated returns over minutes to days, which classic Brownian motion ignores.
These newer frameworks are still active research areas; many are being incorporated into quantitative‑finance toolkits and academic papers published after 2020. They complement, rather than replace, the classic random‑walk and GBM foundations, providing richer, data‑driven descriptions of how stock prices fluctuate.
Promising Future Research Directions in Price‑Fluctuation Mechanisms
| Area | Emerging Questions | Potential Methods |
|---|---|---|
| Explainable Neural‑SDEs | How can we interpret the learned drift and diffusion functions in economic terms? | Symbolic regression on (\mu_\theta(\cdot),\sigma_\phi(\cdot)); attention‑based attribution; hybrid models that embed known factors (e.g., earnings, macro data) as constraints. |
| Multi‑Asset Fractional‑Order Networks | Do long‑memory effects propagate across correlated assets, and how does that affect systemic risk? | Fractional Brownian motion on graph Laplacians; estimation of Hurst exponents per node and edge. |
| Adaptive Regime‑Switching with Real‑Time Signals | Can we detect regime shifts (e.g., from low‑vol to high‑vol) instantly using alternative data (social media, news sentiment, order‑book microstructure)? | Online Bayesian changepoint detection; reinforcement‑learning agents that update transition matrices on the fly. |
| Hybrid Micro‑Macro Simulators | How do high‑frequency order‑flow dynamics aggregate to affect month‑scale volatility surfaces? | Coupled agent‑based order‑book models with stochastic‑volatility SDEs; multiscale Monte‑Carlo that nests fine‑grained simulations inside coarse‑grained paths. |
| Causal Inference for Price Drivers | Beyond correlation, what causal mechanisms (e.g., liquidity shocks, regulatory announcements) drive price paths? | Structural vector autoregressions with exogenous shock identification; Granger‑causality extensions using deep generative models. |
| Privacy‑Preserving Distributed Learning | Can market participants collaboratively train price‑path models without exposing proprietary data? | Federated learning of Neural‑SDE parameters; secure multiparty computation for joint estimation of drift/volatility. |
| Robustness to Model Misspecification | How sensitive are risk metrics (VaR, CVaR) to incorrect assumptions about diffusion or jump intensity? | Stress‑testing via adversarial perturbations of SDE coefficients; Bayesian model averaging across competing path frameworks. |
| Quantum‑Inspired Stochastic Modeling | Do quantum probability concepts (e.g., non‑commutative stochastic calculus) offer new ways to capture market uncertainty? | Development of quantum stochastic differential equations; simulation of price paths using quantum Monte‑Carlo algorithms. |
| Integration of ESG & Climate Risk | How will climate‑related events reshape the stochastic dynamics of affected sectors? | Regime‑switching models where transition probabilities depend on climate‑risk indices; scenario‑based jump processes tied to extreme‑event forecasts. |
| Real‑Time Calibration Pipelines | Can models be continuously re‑calibrated as new tick data arrives, maintaining stability? | Sequential Monte‑Carlo (particle filters) for SDE parameters; Kalman‑filter extensions for stochastic‑volatility states. |
These topics aim to close gaps between theoretical rigor, interpretability, and practical applicability in modeling how stock prices move over time.