196 lines
6.8 KiB
Markdown
196 lines
6.8 KiB
Markdown
# Adaptive Barrier Monitor
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A five-notebook quantitative-finance project connecting random walks, Brownian
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motion, geometric Brownian motion, first-passage times, Brownian bridges, and
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state-dependent monitoring.
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The motivating question is:
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> A stock is monitored for a large move over a short window. Continuous polling
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> is expensive. What probability model describes a hidden barrier crossing, and
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> how can that model inform a sampling schedule?
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The project is written for a mathematically mature reader who wants to see how
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Gaussian processes, conditioning, stochastic calculus, and Monte Carlo methods
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appear in a practical monitoring problem.
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## Core results and scope
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Under geometric Brownian motion,
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$$
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\frac{dS_t}{S_t}=\mu\,dt+\sigma\,dW_t,
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$$
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the relative log-price $X_t=\log(S_t/S_0)$ is arithmetic Brownian motion. A
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10% drop corresponds to the lower log barrier $B=\log(0.9)$.
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For zero drift, the probability of touching the barrier by time $T$ is
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$$
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P(\tau_B\leq T)=2\Phi\!\left(\frac{B}{\sigma\sqrt{T}}\right).
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$$
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At 30% annualised volatility, a 10% move in five trading minutes is roughly a
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49-standard-deviation diffusion event. Pure GBM therefore assigns it probability
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below ordinary floating-point resolution; jumps and market microstructure are
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essential for realistic extreme-move modelling.
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Conditional on two observations $x_0,x_T>B$, the Brownian-bridge probability
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that the hidden path crossed the barrier is
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$$
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P_{\mathrm{cross}}
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=\exp\!\left(
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-\frac{2(x_0-B)(x_T-B)}{\sigma^2\Delta t}
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\right).
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$$
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If both endpoint distances are set equal to $D$, inversion gives
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$$
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\Delta t_{\mathrm{sym}}
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=\frac{2D^2}{\sigma^2\log(1/\varepsilon)}.
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$$
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This inversion is exact **conditional on both endpoints being known**. In a
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live scheduler, the future endpoint is unknown; the implementation substitutes
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the current distance for both endpoints. Thus $\varepsilon$ is a local
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diffusion-design parameter, not an unconditional miss guarantee, and it does
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not control jumps. A hard maximum polling interval remains necessary.
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## Notebooks
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| # | Notebook | Main topics |
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|---|---|---|
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| 01 | [Random walks to Brownian motion](notebooks/01_random_walks_to_brownian_motion.ipynb) | Log returns, the $\min(s,t)$ covariance kernel, Cholesky sampling, Brownian scaling |
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| 02 | [GBM and Itô's lemma](notebooks/02_geometric_brownian_motion_and_ito.ipynb) | Multiplicative prices, exact GBM simulation, Itô correction |
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| 03 | [First passage and reflection](notebooks/03_first_passage_and_reflection_principle.ipynb) | Reflection principle, Bachelier–Lévy formula, hitting-time diagnostics |
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| 04 | [Brownian bridges and hidden crossings](notebooks/04_brownian_bridges_and_miss_probability.ipynb) | Gaussian conditioning, Schur complements, bridge crossing probabilities |
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| 05 | [Adaptive barrier monitoring](notebooks/05_adaptive_barrier_monitor.ipynb) | Unit-consistent scheduler, practical polling cap, controlled jump stress test, model-risk discussion |
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The analytical formulae are checked against Monte Carlo simulation in the
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notebooks.
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## Interactive web application
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The FastAPI/Plotly demo compares two sampling schedules on the **same simulated
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paths**. The adaptive schedule uses the local symmetric-endpoint bridge proxy,
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while the fixed baseline can run in either of two modes:
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- **Equal budget:** the fixed schedule receives exactly the adaptive schedule's
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sample count on each path, isolating where observations are placed.
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- **Fixed cadence:** the fixed schedule samples at a user-selected interval, so
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detection quality, lag, and total observation cost can be compared directly.
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A barrier event counts as detected only if a sampled point remains beyond the
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barrier within a configurable number of simulation steps. The comparison is
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therefore explicit and reproducible rather than based on an unrestricted
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"eventually detected" definition.
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The demo supports GBM and an optional Merton jump-diffusion stress mode. When
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jumps are enabled, the interface explicitly warns that the Brownian diffusion
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parameter $\varepsilon$ does not bound jump-event misses.
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### Run locally
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```bash
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python -m venv .venv
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source .venv/bin/activate
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python -m pip install -e ".[webapp]"
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python -m uvicorn webapp.app:app --host 127.0.0.1 --port 8055
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```
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Open `http://127.0.0.1:8055`.
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### Docker
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```bash
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docker compose -f docker-compose.webapp.yml up --build
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```
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For an existing Caddy Docker network:
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```bash
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docker compose -f docker-compose.webapp.proxy.yml up --build -d
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```
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The container runs as a non-root user and includes an HTTP health check.
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## Run the notebooks
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```bash
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python -m venv .venv
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source .venv/bin/activate
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pip install -e ".[notebooks]"
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jupyter lab notebooks/
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```
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Notebooks that request market data cache successful downloads under
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`data/cache/`. Their analytical and simulation sections remain usable when the
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network fetch is unavailable.
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## Tests
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```bash
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pip install -e ".[dev,webapp]"
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pytest
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```
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The test suite covers:
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- inversion of the Brownian-bridge formula;
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- vectorised interval calculations and input validation;
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- consistent time/volatility units in the adaptive schedule;
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- enforcement of the detection deadline;
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- exact per-path sample-budget equality;
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- equivalence of zero-intensity jump diffusion and GBM;
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- aggregate simulation invariants.
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## Project structure
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```text
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adaptive-barrier-monitor/
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├── notebooks/ # five executed research notebooks
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├── src/adaptive_barrier/
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│ ├── __init__.py
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│ └── engine.py # samplers, closed forms, scheduler, evaluation
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├── tests/
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│ └── test_engine.py
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├── webapp/
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│ ├── app.py # FastAPI API
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│ ├── Dockerfile
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│ └── static/ # vanilla JS, Plotly, CSS
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├── .github/workflows/tests.yml
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├── pyproject.toml
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├── requirements.txt
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├── requirements-webapp.txt
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├── requirements-dev.txt
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├── bibliography.md
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├── LICENSE
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└── webapp.md
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```
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## Model limitations
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- **Online endpoint uncertainty:** the bridge crossing formula is conditional on
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both endpoints; the scheduler uses a local approximation before the next
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endpoint exists.
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- **Jump risk:** diffusion-derived polling cannot guarantee detection of sudden
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jump-and-recovery events.
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- **No market microstructure model:** bid–ask bounce, asynchronous feeds,
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exchange halts, queueing, and packet latency are not represented.
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- **Simulation-grid dependence:** the web demo's detection deadline is measured
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in simulated grid steps; changing `n_steps` changes its physical duration.
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- **Educational calibration:** jump parameters in the sandbox are user-controlled
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stress parameters, not production estimates.
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## Tech stack
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Python, NumPy, SciPy, pandas, SymPy, Matplotlib, FastAPI, Pydantic, Uvicorn,
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Plotly.js, Docker, pytest, and GitHub Actions.
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## License
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MIT — see [`LICENSE`](LICENSE).
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