{
  "lesson": "Introduction to Numerical Methods",
  "slug": "numerical-methods",
  "created": "2026-09-19",
  "source": {
    "title": "Introduction to Numerical Methods",
    "course": "Certificate in Quantitative Finance",
    "user_supplied_filename": "JA253.4 Notes.pdf",
    "pages": 62,
    "author": null,
    "publication_year": null,
    "coverage": {
      "monte_carlo": "4-17",
      "grid_and_derivatives": "18-34",
      "terminal_and_explicit_scheme": "35-48",
      "boundaries": "49-58",
      "method_tradeoffs": "59-62"
    },
    "redistribution": "Original PDF and slide screenshots are excluded. Thai prose, calculations and figures are newly authored."
  },
  "supplementary_sources": [
    {
      "author": "Mike Giles",
      "institution": "University of Oxford",
      "title": "Monte Carlo Lecture 1",
      "url": "https://people.maths.ox.ac.uk/gilesm/mc/mc/lec1.pdf",
      "used_for": "Exact terminal GBM and Box-Muller transformation",
      "accessed": "2026-09-19"
    },
    {
      "author": "Mike Giles",
      "institution": "University of Oxford",
      "title": "Monte Carlo Lecture 9",
      "url": "https://people.maths.ox.ac.uk/gilesm/mc/mc/lec9.pdf",
      "used_for": "Conditional strong and weak convergence orders",
      "accessed": "2026-09-19"
    }
  ],
  "assumptions": {
    "data_status": "Hypothetical, no market data or calibration",
    "contract": "European Call or Put, one unit, no dividends",
    "model": "Constant-parameter risk-neutral GBM / Black-Scholes PDE",
    "S": 100,
    "K": 100,
    "r": 0.03,
    "sigma": 0.2,
    "T": 1,
    "price_unit": "USD",
    "time_unit": "year",
    "r_convention": "Continuously compounded annual rate",
    "sigma_unit": "per square root year"
  },
  "monte_carlo": {
    "method": "IID exact terminal GBM, mean discounted payoff; no antithetics",
    "javascript_seed": 73,
    "javascript_samples": 10000,
    "python_seed": 2530401,
    "python_samples": 20000,
    "normal_generation": {
      "javascript": "Existing seeded normalGenerator with Box-Muller",
      "python": "random.Random(seed).gauss; Python standard-library Normal generator"
    },
    "se": "Sample SD of discounted payoffs divided by sqrt(N)",
    "interval": "Approximate normal 95% interval, mean +/- 1.96 SE; sampling uncertainty only",
    "reproducibility": "Generator-specific seeds; JavaScript and Python samples need not coincide"
  },
  "finite_difference": {
    "scheme": "Forward Euler in tau=T-t; central first and second spatial derivatives",
    "Smax": 400,
    "space_intervals": 80,
    "time_steps": 1000,
    "boundary": "Analytic S=0 and asymptotic Dirichlet Smax conditions for European Call/Put",
    "stability_check": "Sufficient nonnegative coefficient check at every interior node; negative spatial rates are checked independently of dt. Unsafe configurations return no computed price.",
    "interpolation": "Linear in S for price and interior central-difference Greeks",
    "theta": "Calendar-time sign, final two tau layers, USD/year",
    "limitations": "Finite domain, mesh error, payoff kink; positivity does not establish accuracy",
    "python_greeks": "Nearest interior node, with greek_spot explicitly reported; Python helper requires sigma>0 and T>0"
  },
  "editorial_clarifications": [
    "Risk-neutral drift is not a forecast of physical returns",
    "Exact grid-date GBM does not remove continuous monitoring error",
    "Euler weak and strong convergence orders require conditions",
    "Sum of twelve uniforms is not an exact Normal distribution",
    "Increasing N does not reduce model error",
    "Reducing dt cannot repair negative central spatial rates",
    "Alternative gamma-zero upper boundary is not used by the lab"
  ],
  "visuals": {
    "design_system": "QuantCorner / QuantSeras; existing Thai light book layout with optional dark theme",
    "route": "no-image-generator",
    "image_generator_used": false,
    "assets": [
      "assets/images/numerical-monte-carlo.svg",
      "assets/images/numerical-finite-difference.svg",
      "assets/images/numerical-grid-convergence.svg"
    ],
    "source_script": "scripts/make_numerical_methods_figures.py",
    "meaning": "Computed hypothetical examples, not reconstructed source charts"
  },
  "verification": {
    "numerical": "node qa/numerical-methods-checks.mjs",
    "browser": "node qa/numerical-methods-page-checks.cjs",
    "notebook": "python3 scripts/make_numerical_methods_notebook.py",
    "build": "npm run build:pages"
  }
}