{
  "recorded_at": "2026-09-19",
  "chapter": "martingale-pricing.md",
  "source": {
    "provided_filename": "JA253.2 Notes.pdf",
    "title_visible": "CQF Module 3 — Martingales Theory: Application to Option Pricing — Black-Scholes All Over Again",
    "visible_course": "CQF",
    "physical_pages": 120,
    "displayed_slide_numbers": "1–118, followed by two bibliography pages also labeled 118/118",
    "sha256": "2a427782633a35977b3ffd9690f21774ac97e69fb42dc1f3c014dd495ae2df5c",
    "individual_author": "No individual authorship asserted from PDF metadata.",
    "publication_date": "Not established; PDF metadata is not treated as publication date.",
    "access": "User-supplied private PDF, read locally. No PDF, source pages, logos, or page screenshots redistributed.",
    "method": "Local text extraction plus visual verification of key formulas; original Thai explanation and recomputed examples."
  },
  "source_topic_map": [
    {
      "pages": [
        4,
        8
      ],
      "topics": [
        "market assumptions",
        "filtration",
        "stock and money-market account"
      ]
    },
    {
      "pages": [
        9,
        22
      ],
      "topics": [
        "self-financing",
        "admissible strategies",
        "arbitrage"
      ]
    },
    {
      "pages": [
        23,
        27
      ],
      "topics": [
        "discounted stock"
      ]
    },
    {
      "pages": [
        28,
        48
      ],
      "topics": [
        "P and Q",
        "Radon–Nikodym density",
        "Girsanov",
        "Novikov"
      ]
    },
    {
      "pages": [
        49,
        62
      ],
      "topics": [
        "fundamental asset pricing formula",
        "replication",
        "completeness"
      ]
    },
    {
      "pages": [
        63,
        87
      ],
      "topics": [
        "lognormal payoff integral",
        "cash/asset digitals",
        "stock numeraire"
      ]
    },
    {
      "pages": [
        88,
        90
      ],
      "topics": [
        "numeraire pair"
      ]
    },
    {
      "pages": [
        91,
        95
      ],
      "topics": [
        "Feynman–Kac",
        "PDE and expectation"
      ]
    },
    {
      "pages": [
        96,
        99
      ],
      "topics": [
        "continuous dividend yield"
      ]
    },
    {
      "pages": [
        100,
        103
      ],
      "topics": [
        "deterministic time-varying parameters"
      ]
    },
    {
      "pages": [
        104,
        115
      ],
      "topics": [
        "Black-76",
        "options on futures"
      ]
    },
    {
      "pages": [
        116,
        120
      ],
      "topics": [
        "conclusion and bibliography"
      ]
    }
  ],
  "editorial_corrections": [
    "Self-financing is imposed through trading gains; changing holdings does not justify discarding product-rule terms.",
    "Restrict trading strategies to an admissible class, and add true-martingale/integrability conditions before conditional expectation (source p56).",
    "Novikov is sufficient, not necessary or equivalent (source p80).",
    "Complete-market uniqueness is qualified by Brownian filtration, nonzero volatility and the eligible claim class.",
    "Numeraire is a strictly positive traded self-financing asset/portfolio under suitable measure-change assumptions, not any positive process (source p89).",
    "The probabilistic approach does recover Delta through differentiation/Ito; avoid the categorical source p87 claim.",
    "Define physical mu as total expected return in the dividend section; include dividends in discounted gains.",
    "Use the final correct dividend formula, not the inconsistent intermediate reinvested-stock equality on source p99.",
    "Integrated variance A replaces sigma squared times tau, not sigma squared alone (source p102).",
    "Separate option expiry T from futures delivery U; payoff is G(F_T), not G(S_T) (source pp109–115).",
    "Black-76 examples assume an upfront premium and cash payoff at option expiry; deterministic rates permit the forward/futures equality in the specified carry model.",
    "Equivalent measures preserve null events, not physical forecasts; Phi(d2) belongs to Q and Phi(d1) to the stock measure for non-dividend-paying stocks.",
    "The source lessons are paraphrased and reorganized, not reproduced as full slide translations."
  ],
  "examples": {
    "status": "Hypothetical model inputs; no empirical market observations.",
    "units": "Price units per one underlying unit; time in years; continuously compounded rates per year; volatility per square-root year.",
    "base": {
      "S": 100,
      "K": 100,
      "mu": 0.12,
      "r": 0.05,
      "sigma": 0.2,
      "tau": 1,
      "analytic_call": 10.450583572185558,
      "analytic_put": 5.573526022257
    },
    "dividend": {
      "yield": 0.02,
      "call": 9.227005508154036,
      "put": 6.330080627549918
    },
    "term_structure": {
      "segments": [
        {
          "duration": 0.5,
          "r": 0.04,
          "D": 0.01,
          "sigma": 0.1
        },
        {
          "duration": 0.5,
          "r": 0.06,
          "D": 0.03,
          "sigma": 0.3
        }
      ],
      "R": 0.05,
      "Y": 0.02,
      "A": 0.05
    },
    "black76": {
      "F": 100,
      "K": 100,
      "r": 0.05,
      "sigmaF": 0.2,
      "tau": 1,
      "call_and_put": 7.57708214642728
    }
  },
  "web_labs": {
    "mounts": [
      "measure-change-lab",
      "pricing-extensions-lab"
    ],
    "model": "Exact terminal GBM; no Euler discretization.",
    "monte_carlo_samples": 20000,
    "seed": 2532,
    "rng": "Existing project normalGenerator (32-bit linear congruential generator with multiplier 1664525 and increment 1013904223, plus Box–Muller).",
    "coupling": "Common standard Normal draws for physical and direct pricing-measure simulations; this is a variance-reducing comparison, not an assertion that both simulated prices are the same sample path under two measures.",
    "weighted_estimator": "Mean of Z times discounted payoff, without sample normalization; SE computed from the weighted observations.",
    "chart": "Theoretical unconditional discounted stock means, explicitly distinguished from simulated price paths and conditional martingale definition.",
    "limits": "Finite-sample error and model error distinguished; likelihood weighting may have large variance."
  },
  "notebook": {
    "file": "notebooks/martingale-pricing.ipynb",
    "generator": "scripts/make_martingale_pricing_notebook.py",
    "dependencies": "Python standard library",
    "source": "Canonical chapter Markdown with SHA256 recorded in notebook metadata.",
    "rng": "Python random.Random; separate labeled seed and samples, not claimed identical to JavaScript."
  },
  "visuals": {
    "route": "no-image-generator",
    "design_system": "Existing QuantCorner light book with purple/teal chart roles and optional dark mode.",
    "implementation": "Native React/SVG computed from the mathematical model. No new third-party artwork or image generation."
  },
  "supplementary_primary_references": [
    {
      "title": "The Black-Scholes Model",
      "author": "Martin Haugh",
      "url": "https://www.columbia.edu/~mh2078/FoundationsFE/BlackScholes.pdf"
    },
    {
      "title": "Stochastic Calculus: An Introduction with Applications",
      "author": "Gregory F. Lawler",
      "url": "https://www.math.uchicago.edu/~lawler/finbook.pdf",
      "sections": [
        "4.3",
        "5.3",
        "5.4",
        "5.5",
        "5.6",
        "5.7"
      ]
    }
  ],
  "verification_commands": [
    "python3 scripts/make_martingale_pricing_notebook.py",
    "npm test",
    "npm run build:pages",
    "node qa/martingale-pricing-page-checks.cjs",
    "npm run check:site",
    "git diff --check"
  ]
}