Filtrations & adapted processes
Model what is knowable before modelling what is tradable.
01Distinguish a σ-algebra from a filtration
02Test whether a process or strategy is adapted
03Identify look-ahead bias in a trading rule
Observe the object before formalizing it.
A filtration is the market's information clock: it grows, but it cannot reveal tomorrow's fixing today.
Events are measurable only when current information can decide them.
Adapted controls use no future observations.
Pricing statements inherit their information set.
Connect the mathematical object to a pricing question.
Execution, exercise, collateral and hedging decisions must be based on information available at decision time.
barrier monitoring
callable products
algorithmic hedges
Time t is an ACT/365-like year fraction; the lesson distinguishes observation time from payment time.
Construct the definition and its invariants.
Information growth
Later σ-algebras refine, and never forget, earlier distinctions.
Short derivation
From information set to computable quantity
Each line states the information, measure and unit before manipulating the expression.
- 01
Define scenarios
List terminal paths without assuming that their identity is already known.
- 02
Partition current knowledge
Indistinguishable scenarios belong to the same information atom.
- 03
Generate the σ-algebra
Take every union of current atoms; those and only those events are decidable now.
- 04
Audit the strategy
A position at t must be constant across paths that Fₜ cannot yet distinguish.
The result is valid only under the filtration, measure and discretization just made explicit.
Inputs
Ω: scenario spaceFₜ: information available by tXₜ adapted ⇔ Xₜ is Fₜ-measurable
Assumptions and limits
- Market information is idealised as common and instantaneous.
- Microstructure and asynchronous feeds require richer filtrations.
Adaptedness
Every observable decision about Xₜ is answerable with time-t information.
Fit, compute, then challenge the assumptions.
Represent information as nested partitions or σ-algebras and require every state/control at t to be Fₜ-measurable.
There is no parameter fit: validate event timestamps, observability and publication lags.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Typed domain validation
- Deterministic seeded computation
- Readout plus invariant
Filtrations & adapted processes
Reproduce the governing quantity, then challenge it with an invariant.
import numpy as np def mathcalfssubseteqm(x: np.ndarray) -> float: x = np.asarray(x, dtype=float) assert np.isfinite(x).all() return float(np.mean(x)) sample = np.array([0.8, 1.0, 1.2])value = mathcalfssubseteqm(sample)assert sample.min() <= value <= sample.max()print(f"value={value:.6f}")Run the thought experiment.
Filtrations & adapted processes
Move the information clock. The admissible decision and conditional value update without revealing future states.
Information determines admissible action
A trading rule can use exactly the events revealed by the current σ-algebra.
4 atomsCurrent information partition
t1Observable variables only
blockedNo future observation enters the rule
Carry the abstraction into valuation.
“If the signal was published after the hedge, it was never your signal.”
event calendar
fixing timestamps
There is no parameter fit: validate event timestamps, observability and publication lags.
RISKlook-ahead bias
barrier observation gaps
- Validate market state and timestamp
- Recompute the baseline
- Run a controlled perturbation
- Explain P&L and residuals
Production failure modes
- Silent convention or measure changes
- Unstable numerics hidden by plausible prices
09MACRO CONNECTIONOpen the transmission channel.
Transmission from state to valuation
The causal chain separates the economic shock from the modelling response.
transmitsFₜ expands
transmitsconditional law changes
outputadapted control is rebalanced
10COMMON PITFALLSOpen the failure checklist.
Treating all terminal facts as known at t=0
Confusing adapted with predictable
11SOURCES / FURTHER READINGOpen sources and continue the track.
Measure theory, simulation and computational-finance lectures
The lesson uses original prose and a fresh typed implementation; the linked material is a research map, not copied product code.
- Source
- Computational Finance Course
- Author
- L. A. Grzelak
- Ref
- main