Conditional expectations & martingales
Project future cash flows onto today's information.
01Interpret conditional expectation as an information projection
02Apply the tower property
03Test a discounted process for the martingale property
Observe the object before formalizing it.
Conditional expectation replaces an unknown payoff by the best integrable quantity measurable with what is known now.
It is a random variable, not usually a scalar.
It preserves averages on every known event.
The tower removes information in stages.
Connect the mathematical object to a pricing question.
Every risk-neutral price, exposure profile and continuation value is a conditional expectation.
American options
exposure simulation
structured notes
Values are numeraire-denominated and integrable under the stated measure.
Construct the definition and its invariants.
Defining identity
Known events receive the same average mass before and after projection.
Short derivation
From information set to computable quantity
Each line states the information, measure and unit before manipulating the expression.
- 01
Choose the information atoms
Each atom contains states still indistinguishable at t.
- 02
Average within each atom
Weight terminal payoffs by their conditional probabilities.
- 03
Construct the projection
Assign m_A to every scenario inside the same information atom.
- 04
Apply the tower
Project once more onto a coarser σ-algebra and recover the direct projection.
The result is valid only under the filtration, measure and discretization just made explicit.
Inputs
E[X|Fₜ]: Fₜ-measurable projectionMₜ: integrable adapted process
Assumptions and limits
- Nested regression can introduce projection bias.
- Integrability is required; tail models matter.
Martingale condition
Given current information, the future process has no predictable drift.
Fit, compute, then challenge the assumptions.
Compute node-wise conditional averages and verify measurability, integrability and the tower property.
Estimate transition probabilities only when using an empirical tree; pricing trees impose the chosen measure.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Typed domain validation
- Deterministic seeded computation
- Readout plus invariant
Conditional expectations & martingales
Reproduce the governing quantity, then challenge it with an invariant.
import numpy as np def intamathbbexmidmat(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 = intamathbbexmidmat(sample)assert sample.min() <= value <= sample.max()print(f"value={value:.6f}")Run the thought experiment.
Conditional expectations & martingales
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.
“A continuation value is a conditional expectation with an exercise decision attached.”
state variables
transition law
Estimate transition probabilities only when using an empirical tree; pricing trees impose the chosen measure.
RISKregression bias
nested Monte Carlo error
- 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.
transmitsstate atom narrows
transmitsfuture scenarios are reweighted
outputcontinuation estimate moves
10COMMON PITFALLSOpen the failure checklist.
Replacing E[X|Fₜ] with unconditional E[X]
Ignoring the measure in the conditioning operator
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