Equivalent measures & Radon–Nikodym densities
Reweight scenarios without changing which events are impossible.
01Separate physical and pricing probabilities
02Interpret the density process
03Transform expectations safely
Observe the object before formalizing it.
A change of measure changes scenario weights, not the payoff map or the set of null events.
Equivalence preserves zero-probability events.
The density is positive and has unit expectation.
Risk premia migrate from drift into probabilities.
Connect the mathematical object to a pricing question.
Derivative valuation uses pricing probabilities while forecasting, stress and risk management still require the physical distribution.
all derivatives
credit migration
insurance-linked claims
P and Q are explicitly labelled; rates and volatilities are decimals.
Construct the definition and its invariants.
Expectation transfer
The Q expectation is a P expectation with likelihood-ratio weights.
Full derivation
From information set to computable quantity
Each line states the information, measure and unit before manipulating the expression.
- 01
Require absolute continuity
If P assigns zero mass to an event, Q cannot resurrect it.
- 02
Introduce the density
Radon–Nikodym gives a non-negative integrable likelihood ratio.
- 03
Normalize
Unit mass under Q forces Eᴾ[Z]=1.
- 04
Move the expectation
Insert Z into the P integral and preserve the payoff scenario by scenario.
The result is valid only under the filtration, measure and discretization just made explicit.
Inputs
P: physical measureQ: pricing measureZₜ=dQ/dP|Fₜ
Assumptions and limits
- Equivalent martingale measures need not be unique in incomplete markets.
- Density tails can create severe Monte Carlo variance.
Bayes under measure change
Conditional reweighting requires the time-t density in the denominator.
Fit, compute, then challenge the assumptions.
Compute a positive density process and use it as a likelihood ratio for unconditional or conditional expectations.
Infer the market price of risk only from a jointly specified physical model and pricing model.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Typed domain validation
- Deterministic seeded computation
- Readout plus invariant
Equivalent measures & Radon–Nikodym densities
Reproduce the governing quantity, then challenge it with an invariant.
import numpy as np def mathbbemathbbqxmat(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 = mathbbemathbbqxmat(sample)assert sample.min() <= value <= sample.max()print(f"value={value:.6f}")Run the thought experiment.
Equivalent measures & Radon–Nikodym densities
Switch measure and move the physical inputs. Drift, numeraire, martingale and density update together.
Likelihood-ratio weights across the current Brownian state range.
- Zₜ(Wₜ)
Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.
Change weight, drift and martingale together
B₀ Eᴼ[S payoff / Bₜ]
μ=8.50%Forecast and risk-premium distribution
λ=0.227Positive likelihood-ratio process
Sₜ / BₜMoney-market account Bₜ
Carry the abstraction into valuation.
“Q prices the claim; P tells you how often the desk may live through the scenario.”
physical drift
pricing curve
volatility
Infer the market price of risk only from a jointly specified physical model and pricing model.
RISKmeasure mismatch
density-weight variance
- 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.
transmitstransmitsstates are reweighted
outputdiscounted tradables become martingales
10COMMON PITFALLSOpen the failure checklist.
Calling Q the real-world probability
Changing measure without changing drift
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