Distributions, moments & characteristic functions
Describe a payoff law before estimating, transforming, or pricing it.
01Separate a random variable from its probability law.
02Compute expectations, variance, and standardized moments with units intact.
03Use a characteristic function to recover moments and support transform pricing.
04Diagnose when sample estimates do not identify a pricing distribution.
Observe the object before formalizing it.
A random variable maps scenarios to numbers; its law assigns probability to those numbers. Pricing, simulation and risk fail when those two objects—or their measures—are silently conflated.
Expectation is measure-dependent and carries the units of the variable.
Variance measures squared dispersion; volatility restores the original unit.
The characteristic function always exists and turns convolution into multiplication.
Connect the mathematical object to a pricing question.
Terminal payoff laws drive option values, exposure profiles, Monte Carlo estimators, Fourier pricing and tail-risk metrics.
European options
digital options
structured payoffs
portfolio loss distributions
State the probability measure, horizon, currency and whether the object is a return, price, payoff or loss before reporting a moment.
Construct the definition and its invariants.
Expectation
The two integrals describe the same average in scenario space and value space.
Variance
Variance is a second central moment and has squared units.
Characteristic function and moments
Derivatives at the origin recover moments when they exist; transform pricers work directly with φ.
Full derivation
From the law to its Fourier representation
Start from an integrable law, expand only when the relevant moments exist, and keep the probability measure explicit.
- 01
Encode the law
Average the complex exponential over the distribution of X.
- 02
Check normalization
At zero frequency the integrand is one, so total probability fixes the transform.
- 03
Differentiate under the integral
If the corresponding absolute moment is finite, differentiation can pass through the expectation.
- 04
Evaluate at the origin
The exponential disappears and the nth raw moment remains.
The characteristic function is a lossless representation of the law and a computational bridge to convolution and Fourier valuation.
Inputs
X: integrable random variableF_X(x)=P(X≤x): cumulative distributionμ=E[X], σ²=Var(X): first two central summariesφ_X(u)=E[e^{iuX}]: characteristic function
Assumptions and limits
- Finite variance and higher moments are assumptions, not universal properties.
- Sample skew and kurtosis are unstable in small or heavy-tailed samples.
- Risk-neutral and historical laws answer different questions.
Fit, compute, then challenge the assumptions.
Define the law and measure first; use analytical moments when available and deterministic seeded estimators only as approximations with sampling error.
Distribution parameters may be estimated under P or calibrated under Q; the two objectives are not interchangeable.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Pure distribution and moment functions
- Deterministic seed at the experiment boundary
- Analytical references beside sample diagnostics
Normal-law moments and transform
Verify φ(0)=1 and recover the first two moments from an analytical characteristic function.
from __future__ import annotations import cmath def normal_cf(u: float, mean: float, variance: float) -> complex: if variance < 0: raise ValueError("variance must be non-negative") return cmath.exp(1j*u*mean - 0.5*variance*u*u) mean, variance = 0.03, 0.04assert abs(normal_cf(0.0, mean, variance) - 1.0) < 1e-14h = 1e-4first = (normal_cf(h, mean, variance)-normal_cf(-h, mean, variance))/(2*h)second = (normal_cf(h, mean, variance)-2*normal_cf(0, mean, variance)+normal_cf(-h, mean, variance))/(h*h)assert abs(first.imag-mean) < 1e-8assert abs(-second.real-(mean*mean+variance)) < 1e-6print(f"mean={first.imag:.6f} second_moment={-second.real:.6f}")Run the thought experiment.
Distributions, moments & characteristic functions
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 moment without its measure, horizon and unit is not a risk number.”
return or payoff definition
measure and horizon
sample/filter policy
currency and unit
Choose estimation under P for forecasting or calibration under Q for pricing and document the objective.
RISKtail estimation
sampling error
measure mismatch
- define the random quantity
- declare measure and horizon
- compute analytical or sampled summaries
- compare distributional diagnostics
Production failure modes
- silent data filtering
- unstable tail moments
- mixing price and return units
09MACRO CONNECTIONOpen the transmission channel.
From regime uncertainty to a priced law
Economic regimes alter physical outcomes; market prices add state-price weights before a risk-neutral distribution is inferred.
transmitschanges physical frequencies
transmitschange state prices
outputidentify a Q-law
10COMMON PITFALLSOpen the failure checklist.
Calling a random variable its distribution.
Comparing variance with volatility without changing units.
Assuming every distribution has finite moments.
Using historical moments as pricing parameters without a measure argument.
11SOURCES / FURTHER READINGOpen sources and continue the track.
Ch. 1 §§1.1–1.1.3, printed pp. 1–8 (PDF pp. 20–27)
The formulation was checked against the cited sections; prose, examples and code are original to the Academy.
- Source
- Mathematical Modeling and Computation in Finance
- Author
- Cornelis W. Oosterlee and Lech A. Grzelak
- Ref
- First edition (2020)