TheQuantBateman

01

INTUITION

Observe the object before formalizing it.

Changing numeraire is a coordinate change: the relative value V/N is the martingale, not V by itself.

01

Every positive tradable numeraire induces a measure.

02

A T-bond numeraire simplifies T-settled claims.

03

Drifts change when the numeraire changes.

02

WHY MARKETS CARE

Connect the mathematical object to a pricing question.

Caplets, swaptions and multi-curve products are easiest to reason about under payoff-aligned measures.

INSTRUMENTS

caplets

swaptions

CMS coupons

QUOTE CONVENTION

P(t,T) is the OIS discount bond; payment and fixing dates are distinct.

03

MATHEMATICS

Construct the definition and its invariants.

Formula · Full derivation

Numeraire pricing

V_t=N_t\,\mathbb E^{\mathbb Q^N}\!\left[\frac{H_T}{N_T}\mid\mathcal F_t\right]

Relative prices are martingales under the measure induced by N.

Full derivation

From information set to computable quantity

Each line states the information, measure and unit before manipulating the expression.

01
Start under money-market measure
Express value as a Q expectation discounted by B.
$V_t=B_tE^Q[H_T/B_T|F_t]$
02
Form the density ratio
Compare the new numeraire N with B after normalizing initial values.
$\frac{dQ^N}{dQ}|_{F_t}=\frac{N_t/B_t}{N_0/B_0}$
03
Transfer the expectation
Bayes cancels the old discount factor against the density.
04
Align payment date
Set N=P(·,T), so N_T=1 and a T cash flow is undiscounted inside the expectation.
$V_t=P(t,T)E^{Q^T}[H_T|F_t]$

The result is valid only under the filtration, measure and discretization just made explicit.

Inputs

Nₜ: positive tradable numeraire
Qᵀ: T-forward measure

Assumptions and limits

One measure rarely simplifies every cash flow in a multi-date product.
Convexity appears when the natural measure differs from the payment measure.

Formula · Definition

Forward price

F_t^{T}=\frac{S_t}{P(t,T)}

The T-forward price is a martingale under Qᵀ when the asset has no interim cash flows.

05

MODEL / PRICING

Fit, compute, then challenge the assumptions.

METHOD

Match the numeraire to the payoff date, transform the drift and keep every expectation labelled by its measure.

CALIBRATION

Use the same OIS discount curve and forwarding conventions as the Rates track; the measure is not a free parameter.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

Typed domain validation
Deterministic seeded computation
Readout plus invariant

PYTHON 3 · NUMPY / SCIPY

Numeraires & forward measures

Reproduce the governing quantity, then challenge it with an invariant.

REUSABLE EXAMPLE

01import numpy as np

02

03def vtntmathbbemathbbq(x: np.ndarray) -> float:

04 x = np.asarray(x, dtype=float)

05 assert np.isfinite(x).all()

06 return float(np.mean(x))

07

08sample = np.array([0.8, 1.0, 1.2])

09value = vtntmathbbemathbbq(sample)

10assert sample.min() <= value <= sample.max()

11print(f"value={value:.6f}")

07

INTERACTIVE LAB

Run the thought experiment.

MEASURE-CHANGE ENGINE

Numeraires & forward measures

Switch measure and move the physical inputs. Drift, numeraire, martingale and density update together.

SYNTHETIC · EDUCATIONAL

Active drift3.90%forward

Market price λ0.2273(μ−r)/σ

NumeraireZero-coupon bond P(t,T)Tradable / P(t,T)

Likelihood ratio Zₜ by Brownian state Wₜ

Likelihood-ratio weights across the current Brownian state range.

Zₜ(Wₜ)

Brownian state Wₜ: -3.00. Zₜ(Wₜ): 1.927.

Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.

PRICING COORDINATES

Change weight, drift and martingale together

P(0,T) Eᴼᵀ[payoff]

01Physical lawμ=8.50%

Forecast and risk-premium distribution

02Densityλ=0.227

Positive likelihood-ratio process

03QᵀTradable / P(t,T)

Zero-coupon bond P(t,T)

08

FRONT OFFICE

Carry the abstraction into valuation.

ON THE DESK

“State the numeraire before stating the drift.”

VISIBLE INPUTS

discount curve

fixing/payment dates

forward curve

CALIBRATION

Use the same OIS discount curve and forwarding conventions as the Rates track; the measure is not a free parameter.

RISK

payment-delay convexity

measure inconsistency

DAILY WORKFLOW

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.

MACRO CONNECTION

Transmission from state to valuation

The causal chain separates the economic shock from the modelling response.

01Payment datetransmits

selects bond numeraire

02Forward measuretransmits

simplifies matching forward

03Convexityoutput

appears when measures misalign

10COMMON PITFALLSOpen the failure checklist.

01

Mixing a Q drift with a Qᵀ expectation

02

Ignoring delayed payment

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

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

OPEN ORIGINAL SOURCE ↗

From information set to computable quantity

Start under money-market measure

Form the density ratio

Transfer the expectation

Align payment date

Change weight, drift and martingale together

Transmission from state to valuation

Measure theory, simulation and computational-finance lectures