TheQuantBateman

01

INTUITION

Define the exposure before compressing it into a metric.

xVA is a portfolio-level adjustment framework whose ingredients must share exposure, discounting and survival conventions.

01

CVA prices counterparty default loss.

02

DVA reflects own-default benefit under a chosen framework.

03

FVA depends on funding policy and collateral.

02

WHY MARKETS CARE

Fix portfolio, scenarios, horizon, and legal terms.

Uncollateralized derivative prices and new-deal charges require consistent counterparty and funding adjustments.

INSTRUMENTS

OTC derivatives

netting sets

secured funding

QUOTE CONVENTION

Loss given default, marginal default probability and discount factors share one timeline.

03

MATHEMATICS

Aggregate with an explicit measure and convention.

Formula · Full derivation

Discrete CVA

CVA\approx(1-R)\sum_i DF_i\,EE_i\,\Delta PD_i

Discounted expected positive exposure is weighted by marginal default loss.

Full derivation

From information set to computable quantity

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

01
Condition on default interval
Partition first default time into future buckets.
02
Apply close-out loss
Loss equals LGD times positive close-out exposure.
$L_i=(1-R)(V_i-C_i)^+$
03
Weight and discount
Multiply conditional exposure by marginal default probability and discount factor.
04
Reconcile adjustments
Specify close-out, funding and collateral assumptions once to prevent overlap.
Numerical checkWith zero default probability and zero funding spread, simplified xVA is zero.

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

Inputs

LGD=1−R
dPD(t): marginal default probability

Assumptions and limits

FVA is framework- and policy-dependent.
Replacement close-out and own default require governance decisions.

Formula · Short derivation

Funding adjustment

FVA\approx\sum_i DF_i\,s_i^F\,F_i\,\Delta t_i

Funding requirement is integrated against the relevant funding spread.

05

MODEL / PRICING

Reconcile valuation, risk, and model limitations.

METHOD

Integrate discounted exposure profiles against survival/default and funding curves under one close-out convention.

CALIBRATION

Bootstrap hazard from liquid credit instruments and document recovery, wrong-way and funding assumptions.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

Typed domain validation
Deterministic seeded computation
Readout plus invariant

PYTHON 3 · NUMPY / SCIPY

CVA, DVA & FVA

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

REUSABLE EXAMPLE

01import numpy as np

02

03def cvaapproxrsumidfie(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 = cvaapproxrsumidfie(sample)

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

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

07

INTERACTIVE LAB

Move the state. Challenge the equation.

PORTFOLIO RISK LAB

CVA, DVA & FVA

Change scale, volatility, collateral and confidence. Exposure, tail and adjustment metrics respond from one synthetic portfolio.

SYNTHETIC · EDUCATIONAL

EPE0.427mtime-average EE

CVA0.0083mLGD 60% · PD 3.5%

Peak PFE2.548m95.00%

exposure (mm) by future year

Expected and potential future exposure, before and after collateral.

EE
PFE
collateralized EE

future year: 0.0Y. EE: 0.000m. PFE: 0.000m. collateralized EE: 0.000m.

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

View chart data

exposure (mm) by future year
future year	EE	PFE	collateralized EE
0.0Y	0.000m	0.000m	0.000m
0.1Y	0.248m	1.021m	0.248m
0.3Y	0.341m	1.406m	0.341m
0.4Y	0.407m	1.677m	0.350m
0.5Y	0.457m	1.884m	0.350m
0.6Y	0.497m	2.048m	0.350m
0.8Y	0.529m	2.179m	0.350m
0.9Y	0.554m	2.285m	0.350m
1.0Y	0.574m	2.369m	0.350m
1.1Y	0.590m	2.434m	0.350m
1.3Y	0.602m	2.483m	0.350m
1.4Y	0.610m	2.517m	0.350m
1.5Y	0.616m	2.538m	0.350m
1.6Y	0.618m	2.548m	0.350m
1.8Y	0.617m	2.546m	0.350m
1.9Y	0.615m	2.534m	0.350m
2.0Y	0.609m	2.512m	0.350m
2.1Y	0.602m	2.482m	0.350m
2.3Y	0.592m	2.443m	0.350m
2.4Y	0.581m	2.395m	0.350m
2.5Y	0.568m	2.341m	0.350m
2.6Y	0.553m	2.279m	0.350m
2.8Y	0.536m	2.209m	0.350m
2.9Y	0.517m	2.134m	0.350m
3.0Y	0.498m	2.051m	0.350m
3.1Y	0.476m	1.963m	0.350m
3.3Y	0.453m	1.868m	0.350m
3.4Y	0.429m	1.768m	0.350m
3.5Y	0.403m	1.662m	0.350m
3.6Y	0.376m	1.550m	0.350m
3.8Y	0.348m	1.433m	0.348m
3.9Y	0.318m	1.311m	0.318m
4.0Y	0.287m	1.184m	0.287m
4.1Y	0.255m	1.052m	0.255m
4.3Y	0.222m	0.916m	0.222m
4.4Y	0.188m	0.774m	0.188m
4.5Y	0.152m	0.628m	0.152m
4.6Y	0.116m	0.478m	0.116m
4.8Y	0.078m	0.323m	0.078m
4.9Y	0.040m	0.163m	0.040m
5.0Y	0.000m	0.000m	0.000m

RISK AGGREGATION

Scenario → distribution → decision

Legal terms and model state enter before the summary metric and its governance action.

01State18.00% vol

Synthetic market scenarios

02Netting / CSA0.35m

Unsecured exposure boundary

03CVA0.008m

Decision metric with explicit convention

08

FRONT OFFICE

Turn exposure into a controlled decision.

ON THE DESK

“If two adjustments charge the same funding exposure, the spreadsheet is not conservative—it is inconsistent.”

VISIBLE INPUTS

EE/ENE profiles

hazard curves

funding spreads

CALIBRATION

Bootstrap hazard from liquid credit instruments and document recovery, wrong-way and funding assumptions.

RISK

CVA spread

funding basis

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.

01Credit spreadtransmits

moves marginal PD

02Exposuretransmits

sets loss amount

03xVAoutput

changes fair-value adjustment

10COMMON PITFALLSOpen the failure checklist.

01

Using cumulative PD in every bucket

02

Adding FVA without a funding convention

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

Exposure, counterparty credit and xVA notebooks

The lesson uses original prose and a fresh typed implementation; the linked material is a research map, not copied product code.

Source: Financial Engineering: Interest Rates & xVA
Author: L. A. Grzelak
Ref: main

OPEN ORIGINAL SOURCE ↗

From information set to computable quantity

Condition on default interval

Apply close-out loss

Weight and discount

Reconcile adjustments

Scenario → distribution → decision

Transmission from state to valuation

Exposure, counterparty credit and xVA notebooks