APPLIED RESEARCH / 2026

Research library

The notes connect a decision question with a method, stated assumptions and explicit limitations. They distinguish documented facts from research proposals.

Research connected to three engagement questions

The links below provide methodological context. They are not completed client cases or evidence of production readiness. All other research themes remain available in the catalogue.

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New reading pathways: macroeconomics, inference and Greeks

Original syntheses of selected passages from Romer, the two Oxford handbooks and Alòs & Garcia Lorite. Sixteen notes and four synthetic-data demonstrations connect assumptions, numerical checks and model-review questions.

Macroeconomics: mechanisms before forecasts Econometrics: flexibility, identification and numerical evidence Malliavin calculus: sensitivities and volatility dynamics

Corporate Transactions, Private Equity & Legal Risk

A cross-disciplinary programme connecting valuation, contracts, financing and governance.

Explore the programme →The six working papers →

Behavioural finance: six selected notes →

RESEARCH LIBRARY

Global Markets, risk, structuring and transformation

Each note states a question, method, implications, limitations and sources.

Model review notes · four analytical frameworks

Original AHS technical notes · 26 October 2026 · synthetic examples, not peer reviewed. Equations and checks describe the current educational demonstrators; they do not establish production readiness. Find these notes through the site-wide search as well as the links below.

Black–Scholes: price, Greeks and implied volatility

Question and conclusion

Separate price, local sensitivity and an implied parameter. A volatility inferred from one call quotation is not a forecast of realised volatility.

Scope and assumptions

European exercise, constant continuously compounded rates and dividend yield, constant volatility and a frictionless diffusion framework. Positive spot, strike and maturity; nonnegative volatility.

Equations and units

C=S_0e^{-qT}\Phi(d_1)-Ke^{-rT}\Phi(d_2),\quad d_1=\frac{\ln(S_0/K)+(r-q+\sigma^2/2)T}{\sigma\sqrt{T}},\quad d_2=d_1-\sigma\sqrt{T}

C-P=S_0e^{-qT}-Ke^{-rT},\qquad \mathrm{Vega}_{1pp}=0.01\,S_0e^{-qT}\phi(d_1)\sqrt{T}

S₀ and K: currency per asset unit; T: years; r and q: annual rates; σ: annualised return volatility. Formula rates use decimals; input fields use percentages. Φ and φ are the standard normal CDF and density. Vega is the price change derivative per one percentage point of volatility, not per 100 percentage points.

Reproducible checks

Check put–call parity on the same inputs.
Compare delta, gamma and vega with centred finite differences and consistent bump units.
At zero volatility, use the discounted intrinsic limit. At the forward-strike kink, the tool leaves the affected Greeks undefined.

Synthetic example and calculated outputs

Rates and volatilities below are input percentages, not decimal rates. Values are generated by the same current model; the checks above provide separate consistency tests.

S₀: 100
K: 105
T (years): 1
r (%): 3
q (%): 1
σ (%): 22
Call quote (CCY): 8

Calculated illustration · rounded to six decimals
C (CCY / asset)	7.427580
P (CCY / asset)	10.319377
Vega / +1pp (CCY / asset)	0.394887
C − P − S₀ exp(−qT) + K exp(−rT) (CCY / asset)	0.000000

Decision and limitations

Use as a convention and local hedging benchmark. Review smile, jumps, discrete hedging, liquidity and costs outside this model. The IV solver rejects incompatible quotations and caps its search bracket at 500%; a solver failure does not establish a market opportunity.

References and adaptation

Black & Scholes (1973), The Pricing of Options and Corporate Liabilities, JPE 81(3), 637–654.

Foundational references establish the model context. The displayed conventions, simplifications, synthetic examples and review protocol are AHS adaptations.

Open the associated model →

Two flat curves: swap value and three distinct rate shocks

Question and conclusion

Separate discounting, projection and the paid fixed coupon. In this educational flat-curve model, their one-basis-point shocks have different meanings and signs.

Scope and assumptions

Receive floating and pay fixed, constant OIS discount rate and a flat projected rate plus basis. One, two or four payments per year; the last stub is retained. This is not a calibrated market-curve bootstrap.

Equations and units

A=\sum_i\delta_i e^{-r_d t_i},\qquad PV=N(F-K)A,\qquad F=\frac{f_{input}}{100}+\frac{b_{input}}{10000}

\Delta PV_{projection,+1bp}=10^{-4}NA,\quad \Delta PV_{coupon,+1bp}=-10^{-4}NA,\quad \Delta PV_{OIS,+1bp}=N(F-K)\{A(r_d+10^{-4})-A(r_d)\}

N: currency notional; tᵢ and δᵢ: years; r_d, F and K: annual decimal rates; b_input: basis points. A is the discounted accrual sum in years. The interface reports par as a percentage. OIS P&L is a finite +1bp shock; projection DV01 is a sensitivity to the flat projected rate, not a full market-curve risk vector.

Reproducible checks

Verify that accruals sum to maturity, including a non-integer final stub.
Set the fixed coupon equal to the projected rate plus basis: model NPV must vanish.
Bump projection, coupon and OIS separately; reconcile signs and monetary units.

Synthetic example and calculated outputs

Rates and volatilities below are input percentages, not decimal rates. Values are generated by the same current model; the checks above provide separate consistency tests.

N (CCY): 10000000
T (years): 5.25
K (%): 3.1
OIS (%): 2.6
Projection (%): 2.95
Basis (bp): 10
Payments / year: 2

Calculated illustration · rounded to six decimals
PV (CCY)	-24381.643708
Par (%)	3.050000
Projection +1bp (CCY)	4876.328742
OIS +1bp P&L (CCY)	6.851479

Decision and limitations

Use to attribute a simplified valuation change, then review actual calendars, collateral terms, tenor-specific projection and market calibration. Do not interpret the three shocks as interchangeable hedge ratios.

References and adaptation

Bianchetti, Two Curves, One Price, arXiv:0905.2770v4 (2012). Context for distinct forwarding and discounting curves; the flat-rate implementation above is an AHS simplification.

Foundational references establish the model context. The displayed conventions, simplifications, synthetic examples and review protocol are AHS adaptations.

Open the associated model →

Structural Merton: distinguish physical and pricing probabilities

Question and conclusion

Equity and risky debt share the same asset value, but default probabilities depend on the probability measure. Changing the physical drift must not change risk-neutral equity or debt values.

Scope and assumptions

One zero-coupon debt payment at maturity, lognormal firm assets, constant rate and asset volatility, no asset payouts. Default occurs at maturity when assets are below face debt. This is the 1974 structural model, not the 1976 jump model.

Equations and units

E=V\Phi(d_1^Q)-Fe^{-rT}\Phi(d_2^Q),\quad D=V-E,\quad d_2^Q=\frac{\ln(V/F)+(r-\sigma_V^2/2)T}{\sigma_V\sqrt{T}}

PD^Q=\Phi(-d_2^Q),\quad PD^P=\Phi\!\left(-\frac{\ln(V/F)+(\mu-\sigma_V^2/2)T}{\sigma_V\sqrt{T}}\right),\quad s_{bp}=10000\left[-\frac{\ln(D/F)}{T}-r\right]

V, F, E and D: common currency amounts; T: years; r and μ: annual decimal rates; σ_V: asset-return volatility. d₁^Q = d₂^Q + σ_V√T. The interface multiplies both PDs by 100 and reports the spread in basis points. μ is entered, not estimated from market data.

Reproducible checks

Verify V = E + D and D ≤ F exp(−rT) within numerical tolerance.
Change μ alone: physical PD changes; prices and risk-neutral PD stay fixed.
Inspect leverage and asset-volatility scenarios without substituting equity volatility.

Synthetic example and calculated outputs

Rates and volatilities below are input percentages, not decimal rates. Values are generated by the same current model; the checks above provide separate consistency tests.

V (CCY): 150
F (CCY): 100
T (years): 1
r (%): 3
μ (%): 6
σV (%): 25

Calculated illustration · rounded to six decimals
E (CCY)	53.450283
D (CCY)	96.549717
PD(P) (%)	4.120590
PD(Q) (%)	5.295421
Spread (bp)	51.121064

Decision and limitations

Use to examine a capital-structure scenario. A quoted credit spread is not directly a physical PD. No early default, multiple debt seniorities, liquidity premium or asset-value calibration is represented here.

References and adaptation

Merton (1974), On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29(2), 449–470, sections II–III.

Foundational references establish the model context. The displayed conventions, simplifications, synthetic examples and review protocol are AHS adaptations.

Open the associated model →

FX vanilla: quote convention, notional and hedge units

Question and conclusion

State which currency is domestic before reading a price or delta. The displayed delta is per foreign unit; the hedge quantity scales that delta by the entered foreign-currency notional.

Scope and assumptions

European exercise, domestic and foreign continuously compounded rates, constant volatility and positive foreign notional. No stochastic cross-currency basis, premium-adjusted market delta or transaction costs.

Equations and units

C=N_f\{S_0e^{-r_fT}\Phi(d_1)-Ke^{-r_dT}\Phi(d_2)\},\quad d_1=\frac{\ln(S_0/K)+(r_d-r_f+\sigma^2/2)T}{\sigma\sqrt T},\quad d_2=d_1-\sigma\sqrt T

\Delta_{unit}=e^{-r_fT}\Phi(d_1),\quad H_f=N_f\Delta_{unit},\qquad C-P=N_f\{S_0e^{-r_fT}-Ke^{-r_dT}\}

S₀ and K: domestic currency per one foreign unit (DOM/FOR); N_f: foreign units; C and P: DOM; H_f: FOR. Rates and volatility enter as percentages and are converted to decimals. The tool scales gamma and vega by notional, but keeps delta per foreign unit; doubling notional doubles premium, gamma, vega and hedge, not unit delta. Notional gamma has units FOR²/DOM; notional vega is DOM per volatility percentage point.

Reproducible checks

Check domestic-currency put–call parity on consistent cash-flow units.
Double foreign notional and test which outputs scale.
Compare with the equity formula using q = foreign rate and r = domestic rate.

Synthetic example and calculated outputs

Rates and volatilities below are input percentages, not decimal rates. Values are generated by the same current model; the checks above provide separate consistency tests.

S₀ (DOM/FOR): 1.08
K (DOM/FOR): 1.1
T (years): 1
rDOM (%): 3
rFOR (%): 1.5
σ (%): 10
NFOR (FOR): 1000000

Calculated illustration · rounded to six decimals
C (DOM)	40737.005862
P (DOM)	44306.197994
Unit delta	0.499044
Spot hedge (FOR)	499043.606434
Vega / +1pp (DOM)	4243.851942

Decision and limitations

Use for a clearly defined spot-delta convention. Verify pair orientation, premium currency, settlement and the desk’s actual delta convention before preparing a hedge. Inverting a currency pair is not achieved by replacing S alone.

References and adaptation

Garman & Kohlhagen (1983), Foreign currency option values, JIMF 2(3), 231–237.

Haugh (2013), Foreign Exchange, ADRs and Quanto-Securities, section 1.

Foundational references establish the model context. The displayed conventions, simplifications, synthetic examples and review protocol are AHS adaptations.

Open the associated model →

DOCUMENTATION & GOVERNANCE

Research on documents, model governance and accountable decisions.

Read documentation and governance notes →

Research pathways

Explore the Banking Management Lab →

Strategy, process improvement and change management

Five frameworks connecting a markets problem to a decision, a method, evidence and sustained control. Assumptions, limits and sources are stated in every method.

Behavioural finance: clarify decisions and control biases

Six selected notes connect descriptive models with portfolio management, trading, risk and management. Assumptions remain explicit; proposed controls are not empirical results.

Reference points, loss aversion and probability weighting

The same portfolio can be perceived as a gain or a loss depending on its benchmark. Prospect theory describes relative evaluation; it supplies neither an arbitrage-free price nor a market forecast. Subjective value and decision weights must remain separate from contractual cash flows and pricing probabilities.

Framework and reference formula

x=\frac{W-r}{s},\qquad v(x)=\begin{cases}x^{\alpha},&x\geq0\\-\lambda(-x)^{\beta},&x<0\end{cases}

W: wealth in one currency; r: reference wealth in the same currency; s > 0: a fixed monetary scale in that currency; x and v(x): dimensionless. α and β ∈ (0,1]: curvature; λ > 1: relative loss sensitivity. This normalised power form is an AHS illustration of the properties discussed in the book, with no parameter estimation.

Application and synthesis

Application: present a scenario with a common horizon and currency, then compare interpretations against initial capital, the objective and the benchmark. Synthesis: reframing can change preference without changing payoff; present gains and losses symmetrically.

Assumptions and limits

Assumptions: fix the reference and scale before comparison; distinguish gains from losses; allow diminishing sensitivity. A subjective weight w(p) is not an observed frequency. For multiple outcomes, cumulative prospect theory requires rank-dependent weights; simply summing w(p) is insufficient. A past loss does not mechanically justify taking more risk.

Source and editorial status

Daxhammer, R. J.; Facsar, M.; Papp, Z. Behavioral Finance: Limited Rationality in Financial Markets, 3rd ed., UVK, 2023. ch. 6, pp. 180–191. DOI

Original synthesis and applications proposed by AHS. No behavioural calibration, control effectiveness or performance is established here.

Portfolio: aggregate diversification and rebalancing discipline

Mental accounting can encourage evaluating each sleeve in isolation; familiarity can favour domestic or employer shares; recency can turn a recent rise into an assumed persistent trend. These mechanisms explain possible decisions, but do not automatically diagnose an investor’s behaviour.

Framework and reference formula

\sigma_P^2=\mathbf{w}^{\mathsf T}\Sigma\mathbf{w},\qquad \sum_i w_i=1

wᵢ: exposure weight, dimensionless; Σ: return covariance matrix over a common horizon; σ²P: portfolio return variance, in squared return units. With decimal returns, σP is decimal. This classical portfolio formula is used as a consistency check, not as a behavioural model.

Application and synthesis

Proposed application: consolidate sleeves, measure concentrations, document horizon and liquidity needs, and set review rules before market moves. Compare each transaction with holding the position, after costs and applicable taxes. Synthesis: familiarity is no substitute for diversification; rebalancing should follow the mandate rather than a recent impulse.

Assumptions and limits

Assumptions: aggregate exposures in a common currency, use a symmetric positive-semidefinite covariance matrix and a consistent horizon. Correlations can change in a crisis; variance captures neither all liquidity risk nor extreme losses. Goal-based sleeves remain possible if risks are consolidated.

Source and editorial status

Daxhammer, R. J.; Facsar, M.; Papp, Z. Behavioral Finance: Limited Rationality in Financial Markets, 3rd ed., UVK, 2023. ch. 8–10, especially pp. 277–282. DOI

Original synthesis and applications proposed by AHS. No behavioural calibration, control effectiveness or performance is established here.

Trading and risk: confidence, disposition and challenge

Overconfidence can increase turnover and understate uncertainty. The disposition effect can encourage selling gains quickly while retaining losses. Hindsight can then make the outcome appear inevitable. Review decisions and their evidence, rather than realised P&L alone.

Framework and reference formula

\mathrm{P\&L}_{\mathrm{net}}=\mathrm{P\&L}_{\mathrm{gross}}-C_{\mathrm{execution}}-C_{\mathrm{funding}}-C_{\mathrm{other}}

All terms are amounts in the same currency over the same horizon. Cexecution includes measured fees and slippage; Cfunding is funding cost; Cother covers other identified costs. Do not count costs already included in gross P&L twice. This is an AHS monitoring identity, with no assumed costs or performance.

Application and synthesis

Proposed application: record the thesis, falsifying evidence, adverse scenarios, position size and next review. Assess winners and losers with the same criteria and independently challenge material decisions. Synthesis: distinguish process quality, accepted risk and outcome; a gain alone does not establish a sound decision.

Assumptions and limits

Assumptions: timestamp the decision log before execution, trace costs and define limits before the scenario. Increasing exposure after a fall can follow a documented strategy and does not prove bias. Scenarios are conditional tests: a 3D surface assigns no probabilities to displayed regions and does not predict their occurrence.

Source and editorial status

Daxhammer, R. J.; Facsar, M.; Papp, Z. Behavioral Finance: Limited Rationality in Financial Markets, 3rd ed., UVK, 2023. ch. 8–10, pp. 228–264 and 279–282. DOI

Original synthesis and applications proposed by AHS. No behavioural calibration, control effectiveness or performance is established here.

Management: sunk costs, governance and choice architecture

A funded project may be continued to defend a past decision. Selective perception and self-attribution can make successes feel personal and failures external. The book also connects incentives with governance and presents nudging as choice architecture that should preserve freedom and transparency.

Framework and reference formula

\Delta\mathrm{NPV}=\sum_{t=0}^{T}\frac{\Delta CF_t}{(1+k)^t}

ΔCFt: difference in future cash flows between continuing and stopping, in a common currency; t: periods; T: horizon; k > −1: discount rate per period; ΔNPV: amount in that currency. Exclude irrecoverable past costs from incremental flows; include future exit costs. This is a proposed economic framework, not a behavioural equation from the book.

Application and synthesis

Proposed application: independent reviews, prewritten continuation and stopping criteria, comparable option presentation and recorded exceptions. For UI: explain defaults, allow changes and disclose costs alongside risks. Synthesis: make challenge practicable and measure the effects of presentation changes before claiming effectiveness.

Assumptions and limits

Assumptions: comparable options, estimated future flows and explicit risks; test alternative costs, delays and benefits. Positive NPV does not replace capital, liquidity or compliance constraints. Biases are review hypotheses, not evidence of wrongdoing or individual motives.

Source and editorial status

Daxhammer, R. J.; Facsar, M.; Papp, Z. Behavioral Finance: Limited Rationality in Financial Markets, 3rd ed., UVK, 2023. ch. 11–13, pp. 298–301, 307–331 and 333–335. DOI

Original synthesis and applications proposed by AHS. No behavioural calibration, control effectiveness or performance is established here.

Transformation

CHECKPOINT 01 · TRANSFORMATION FINANCE

Transformer la finance de marché : de la livraison de projets à la plateforme de décision

La transformation réussie n’est ni une migration informatique isolée, ni une succession de comités.

1. Partir de la décision, pas du système

Une architecture cible devient utile lorsqu’elle précise qui décide, sur quelle donnée, avec quel modèle, à quel horizon et sous quel contrôle.

2. Contractualiser les interfaces critiques

Front Office, Risk, Finance et Technologie doivent partager des contrats explicites.

3. Piloter par la preuve

01Baseline

Temps de cycle, ruptures, contrôles manuels, incidents et coûts observables.

02Cible testable

Critères d’acceptation métier, data lineage et contrôles de modèle.

03Incréments

Une chaîne de valeur utilisable à chaque étape, avec retour utilisateur.

04Contrôle durable

Ownership, indicateurs, seuils et processus de changement après mise en service.

4. Les anti-patterns à éliminer

Un programme défini par une liste d’applications plutôt que par des capacités métier.
Une donnée dite “golden source” sans propriétaire, horodatage ni règle de qualité.
Une UAT qui vérifie les écrans sans réconcilier cash-flows, Greeks, P&L et comptabilité.
Un comité qui suit l’activité mais ne tranche ni risque, ni priorité, ni dérogation.

Taux

CHECKPOINT 02 · ARCHITECTURE DE COURBES

Valoriser les taux après la courbe unique : actualisation, projection et risque de base

Une architecture multi-courbes sépare actualisation et projection.

1. Le noyau mathématique

PV_swap = N Σ_i P_d(0,T_i) δ_i [F_p(0;T_i−1,T_i) − K]

P₍d₎ actualise ; F₍p₎ projette les coupons de l’indice.

2. Architecture et validation

Affecter chaque instrument à sa courbe.
Bootstrapper avec conventions exactes.
Repricing et risk reconciliation.

3. Limites

Le démonstrateur illustre la séparation des courbes, pas un bootstrap de production.

SABR

CHECKPOINT 03 · VOLATILITY SMILE

SABR : calibrer un smile sans confondre qualité d’ajustement et qualité de couverture

SABR couple un forward et sa volatilité stochastique.

1. Dynamique et paramètres

dF_t = α_tF_t^βdW_t, dα_t = ν α_tdZ_t, d⟨W,Z⟩_t = ρdt

Les quatre paramètres interagissent.

2. Calibration falsifiable

01Documenter β.
02Calibrer niveau, skew et courbure.
03Publier les résidus.
04Tester stabilité et couverture.

3. Limites

Une bonne RMSE ne suffit pas à valider le modèle.

Heston

CHECKPOINT 04 · STOCHASTIC VOLATILITY

Heston : de la fonction caractéristique à un pricer numériquement contrôlé

Le défi essentiel est autant numérique que stochastique.

1. Modèle risque-neutre

dS_t=(r−q)S_tdt+√v_tS_tdW_t^S
dv_t=κ(θ−v_t)dt+σ_v√v_tdW_t^v, d⟨W^S,W^v⟩_t=ρdt

Le prix européen provient de la fonction caractéristique.

2. Risques numériques et tests

Branches	Little Heston Trap.
Bornes	max(S₀e^−qT−Ke^−rT,0) ≤ C ≤ S₀e^−qT
Parity	C−P=S₀e^−qT−Ke^−rT
Limite	σ_v→0 → Black–Scholes à variance intégrée
Convergence	\|C_N−C_2N\| < ε

3. Limites

La calibration de vanilles ne valide pas toute la dynamique.

xVA

CHECKPOINT 05 · COUNTERPARTY RISK

xVA : transformer un prix sans défaut en mesure économique du contrat

La difficulté réside dans la cohérence de toute la chaîne.

1. Décomposition minimale

CVA ≈ (1−R_C) Σ_i DF_iEPE_iΔPD_C,i
DVA ≈ (1−R_B) Σ_i DF_iENE_iΔPD_B,i

EPE et ENE doivent être calculées après netting et collatéral.

2. Chaîne de calcul et contrôles

01Simuler les facteurs corrélés.
02Revaloriser et agréger.
03Intégrer crédit, funding et marge.
04Contrôler convergence et attribution.

3. Limites

Le wrong-way risk et les conventions internes sont déterminants.

Quantum Finance

CHECKPOINT 06 · QUANTUM FINANCE

Finance quantique : passer de la promesse algorithmique à une preuve de valeur

La bonne question n’est pas « quel problème envoyer sur un ordinateur quantique ? ».

1. La chaîne de preuve quantique

05Décision économiqueValeur, risque, délai, coût

04BenchmarkExact, heuristique et classique

03ExécutionCircuit, annealing, bruit, échantillons

02EncodageQUBO · Ising · amplitude · kernel

01Problème financierObjectif, contraintes, précision

2. Trois familles de cas d’usage

QUBO

Optimisation discrète de portefeuille

Budget, lots entiers, cardinalité et coûts de transaction peuvent être encodés dans une fonction quadratique binaire.

QAE / QMC

Estimation d’espérance

Le pricing et le risque reposent souvent sur des espérances.

QML

Classification et génération

Boosting quantique, kernels et modèles génératifs peuvent être étudiés pour le crédit ou les scénarios de marché.

3. Le protocole minimum d’une expérimentation crédible

01Définir une instance financière reproductible et sa métrique économique.
02Construire au moins un benchmark classique exact et un benchmark heuristique.
03Comptabiliser encodage, prétraitement, exécution, échantillonnage et post-traitement.
04Répéter les expériences, publier les distributions de résultats et analyser le bruit.
05Ne conclure à un avantage que sur une métrique définie ex ante et vérifiée hors échantillon.

4. Ce que cette note ne prétend pas

Cette note ne confirme aucun avantage quantique pratique en production. Elle décrit une méthode d’évaluation.

5. Explore, calculate, challenge

Original research syntheses · 19 October 2026 · Not peer-reviewed

Three supplied books, twelve focused notes and three reproducible teaching experiments. All calculations run classically in your browser. No quantum device, current hardware benchmark or production advantage is claimed.

Search these twelve notes

Portfolio selection: encode cardinality

Need, inputs and proposed deliverable

Select exactly M instruments from an eligible universe. Gather eligibility, selection scores, pair interactions and the required count. Proposed output: an auditable binary objective and feasible reference selections.

Method and assumptions

Use qᵢ∈{0,1}; add a squared cardinality penalty to linear scores and pair interactions. Inspect every coefficient and the constant offset. The four-instrument example below uses synthetic dimensionless scores.

C(q)=\sum_i a_iq_i+\sum_{i<j}b_{ij}q_iq_j+P(\sum_iq_i-M)^2

Limits and controls

Binary selection is not a full allocation model. No realised returns, calibrated covariance matrix, execution costs or investment recommendation are supplied.

Jacquier & Kondratyev (2024), Quantum Machine Learning and Optimisation in Finance, 2nd ed., Packt, ch. 3, pp. 65–69, 74–78 (PDF 102–106, 111–115).

QUBO → Ising: preserve the objective

Need, inputs and proposed deliverable

Check that an encoding change does not alter rankings or the optimum. Required inputs: upper-triangular QUBO coefficients and constant. Output: an equivalence check over the small reference instance.

Method and assumptions

Substitute qᵢ=(1+sᵢ)/2 with sᵢ∈{−1,+1}. Keep the constant for numerical equality; count each pair once. The published tests compare both energies for all 16 states.

J_{ij}=b_{ij}/4,\quad h_i=a_i/2+\tfrac14\sum_{j\ne i}b_{\min(i,j),\max(i,j)}

Limits and controls

Hardware conventions for bit order, spin sign and coefficient scaling must be verified separately. No device embedding is performed here.

Jacquier & Kondratyev (2024), Quantum Machine Learning and Optimisation in Finance, 2nd ed., Packt, ch. 3, pp. 65–69, 74–78 (PDF 102–106, 111–115).

Penalty strength: feasibility before optimum

Need, inputs and proposed deliverable

A solver returns an attractive score but violates the required number of holdings. Output: a penalty-sensitivity table separating unconstrained and feasible optima.

Method and assumptions

Evaluate the constraint residual explicitly. Sweep P while keeping scores and interactions fixed. Compare the penalised minimiser with the exact constrained reference.

r(q)=\sum_iq_i-M,\quad C(q)=C_0(q)+P r(q)^2

Limits and controls

A large P is not a universal certificate: hardware precision and coefficient scaling matter. The demonstrator is exact classical enumeration, not an annealer.

Jacquier & Kondratyev (2024), Quantum Machine Learning and Optimisation in Finance, 2nd ed., Packt, ch. 3, pp. 65–69, 74–78 (PDF 102–106, 111–115).

QAOA: inspect probabilities, not a promise

Need, inputs and proposed deliverable

Study how a candidate circuit concentrates probability on feasible or minimum-cost selections. Output: the full 16-state distribution and exact-reference comparison.

Method and assumptions

Apply one cost phase followed by four X mixers to the uniform state. β and γ are user-selected radians. Report expected penalised cost, feasible mass and global-minimum mass.

|\psi\rangle=e^{-i\beta\sum_iX_i}e^{-i\gamma C(Z)}|+\rangle^{\otimes4}

Limits and controls

Ideal state-vector simulation on a classical computer; depth p=1, no parameter training, shots, device noise or hardware run. A low expectation does not ensure a feasible measured solution.

Jacquier & Kondratyev (2024), 2nd ed., ch. 11, pp. 299–304 (PDF 336–341).

Data encoding: define the information contract

Need, inputs and proposed deliverable

Translate financial features into circuit parameters. Gather feature units, missing-value policy, training bounds and out-of-range treatment. Output: a versioned encoding specification.

Method and assumptions

A possible angle map rescales a feature to [0,π]. Fit scaling on training data only and define the constant-feature case. The single-qubit demo accepts angles directly, without fitting any dataset.

\theta(x)=\pi\frac{x-x_{\min}}{x_{\max}-x_{\min}},\quad x_{\max}>x_{\min}

Limits and controls

Encoding is a modelling choice, not evidence of predictive information. Periodicity, clipping and train/test leakage need controls.

Jacquier & Kondratyev (2024), 2nd ed., ch. 7, pp. 195–199 (PDF 232–236).

Quantum kernel: test the similarity definition

Need, inputs and proposed deliverable

Explore a similarity measure for a later classification experiment. Output: a feature-map specification and comparisons with classical kernels on identical data splits.

Method and assumptions

Use the squared state overlap. For a single Ry rotation this reduces to cos²((θ−φ)/2), evaluated analytically below. It is deliberately simple enough to verify classically.

k(x,y)=|\langle\psi(y)|\psi(x)\rangle|^2

Limits and controls

No classifier, credit decision, trained model or quantum advantage is demonstrated. A kernel formula alone establishes none of these.

Jacquier & Kondratyev (2024), 2nd ed., ch. 12, pp. 316–319 (PDF 353–356).

Measurement budget: precision has a cost

Need, inputs and proposed deliverable

Choose a number of repeated measurements for an overlap estimate. Output: a shot-budget sensitivity study and an explicit error definition.

Method and assumptions

For independent ideal Bernoulli outcomes, the frequency estimates p and has standard deviation √(p(1−p)/N). The demo displays this theoretical sampling standard error using the exact p.

\hat p=K/N,\quad \mathrm{SE}(\hat p)=\sqrt{p(1-p)/N}

Limits and controls

No shots are actually sampled. Zero variance at p=0 or 1 assumes an ideal known probability and says nothing about readout bias or hardware errors.

Jacquier & Kondratyev (2024), 2nd ed., ch. 12, pp. 316–319 (PDF 353–356).

Benchmark: compare the same financial problem

Need, inputs and proposed deliverable

Decide whether a research prototype deserves further evaluation. Output: a pre-agreed benchmark protocol including objective, feasibility, error and total cost.

Method and assumptions

Start with exact enumeration for the small instance. For larger instances, add suitable classical heuristics and record preparation, optimisation, execution and post-processing separately. This expanded protocol is our proposed adaptation.

T_{\mathrm{total}}=T_{\mathrm{data}}+T_{\mathrm{encode}}+T_{\mathrm{train}}+T_{\mathrm{run}}+T_{\mathrm{readout}}

Limits and controls

Browser timings are not hardware comparisons. We publish no speedup, business gain or production validation from these teaching examples.

Jacquier & Kondratyev (2024), Quantum Machine Learning and Optimisation in Finance, 2nd ed., Packt, ch. 3, pp. 65–69, 74–78 (PDF 102–106, 111–115).

Quantum-inspired finance ≠ quantum hardware

Need, inputs and proposed deliverable

Clarify whether a proposal models markets with quantum-inspired equations or actually executes an algorithm on qubits. Output: an explicit model and execution taxonomy.

Method and assumptions

Lee develops a financial interpretation of Schrödinger-type and anharmonic-oscillator models. Numerical finite-difference calculations on a conventional computer remain classical computations.

\widehat H\phi=E\phi\quad\text{(model equation, not an empirical proof)}

Limits and controls

A physics analogy does not establish that markets are quantum systems. The book’s forecasts and empirical generalisations have not been independently reproduced here.

Raymond S. T. Lee (2020), Quantum Finance: Intelligent Forecast and Trading Systems, Springer, ch. 5, §§5.3–5.9, pp. 94–101 (PDF 121–128), DOI 10.1007/978-981-32-9796-8.

Density, amplitude and return conventions

Need, inputs and proposed deliverable

Review a wave-function-based market model before reusing its output. Output: a notation audit and a falsifiable validation plan.

Method and assumptions

Distinguish a density ρ from an amplitude ψ with ρ=|ψ|² and specify normalisation. Lee §5.3 uses the gross price ratio Pₜ/Pₜ₋₁; it must not be silently treated as a simple or log return.

\rho(r)=|\psi(r)|^2,\quad\int\rho(r)\,dr=1,\quad R_t=P_t/P_{t-1}

Limits and controls

A return histogram does not identify a unique complex phase or validate a Hamiltonian. Normality, symmetry and forecasting power require separate empirical tests.

Raymond S. T. Lee (2020), Quantum Finance: Intelligent Forecast and Trading Systems, Springer, ch. 5, §§5.3–5.9, pp. 94–101 (PDF 121–128), DOI 10.1007/978-981-32-9796-8.

Technology readiness: classify before procuring

Need, inputs and proposed deliverable

Frame an exploratory technology assessment for Risk, IT and a business sponsor. Output: a decision record identifying the algorithm, execution mode and evidence still needed.

Method and assumptions

Use the 2019 chapter as historical context for annealing versus gate-based computing. Our proposed checklist distinguishes classical solver, ideal simulator, noisy simulator and physical device.

\text{decision evidence}=\text{instance}+\text{execution mode}+\text{metric}+\text{date}

Limits and controls

Hardware counts, vendor forecasts and speed claims from 2019 are not current evidence. No claim about today’s device capabilities is made here.

Jason Kang, ch. 11 in Schulte & Lee (2019), AI & Quantum Computing for Finance & Insurance, World Scientific, pp. 311–312, 317–321 (PDF 336–337, 342–346). Historical source.

Governance: establish the experiment mandate

Need, inputs and proposed deliverable

Separate scientific curiosity from a client decision. Gather purpose, data rights, confidentiality constraints, owner and acceptance criteria. Output: a scoped evaluation mandate.

Method and assumptions

The introduction argues for evidence and ethical context. We adapt that principle into a proposed workflow: approve data, record assumptions, reproduce baselines, review evidence and decide whether to stop or investigate further.

\text{mandate}\rightarrow\text{baseline}\rightarrow\text{evidence review}\rightarrow\text{decision}

Limits and controls

This is a proposed advisory method, not a completed client engagement, legal opinion, certification or independent validation. Do not send confidential data through contact links.

Schulte & Lee (2019), AI & Quantum Computing for Finance & Insurance, World Scientific, introduction, p. xvii (PDF 18). Historical source.

Educational demonstrator · classical execution

Exact selection benchmark

Four synthetic instruments A–D. Linear scores: −3, −2, −1.5, −1. Pair scores AB, AC, AD, BC, BD, CD: 1.2, 0.4, 0.2, 0.6, 0.3, 0.8, multiplied by the pair-score parameter. Lower is better; all scores are dimensionless. Enumerate all sixteen binary selections and compare with the exact feasible minimum.

Educational demonstrator · classical execution

QAOA p=1 · ideal classical simulation

Same synthetic objective, uniform initial state and one phase/mixer layer. Angles are supplied, not optimised. The chart and table show exact ideal probabilities; no measurement sampling, noise or quantum device execution. Bit order is A, B, C, D from left to right.

Educational demonstrator · classical execution

One-qubit overlap and measurement precision

States Ry(θ)|0⟩ and Ry(φ)|0⟩. The overlap and ideal Bernoulli standard error are calculated analytically. N is a hypothetical measurement budget; no random shots or hardware calls are made.

STRATEGIC MANAGEMENT

OPERATING MODEL / MODEL RISK

Industrialiser la recherche quantitative sans diluer la responsabilité

La transformation construit une chaîne de décision et de preuve.

Architecture de contrôle

01Spécification économique versionnée.
02Données sourcées et rejouables.
03Benchmark indépendant.
04Surveillance et escalade.

Arbitrage managérial

Séparer composants certifiés et espace de recherche.

CAPITAL ALLOCATION

ACTIVE CREDIT PORTFOLIO MANAGEMENT

Passer du suivi des limites à l’allocation active du capital crédit

Relier origination, prix, concentration et capital.

Cadre de décision

RAROC = (revenus − coûts − EL) / capital économique
EL = EAD × PD × LGD

La mesure doit intégrer calibration, concentration et coûts.

Risques de transformation

Un RAROC mal spécifié peut déplacer le risque.

TRANSFORMATION

FRONT-TO-RISK ARCHITECTURE

Construire une source de vérité sans confondre standardisation et uniformité

Partager les faits ; gouverner les représentations.

Cinq couches

Contract	Taxonomie et conventions.
Market	Données de marché.
Model	Modèle et méthode numérique.
Risk	Mesures de risque.
Control	Contrôles et audit.

Indicateurs de réussite

Mesurer qualité, contrôle et délai.

FURTHER READING · THREE TAX RESEARCH PATHS

Tax, financial engineering and AI

Nine complementary notes in the Labs, based on three supplied books. These reading paths are separate from the existing filtered catalogue.

Open seven tax demonstrators

Research library

Research connected to three engagement questions

Valuation and model review

Front Office–Risk–Finance diagnostic

Model and risk governance

New reading pathways: macroeconomics, inference and Greeks

Corporate Transactions, Private Equity & Legal Risk

Question and conclusion

Scope and assumptions

Equations and units

Reproducible checks

Decision and limitations

References and adaptation

Question and conclusion

Scope and assumptions

Equations and units

Reproducible checks

Decision and limitations

References and adaptation

Question and conclusion

Scope and assumptions

Equations and units

Reproducible checks

Decision and limitations

References and adaptation

Question and conclusion

Scope and assumptions

Equations and units

Reproducible checks

Decision and limitations

References and adaptation

Research on documents, model governance and accountable decisions.

Strategy, process improvement and change management

Behavioural finance: clarify decisions and control biases

Reference points, loss aversion and probability weighting

Framework and reference formula

Application and synthesis

Portfolio: aggregate diversification and rebalancing discipline

Framework and reference formula

Application and synthesis

Trading and risk: confidence, disposition and challenge

Framework and reference formula

Application and synthesis

Management: sunk costs, governance and choice architecture

Framework and reference formula

Application and synthesis

1. Partir de la décision, pas du système

2. Contractualiser les interfaces critiques

3. Piloter par la preuve

4. Les anti-patterns à éliminer

1. Le noyau mathématique

2. Architecture et validation

3. Limites

1. Dynamique et paramètres

2. Calibration falsifiable

3. Limites

1. Modèle risque-neutre

2. Risques numériques et tests

3. Limites

1. Décomposition minimale

2. Chaîne de calcul et contrôles

3. Limites

1. La chaîne de preuve quantique

2. Trois familles de cas d’usage

Optimisation discrète de portefeuille

Estimation d’espérance

Classification et génération

3. Le protocole minimum d’une expérimentation crédible

4. Ce que cette note ne prétend pas

5. Explore, calculate, challenge

Need, inputs and proposed deliverable

Method and assumptions

Limits and controls

Need, inputs and proposed deliverable

Method and assumptions

Limits and controls

Need, inputs and proposed deliverable

Method and assumptions

Limits and controls

Need, inputs and proposed deliverable