Valuation and conventions
Contracts, cash flows and valuation assumptions.
QUANTITATIVE METHODS / INTERACTIVE DEMONSTRATORS
Quantitative methods.
Explicit assumptions.
A selected library of valuation and risk methods running locally in your browser. Inputs are neither transmitted nor stored.
Educational researchScope and decision framework
Why this laboratory?
A price becomes useful when its underlying assumptions can be explained. This laboratory connects contracts, models, valuation and sensitivities to examine a result and its weaknesses. It complements the Risk and Banking Management Labs with a quantitative reading of exposures.
Ambition with discipline
The demonstrators illustrate methods under simplified assumptions. Reproducibility establishes neither market calibration nor independent validation. Professional use requires checks of data, contractual conventions, numerical errors and model controls.
Sensitivities and scenarios
Parameter effects and comparison of exposures.
Numerical methods
Simulation, convergence and approximation checks.
All methods and sections
Related methods and sections
Research pathways
SELECTED METHODS
Explore a documented method by risk family
Documented methods available in this selection.
01 Contrat→
02 Données→
03 Modèle→
04 Valorisation→
05 Risque & contrôles
RATES / 01
Multi-curve Interest Rate Swap
Actualisation OIS et projection séparée du taux forward avec basis configurable.
NPV receive float—
Par swap rate—
Fixed coupon PV01 magnitude (€)—
Discount annuity—
Received float-leg PV (€)—
Paid fixed-leg PV (€)—
Projection +1bp P&L (€)—
OIS +1bp P&L (€)—
Illustrative scenarios
Presets change assumptions; they are not market observations. In BSM, non-reference presets also set an illustrative call price for an inversion check.
Model, conventions and interpretation
Mechanism
Separate deterministic discounting from projected coupons. Year-fraction periods end exactly at T, including a final stub. With a constant projected coupon, the par rate is independent of OIS discounting; an off-market swap NPV still depends on it.
Reference formulas
\begin{aligned}A_d&=\sum_i\delta_i e^{-r_dt_i},\quad L_b=L+10^{-4}b\\PV_{float}&=NL_bA_d,\quad PV_{fixed}=NKA_d\\NPV_{rec.float}&=N(L_b-K)A_d,\quad K_{par}=L_b\end{aligned}
N: EUR; K,L,r_d: decimal annual rates; b: basis points; t_i,δ_i: years; A_d: discounted annuity in years. Fixed coupon PV01 magnitude is N A_d×10⁻⁴; receiving float has the opposite fixed-coupon P&L sign. OIS P&L bumps discounting only.
Assumptions, limits and primary reference
Flat OIS and projected rates, one simplified schedule for both legs, no principal exchange. No bootstrap, market instruments, calendars, actual day-count rules, CSA specifics, fixing lag or stochastic basis. This develops the entered flat-rate model, not a production curve builder.
Ametrano & Bianchetti (2013), Everything You Always Wanted to Know About Multiple Interest Rate Curve Bootstrapping but Were Afraid to Ask. Flat-rate schedule is an AHS teaching specialization.
Ametrano & Bianchetti — original reference ↗Inspect the cash-flow schedule
Methodology & assumptions
Courbes plates déterministes d’actualisation OIS et de projection. Les flux sont valorisés par Σ DF(t) × accrual × taux.
RATES / 02
HJM: forward-bond option
European bond call and put under a payment-date forward measure, with effective lognormal bond volatility.
Illustrative cases
Cases change strike relative to P(0,T₂)/P(0,T₁) and restore example volatility 12%, except the zero case. Your discount factors, expiry and notional stay fixed.
Bond call PV (€)—
Bond put PV (€)—
Forward bond F (ratio)—
Forward delta / €1 notional—
Vega per +1 volatility point (€)—
Call–put parity residual (€)—
3D price and risk diagnostics
Axes vary relative strike K/F and annual volatility σB. P(0,T₁), P(0,T₂), T₁ and N remain fixed. These are input sensitivities, not a calibrated HJM term structure.
Bond call value
Bond option vega
The HJM framework
df(t,U)=\alpha(t,U)dt+\sigma_f(t,U)dW_t,\quad\alpha(t,U)=\sigma_f(t,U)\int_t^U\sigma_f(t,s)ds
Under the money-market risk-neutral measure, drift is constrained by forward-rate volatility. This panel prices a reduced European bond option; it does not simulate the full forward curve.
Model and method references
HJM is the model framework; the implemented bond-option proxy uses Black pricing, not an HJM simulation.
David Heath, Robert Jarrow, Andrew Morton (1992). Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation. Econometrica 60(1), 77.
Fischer Black (1976). The pricing of commodity contracts. Journal of Financial Economics 3(1-2), 167-179.
Effective volatility
F=\frac{P(0,T_2)}{P(0,T_1)},\quad\sigma_B^2 T_1=\int_0^{T_1}\left(\int_{T_1}^{T_2}\sigma_f(u,s)ds\right)^2du
T₂ is the underlying bond payment date, after exercise T₁; its discount factor is entered directly. σB is an effective annual volatility (%), F and P are ratios. At T₁=0 use the deterministic payoff.
Call, put and parities
C=NP_1[F\Phi(d_1)-K\Phi(d_2)],\quad C-P=NP_1(F-K)
P₁=P(0,T₁); N is in euros. d₁=[ln(F/K)+σB²T₁/2]/(σB√T₁), d₂=d₁−σB√T₁. Φ is the standard normal CDF. The put is evaluated separately.
Interpretation and limits
\mathrm{Vega}_{1\,pt}=NP_1 F\phi(d_1)\sqrt{T_1}\times0.01
Delta is per €1 notional and per unit forward-bond ratio. Vega is euros per +1 volatility point. No curve bootstrap, multi-factor calibration, coupon-bond or Bermudan pricing is supplied. At zero volatility/expiry the ATM delta uses the engine’s zero convention at the kink.
Assumptions and source
A positive-forward Black reduction consistent with deterministic-volatility HJM. Discount factors are supplied; volatility is effective annual σB, not integrated volatility. Heath, Jarrow & Morton (1992); Henrard, Explicit bond option and swaption formula in HJM one-factor model.
RATES / 03
BGM / LMM: caplet and floorlet
Black-76 valuation under the payment-date forward measure. Explore price, risk and three 3D diagnostics, then read how this building block connects to a multi-tenor LMM.
Illustrative cases
Cases change strike relative to your current forward and restore the example volatility of 24%, except the zero-volatility case. Notional, accrual, expiry and discount factor remain your inputs. These are examples, not market quotes.
Caplet PV (€)—
Floorlet PV (€)—
Forward delta per +1 bp (€)—
Vega per +1 vol point (€)—
Discounted caplet intrinsic (€)—
Caplet–floorlet parity residual (€)—
Sensitivity curves and CSV exports use unrounded calculated values. Result cards are rounded to the nearest euro.
3D valuation and sensitivity diagnostics
Relative strike K/F and reset expiry T vary. Forward, volatility, accrual, notional and payment discount factor stay fixed. Keeping D fixed across T is an illustrative sensitivity convention, not a calibrated yield curve.
Caplet present value
Caplet vega per volatility point
Floorlet present value
BGM / LMM: dynamics, valuation and validation
Forward measure and the caplet
dF_i(t)=F_i(t)\sigma_i(t)\,dW_t^{T_{i+1}}
Under its own payment-date measure, a lognormal forward has zero drift. The displayed single-period calculation takes constant volatility. Different forwards are not all driftless under one common measure: a multi-tenor LMM introduces measure-dependent drift and a calibrated covariance structure.
F_i: forward for [T_i,T_i+1], decimal annual rate; t: years; σ_i: annualised volatility, year⁻¹/²; W: Brownian motion under the payment-date measure.
Model and method references
BGM is the reference framework; this implementation prices a single lognormal caplet.
Alan Brace, Dariusz G¸atarek, Marek Musiela (1997). The Market Model of Interest Rate Dynamics. Mathematical Finance 7(2), 127-155.
Fischer Black (1976). The pricing of commodity contracts. Journal of Financial Economics 3(1-2), 167-179.
Black-76 value and contractual payoff
\begin{aligned}C&=N\delta D\,[F\Phi(d_1)-K\Phi(d_2)]\\d_1&=\frac{\ln(F/K)+\tfrac12\sigma^2T}{\sigma\sqrt T},\quad d_2=d_1-\sigma\sqrt T\end{aligned}
At the payment date, the caplet pays Nδ(F_T−K)⁺ and the floorlet Nδ(K−F_T)⁺. N is euro notional; δ the accrual fraction in years; D the dimensionless payment discount factor; T the reset expiry in years. F, K and σ enter the formula as decimals; Φ is the standard normal CDF. For zero volatility or zero expiry, the engine uses discounted intrinsic value.
Risk, parity and boundary checks
\begin{aligned}C-P&=N\delta D(F-K)\\\Delta_{1\mathrm{bp}}&=N\delta D\Phi(d_1)\,10^{-4}\\\mathrm{Vega}_{1\mathrm{pt}}&=N\delta DF\varphi(d_1)\sqrt T\,0.01\end{aligned}
Delta and vega are first-order sensitivities, not full stressed profit and loss. φ is the standard normal density. The signed parity residual compares two independently evaluated Black-76 expressions with the forward payoff; it checks arithmetic consistency, not market calibration. At the deterministic ATM kink, the reported delta follows the engine’s one-sided convention and should not be treated as a smooth derivative.
From one caplet to a multi-tenor LMM
A full LMM specifies a tenor schedule, forward curve, time-dependent volatility loadings, correlation matrix, numeraire and discretisation. Caplet quotes constrain marginal variances; swaption prices also depend on joint covariance. Check positive-semidefinite correlation, curve conventions, martingale consistency and numerical convergence. Repricing vanillas alone does not validate Bermudan exercise or hedging.
This page implements the single-period caplet/floorlet and illustrative surfaces. It does not simulate correlated forward paths, fit a multi-tenor LMM, or price Bermudan swaptions. For non-positive rates, use an explicitly shifted or normal specification rather than this unshifted lognormal formula.
Rebonato, McKay & White, The SABR LIBOR Market Model, ch. 4–5; Black (1976), The Pricing of Commodity Contracts. Educational single-period implementation.
MODEL INTERPRETATION
BGM/LMM: a caplet is a building block
A forward rate is driftless under its own payment-date forward measure. A joint LMM uses one common simulation measure, so its other forward rates generally acquire drift terms. Caplets constrain marginal integrated variances; swaptions also depend on covariance between tenors. Freezing swap weights converts a nonlinear swap into an approximate basket and must be labelled as an approximation. The current single-period caplet/floorlet remains a building block: it does not establish a calibrated multi-tenor simulator or a Bermudan exercise engine.
Reference relation and assumptions
- L_i(t): simple annualized forward rate for [T_{i−1},T_i], in decimal notation.
- T_i: payment date; t and dates measured in years.
- σ_i(t): deterministic factor-loading vector, units year⁻¹ᐟ².
- W^{T_i}: Brownian motion under the forward pricing measure with P(t,T_i) as numeraire, units year¹ᐟ².
- Positive unshifted lognormal forwards and consistent covariance are assumptions. The zero drift is stated under each rate’s own measure.
Scope, limitations and source
Each rate’s own forward measure is distinct. The current single-period tool is not a calibrated multi-tenor LMM or a Bermudan simulator.
Pierre Henry-Labordère, Analysis, Geometry, and Modeling in Finance (2009), §8.2–8.2.1, printed pp.207–213 / PDF226–232.
CREDIT / 04
Reduced-Form CDS
Jambes de prime et de protection sous intensité de défaut constante, avec accrual-on-default.
Buyer upfront PV—
Par CDS spread—
Protection leg—
Premium leg—
Methodology & assumptions
Q(τ>t)=exp(−λt), recouvrement du pair et actualisation déterministe.
Model and method references
Reduced-form credit foundations; the calculator uses a constant intensity and deterministic recovery.
ROBERT A. JARROW, STUART M. TURNBULL (1995). Pricing Derivatives on Financial Securities Subject to Credit Risk. The Journal of Finance 50(1), 53-85.
Darrell Duffie, Kenneth J. Singleton (1999). Modeling Term Structures of Defaultable Bonds. Review of Financial Studies 12(4), 687-720.
COUNTERPARTY RISK / 05
CVA · DVA · FVA
Profil d’exposition analytique et intégration discrète des ajustements de valorisation.
CVA—
DVA—
FVA—
Total valuation adjustment—
Methodology & assumptions
Le MtM futur est approché par un processus gaussien sans dérive.
Model and method references
Credit valuation foundations only. Exposure and funding approximations are specified by AHS, not reproduced from these papers.
ROBERT A. JARROW, STUART M. TURNBULL (1995). Pricing Derivatives on Financial Securities Subject to Credit Risk. The Journal of Finance 50(1), 53-85.
Darrell Duffie, Kenneth J. Singleton (1999). Modeling Term Structures of Defaultable Bonds. Review of Financial Studies 12(4), 687-720.
PORTFOLIO / 06
Parametric VaR & Expected Shortfall
Risque gaussien d’un portefeuille à trois facteurs avec matrice de corrélation contrôlée.
Portfolio volatility—
Value at Risk—
Expected Shortfall—
Diversification benefit—
Surface 3D de VaR du portefeuille
La VaR varie selon la volatilité du facteur 1 et la corrélation entre les facteurs 1 et 2.
VaRα = zα × √(h/252 × xᵀΣx)
xᵢ : exposition × volatilité annualisée du facteur i (€) ; Σ : matrice de corrélation ; h : horizon en jours ouvrés ; zα : quantile gaussien.
Methodology & assumptions
Agrégation delta-normale à moyenne nulle et rendements gaussiens multivariés.
Model and method references
Reference for the Gaussian delta-normal risk framework.
J.P. Morgan / Reuters (1996). RiskMetrics — Technical Document, fourth edition.
PORTFOLIO / RISK ALLOCATION
Contribution Euler à la VaR
Répartir la VaR paramétrique entre trois facteurs corrélés.
Factor 1 risk contribution—
Factor 2 risk contribution—
Factor 3 risk contribution—
Total portfolio VaR—
3D contribution sensitivity
Contribution du facteur 1 selon sa volatilité et la corrélation ρ₁₂.
Contribution de risque Euler : RCᵢ = VaR × contribution marginale
Methodology & assumptions
Approximation gaussienne delta-normale ; contributions de couverture potentiellement négatives.
Model and method references
Euler allocation applied to the implemented Gaussian VaR.
D. Tasche (2007). Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle. arXiv:0708.2542.
J.P. Morgan / Reuters (1996). RiskMetrics — Technical Document, fourth edition.
PORTFOLIO / ACTIVE EXTENSION
130/30 active extension & tracking error
Séparer cœur indiciel et overlay actif long/short.
Net / gross exposure—
Expected active return (bp)—
Tracking error (%)—
Information ratio—
Methodology & assumptions
Risk and return estimates are assumption-sensitive and omit implementation costs.
Model and method references
Long/short portfolio framework; the three-sleeve convention is an AHS specification.
M. L. Leibowitz, S. Emrich & A. Bova (2009). Modern Portfolio Management: Active Long/Short 130/30 Equity Strategies. Wiley.
Harry Markowitz (1952). PORTFOLIO SELECTION. The Journal of Finance 7(1), 77-91.
EXOTICS / 07
Arithmetic Asian Call Monte Carlo
Simulation GBM antithétique, erreur standard et delta par common random numbers.
Asian call price—
Standard error—
95% confidence interval—
Pathwise bump delta—
Methodology & assumptions
Mouvement brownien géométrique risque-neutre et variables antithétiques.
Model and method references
Monte Carlo methodology; discrete arithmetic averaging and sampling conventions belong to this implementation.
Phelim P. Boyle (1977). Options: A Monte Carlo approach. Journal of Financial Economics 4(3), 323-338.
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability.
EQUITY / 08
Black–Scholes · Greeks · Implied Vol
Valorisation call/put, sensibilités analytiques et inversion de la volatilité par dichotomie.
Call price—
Put price—
Call delta—
Gamma—
Vega +1 vol pt—
Implied volatility—
Put spot delta—
Call rho / +1 rate point (CCY)—
Put–call parity residual (CCY)—
Forward F₀,T (CCY)—
Illustrative scenarios
Presets change assumptions; they are not market observations. In BSM, non-reference presets also set an illustrative call price for an inversion check.
Model, conventions and interpretation
Mechanism
One constant volatility prices each European option. Changing strike or maturity produces a price/Greek sensitivity surface; it does not generate a calibrated volatility smile. Entered call price is used only for implied-volatility inversion.
Reference formulas
\begin{aligned}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&=S_0e^{-qT}\Phi(d_1)-Ke^{-rT}\Phi(d_2)\\P&=Ke^{-rT}\Phi(-d_2)-S_0e^{-qT}\Phi(-d_1)\end{aligned}
S₀,K,C,P,F: one asset unit in CCY; T: years; r,q: annual continuously compounded decimals; σ: annual decimal volatility; Φ: standard normal CDF. Vega and rho are per +1 percentage point, not per decimal unit.
Assumptions, limits and primary reference
No jumps, stochastic volatility or early exercise. At zero volatility the discounted payoff is deterministic; spot Greeks at the forward strike are not assigned a smooth value. IV at the lower bound is zero; an upper-bound price has no finite IV.
Black & Scholes (1973), The Pricing of Options and Corporate Liabilities; dividend-yield extension and analytical controls are AHS specializations.
Black & Scholes — original reference ↗Methodology & assumptions
Options européennes sous diffusion lognormale à paramètres constants.
EQUITY / 09
American Put · CRR Tree
Arbre binomial recombining avec décision d’exercice anticipé à chaque nœud.
American put—
European put—
Early-exercise premium—
Tree delta—
Methodology & assumptions
Arbre Cox–Ross–Rubinstein et induction rétrograde.
Model and method references
Binomial exercise-versus-continuation methodology.
John C. Cox, Stephen A. Ross, Mark Rubinstein (1979). Option pricing: A simplified approach. Journal of Financial Economics 7(3), 229-263.
FIXED INCOME / 10
Bond Price · Duration · Convexity
Analytics complètes d’une obligation à coupon fixe sous rendement actuariel constant.
Dirty price—
Macaulay duration—
Modified duration—
DV01—
Convexity—
Methodology & assumptions
Échéancier régulier et rendement actuariel plat.
Model and method references
Original duration reference; cash-flow discounting and convexity conventions are stated by the calculator.
F. R. Macaulay (1938). Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856. NBER.
ALLOCATION / 11
Mean–Variance Optimizer
Portefeuille global minimum variance et portefeuille tangent pour trois actifs corrélés.
GMV weights—
GMV return / vol—
Tangency weights—
Tangency Sharpe—
Methodology & assumptions
Optimisation de Markowitz non contrainte en forme fermée.
Model and method references
Mean-variance framework; this implementation is unconstrained and uses entered return/covariance assumptions.
Harry Markowitz (1952). PORTFOLIO SELECTION. The Journal of Finance 7(1), 77-91.
CREDIT / 12
Gaussian Credit Portfolio
Distribution de pertes d’un portefeuille homogène sous facteur systémique gaussien.
Expected loss—
Unexpected loss—
99% loss VaR—
99% Expected Shortfall—
Tranche expected loss—
Analytical EL check—
Methodology & assumptions
Modèle gaussien à un facteur latent et défauts conditionnellement indépendants.
Model and method references
One-factor conditional independence is the foundation; the calculator simulates a finite pool.
O. Vasicek (2002). Loan portfolio value. Risk, December.
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability.
RATES VOL / 13
SABR Implied Volatility Smile
Approximation asymptotique de Hagan de la volatilité Black.
Black implied vol—
Caplet price—
ATM vol—
25bp skew—
Vega +1 vol pt—
β regime—
SABR : dynamique, sourire et calibration
Lecture des trois axes : X = K/F, Y = T, Z = σᴮ(K/F,T).
Variables d’état et dynamique
dFₜ = αₜ Fₜᵝ dWₜ¹
dαₜ = ν αₜ dWₜ²
d⟨W¹,W²⟩ₜ = ρ dt
F est le taux forward sous sa mesure forward associée.
Model and method references
SABR implied-volatility asymptotics followed by Black valuation.
P. S. Hagan, D. Kumar, A. S. Lesniewski & D. E. Woodward (2002). Managing Smile Risk. Wilmott, September, 84–108.
Fischer Black (1976). The pricing of commodity contracts. Journal of Financial Economics 3(1-2), 167-179.
Paramètres et lecture de marché
- F, K, T
- Forward, strike et échéance de l’option.
- α
- Échelle de volatilité initiale.
- β ∈ [0,1]
- Élasticité du modèle.
- ρ, ν
- ρ pilote le skew ; ν pilote la courbure.
Approximation de volatilité implicite de Hagan
Approximation asymptotique, avec limite continue à l’ATM.
σᴮ(F,K,T) ≈ α / ((FK)^((1−β)/2) A(F,K)) · z/x(z) · [1 + T·B(F,K)]
A et B regroupent les corrections de log-moneyness et de temps.Calibration, risques et limites
- Ajuster le sourire à des cotations liquides et contrôler les résidus par strike.
- Les paramètres peuvent se compenser : tester stabilité et sensibilités.
- Forward négatif ou nul : préférer une variante normale ou décalée.
- Contrôler les ailes, l’arbitrage et l’extrapolation avant les exotiques.
Methodology & assumptions
Approximation lognormale non décalée de Hagan et al.
MODEL INTERPRETATION
SABR: a smile approximation needs a domain of validity
SABR links the forward-rate level to a stochastic volatility factor. Correlation affects the direction of the smile, while volatility of volatility influences curvature; these are model mechanisms, not independently observed causal estimates. A short-time implied-volatility approximation should be checked across strike and maturity against a suitable numerical benchmark. Extreme strikes, long maturities and the treatment of a zero-rate boundary require particular care. The book’s mean-reverting λ-SABR extension is a research comparison, not a capability implemented by this page.
Scope, limitations and source
The existing Hagan approximation is preserved. λ-SABR, exact densities and geometric solvers are not implemented by this addition.
Pierre Henry-Labordère, Analysis, Geometry, and Modeling in Finance (2009), §6.3, printed pp.164–167 / PDF183–186; Figs.6.3–6.5, pp.174–176 / PDF193–195.
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EQUITY VOL / 14
Heston Stochastic Volatility
Prix européen semi-analytique par fonction caractéristique.
Call price—
Put price—
BS implied vol—
Delta—
Feller ratio—
Quadrature check—
Heston : dynamique de variance, valorisation et calibration
Lecture des trois axes : X = K/S₀, Y = T, Z = σᵢₘₚ(K/S₀,T).
Dynamique sous mesure risque-neutre
dSₜ = (r−q)Sₜ dt + √vₜ Sₜ dWₜˢ
dvₜ = κ(θ−vₜ)dt + σᵥ√vₜ dWₜᵛ
d⟨Wˢ,Wᵛ⟩ₜ = ρ dt
Le processus CIR mean-revert la variance instantanée.
Model and method references
Original stochastic-volatility model; the calculator uses finite characteristic-function integration.
Steven L. Heston (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies 6(2), 327-343.
Cinq paramètres fondamentaux
- v₀
- Variance initiale (volatilité initiale √v₀).
- θ
- Variance de long terme.
- κ
- Vitesse de retour à la moyenne.
- σᵥ, ρ
- σᵥ et ρ façonnent courbure et skew.
r et q : taux sans risque et rendement du dividende.
Valorisation par fonction caractéristique
La fonction caractéristique donne accès aux probabilités risque-neutres.
C = S₀·e^(−qT)·P₁ − K·e^(−rT)·P₂; Pⱼ = ½ + (1/π)∫₀∞ Re[e^(−iu ln K) φⱼ(u)/(iu)]du
La borne d’intégration et la grille influencent la précision.Calibration, diagnostics et limites
- Calibration conjointe, pondération et contrôle de l’identifiabilité.
- Le critère de Feller est un diagnostic du processus continu.
- Convergence numérique, parités et bornes doivent être contrôlées.
- Une bonne calibration vanille ne suffit pas à valider les exotiques.
Methodology & assumptions
Le moteur utilise la fonction caractéristique et une quadrature de Simpson.
MODEL INTERPRETATION
Heston: analytical structure, numerical implementation
Heston models instantaneous variance as a mean-reverting square-root process and couples its shocks to the asset. Initial variance and long-run variance are different quantities; their square roots are volatility levels. An analytical characteristic function still requires numerical Fourier integration for option values. Integration cutoffs, quadrature and complex-function conventions therefore matter. The variance boundary also affects simulation and short-time expansions. A displayed surface is an output of the entered assumptions, not proof of calibration or of hedge reliability.
Scope, limitations and source
Risk-neutral parameters differ from historical estimates. Feller-condition failure alone does not invalidate Heston; boundary treatment and numerical method still require review.
Pierre Henry-Labordère, Analysis, Geometry, and Modeling in Finance (2009), §6.5–6.5.2, printed pp.181–183 / PDF200–202.
STRUCTURED PRODUCTS / 15
Capital-Protected Participation Note
Décomposition en zéro-coupon et call européen avec participation configurable.
Theoretical value—
Zero-coupon component—
Option component—
Structure delta—
Methodology & assumptions
Payoff à l’échéance N[1+p·max(Sₜ/K−1,0)]. Valeur par décomposition zéro-coupon + calls européens.
Model and method references
AHS bond-plus-call payoff decomposition; this paper supplies the option model, not the note’s contract.
Fischer Black, Myron Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637-654.
STRUCTURED PRODUCTS / 16
European-Barrier Reverse Convertible
Coupon fixe contre exposition conditionnelle à la baisse du sous-jacent.
Monte Carlo value—
Risk-neutral loss probability—
Total coupon—
MC standard error—
Methodology & assumptions
À l’échéance : nominal au-dessus de la barrière, participation linéaire à la baisse en dessous.
Model and method references
GBM and Monte Carlo foundations; coupon and terminal-barrier payoff are the tool’s stated contract.
Fischer Black, Myron Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637-654.
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability.
RATES / STRUCTURED COUPONS
CMS coupon · convexity scenario
Valeur illustrative d’un coupon, avec ajustement de convexité fourni.
Adjusted CMS rate—
Coupon cash flow—
Discounted PV—
PV = N · δ · D · (F_CMS + CA)
Methodology & limits
CA est saisi, non calibré par cet outil. Le calcul d’un coupon isolé n’est pas une valorisation CMS complète.
Model and method references
CMS methodological reference; the convexity adjustment is entered, not solved from the full model.
D. Brigo & F. Mercurio (2006). Interest Rate Models — Theory and Practice, second edition. Springer.
STRUCTURED PRODUCTS / 17
Autocall Phoenix · Memory Coupon
Structure path-dependent avec observations annuelles, coupon mémoire et protection conditionnelle à maturité.
Monte Carlo value—
Early redemption probability—
Principal loss probability—
Expected life—
Methodology & assumptions
Observations annuelles, coupon mémoire, remboursement anticipé et protection conditionnelle.
Model and method references
Model and simulation references only; annual coupon/call/protection conditions are defined by this tool.
Fischer Black, Myron Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637-654.
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability.
VOLATILITY / 18
Cash-or-Nothing Digital Option
Option binaire européenne, discontinuité de payoff et sensibilité localisée près du strike.
Digital call—
Digital put—
Risk-neutral P(Sₜ>K)—
Call delta—
Discounted cash—
Parity residual—
Methodology & assumptions
Formules fermées BSM et contrôle de parité digitale.
Model and method references
Cash-or-nothing payoff derived under the lognormal terminal distribution.
Fischer Black, Myron Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637-654.
VOLATILITY / 19
Variance Swap · Forward Variance
Exposition linéaire à la variance réalisée, avec strike et variance forward explicites.
Fair strike vol—
Model PV—
Realised settlement—
Forward variance spread—
+1 vol-point sensitivity—
Observation grid—
Methodology & assumptions
Payoff linéaire en variance et valorisation par variance forward exogène.
Model and method references
Reference for variance/volatility derivatives; this tool computes entered-variance settlement, not option-strip replication.
K. Demeterfi, E. Derman, M. Kamal & J. Zou (1999). More Than You Ever Wanted to Know About Volatility Swaps.
MULTI-ASSET / 20
Two-Asset Arithmetic Basket Call
Call européen sur panier arithmétique, simulation corrélée et diagnostic de diversification.
Basket call—
MC standard error—
Risk-neutral P(Bₜ>K)—
Parallel delta—
Effective basket vol—
Diversification benefit—
Methodology & assumptions
GBM corrélés, variables antithétiques et nombres aléatoires communs.
Model and method references
Correlated Monte Carlo methodology; weights and payoffs are specified by the calculator.
Phelim P. Boyle (1977). Options: A Monte Carlo approach. Journal of Financial Economics 4(3), 323-338.
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability.
RATES / 21
Hull–White Zero-Coupon Bond Option
Option européenne sur zéro-coupon sous modèle gaussien à taux court avec retour à la moyenne.
Bond call—
Bond put—
Forward bond—
Effective bond vol—
Vega / +1 bp σ—
Parity residual—
Methodology & assumptions
Formule fermée Hull–White sous mesure forward.
Model and method references
Original one-factor Gaussian interest-rate model.
John Hull, Alan White (1990). Pricing Interest-Rate-Derivative Securities. Review of Financial Studies 3(4), 573-592.
CREDIT / 22
Merton Structural Credit Model
Capitaux propres comme call sur les actifs, dette risquée et distance au défaut.
Equity value—
Risky debt value—
Physical default probability—
Risk-neutral default probability—
Distance to default—
Implied credit spread—
Debt face / assets F/V₀ (%)—
Asset − equity − debt (CCY)—
Illustrative scenarios
Presets change assumptions; they are not market observations. In BSM, non-reference presets also set an illustrative call price for an inversion check.
Model, conventions and interpretation
Mechanism
Shareholders hold the residual payoff max(V_T−F,0); the single debt claim receives min(V_T,F). Asset volatility and leverage link prices and default probabilities. The physical drift is entered, not estimated. This is structural Merton credit, not Merton jump-diffusion.
Reference formulas
\begin{aligned}d_2^Q&=\frac{\ln(V_0/F)+(r-\sigma_V^2/2)T}{\sigma_V\sqrt T},\quad d_1^Q=d_2^Q+\sigma_V\sqrt T\\E&=V_0\Phi(d_1^Q)-Fe^{-rT}\Phi(d_2^Q),\quad D=V_0-E\\DD_P&=\frac{\ln(V_0/F)+(\mu-\sigma_V^2/2)T}{\sigma_V\sqrt T}\\PD_P&=\Phi(-DD_P),\quad PD_Q=\Phi(-d_2^Q)\end{aligned}
V₀,F,E,D: same currency; T: years; σ_V: decimal annual asset volatility; r,μ: decimal annual continuous rates; DD_P: standard-deviation units; PD: probability. Spread = −ln(D/F)/T−r, displayed in bp. μ affects physical PD, not risk-neutral debt/equity valuation.
Assumptions, limits and primary reference
One zero-coupon liability, constant rates/asset volatility, default only at maturity and no dividends. Firm asset value/volatility are supplied rather than inferred from equity quotes. No recovery calibration, default barrier, rating mapping, market spread fit or realised default forecast.
Robert C. Merton (1974), On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29, 449–470.
Robert C. Merton — original reference ↗Methodology & assumptions
Modèle structurel à dette zéro-coupon unique.
CREDIT / 23
One-Year Rating Migration Bond
Valeur d’une obligation sous scénarios de migration, spreads d’état et recouvrement.
Current risky price—
Migration-adjusted value—
Migration P&L—
Default-state PV contribution—
Conditional expected spread—
Probability sum—
Methodology & assumptions
Valorisation par états de migration à horizon un an.
Model and method references
Reference for migration-based revaluation; the tool uses entered one-year state probabilities.
J.P. Morgan (1997). CreditMetrics — Technical Document.
RATES / 24
European Swaption — Black‑76
Option européenne sur un swap de taux.
Payer value—
Receiver value—
Forward intrinsic—
Vega / +1 vol point—
d₁—
Parity residual—
Methodology & assumptions
Formule Black‑76 sous mesure annuité.
Model and method references
Black pricing with a deterministic discounted swap annuity.
Fischer Black (1976). The pricing of commodity contracts. Journal of Financial Economics 3(1-2), 167-179.
D. Brigo & F. Mercurio (2006). Interest Rate Models — Theory and Practice, second edition. Springer.
RATES / 25
Callable Bond — Short‑Rate Tree
Obligation callable sur arbre pédagogique.
Callable value—
Straight-bond value—
Embedded call cost—
DV01—
Exercise nodes—
Dominance residual—
Methodology & assumptions
Induction rétrograde sur arbre de taux courts.
Model and method references
Related rate-model and backward-induction references; the one-call-date tree is an AHS specialization.
John Hull, Alan White (1990). Pricing Interest-Rate-Derivative Securities. Review of Financial Studies 3(4), 573-592.
John C. Cox, Stephen A. Ross, Mark Rubinstein (1979). Option pricing: A simplified approach. Journal of Financial Economics 7(3), 229-263.
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EXOTICS / 26
Up‑and‑Out Call — Brownian Bridge MC
Option barrière et biais de monitoring.
Brownian-bridge value—
Discrete-monitoring value—
Monitoring correction—
MC standard error—
Mean survival weight—
European call upper bound—
Methodology & assumptions
Monte‑Carlo avec correction de franchissement intra‑pas.
Model and method references
Brownian-bridge Monte Carlo method reference; not a claim of a unique originating paper.
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability.
RATES / 27
Shifted-Black Caplet
Caplet shifted-lognormal.
Caplet value—
Delta / 1 bp—
Vega / 1 pt—
Intrinsic—
Methodology & assumptions
Shifted Black‑76.
Model and method references
Black lognormal foundation; the exogenous displacement is an extension, not the original unshifted formula.
Fischer Black (1976). The pricing of commodity contracts. Journal of Financial Economics 3(1-2), 167-179.
D. Brigo & F. Mercurio (2006). Interest Rate Models — Theory and Practice, second edition. Springer.
CREDIT / 28
Synthetic CDO Tranche — LHP
Tranche loss under Gaussian copula.
Expected tranche loss—
Expected loss (%)—
Portfolio EL—
Tranche notional—
Methodology & assumptions
LHP Gaussian one-factor model.
Model and method references
Large homogeneous one-factor loss framework; terminal tranche loss is an AHS payoff application.
O. Vasicek (2002). Loan portfolio value. Risk, December.
ALM / PRICING
Loan pricing & funds transfer pricing
Mesurer la marge économique après transfert de liquidité, risque de crédit et consommation de capital.
Risk-adjusted margin (bp)—
Annual contribution (€)—
Break-even coupon (%)—
Capital return proxy (%)—
Surface de marge FTP
Contribution annualisée selon le taux FTP et la perte attendue.
m_bp = 100 (coupon − FTP) − EL − OPEX − liquidité − capital ; Π = nominal × marge / 10 000
Taux en %, charges en points de base, nominal en euros.
Methodology & assumptions
Risk-adjusted contribution proxy.
Model and method references
Management reference for the framework; the static contribution equation is specified by AHS.
J. Dermine & Y. F. Bissada (2002). Asset & Liability Management: A Guide to Value Creation and Risk Control.
ALM / DEPOSIT FTP
Tarification FTP des dépôts
Estimer la contribution d’un dépôt après FTP et coûts.
Risk-adjusted margin (bp)—
Annual contribution (€)—
Break-even customer rate (%)—
Market shock pass-through (bp)—
FTP margin sensitivity
Contribution selon le taux FTP et les coûts opérationnels.
m = 100 (FTP − taux client) + liquidité − coûts − capital ; contribution = encours × m / 10 000
Les taux sont en %, les coûts en pb et le nominal en euros.
Methodology & assumptions
Approximation statique ; pas de modèle comportemental des dépôts.
Model and method references
Management reference for the framework; entered pass-through and liquidity benefit are AHS scenario assumptions.
J. Dermine & Y. F. Bissada (2002). Asset & Liability Management: A Guide to Value Creation and Risk Control.
ALM / BALANCE SHEET
Repricing gap, NII & EVE sensitivity
Lecture simplifiée des écarts de repricing et de la sensibilité de valeur économique.
ΔNII proxy (€)—
Duration gap (years)—
ΔEVE proxy (€)—
Opening surplus (€)—
Surface de sensibilité du bilan
Sensibilité de valeur économique aux chocs de taux et à la durée des actifs.
ΔNII ≈ (RSA − RSL) Δr H ; ΔEVE ≈ −(D_A A − D_L L) Δr
Choc en points de base, durées en années et valeurs en euros.
Methodology & assumptions
Gap and first-order duration approximations.
Model and method references
IRRBB context and duration foundations; this first-order EVE/NII illustration is not the full regulatory calculation.
Basel Committee on Banking Supervision (2016). Interest rate risk in the banking book.
F. R. Macaulay (1938). Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856. NBER.
OPTIONS / FOURIER
Jump pricer: Merton / Variance Gamma
European call and put under exponential jump dynamics. Compare illustrative cases, parameter surfaces and numerical refinement.
Illustrative jump cases
Cases restore example parameters of the selected process; frequent jumps sets λ=1.5/year, diffusion-only sets λ=0. Spot, strike, maturity, rate and dividend remain your inputs.
Call value (€ per unit)—
Put value by parity (€ per unit)—
Discounted intrinsic lower bound (€)—
Grid-refinement difference (€)—
Call–put parity residual (€)—
Integration settings—
Price and jump-parameter surfaces
Surface de prix sous dynamique de Lévy
Prix Fourier recalculé selon le moneyness et l’échéance.
Carr–Madan damped Fourier integral for call value.
Merton jump-diffusion characteristic function.
Variance Gamma characteristic function and martingale correction.
Paramètres risque-neutres ; précision contrôlée par convergence de la quadrature.
Jump-parameter sensitivity
The second surface varies K/S₀ and λ for Merton, or ν for VG, holding maturity and other inputs fixed. Surface quadrature uses 320 intervals; invalid moment-domain nodes are omitted.
Risk-neutral jump compensation
\kappa_J=e^{\mu_J+\sigma_J^2/2}-1,\quad b=r-q-\tfrac12\sigma^2-\lambda\kappa_J
Merton log-jumps are normally distributed with decimal mean μJ and standard deviation σJ. λ is expected jumps per year; the compensator preserves the forward expectation.
Model and method references
Carr–Madan transform with Merton or Variance Gamma. The tool uses Simpson quadrature, not an FFT implementation.
Peter Carr, Dilip Madan (1999). Option valuation using the fast Fourier transform. The Journal of Computational Finance 2(4), 61-73.
Robert C. Merton (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1-2), 125-144.
Dilip B. Madan, Peter P. Carr, Eric C. Chang (1998). The Variance Gamma Process and Option Pricing. Review of Finance 2(1), 79-105.
VG moments and damping
\omega=\nu^{-1}\ln(1-\theta\nu-\tfrac12\sigma_{VG}^2\nu),\quad 1-p\theta\nu-\tfrac12p^2\sigma_{VG}^2\nu>0
The risk-neutral compensator requires the first moment. Carr–Madan damping here also requires p=α+1=2.5. The panel rejects VG parameters outside either domain.
Numerical controls
\varepsilon_{grid}=|C_{3600}-C_{1800}|
The main price uses Simpson quadrature over [0,160], 1800 intervals; refinement doubles intervals at the same cutoff. This difference is a grid diagnostic, not a confidence interval or a bound on truncation error. Surface prices are coarser diagnostics.
Interpretation and limits
C-P=S_0e^{-qT}-Ke^{-rT}
Put value uses parity and the residual checks that identity only; it does not independently validate Fourier accuracy. Parameters are illustrative, not calibrated. No stochastic rates, barriers, issuer risk or market feed is implemented.
Methodology & assumptions
Carr–Madan damped transform; numerical convergence remains a required check.
OPTIONS / LÉVY DIGITAL
Call digital sous dynamique de Lévy
Prix d’un call digital par dérivée en strike du call Fourier.
Digital call value (unit payout)—
Finite-difference step (spot bp)—
Put digital (parity)—
Digital price surface
Paiement unitaire si Sₜ > K ; dérivée approximée par différence centrée.
Vdig ≈ −∂C/∂K (différence centrée)
La sensibilité au pas ε et à la convergence numérique doit être contrôlée.
Methodology & assumptions
Modèle indicatif, dérivée numérique et paramètres non calibrés.
Model and method references
Digital value is an AHS finite-strike derivative of the Fourier call.
Peter Carr, Dilip Madan (1999). Option valuation using the fast Fourier transform. The Journal of Computational Finance 2(4), 61-73.
Robert C. Merton (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1-2), 125-144.
Dilip B. Madan, Peter P. Carr, Eric C. Chang (1998). The Variance Gamma Process and Option Pricing. Review of Finance 2(1), 79-105.
FX / 29
FX Forward & Points
Parité couverte des taux.
Fair forward—
Forward points—
MTM domestic—
Cash at maturity—
Methodology & assumptions
Covered interest parity.
Model and method references
Covered-interest-parity reference, section 1, equation (1); not a claim of original authorship of the parity identity.
M. Haugh (2013). Foreign Exchange, ADRs and Quanto-Securities. Financial Engineering: Continuous-Time Models, Columbia University.
FX / 30
FX Vanilla — Garman–Kohlhagen
FX vanilla.
Call value DOM—
Put value DOM—
Spot delta—
Vega / 1 vol pt—
Forward F₀,T (DOM/FOR)—
Call gamma (DOM per quote-unit²)—
Call hedge amount (FOR)—
Put–call parity residual (DOM)—
Illustrative scenarios
Presets change assumptions; they are not market observations. In BSM, non-reference presets also set an illustrative call price for an inversion check.
Model, conventions and interpretation
Mechanism
The quote is domestic currency for one foreign unit. The foreign rate acts as a continuous carry yield. A call buys the foreign currency at K; total premium scales with foreign notional. Both rates and notional stay fixed across the strike–maturity surfaces.
Reference formulas
\begin{aligned}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\\C&=N_f[S_0e^{-r_fT}\Phi(d_1)-Ke^{-r_dT}\Phi(d_2)]\\C-P&=N_f[S_0e^{-r_fT}-Ke^{-r_dT}]\end{aligned}
S₀,K,F: DOM per FOR; N_f: foreign-currency amount; C,P: domestic-currency amount; T: years; r_d,r_f: annual continuous decimals; σ: decimal annual FX volatility. Delta is unadjusted spot delta per unit N_f; N_f×delta is a FOR hedge amount.
Assumptions, limits and primary reference
Constant volatility and rates, European exercise. No premium-adjusted delta, smile/RR/BF calibration, currency-specific cut, settlement lag or cross-currency basis. DOM/FOR are unit conventions, not an asserted live currency pair. Zero-volatility kink Greeks are undefined.
Garman & Kohlhagen (1983), Foreign currency option values, Journal of International Money and Finance 2(3), 231–237.
Garman & Kohlhagen — original reference ↗Methodology & assumptions
Garman–Kohlhagen.
FX / 31
FX Digital — Domestic Cash-or-Nothing
Digital FX.
Digital call—
Digital put—
Exercise probability—
Parity check—
Methodology & assumptions
FX digital.
Model and method references
Digital payoff derived under the Garman–Kohlhagen terminal distribution.
Mark B. Garman, Steven W. Kohlhagen (1983). Foreign currency option values. Journal of International Money and Finance 2(3), 231-237.
RATES / BERMUDAN OPTIONALITY
Swap de taux cancellable
Swap payeur fixe augmenté d’un panier indicatif de swaptions receveur aux dates d’annulation prévues.
Valeur du swap payeur—
Approximation de l’option d’annulation—
Valeur du swap cancellable—
Échéancier d’exercice—
V_cancel ≈ N·A₀(F−K) + N·Σᵢ AᵢD(tᵢ)P₇₆(F,K,tᵢ,σ)
Méthode et limites
Décomposition illustrative ; une valorisation complète exige un modèle bermudéen calibré et les conventions contractuelles.
Model and method references
Option-pricing references only; the sum-of-European-options approximation is not an exact Bermudan cancellation model.
Fischer Black (1976). The pricing of commodity contracts. Journal of Financial Economics 3(1-2), 167-179.
D. Brigo & F. Mercurio (2006). Interest Rate Models — Theory and Practice, second edition. Springer.
STRUCTURED PRODUCTS / ATHENA
Autocall Athena · coupon mémoire
Observations annuelles, coupon conditionnel à mémoire, barrière de rappel dégressive et barrière de protection à maturité.
Valeur actualisée risque-neutre—
Probabilité de rappel anticipé—
Probabilité de perte en capital à maturité—
Durée de vie moyenne—
Erreur standard Monte-Carlo—
V = Eᵠ[Σᵢ Dᵢ Cᵢ 1(Sᵢ/S₀ ≥ Bc) + DτRτ]
Méthode et limites
Term sheet pédagogique stylisé ; les barrières et conventions réelles dépendent du contrat.
Model and method references
GBM/Monte Carlo foundations; memory coupon and step-down conditions are specified by the tool, not these publications.
Fischer Black, Myron Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637-654.
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability.
RATES / CONDITIONAL COUPON
Range accrual CMS
Valeur attendue du coupon à partir de la probabilité que chaque fixing reste dans un corridor de taux défini.
Valeur actuelle des coupons attendus—
Fraction attendue en corridor—
Taux de coupon attendu—
Nombre d’observations—
PV=N·c·δ·ΣᵢDᵢ[Φ((U−Fᵢ)/σᵢ)−Φ((L−Fᵢ)/σᵢ)]
Méthode et limites
Approximation marginale ; le calendrier, la dépendance et les conventions CMS doivent être définis par la transaction.
Model and method references
Normal-distribution and rate-method references; corridor coupon conventions are specified by the tool.
L. Bachelier (1900). Théorie de la spéculation. Annales scientifiques de l’École Normale Supérieure, 17, 21–86.
D. Brigo & F. Mercurio (2006). Interest Rate Models — Theory and Practice, second edition. Springer.
CREDIT / FUNDED PROTECTION
Note liée au crédit (CLN)
Note financée valorisée par ses coupons et son nominal conditionnels à la survie, plus le recouvrement en cas de défaut.
Valeur risque-neutre indicative—
Intensité implicite—
Probabilité cumulée de défaut—
Perte en principal attendue non actualisée—
λ≈s/(1−R); V=N[c(1−e⁻⁽ʳ⁺λ⁾ᵀ)/(r+λ)+e⁻⁽ʳ⁺λ⁾ᵀ+Rλ(1−e⁻⁽ʳ⁺λ⁾ᵀ)/(r+λ)]
Méthode et limites
La prime de risque, la liquidité, les clauses de crédit et le risque de l’émetteur restent hors modèle.
Model and method references
Reduced-form credit foundations; coupon, recovery and principal conventions are stated by the tool.
ROBERT A. JARROW, STUART M. TURNBULL (1995). Pricing Derivatives on Financial Securities Subject to Credit Risk. The Journal of Finance 50(1), 53-85.
Darrell Duffie, Kenneth J. Singleton (1999). Modeling Term Structures of Defaultable Bonds. Review of Financial Studies 12(4), 687-720.
NUMERICAL METHODS ATLAS
De l’équation à l’erreur mesurable
La méthode détermine aussi les erreurs à mesurer.
Formules fermées
Évaluation analytique directe.
- Erreur
- Modèle
- Contrôle
- Parités
Arbres recombinants
Discrétisation et induction rétrograde.
- Erreur
- Pas temporel
- Contrôle
- Convergence
Intégration numérique
Intégrale de fonction caractéristique.
- Erreur
- Troncature
- Contrôle
- Doublement
Monte‑Carlo
Estimation statistique reproductible.
- Erreur
- SE = s/√N
- Contrôle
- IC et convergence
Résolution et calibration
Problème inverse sous contraintes.
- Erreur
- Identifiabilité
- Contrôle
- Repricing
Optimisation de portefeuille
Exactitude mathématique et robustesse économique sont distinctes.
- Erreur
- Estimation
- Contrôle
- Stress des entrées
Budget d’erreur total
εtotal = εdata + εmodel + εcalibration + εnumerical + εimplementation
Décomposition conceptuelle, non estimation statistique.
SCIENTIFIC CONTROL PLANE / BUILD 2026-09-14
Des résultats accompagnés de critères de réfutation.
Les tests structurels ci-dessous vérifient les propriétés que les moteurs doivent respecter indépendamment d’un scénario de marché particulier.
Parité put-call
|C − P − Se−qT + Ke−rT| < 10−8
Équilibre au spread par
|PVpremium − PVprotection| < ε
Dominance américaine
VAmerican ≥ VEuropean
Cohérence de queue
Expected Shortfall ≥ Value at Risk
Admissibilité de covariance
λmin(Σ) > 0 · Σ = ΣT
Reproductibilité Monte Carlo
Graine fixe · mêmes entrées, mêmes sorties
Continuité ATM
limK→F σSABR(F,K) = σATM
Bornes et convergence
Cintrinsic ≤ C ≤ Se−qT · |ΔN| monitored
Plancher de protection
Payoffprotected(ST) ≥ N
Reproductibilité des structures
Graine fixe · mêmes entrées, mêmes sorties
Parité digitale
Cdigital + Pdigital = Qe−rT
Strike de variance forward
Fvar = Kvar ⇒ PV = 0
Bornes de corrélation
−1 < ρ < 1 · SEMC reported
Parité option obligataire
C − P = P(0,T) − KP(0,S)
Conservation de valeur
Equity + Risky Debt = Firm Assets
Masse de probabilité
Σstates pi = 100%
Parité payeur-receveur
Payer − Receiver = N·A·(F−K)
Dominance obligataire
Vcallable ≤ Vstraight
Borne barrière
0 ≤ Vup-and-out ≤ VEuropean call
Portée : tests unitaires et invariants de démonstrateurs. Ils ne remplacent ni benchmark externe, ni revue de code indépendante, ni validation de modèle de production.
Architecture & protocole complet ↗Explore the Gaussian animations →
APPLIED RESEARCH · READING PATHWAY
Malliavin calculus: sensitivities and volatility dynamics
Four research notes and one restricted Black–Scholes experiment distinguish a numerical Greek, a hedging representation and an asymptotic model diagnostic.
Estimate a digital Delta without differentiating its indicator
Question
How do integration-by-parts weights compare with a common-random-number bump?
Inputs to gather
Spot, strike, risk-free rate, volatility, maturity, sample size, bump and seed.
Method and assumptions
Simulate a cash-or-nothing payoff under constant-volatility Black–Scholes with no dividends. Multiply by Z/(Sσ√T), and compare with central finite differences on the same draws and the analytical digital Delta.
\Delta=e^{-rT}E\!\left[\mathbf1_{\{S_T>K\}}\frac{Z}{S\sigma\sqrt T}\right],\quad Z\sim N(0,1)
Interpretation and proposed use
A unit cash payoff gives Delta in inverse spot units. Display both Monte Carlo standard errors; changing the bump changes the bias–variance trade-off of finite differences.
Limits and controls
The weight coincides with a likelihood-ratio score in this simple model. This is not a Heston, rough-volatility or general Malliavin engine. Rare payoff hits can make a small estimated standard error misleading.
Reading reference
Elisa Alòs & David Garcia Lorite, Malliavin Calculus in Finance: Theory and Practice, 2nd ed., CRC Press, 2025, §§4.2.1–4.2.2, printed pp. 86–89 / PDF pp. 113–116. DOI: 10.1201/9781032636382. The digital-payoff comparison is an original Black–Scholes teaching application of the integration-by-parts weight.
Discuss model review and governanceA martingale representation needs a market model
Question
When does a conditional Malliavin derivative describe a hedge?
Inputs to gather
Payoff, filtration, traded assets, discounting and integrability assumptions.
Method and assumptions
Write the Brownian martingale representation and identify the conditional derivative integrand. Translate it into traded units only after the asset dynamics and completeness assumptions are specified.
F=E[F]+\int_0^T E[D_tF\mid\mathcal F_t]\,dW_t
Interpretation and proposed use
For the complete one-stock Black–Scholes setting, the resulting hedge connects to classical Delta. In an incomplete model, a representation alone does not establish exact replication.
Limits and controls
Requires the relevant Malliavin regularity and conditional expectations. No executable hedging strategy, transaction-cost model or incomplete-market optimisation is supplied.
Reading reference
Alòs & Garcia Lorite (2025), §§4.1.1–4.1.2, printed pp. 75–80 / PDF pp. 102–107.
Discuss model review and governanceShort-end skew is a model diagnostic, not a model verdict
Question
What short-maturity behaviour can a rough volatility kernel produce?
Inputs to gather
Maturity convention, log-strike, correlation, kernel form and Hurst parameter H.
Method and assumptions
Study the stated fractional-model assumptions and the order of the ATM implied-volatility skew as maturity shrinks. Compare orders only with consistent strike and volatility conventions.
\partial_k I(T,k_{ATM})=O\!\left(T^{H-1/2}\right)\quad\text{in the stated fractional setting}
Interpretation and proposed use
For H<1/2 the stated order allows a growing short-end skew when the leading coefficient is nonzero. This motivates diagnostics beyond a single smile fit.
Limits and controls
An asymptotic order does not calibrate H, ensure a nonzero skew or establish absence of arbitrage. No rough-model prices are computed in this module.
Reading reference
Alòs & Garcia Lorite (2025), §7.6.4, printed pp. 205–206 / PDF pp. 232–233.
Discuss model review and governanceA vanilla smile does not determine every forward smile
Question
What changes when the strike is fixed at a future start date?
Inputs to gather
Start date s, expiry T, forward moneyness and joint spot–volatility dynamics.
Method and assumptions
Write the random-strike payoff explicitly. Use constant volatility as a benchmark, then distinguish spot-smile calibration from assumptions on the future joint distribution.
\mathrm{payoff}=(S_T-e^{\alpha}S_s)^+,\qquad 0<s<T
Interpretation and proposed use
A proposed model review compares forward-start sensitivities across models that fit similar vanilla surfaces.
Limits and controls
Matching vanilla prices alone does not validate forward dynamics. This is a research protocol, not a newly implemented forward-start pricer.
Reading reference
Alòs & Garcia Lorite (2025), §9.2, printed pp. 259–260 / PDF pp. 286–287.
Discuss model review and governanceNo matching note. Clear the search to restore all notes.
Digital Delta: integration-by-parts weight versus finite differences
Unit cash payoff, constant-volatility Black–Scholes, no dividends. Rate is continuously compounded; volatility is annualised. Both estimators use the same seeded pseudo-random Gaussian draws. This is a restricted teaching implementation, not a general Malliavin engine.
Elisa Alòs & David Garcia Lorite, Malliavin Calculus in Finance: Theory and Practice, 2nd ed., CRC Press, 2025, §§4.2.1–4.2.2, printed pp. 86–89 / PDF pp. 113–116. DOI: 10.1201/9781032636382. The digital-payoff comparison is an original Black–Scholes teaching application of the integration-by-parts weight.
Reading scope: selected passages and tables of contents were consulted, not the complete books. Notes and numerical examples are original syntheses. The books provide the analytical framework; no author endorsement or independent model validation is implied.
RESEARCH FIGURES / 3D
Graphiques de recherche interactifs
Price, volatility and risk surfaces are shown in 3D. Rotate them to examine interactions between two parameters.
Les graphiques utilisent des paramètres synthétiques. Les équations et leurs limites figurent sous chaque graphique. La couleur représente la valeur calculée ; les échelles sont propres à chaque figure. Le réglage de hauteur concerne toutes les surfaces de cette galerie.
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
S/K : 0,6–1,4 ; T : 0,1–5 ans ; K=100, σ=22 %, r=3 %, q=1 %. Z=C.
Equation
C=S e^(−qT)Φ(d₁)−K e^(−rT)Φ(d₂); d₁=[ln(S/K)+(r−q+σ²/2)T]/(σ√T); d₂=d₁−σ√TInterprétation
Valeur du call européen en unités monétaires du strike. La pente selon S/K vaut K fois le delta ; la courbure vaut K² fois le gamma. La courbure temporelle ne mesure pas le vega.
Limites et contrôles à réaliser
Volatilité constante, taux et dividendes déterministes. Sans smile, sauts ou exercice anticipé. Vérifier max(Se^(−qT)−Ke^(−rT),0) ≤ C ≤ Se^(−qT).
Référence
Black & Scholes (1973), The Pricing of Options and Corporate Liabilities
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
τ : 0,1–10 ans ; a : 0,05–1 par an ; σ₀=0,012. Z=ΣP.
Equation
ΣP(τ,a)=σ₀(1−exp(−aτ))/aInterprétation
Intégrale sur la durée résiduelle d’un noyau exponentiel de volatilité des taux forward. Il s’agit de l’amplitude d’un coefficient de volatilité obligataire, pas d’une courbe de taux simulée ou d’un prix obligataire.
Limites et contrôles à réaliser
Un facteur déterministe ; sans calibration de courbe ni simulation de la dérive HJM. Lorsque a→0, ΣP→σ₀τ ; lorsque τ augmente, ΣP tend vers σ₀/a.
Référence
Heath, Jarrow & Morton (1992), Bond Pricing and the Term Structure of Interest Rates
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
Z : −3–3 ; ρ : 0,01–0,60 ; PD inconditionnelle=2 %. Sortie en fraction de probabilité.
Equation
PD(Z,ρ)=Φ((Φ⁻¹(0.02)−√ρ Z)/√(1−ρ))Interprétation
Les valeurs faibles du facteur systémique correspondent aux états défavorables. La corrélation module la sensibilité à cet état ; son effet n’est pas croissant pour toutes les valeurs de Z.
Limites et contrôles à réaliser
Modèle latent gaussien ; sans instant de défaut ni dynamique du recouvrement. Résultats dans [0,1] ; l’intégration sur Z normal standard restitue 2 %.
Référence
Vasicek (2002), Loan Portfolio Value
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
S : 0,25–5 ans ; a : 0,01–0,60 par an ; T=10 ans, σ=1 %. Z=volatilité totale de l’option obligataire.
Equation
σP(S,T)=σ·B(S,T)·√((1−exp(−2aS))/(2a)); B(S,T)=(1−exp(−a(T−S)))/aInterprétation
S représente l’échéance de l’option, et non un spot ou un taux court initial. La sortie combine variance accumulée avant l’échéance et duration obligataire résiduelle après cette échéance.
Limites et contrôles à réaliser
Modèle gaussien à un facteur et coefficients constants. Ce n’est ni une cotation annualisée ni un prix. La limite a→0 est σ(T−S)√S.
Référence
Hull & White (1990), Pricing Interest-Rate-Derivative Securities
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
V/F : 0,55–2,20 ; σV : 8–70 % ; μ=6 %, T=1 an. Sortie en fraction de probabilité.
Equation
PD=Φ(−[ln(V/F)+(μ−σV²/2)T]/(σV√T))Interprétation
Le défaut correspond à des actifs inférieurs à la dette à l’échéance. Avec μ fixé comme dérive des actifs, il s’agit d’un scénario sous mesure physique, pas d’une valorisation risque-neutre de dette. L’effet de la volatilité dépend du levier.
Limites et contrôles à réaliser
Dette zéro-coupon unique, actifs lognormaux ; sans défaut anticipé ni cascade de passif. Vérifier les bornes [0,1] et la baisse de PD avec V/F.
Référence
Merton (1974), On the Pricing of Corporate Debt
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
PD : 0,001–0,20 ; R : 0,10–0,80. Z=perte par unité d’exposition.
Equation
EL/EAD=PD(1−R)Interprétation
Multiplier par l’EAD pour obtenir une perte attendue monétaire. Cette figure ne contient aucune matrice de transition de notation malgré son identifiant interne migration.
Limites et contrôles à réaliser
PD et recouvrement déterministes ; sans actualisation ni dépendance défaut-recouvrement. EL s’annule si PD=0 ou R=1.
Référence
Schönbucher, Credit Derivatives Pricing Models
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
K/F : 0,5–1,5 ; T : 0,1–10 ans ; F=3 %, σ=22 %. Z=v.
Equation
v=FΦ(d₁)−KΦ(d₂); d₁=[ln(F/K)+σ²T/2]/(σ√T); d₂=d₁−σ√TInterprétation
Option payeuse européenne par unité de notionnel et d’annuité de swap. La valeur monétaire nécessite une multiplication par N et l’annuité actualisée A, absents de la figure.
Limites et contrôles à réaliser
Forward et strike positifs, volatilité lognormale constante. Ce n’est pas une surface de smile. Vérifier payeur−receveur=F−K avant multiplication par l’annuité.
Référence
Black (1976), The Pricing of Commodity Contracts
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
r₀ : 0,5–8 % ; σr : 0–4 % ; nominal=100, coupon annuel=5, maturité=7 ans, rappel unique à 3 ans.
Equation
Continuation(i,j)=0.5[V(i+1,j)+V(i+1,j+1)]exp(−[r₀+(2j−i)σr]); call at i=3: min(continuation,100)Interprétation
Induction rétrograde sur sept pas ; le coupon est ajouté après comparaison avec le prix de rappel à la date de call. Arbre de taux stylisé à probabilités égales, non calibré à Hull–White.
Limites et contrôles à réaliser
Sans ajustement à une courbe de marché, spread de crédit ou calendrier de rappels multiples. Comparer au même arbre sans plafond de rappel : valeur callable ≤ valeur non callable.
Référence
Numerical reference: Cairns, Interest Rate Models; local seven-step illustrative implementation
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
H/S : 1,02–1,80 ; σ : 8–60 % ; Δt=0,02 an. Z=probabilité conditionnelle de franchissement.
Equation
p(hit | S₀=S₁=S)=exp(−2[ln(H/S)]²/(σ²Δt))Interprétation
Deux extrémités identiques sous une barrière haute fixe. La trajectoire peut franchir entre observations malgré des extrémités sous H. Ce n’est pas la probabilité de toucher sur toute la vie de l’option.
Limites et contrôles à réaliser
Diffusion lognormale continue ; sans sauts ni barrière mobile. La probabilité tend vers un lorsque H/S→1 et vers zéro lorsque la volatilité s’annule.
Référence
Glasserman, Monte Carlo Methods in Financial Engineering — Brownian bridge
Research · documentation v1 · 2026-09-19
Comprendre le modèle · Démonstrateur de recherche
Axes, unités et hypothèses fixes
S/K : 0,7–1,3 ; T : 0,05–5 ans ; K=1, rd=3 %, rf=1,5 %, σ=10 %.
Equation
C=S e^(−rf T)Φ(d₁)−K e^(−rd T)Φ(d₂); d₁=[ln(S/K)+(rd−rf+σ²/2)T]/(σ√T)Interprétation
Prix du call en devise domestique par unité de notionnel étranger. Le taux étranger joue le rôle de rendement continu ; la valeur est normalisée avec un strike de un.
Limites et contrôles à réaliser
Taux plats, volatilité constante ; sans delta ajusté de la prime, smile, basis ou délai de règlement. Vérifier C−P=Se^(−rf T)−Ke^(−rd T).
Référence
Garman & Kohlhagen (1983), Foreign Currency Option Values
Research · documentation v1 · 2026-09-19
MODEL REFERENCES
Cadres théoriques mobilisés
- Andrew J. G. CairnsInterest Rate Models — HJM (ch. 5–6), BGM/LMM (ch. 9), numerical methods (ch. 10).
- Philipp J. SchönbucherCredit Derivatives Pricing Models — survival curves, CDS valuation and intensity models (ch. 3 & 7).
- Antoine Savine & Jesper AndreasenModern Computational Finance — scripting, Monte Carlo and application to xVA (Part V).
- Frans de WeertExotic Options Trading — payoff mechanics, hedging and exotic risk.
- Christian FriesMathematical Finance — Black–Scholes, Greeks, Monte Carlo sensitivities and Bermudan optimal stopping.
- Gérard Cornuéjols & Reha TütüncüOptimization Methods in Finance — mean–variance and quadratic portfolio optimization (ch. 8).
- Jean-Luc PrigentPortfolio Optimization and Performance Analysis — mean–variance allocation, benchmark tracking, performance measures and portfolio insurance (ch. 3–5, 9–10).
- Justin LondonModeling Derivatives in C++ — pricing fundamentals, Monte Carlo, CRR trees and implied volatility.
- Christian Crispoldi, Gérald Wigger & Peter LarkinSABR and SABR LIBOR Market Models in Practice — SABR parameters, Hagan lognormal and normal approximations, calibration, sensitivities, approximation limits and shifted SABR.
- Fabrice D. RouahThe Heston Model and Its Extensions — European pricing (ch. 1), integration and Little Heston Trap (ch. 2), numerical integration (ch. 5).
- B. P. Lancaster, G. M. Schultz & F. J. FabozziStructured Products and Related Credit Derivatives — structuring, cash-flow waterfalls, credit derivatives and scenario analysis.
- Uwe WystupFX Options and Structured Products — currency options, technical conventions, vanilla strategies, exotics, structured products and Vanna–Volga (ch. 1, 2 & 4).
- Mhamed BettaiebOptions Pricing with Python — payoff construction, Black–Scholes, Greeks, parity tests and Monte Carlo reproducibility (ch. 2, 4 & 6).
- Yves HilpischListed Volatility and Variance Derivatives — model-free variance replication, realised variance and variance swaps (ch. 3 & 9).
- Lixin WuInterest Rate Modeling: Theory and Practice, 3rd ed. — HJM, Hull–White / extended Vasicek and zero-coupon bond options (ch. 6, §6.5.2).
- Leif B. G. Andersen & Vladimir V. PiterbargInterest Rate Modeling, Vol. II — one-factor Gaussian short-rate models, European option pricing and swaption calibration (ch. 10).
- Gunter Löffler & Peter N. PoschCredit Risk Modeling using Excel and VBA — structural default valuation and Merton implementation (ch. 2); transition matrices and multi-period transitions (ch. 3).
- Philipp J. SchönbucherCredit Derivatives Pricing Models — intensity models, rating transitions, firm-value models and portfolio default dependence (ch. 5, 8–10).
- Howard CorbInterest Rate Swaps and Other Derivatives — swap valuation and risk (ch. 1–3), caps/floors (ch. 4) and swaptions (ch. 5).
- Donald J. SmithValuation in a World of CVA, DVA, and FVA — callable debt (ch. IV), bilateral-credit swaps (ch. V) and funding valuation adjustment (ch. VI).
- Pricing and Hedging Financial DerivativesOption replication and risk-neutral valuation (ch. 5), Greeks and implied volatility (ch. 6), volatility structures (ch. 7) and exotic/barrier options (ch. 8).
- Patrick S. Hagan“Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors”, Wilmott (2003), DOI 10.1002/wilm.42820030211.
- ISDA2021 Interest Rate Derivatives Definitions — floating-rate options, conventions and fallback architecture.
- Egger et al.“A Threshold for Quantum Advantage in Derivative Pricing” — explicit TARF path payoff used as a computational benchmark.
- Iain J. ClarkCommodity Option Pricing — Black-76, Asian options, commodity swaptions, calendar/Asian spread options, convenience-yield models, swing, spark and crack spread options (ch. 2, 5–9).
- Viviana FanelliFinancial Modelling in Commodity Markets — spot/forward models, seasonality, jump diffusion, Monte Carlo, electricity and spark-spread applications (ch. 1–5).
- Raoul PieterszPricing Models for Bermudan-Style Interest Rate Derivatives — early-exercise modelling and numerical methods for cancellable and Bermudan rate products.
Ces outils utilisent des hypothèses simplifiées et des données saisies par l’utilisateur. Ils ne constituent ni une valorisation indépendante, ni un conseil en investissement, ni un système de production. Toute utilisation professionnelle exige calibration aux données de marché, conventions contractuelles, validation indépendante et contrôles de modèle.
Hybrid products and models
Structured
Rates
FX
Portfolio
INVESTMENT MANDATE
The investment mandate before active or passive allocation
Start with the mandate rather than a preferred investment style. Record the client's return objective, loss tolerance, liquidity needs, horizon, investment restrictions and benchmark. Translate these into portfolio constraints and a review schedule before selecting active, passive or blended exposures. Compare alternatives after fees and implementation costs, using the same risk and benchmark definitions. Index tracking retains the risks of its underlying market; active management adds discretion without guaranteeing outperformance. Rebalancing and mandate exceptions should follow an agreed decision process, with deviations explained to the investment committee rather than hidden by an aggregate ESG label.
Decision, data and controls
Decision: approve allocation only within the documented mandate. Data: IPS, benchmarks, exposure limits, costs and liquidity assumptions. Control: monitor deviations and reapprove material changes.
Scope, limitations and source
A mandate framework, not automatically enforced restrictions in the existing simulators. Neither active nor passive labels guarantee returns.
Sabrina Failo, Sustainable Asset Management (eds. Karen Wendt & Marta Ra, 2026), ch.4 §4.4, printed pp.60–62 / PDF67–69. DOI10.1007/978-3-032-05659-7_4.
INVESTMENT MANDATE
Engagement milestones, escalation and attribution
An engagement programme needs a specific issue, a baseline, a requested action and a review date. Record contact with the issuer separately from the change sought: a meeting or a favourable vote is an activity, not proof of an improved outcome. Decide in advance what would trigger further dialogue, a different voting instruction or an investment review, subject to the mandate and applicable rights. Retain unsuccessful cases as well as completed actions. If an issuer changes its practice, distinguish observed progress from the claim that the investor caused it. Report evidence gaps without converting engagement counts into an impact score.
Decision, data and controls
Decision: continue, escalate or review the holding. Data: issue-level baseline, requests, responses, voting rationale and outcomes. Control: separate activities from outcomes and test consistency with the investment mandate.
Scope, limitations and source
Track activity and observed outcomes separately. Meetings and votes do not establish causality, achieved impact or a permission to act beyond the mandate.
Emma Farrell, Sustainable Asset Management (eds. Karen Wendt & Marta Ra, 2026), ch.6 §6.4.3, printed pp.93–94 / PDF98–99; §6.5–6.6, pp.97–98 / PDF102–103. DOI10.1007/978-3-032-05659-7_6.
Contingent payments and decisions
Energy
Interactive research · Synthetic scenarios
Explore carbon-inclusive generation margin
Energy and carbon cost
Business valuation
VALUE BRIDGE
Acquisition value: separate the asset from the execution case
An acquisition thesis should explain why the target advances the buyer's strategy, which assumptions support its standalone value, and what additional value depends on combining the businesses. Revenue opportunities and cost savings belong in a separate synergy case: they require decisions, resources and execution. The purchase premium consumes part of that potential value before integration begins. A decision paper should therefore compare the acquisition with feasible alternatives and identify the assumptions that would invalidate the case. The value bridge below is a financial value identity between consistently defined transaction values, not a forecast that synergies will be realised. Include integration and transaction costs in the combined-value estimate or disclose them separately; never subtract them twice.
Reference relation and assumptions
- V_created: incremental value to buyer shareholders, measured at the valuation date.
- V_combined: target standalone value plus the value of combination effects on the same scope; costs included consistently.
- P_paid: purchase price on the same value basis.
- All three amounts use one currency and date; this is a value bridge, not an acquisition-accounting journal entry.
Scope, limitations and source
Use a common scope, date and currency. Combined value here is target standalone value plus combination effects, not the whole acquirer’s enterprise value; count costs once.
William J. Gole & Paul J. Hilger, Due Diligence: An M&A Value Creation Approach (2009), ch.1, printed pp.17–20 / PDF41–44; Exhibit1.4, p.18 / PDF42.
LIRE UNE MÉTHODE DE VALORISATION
Un modèle est une chaîne d’hypothèses, pas un chiffre isolé.
Ce schéma décrit le raisonnement des démonstrateurs interactifs. Il ne décrit pas un processus de valorisation de production.
Profil économique
Préciser la question contractuelle et la convention de paiement.
Paramètres
Expliciter courbes, volatilités, dates et hypothèses de scénario.
Mécanisme
Choisir une représentation de l’incertitude et des conventions de marché.
Résultat illustratif
Interpréter la sensibilité et la dépendance aux scénarios, pas seulement un résultat unique.
Frontière de décision
Identifier ce que le démonstrateur ne teste ni ne valide.
Portfolio: complement the model with decision 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.
Application to the workflow
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.
DECISION FRAMEWORK
Risk preferences are not pricing probabilities
Reference-dependent choice evaluates gains and losses relative to a stated benchmark. Loss aversion and probability weighting describe preferences; they do not change the contractual payoff or supply risk-neutral probabilities. A portfolio discussion should therefore identify the reference point, separate financial loss from subjective value, and test how conclusions change under alternative preferences. Parameters taken from a textbook illustration are not estimates for a client, a bank or the market. Broad portfolio framing should be distinguished from evaluating each position in isolation.
Scope, limitations and source
Preference parameters are illustrative, not estimated client traits or pricing probabilities. Loss aversion alone is not complete cumulative prospect theory.
Edward Cartwright, Behavioral Economics, 3rd ed. (2018), §3.3–3.3.2, printed pp.115–120 / PDF146–151; §2.2.5–2.2.7, pp.52–56 / PDF83–87.
Behavioural finance: pricers and decisions
Jump pricing
Portfolio
Explore the entered carbon pathway
Portfolio carbon pathway
Review discrete CDS cash-flow conventions
Credit cash-flow conventions
MBS: cash flows and valuation
Deterministic mortgage and passthrough scenarios: CPR, PSA, IO/PO, principal life, static spread and fixed-flow duration. No calibrated OAS or market feed.
Review MBS assumptions and controls →
ALM
Equities
Quanto
Fractal
MODEL INTERPRETATION
Time averages need a justified probabilistic setting
Stationarity describes invariance of specified distributional properties over time; ergodic reasoning connects long-run time averages with population expectations under additional conditions. Neither follows automatically from converting prices to returns. Before using a historical average as a prospective trading input, specify the process, sampling interval, dependence structure and window over which the assumptions are plausible. A test can challenge a chosen model but cannot establish that every future regime follows it. Keep physical statistical forecasts distinct from the risk-neutral expectations used for derivative valuation.
Decision, data and controls
State the process and probability measure; inspect window and dependence sensitivity; do not infer guaranteed trading profit from a stationary-process theorem.
Scope, limitations and source
Companion synthesis across the two books; measure distinction is explanatory editorial guidance, not an additional theorem claimed from Wang/Zheng. Do not reproduce the chapter6 heading “Logarithmic Return is Stationary” as a universal empirical fact or the preface’s profitability claims.
Wang & Zheng (2015), §5.2.2–5.2.3 printed pp93–98 / PDF108–113; Wooldridge (7th ed., 2020), §11-1 printed pp367–370 / PDF390–393.