Mathematics & methods

What is a claim worth today, or at an earlier portfolio horizon?

H is the valuation horizon; CF is a future contractual payment value (including a valued delivery leg where applicable); D discounts from its payment date to H. F_H means only information available at H. Q is the selected pricing measure.

Price the remaining claim using its state at H. Already paid cash belongs to the ledger. A realised payoff later on a physical path cannot replace this conditional value. Physical expectations and probabilities use P; a client probability view does not, by itself, change today’s Q quote.

Analytic vanilla special case

For a European call with constant volatility, constant rate and no dividends, the Black–Scholes reference is:

K is strike, T time to expiry and Φ the standard normal CDF. Dividends, exercise style and currency conventions must select the matching qualified adapter; this special case does not cover them automatically.

Algorithm and inputs

1. Resolve contract state, observations, payment dates, curves, dividends, volatility and the pricing convention.
2. Use an analytic vanilla adapter where qualified; otherwise use the supported payoff evaluator and a declared Monte Carlo continuation method.
3. Return value, currency, coverage and numerical precision with the resolved context.

The prototype has QuantLib analytic vanilla controls. The full Monte Carlo research spike exercises conditional continuation for its declared cases. The general shared pricing service is proposed. These prototype values contain no issuer-default model.

Method foundation: Black & Scholes (1973). Existing control: engine.py · vanilla_check. Contract: P/Q compatibility and residual value.

How does the prototype turn a volatility smile into simulated pricing paths?

S is the native asset price, r the pricing rate, q the continuous dividend yield and σ_loc a time/price-dependent volatility. Correlated Brownian increments link the assets.

1. Build a synthetic strike smile from ATM, skew and curvature inputs, with implied volatility floored at 12%.
2. Pass the calibration set to QuantLib’s AndreasenHugeVolatilityInterpl and AndreasenHugeLocalVolAdapter.
3. Sample a local-volatility grid, clip it to 5%–200%, then simulate correlated log-Euler steps with antithetic pairs.
4. Compare selected vanilla prices with independent analytic controls.

This is the method actually called in the pinned prototype. It uses 252 steps per year and 21 steps per “month”, synthetic smiles and flat curves. These choices are prototype conventions, not actual contractual calendars.

The interpolation paper does not establish that the clipped grid, discretised paths, correlations or synthetic inputs are arbitrage-free or market-calibrated. Repricing and convergence need their own qualification.

Andreasen & Huge, Volatility Interpolation; smile, calibrate_local_vol, local_vol_paths.

Which events produce coupons, early redemption or physical conversion—and what coupon meets a new issue’s value target?

The selected prototype BRC pays monthly coupons while alive, checks autocall from month two, and converts only if it survives, has the selected knock-in event and finishes below strike. In that conversion branch, redemption as a fraction of notional is terminal worst performance divided by strike fraction. This is one contract preset; FCN, physical accumulation and net-cash variants have separate rules.

Algorithm and assumptions

Process observations in contract order, retain prior knock-in/call state, emit dated gross cash and asset legs, then distinguish termination from payment completion. A coupon solve needs a positive annuity, fixed remaining terms, explicit fee/value target and valid domain. Root searches must disclose bounds, residuals and incomplete evaluations.

Coupon-linear solving and BRC payoff code exist. Corrected held-state and settlement cases have bounded independent proofs. Their generic dated evaluator is proposed. A new-issue solve must never reset the factual coupon on a held note.

Contract rules—not a paper—define these events: Products and lifecycle specification. Pinned KoLeg and brc_cell; independent product cases T-AC-203/204/205/218/219.

How can a stock, a note on that stock, cash and FX share one coherent future?

Generate one joint state process on the union of required observation, valuation, cash and view-event dates. Each factor has declared units and a transform: native total return, FX or a qualified yield state. Direct holdings and contract underlyings reuse the same state.

Inputs include ordered factors, dependence, physical drifts, distribution rules and random-stream identity. Validate covariance and exact dependencies; a material negative eigenvalue is a failure, not a silent matrix repair. Missing positions cannot be removed to make a complete-book result.

Prototype joint simulation and narrower research cases exist. The proposed Gaussian-factor and supplied-weighted-path profiles generalise their date, state and coverage contracts. Current v2 placeholder means do not become calibrated forward expectations.

RealWorld, RealWorldUniverse; RULE-103. The displayed process is the selected model definition, not a claim that real returns are Gaussian.

What changes if an event’s probability moves from a baseline p₀ to a client-specified q?

pᵢ are normalised prior scenario probabilities; p₀ is the prior probability of the event. The displayed case requires 0 < p₀ < 1. A requested event cannot acquire support from paths that do not exist.

This is the single-event minimum-relative-entropy solution: change event mass while preserving relative probabilities within each group. A neutral view leaves the distribution exactly unchanged.

For an event after the risk horizon, retain the full joint event information when weighting earlier P outcomes. The earlier Q continuation still conditions only on the state known at that earlier horizon. A future realised event label cannot choose an earlier option payoff.

The binary special case and an eight-path later-event/earlier-horizon oracle were exercised. A general multi-view optimiser is not claimed. Views do not alter bank thresholds or contractual terms.

Meucci (2008) supplies the general entropy-pooling framework; this closed case is narrower. RULE-104 and T-AC-102.

How much might the complete portfolio lose, including losses beyond a percentile?

Sum the portfolio per scenario first; then compute its loss distribution. For return loss, L = −R. Use the inverse empirical CDF and include only the needed fraction of probability at the tail boundary. Averaging component VaRs or just taking all observations beyond a threshold gives a different result.

One explicit four-atom acceptance example gives mean return 10.4%, VaR₉₅ 10% and ES₉₅ 22%. These are hand-checkable test values, not a portfolio forecast or default tail setting.

The prepared-context research code uses common weighted population statistics. Older prototype estimators differ. The proposed service standardises this convention with an explicitly supplied α, horizon, loss basis and denominator.

Outcome volatility is not sampling error. Weighted or nested simulation needs a qualified uncertainty method; neither effective sample size nor σ/√N is a universal confidence interval.

Rockafellar & Uryasev, general loss distributions, including discrete tails. RULE-109 and T-AC-105.

Is there enough usable cash, and how does a funded action change wealth?

Each signed component is counted once; pending net value includes receivables and payables. Funding a put subtracts premium and costs from cash and adds the option, without inventing another stock position. Income, sales proceeds, principal repayment and external contributions have different roles.

The ledger records fixing, entitlement and settlement dates separately. Reinvestment changes quantities and cash together. A retained-dividend hand path finishes at wealth 120 / cash 15; its reinvested counterpart finishes at wealth 121 / cash 0. Adding distributions to already reinvested wealth would falsely give 136.

Flow-neutral return requires valid values immediately before and after external flows and positive unitisation bases. Without them, that return is unavailable while valid cash and wealth figures remain visible. Paying an already recognised liability is not a new external investment flow.

Exact small ledgers and period-target counterexamples were executed. A generic analytical ledger is proposed; it does not replace the bank’s operational ledger. Joint success cannot be inferred from marginal monthly probabilities.

Product accounting rules; T-AC-206–214. The conservation identities and contract cases are the references here.

How could compatible benchmark expectations become instrument forecasts?

Under the proposed zero-forward-alpha profile, αᵢ = 0. B is fitted jointly in a declared independent benchmark basis; λ contains compatible excess-return expectations. Residual covariance remains part of risk.

1. Resolve return type, currency, horizon and cash reference. Arithmetic means and compounded growth targets are not interchangeable.
2. Validate declared aliases/composites and the target system’s rank and consistency.
3. Use the selected reduced joint OLS or constrained prediction-space shrinkage method; require an explicit completion policy for unidentified expectations.

The factor reconciliation experiment was executed. The new forecast policy is proposed and not activated by the current v2 data. Summing pairwise betas, treating a pseudoinverse as identification, or silently choosing a shrinkage penalty is outside this design.

RULE-102 and T-AC-101. This equation is the proposed model contract; it is not an empirical finding that future alpha is zero.

How do coupons, remaining value and yield changes fit together without counting risk twice?

Use actual remaining contractual flows and the selected discount/spread state. Settled coupons live in cash, and pending entitlements retain their payment status. A clean market quote and a dirty continuation value require an explicit accrued-interest convention.

The profile must say whether its base discount curve already includes the bond spread. Add a baseline spread only when the curve excludes it. If price is revalued under yield/spread states, do not add a second native bond total-return shock.

A two-bond reference checks cash/value partition and local finite-difference sensitivities. It does not identify a unique physical rate/spread process from v2 return covariance. Full probability forecasts require that separately specified joint law.

This ordinary-bond formula is not a callable-bond model. Source-enriched maturity, coupon steps and call rights require a coherent new contract revision; they cannot be attached piecemeal to invented demo schedules.

RULE-107 and T-AC-104; inspect curve source roles and stored points.

Is a foreign asset converted at future FX, or paid using a fixed contractual conversion?

F is domestic currency per one foreign unit. The second equation is the selected constant-volatility reference: foreign asset drift under domestic pricing measure Q_d, with asset/FX correlation ρ.

Native, floating-FX and fixed-quanto payoffs are distinct contracts. Resolve payout currency, numeraire, quantity-fixing rule, FX fixing date and settlement date before choosing a pricer. A reporting conversion is not an FX trade or cash funding.

Deliverable FX hedges retain both signed currency legs. Option currency exposure is a qualified sensitivity; premium, notional or equity delta is not a universal substitute. The conversion happens once at the specified date.

Independent analytical/QuantLib currency cases and signed-leg controls exist for their declared conventions. They do not qualify arbitrary path-dependent quanto baskets.

Currency contract rules and T-AC-210; RULE-106 and T-AC-110. The source terms choose the currency mode.

Does the proposed portfolio meet the client’s original requirements?

Resolve the required figure and its scope, date, unit, metric basis, model coverage and precision. Only then compare unrounded values with the supplied bound. Missing or incompatible primary/supporting evidence is unknown; it cannot close an existing gap.

Shared producers supply allocation, issuer concentration, signed FX, income, risk and liquidity. Health does not implement another valuation model. Collateral composition uses one eligibility gate: unknown eligibility yields unavailable policy collateral; known ineligibility yields zero; eligible zero remains a different state. Model collateral, bank recognition and drawable credit stay separate.

Corrected deterministic contract and collateral fixtures passed their bounded controls. The shared backend/API/MCP service is proposed. Client views cannot relax lender policy or replace the original requirement set.

Health measurement contracts and support/eligibility acceptance cases. These are policy and evidence rules, not conclusions supplied by a scientific paper.