Intermediate analysis: map paper techniques to ORE asset classes

A Gaussian Mixture Model (GMM) is fitted to historical daily market data via Expectation-Maximisation. The mixture captures multiple market regimes (e.g. bull/bear, steep/inverted curve) as K Gaussian components, each with its own mean vector and covariance matrix. Generation is a two-step draw: sample a component index from the mixing weights, then sample from that component's multivariate normal. Closed-form marginals and conditionals make conditional generation tractable without MCMC. The method operates entirely under the real-world measure P: it generates plausible market snapshots but does not enforce risk-neutral no-arbitrage constraints on option surfaces.

Given a finite set of calibrated expiry-pillar smiles expressed as normal/lognormal mixture densities, this method interpolates to any intermediate expiry by linearly blending mixture weights while holding all component locations and widths frozen (the "frozen pool"). Because the interpolated call price is a convex combination of arbitrage-free pillar prices, calendar-spread arbitrage (non-decreasing total variance) and butterfly arbitrage (non-negative density) are enforced by construction at zero additional cost. The technique operates in Q-measure: it takes calibrated risk-neutral pillar smiles as input and outputs a complete, arbitrage-free vol surface. It is downstream of any P-measure generation step and requires pillars to already be in mixture form (SABR/SVI pillars need a conversion pre-step).

A pretrained Mixture Density Network (MDN) maps GJR-GARCH model parameters and maturity to a Gaussian mixture representation of the terminal return density; European option prices follow from a closed-form weighted sum of Black formulas. The surrogate is ~400,000× faster than matched-accuracy Monte Carlo and is error-certified against the Monte Carlo noise floor. Like the frozen-pool method, it operates in Q-measure and produces arbitrage-free surfaces directly from model parameters. It is an alternative route to option vol surfaces for equity, FX, and commodity underliers, bypassing the need to generate a P-measure path first and then run a separate calibration. It does not cover swaption, cap/floor, or credit vol surfaces.

What is missing: No paper provides a validated generation procedure for credit spread curves or hazard rate term structures. GMM is structurally applicable, but the paper has no credit experiment, and joint generation across many reference names becomes computationally intractable (n² covariance) without dimensionality reduction.

Interim approach: Use historical CDS spread curves from ORE example data (e.g. ExposureWithCollateral/marketdata.csv) with random date offsets or additive Gaussian noise on the log-spread vector. Alternatively, apply a two-factor Nelson-Siegel parameterisation (level + slope) fitted to the historical distribution and draw random (level, slope) pairs.

Proper solution: A sector-level GMM fitted on log-CDS-spreads per rating bucket (IG, HY, sub-IG), with a copula layer to reintroduce name-level correlation. This is a known technique outside the three papers (e.g. factor copula credit models). Credit migration matrices (RATING/TRANSITION_PROBABILITY) require a separate Markov chain model.

What is missing: Recovery rates are bounded [0, 1], sparsely updated, and the GMM paper explicitly excludes them as unsuitable. The GJR-GARCH NN and frozen pool do not cover credit parameters.

Interim approach: Static values from ORE examples (40% senior, 20% subordinated) are sufficient for most generation scenarios. A trivial perturbation is to draw uniformly from [35%, 45%] per senior name.

Proper solution: Empirical recovery distribution per seniority and rating tier. The CDS spread and recovery must be co-generated to satisfy the CDS_spread ≈ (1 − R) × hazard_rate linkage (see section 5, constraint 9).

What is missing: No paper covers CDX/iTraxx tranche base correlation generation. These are structurally correlation surfaces in (maturity, detachment) space.

Interim approach: Copy static slices from ORE example data (MarketRisk/marketdata.csv has CDX.NA.IG base correlations). Apply a small uniform random perturbation (±2%) while preserving monotonicity in the detachment dimension.

Proper solution: Derive base correlations from the generated index CDS spread using standard Gaussian copula base correlation bootstrapping. Requires a consistent underlying constituent spread curve.

What is missing: No paper covers equity dividend yield curve generation. The dividend yield determines the equity forward (section 5, constraint 2) and is a direct input to the GJR-GARCH NN forward F₀.

Interim approach: Flat dividend yield from ORE example data per equity name (EQUITY_DIVIDEND/RATE/SP5/USD/1Y at ~1.8% is a common reference). A small random offset (±0.3%) per generated scenario keeps the dividend yield plausible without a full curve model.

Proper solution: Fit a GMM on the term structure of dividend yields from historical data or equity option put-call parity stripping. This is a straightforward GMM extension but is not demonstrated in the papers.

What is missing: GMM covers ZC/YY inflation swap rates (the underlying curve); none of the three papers covers inflation cap/floor vol surfaces.

Interim approach: Static vol surface from ORE example data (Exposure/market_inflation.txt provides ZC_INFLATIONCAPFLOOR prices). Apply a proportional random scaling factor (e.g. multiply all strikes by a lognormal draw with σ = 0.1) to produce varied but plausible scenarios.

Proper solution: Apply GMM to the ZC inflation cap/floor vol matrix in the same way the GMM paper applies it to equity implied-vol surfaces. Arbitrage enforcement via frozen pool is applicable in principle but no inflation-specific mixture calibration method is available in the papers.

What is missing: GMM can generate pillar implied vols for swaptions and cap/floors. The frozen pool can interpolate between pillars. However, the frozen pool requires mixture density inputs, and ORE's swaption cube is typically calibrated with SABR or stored as a raw implied-vol grid. Converting a SABR smile (or a raw vol grid) to a normal-kernel Gaussian mixture is not addressed by any of the three papers.

Interim approach: Skip the frozen pool for swaptions in the first generation pass. Use GMM to generate ATM normal vols and a smile-parameterisation approximation (e.g. SABR α, ρ, ν from historical distribution). Write directly to SWAPTION/RATE_NVOL keys. No cross-expiry arbitrage checking is then enforced, which is acceptable for scenario generation if not for pricing calibration.

Proper solution: Fit a normal Gaussian mixture to the density ∂²C_SABR/∂K² at each pillar via EM, using the implied density as training data. This is a well-posed numerical optimisation problem but requires implementation work (3–5 mixture components per pillar is sufficient). Once done, the frozen pool applies exactly as described.

What is missing: ORE's FX_OPTION/RATE_LNVOL quote keys use delta conventions (ATM, 25RR, 25BF, 10RR, 10BF). The frozen pool and GJR-GARCH NN work in strike space. The conversion requires the generated FX forward and the domestic/foreign zero rates — which are available if generation follows the stack order in section 6 — but the conversion itself (Garman- Kohlhagen delta inversion) is not discussed in the papers.

Interim approach: Implement a standard delta-to-strike conversion using QuantLib's BlackDeltaCalculator before populating the frozen-pool pillars. This is a one-day implementation task given available IR curve and FX spot inputs.

Proper solution: The same delta-to-strike conversion as above; there is no research gap here, only an implementation task.

What is missing: No paper covers bond option vol generation. Bond option vols are related to swaption vols via approximate duration mapping.

Interim approach: Derive from the generated swaption vol surface using the ORE catalogue's suggested approach: BOND_OPTION/RATE_LNVOL ≈ SWAPTION/RATE_LNVOL at (bond option expiry, modified duration × rate sensitivity). Apply the same duration-based mapping used in ORE's sensitivity framework.

Proper solution: Joint generation with swaption vols; the mapping is an approximation that breaks down for long-dated or convex bond structures.

What is missing: Conditional prepayment rates for MBS/ABS and BalanceGuaranteedSwap products are not covered by any paper.

Interim approach: Static value from ORE example data. CPR is only required for a narrow product set; skip for the initial generation scope.

Proper solution: Econometric prepayment model depending on rate level and spread. Out of scope for this sprint.