Krylo — Solve the model once

The problem

The autocallable loss cycle is computational

Worst-of and knock-in books broke their holders three times in six years — and the failure point is the same every cycle. The industry risk-manages these products by re-simulating them, and simulation-based risk numbers are at their noisiest exactly at the loss events: second differences across a payoff discontinuity amplify Monte-Carlo noise catastrophically. On a representative book of eight step-down notes, the industry-standard estimate of correlation risk came out at −0.06 ± 0.06 — the sign is unreadable on half the book, on the day it matters most.

€260M

Natixis, Korean worst-of books, 2018

€673M

SocGen + Natixis hedging losses, COVID quarter 2020

$4.6B

HSCEI knock-in wave, 2024 — plus $1.2B compensation reserves

$538B

callable / autocallable issuance in 2025 — a record, increasingly held outside dealers

How it works

The semigroup, computed — not approximated around

Monte-Carlo samples the transition operator one random path at a time. Finite differences re-step through it per product and per scenario. Neither ever constructs the reusable object itself. Krylo does — three components matter.

Analytic kernel

The short-time transition kernel is evaluated in closed form and moment-corrected to sixth order in space — with the correlation cross-term handled exactly, the term that forces operator-splitting compromises in ADI schemes. A one-year horizon is a handful of large, high-order steps.

Compression

The solved operator is projected to a small subspace where any horizon is a small-matrix computation: applying thirty years costs the same as applying one day, and a whole maturity surface comes from one build.

Products as projections

Autocall dates, memory coupons, and discrete knock-ins apply between marches as exact projections — discrete monitoring priced with no continuity correction. Run the same operator in reverse and it produces counterparty-exposure profiles with no nested simulation.

Krylo architecture: the model is solved once per underlier set; products, risk, scenarios, maturities and exposure are reads of the solved object. — **The engine in one picture.** The model is solved once per underlier set; the next product on the same underliers is a projection, not a re-solve.

The operator library: a lattice of solved models interpolated across market state, repricing the book intraday without a rebuild. — **The operator library.** A solved model is a reusable object: store a lattice of them, interpolate across market state intraday. One tangent step — “yesterday’s solution + a derivative” — reprices the book across an overnight move, no rebuild. No incumbent tool has this primitive.

Evidence

Exact where simulation is noisiest — and hundreds of times faster

Every number here regenerates from a provenance-stamped, gated benchmark: one command rebuilds the full evidence base, a numeric pass gate lives inside each benchmark, and a regression flips the report on its own. The comparisons run against closed forms, QuantLib, and 10M-path Monte-Carlo anchors — nulls published as prominently as the wins.

faster than full-path Monte-Carlo at T = 30y — unbiased, cost flat in maturity

converged price of a real step-down term sheet at the coarsest production grid

/15

independently gated benchmarks vs closed forms, QuantLib, and 10M-path Monte-Carlo anchors

numerical failures across a 26-rung certification campaign on deployment-class hardware

Simulation cost grows with maturity; the solved-operator cost is flat — 306× at thirty years, without bias. — **Cost flat in maturity.** Simulation pays per path, per year — 306× slower at T = 30. One solved operator applies thirty years for the cost of one day, exactly and without bias; a whole maturity surface comes from one build.

Through the breach. Correlation risk of a real step-down note (memory coupons, discrete knock-in, local vol + correlation + jumps) as the market slides toward the knock-in. Blue: the exact profile, read from one solved model. Red: re-simulation with nudged inputs, with its measured noise. The independent cross-check agrees: 4.68 ± 0.17 vs 4.78 at the knock-in.

Deterministic is not converged: ±3bp grid-dependent error removed by sixth-order contract loading. — **Deterministic ≠ converged.** The same term sheet on three production grids: standard contract loading carries ±3 bp that changes sign with the grid. With sixth-order loading: 0.04 bp at the coarsest grid — a two-second march.

Exposure without the farm. The solved model run forward, paired with its backward values: EPE/PFE with the autocall runoff staircase, ≤ 0.12% of brute-force nested simulation at every date — in about four minutes on one GPU.

The living calibration

Calibration as state estimation, not snapshot fitting

Because the value flow of the pricing model is a linear semigroup, streaming quotes can be assimilated into the model’s coefficients continuously, with calibrated uncertainty — and every innovation decomposed into noise, drift, or an attributed shock. Validated on live exchange options: exactly silent through quiet weeks, 38 named and sized events through a real volatility episode. Every commercial surface product either fits snapshots or reprices fast; none tracks a model-consistent trajectory, carries calibrated uncertainty, or attributes its misfit.

Attributed shock detection on real exchange volatility data: silent on quiet weeks, named and sized events through a real episode. — **Attributed shocks on real market data.** The assimilation channel through a real volatility episode: discrete repricings detected, named, and sized — with the quiet-week null reported just as prominently.

Validation

A falsification program, with the nulls published

Claims about a pricing method should be cheap to check and expensive to fake. The comparisons run against closed forms, QuantLib, and converged Monte-Carlo — and the null results are published as prominently as the wins.

Claim	Reference	Result
Real step-down term sheet (memory, discrete KI)	identical-logic Monte-Carlo	within MC noise (±0.02%)
Correlation risk through the breach	matched-seed MC bump	4.78 vs 4.68 ± 0.17
2-asset worst-of	Stulz (1982) closed form	10⁻⁸
Whole implied-vol surface to 30y	Richardson-extrapolated fine CN	0.006 bp mean · 1.6 s
Counterparty exposure profile	brute-force nested simulation	≤ 0.12% at every date
Bermudan exercise	QuantLib finite differences	< 0.5%
vs Craig–Sneyd ADI / sparse CN, same grid	Stulz closed form	~230× (price and corr. risk)

Published nulls, for the avoidance of doubt: in 1D on raw payoffs, tridiagonal Crank–Nicolson wins wall-clock at equal accuracy; and simulation deltas are fine — we do not sell better deltas. The value is in the numbers simulation cannot read.

Engagement

A four-to-six week pilot, on your book

Second-opinion risk engine — beside your stack, nothing ripped out.

What we run

5–10 of your term sheets — public or synthetic equivalents acceptable, no MNPI required
Daily marks, breach risk profiles, the full per-note risk set
Optionally: counterparty-exposure profiles
On our hardware or a GPU box in your environment

Success criteria, agreed up front

Prices inside your own simulation noise
Risk numbers stable through knock-in scenarios
Full-book pack inside the daily window
The business test: the pack is in your morning risk meeting by week three

andrew@kryloquant.com White paper (PDF)

Fixed fee · week-two review · either side stops.