Schrödinger bridges for tail-risk scenarios
SPY return paths under stochastic volatility and jumps.
7
generative models benchmarked
3–13×
baseline error on jumps
5,068
daily windows
MathsFI Lab
Generative models
with Prof. Huyên Pham
IMC Prosperity
Top 1.5%
280 / 18,803 teams
QRT Data Challenge
#1
public leaderboard
Publications
3 journals
incl. first author, NST 2024
01 / Lab
See, don't read.
click: informed flow · drag: rotate
Hawkes flow & adverse selection
\[\lambda^{\pm}_t=\mu+\mu^{I,\pm}_t+\!\sum_{t_i\lt t}\!\alpha\,e^{-\beta(t-t_i)}\]\[\delta_t=\gamma\sigma_t^2+\tfrac{2}{\gamma}\ln\!\big(1+\tfrac{\gamma}{k}\big)+\psi\,\mathrm{VPIN}_t\]
spread (ticks)
—
VPIN
—
spread capture
—
adverse selection
—
market-maker P&L, ticks · 12-tick markouts
drag: rotate
Multi-venue execution & impact
\[I_t=A\sum_{s\le t}q_s\,(1+\tfrac{t-s}{\tau_0})^{-\beta}\]\[f_i+\tfrac32\,c_i\sqrt{x_i/V_i}=\mu\]
shortfall
—
fees + slippage
—
peak impact
—
residual impact
—
square-root law · ○ \(Y\sigma\sqrt{Q/V}\) ● simulated peak
drag: rotate · hover: smile
σ 5% → 65%
SSVI volatility surface
\[w=\tfrac{\theta_T}{2}\Big(1+\rho\varphi k+\sqrt{(\varphi k+\rho)^2+1-\rho^2}\Big)\]
ATM 1M
—
ATM 2Y
—
1Y skew \(\partial\sigma/\partial k\)
—
3M 80% put − ATM
—
pathsfitted uₙexact
Deep backward scheme · 100-d HJB
\[Y_n=\mathbb{E}\big[\,Y_{n+1}-\lambda\lVert\nabla u_{n+1}\rVert^2\Delta t\;\big|\;X_n\big]\]
scheme \(Y_0\)
—
exact \(u(0,0)\)
—
rel. error
—
runtime
—
noise → data
particlestarget
Schrödinger bridge
\[X_t=(1-t)\,x_0+t\,x_1+\sqrt{\varepsilon}\,(W_t-tW_1)\]
Sinkhorn iters
—
\(\lVert\pi\mathbf{1}-a\rVert_1\)
—
\(\mathbb{E}\lVert X_1-X_0\rVert^2\)
—
\(\varepsilon\)
—
IPF convergence · \(\log_{10}\) marginal error
highlighted marketnoise flow
Kyle–Back equilibrium
\[dP_t=\lambda\,(dX_t+dZ_t),\quad \lambda=\frac{\sigma_v}{\sigma_u}\]
impact \(\lambda\)
—
\(\mathbb{E}[\pi]=\sigma_v\sigma_u\)
—
price vol
—
info by \(t=\tfrac12\)
—
02 / Work
Selected work
Cross-sectional alpha, CSI 300
Nonlinear temporal features, point-in-time OOS, net of costs.
.031→.038
rank IC
.78→.91
net IR
Probabilistic traffic forecasting
Forecasts tied to SLA cost; root-cause search on a network digital twin.
1.68→0.48
MASE
4.43→1.28
CRPS
13.8→1.6%
delay error
Competitions
Algorithmic trading, allocation forecasting, mathematical modelling.
Top 1.5%
IMC Prosperity
#1
QRT public LB
0.17%
CSIAM national 2nd prize
Queue-reactive order book simulator
Huang–Lehalle–Rosenbaum: queue-size-dependent limit, cancel and market order intensities.
HLR
queue-reactive model
Surrogates for inverse problems
SVD-autoencoder latent state, kNN surrogate, coarse-to-fine search.
<0.1 s
per inference
~0.3%
L2 error
3
journal papers
03 / Principles
How I see the world
\[p(\theta \mid \mathcal{D}) \;\propto\; p(\mathcal{D} \mid \theta)\, p(\theta)\]
Uncertainty first
“Probability theory is nothing but common sense reduced to calculation.”Pierre-Simon Laplace, 1819
Never trust a single deterministic point when the data is noisy. Keep a posterior, and update it as evidence arrives. That discipline is behind my forecasts in the QRT Data Challenge, ranked 1st on the public leaderboard.
\[dX_t = b(t, X_t)\,dt + \sigma(t, X_t)\,dW_t + dJ_t\]
Stochastic processes & jumps
“The market, unbeknownst to itself, obeys a law which governs it: the law of probability.”Louis Bachelier, Théorie de la spéculation, 1900
Prices are not smooth deterministic curves but stochastic paths: Brownian diffusion plus sudden jumps. With Prof. Huyên Pham I use Schrödinger bridges, an entropic stochastic-control problem, to generate SPY tail-risk scenarios under stochastic volatility and jumps.
\[\partial_t v(t,x) + \sup_{u}\big\{\mathcal{L}^u v(t,x) + f(t,x,u)\big\} = 0\]
Dynamic optimization
“An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.”Richard Bellman, 1957
Executing and splitting an order is not a static fit but a Bellman problem: trading off transient market impact against inventory risk, step by step, over a multi-dimensional state.
\[I_t = Y \sigma \sqrt{\frac{Q}{V}}\]
Market friction & net alpha
“Noise trading provides camouflage which conceals his trading from market makers.”Albert S. Kyle, Continuous Auctions and Insider Trading, 1985
Every trade leaves a footprint. Impact grows like the square root of size (the square-root law, Tóth, Bouchaud et al.), so alpha measured before friction is an illusion; net-of-costs performance is the only criterion that survives.
04 / Publications
Peer-reviewed
- 2025Decision tree based parameter identification and state estimation: application to Reactor Operation Digital TwinHong, L. et al. · Nuclear Engineering and Technology
- 2024Optimizing near-carbon-free nuclear energy systems: advances in reactor operation digital twin through hybrid machine learning algorithmsHong, L. et al. (first author) · Nuclear Science and Techniques
- 2024A noise and vibration tolerant ResNet for field reconstruction with sparse sensorsHong, L. et al. · Communications in Computational Physics
05 / Stack
Toolkit
codePythonPyTorchNumPyPandasscikit-learnSQLJavaMATLABGitLinux
mathSDEsStochastic calculusMonte CarloRare eventsOptimal transportBayesian inferenceNumerical optimisation
mlGenerative modelsDeep learningTime-series forecastingLLM agents
quantAlpha researchFactor modelsBacktestingOOS validationMicrostructurePortfolio optimisationCovariance estimation
lang中文EnglishFrançais
06 / About
Desk and iron.
SJTU computer science → École Polytechnique applied mathematics. Head of Quant at X-Finance.
2023 Shanghai intercollegiate bodybuilding champion. Rugby, American football, Hyrox.
Writing →