First-year coursework: Linear Algebra, Discrete Mathematics, Data Structures and Algorithms, Introduction to Programming, Analysis I, Algorithms and Probability, Parallel Programming, Digital Design and Computer Architecture.
Turkish (native), German (C2), English (C1)
Each project opens into a full write-up: the mathematics, live simulations, and the engineering behind it.
I built a corpus of about 1.8 million fitted sine networks and an evaluation ladder for models that read their weights, then proved the complete function-preserving symmetry group, D∞ ≀ Sn per layer. That group alone costs 79.1 of 80.4 accuracy points; the exactly invariant reader I designed recovers 0.917 of them and ships as the phasorkit library.∎
I designed a framework-agnostic forecasting API, with an exact backend and an MLX backend for pretrained models, that predicts the across-seed spread of RL post-training from one stored run through a covariance recursion and an adjoint pass: 1.11x median error over 192 prospective settings, with its failures on Qwen2.5 measured and published.
I proved that the confounding gap between estimated and structural cross-impact has rank at most K + rank(B) and that its cost error is a low-rank quadratic form, then built a sharded known-truth simulator, 107 observations in 100 frozen shards with byte-for-byte replay, that verifies every derivation: an index basket is mispriced by 54.23%, a dollar-neutral one by nothing.∎
I engineered a validated ingestion pipeline for the OCC's daily clearing reports, producing a 13-month panel of the unlisted FLEX market (9.06m series-days, $596bn of mark value), with leg grouping that corrects 49x notional errors. A placebo-tested fixed-effects regression shows the hidden inventory does not predict next-day volatility.
I built a known-truth Monte Carlo harness, in Python with a Rust indicator library, that measures the real size of forecast-comparison tests: at a nominal 5%, Diebold-Mariano on panel rows rejects a true null 79.13% of the time on 100 correlated stocks. I derived the closed form, rebuilt the apparatus, and made the whole run replay byte for byte.∎
I built a leak-proof walk-forward evaluation, with purging, embargo and in-fold scaling enforced by tests, and used it to test 18 machine-learning settings for BTC, ETH and SOL with the correct test for nested models. 5 reject no-predictability, 2 survive false-discovery control, none survive family-wise control or costs.
Eight theorems about random walks and Brownian motion, each simulated in the browser until it happens: Donsker's invariance principle, the arcsine law, the law of the iterated logarithm, optional stopping, quadratic variation, Itô against Stratonovich, Girsanov's change of measure, and Black-Scholes as a Feynman-Kac expectation. The same processes run quietly through this page.
TÜBİTAK National Informatics Olympiad, 2025. Qualified as one of 511 from 18,463 students.
TÜBİTAK National Mathematics Olympiad, 2024. Advanced to the proof-based national round.
Parameter symmetry and weight-space learning on sine-network implicit neural representations
Start here. A real sine network, one input, two hidden layers of four neurons, one output, evaluated in your browser. Every button changes the parameters; none changes the function on the right. This page proves those are the only such changes, and measures what they cost a model that reads weights.
Abstract. An implicit neural representation stores a signal as the weights of a small network fitted to it. Classifiers that read those weights work when every network starts from a shared initialization and collapse when each starts from its own: on MNIST, 80.4 accuracy points separate the two, with the same images, architecture and reader. I characterise the complete function-preserving symmetry group of sine networks, show by intervention that it alone reproduces 79.1 of the 80.4 points, and build a reader that is exactly invariant to it.
Each image is stored as a SIREN with two hidden layers of width 32, fitted to map a pixel coordinate to its intensity:
\[ \Phi_\theta(x) = W_3 \sin\!\big(W_2 \sin(W_1 x + b_1) + b_2\big) + b_3, \qquad \theta = (W_\ell, b_\ell)_{\ell=1}^{3}. \tag{1} \]I fitted about 1.8 million such networks across MNIST, FashionMNIST and CIFAR-10, on one laptop, under four protocols that differ in exactly one source of variation. A corpus is admitted only if a classifier trained on renders of its networks matches one trained on real pixels, so no result can come from information lost in fitting.
This is not a picture. It is a SIREN with two hidden layers of 80 neurons, fitted to the text of my name and evaluated pixel by pixel in your browser, assembled here one neuron at a time. Now apply a random element of the group \(G\) from Section 2 to all of its parameters.
An INR you can inspect. Thousands of parameters change and not one pixel does, up to floating-point rounding: Theorem 1 says that for a generic network this is the only way it can happen.
Two identities of the sine do all the work:
\[ \sin(-z) = -\sin z, \qquad \sin(z + \pi) = -\sin z. \tag{2} \]For neuron \(i\) of a hidden layer, with incoming row \(w_i\), bias \(b_i\) and outgoing column \(v_i\), they give two function-preserving moves:
\[ \sigma_i : (w_i, b_i, v_i) \mapsto (-w_i, -b_i, -v_i), \qquad \tau_i : (b_i, v_i) \mapsto (b_i + \pi, -v_i). \tag{3} \]Since \(\sigma_i^2 = 1\) and \(\sigma_i \tau_i \sigma_i = \tau_i^{-1}\), each neuron carries a copy of the infinite dihedral group, and reordering the neurons of a layer adds a wreath product:
\[ \langle \sigma_i, \tau_i \rangle \cong D_\infty, \qquad G = \prod_{\ell} D_\infty \wr S_{n_\ell} = \prod_{\ell} D_\infty^{\,n_\ell} \rtimes S_{n_\ell}. \tag{4} \]Theorem 1 (completeness)
For generic parameters of a sine network with one or two hidden layers,
\[ \Phi_\theta = \Phi_{\theta'} \iff \theta' \in G \cdot \theta. \]No function-preserving change exists outside \(G\). Proof in the manuscript; the depth-two case is also checked numerically in the repository.∎
Write \(a_r\) for the test accuracy of rung \(r\) of the ladder in Figure 1. The share of the gap a rung recovers is
\[ f(r) = \frac{a_r - a_{\mathrm{W3}}}{a_{\mathrm{W1}} - a_{\mathrm{W3}}}, \tag{5} \]with \(a_{\mathrm{W1}} = 94.36\) for weights from a shared initialization and \(a_{\mathrm{W3}} = 13.92\) from independent ones.
The causal question is answered by intervention: applying a random element of \(G\) to every network of the shared corpus changes no function, yet costs 79.1 of the 80.4 points. A factorial design over the four kinds of change \(N = \{\text{reorder}, \text{sign}, \pi, 2\pi\}\) splits that loss by Shapley value,
\[ \phi_j = \sum_{S \subseteq N \setminus \{j\}} \frac{|S|!\,\big(|N| - |S| - 1\big)!}{|N|!}\,\big[v(S \cup \{j\}) - v(S)\big], \tag{6} \]where \(v(S)\) is the accuracy lost when only the changes in \(S\) are applied. A scramble that does change the function costs only 0.7 to 2.0 points, so the damage comes from the group and not from reordering as such.
Encoding each bias on the circle removes the infinite part of the group:
\[ b \mapsto z = (\cos b, \sin b) \in S^1, \qquad \tau : z \mapsto -z, \qquad \sigma : z \mapsto \bar z. \tag{7} \]The \(2\pi\) winding acts trivially on \(z\), so on the encoded parameters the group acts through a finite group of sign changes, and a reader built on sign-invariant combinations is exactly invariant on the raw parameters. At the same parameter count it recovers 0.917 of the gap, against 0.628 for the best alignment, and scores 93.51, 74.81 and 44.20 on the published INR benchmarks: third, third and first among readers that use weights alone.
Not part of the repository. How the same problem reads through stochastic calculus, pricing theory and learning theory.
Near a minimum, stochastic gradient descent behaves like Langevin dynamics, whose stationary law is the Gibbs measure \(\pi\). Because the loss is invariant, \(L(g\theta) = L(\theta)\) for every \(g \in G\), and the group acts by volume-preserving maps, \(\pi\) is invariant too: \(\pi(g\theta) = \pi(\theta)\). Independent runs therefore cannot prefer one member of an orbit over another. That is the random-initialization corpus, seen as a diffusion.
The Reynolds operator turns any reader into an invariant one, but only when the group is finite. The phasor encoding is what makes the infinite group \(G\) act through a finite group \(G^{\prime}\) of sign changes, so exact invariance becomes a finite computation rather than an approximation.
By Bochner's theorem every shift-invariant kernel is the expectation of a cosine, so a first layer \(\sin(w^{\top}x + b)\) with random \(w\) is a random Fourier feature map (Rahimi and Recht). A SIREN learns its Fourier basis, and the \(\pi\) shift in the symmetry group is the phase ambiguity of that basis.
import torch import phasorkit as pk def build_corpus(n: int = 256, seed: int = 0) -> pk.INRCorpus: arch = pk.Architecture(dims=[2, 32, 32, 1]) g = torch.Generator().manual_seed(seed) weights = torch.randn(n, arch.n_params, generator=g) * 0.3 labels = torch.randint(0, 2, (n,), generator=g) weights[labels == 1] *= 2.0 return pk.INRCorpus.from_flat(weights, arch, labels=labels) report = pk.compare(build_corpus(), seed=0) # probe vs weight reader, matched FLOPs
compare() trains a weight reader and a function probe at matched inference cost; certify() audits any model's invariance on sampled group elements and returns inconclusive, not guilty, when float32 corrupts the phase shift \(b + \pi j\).Group theory, weight-space learning, invariant architectures, Shapley decomposition, factorial experiments, preregistration, PyTorch.
Forecasting the across-seed spread of reinforcement learning post-training from a single stored run
The idea in one picture. Forty runs of noisy gradient descent on a quadratic, differing only in their noise, simulated in your browser. The band is ±2 standard deviations predicted by recursion (2) below, computed without looking at the runs. Remove the learning signal and the contraction disappears. An illustration of the mechanism, not the paper's data.
Abstract. A score reported from RL post-training is the output of a stochastic procedure, and the only instrument the field has for its spread is to train again with new seeds. Gradient noise injected at one update is filtered by the updates that follow rather than accumulated, so the covariance of a run about its mean obeys a linear recursion. Carrying the gradient of the reported metric backwards along one stored trajectory turns that recursion into an error bar. Tested prospectively it holds to 1.11x over 192 settings; on pretrained models it fails, and I report by how much.
With \(P\) prompts per update, step size \(\eta\) and a group-based advantage estimator, the parameters follow a noisy gradient step, and linearizing around the mean trajectory gives the covariance of a run about its mean:
\[ \theta_{t+1} = \theta_t + \eta\,\hat g_t, \qquad \mathbb{E}\,\hat g_t = g(\theta_t), \qquad \operatorname{Cov}\hat g_t = \Sigma_t / P, \tag{1} \] \[ S_{t+1} = A_t S_t A_t^{\top} + \frac{\eta^2}{P}\,\Sigma_t, \qquad A_t = I + \eta J_t, \qquad J_t = \nabla g(\theta_t). \tag{2} \]When the eigenvalues of \(A_t\) sit inside the unit disc, noise injected early is filtered out by the updates that follow. Against exact Monte Carlo over twelve settings the recursion is accurate to 1.09x at the median.
Unrolling (2) and projecting onto the reported metric \(M\) gives the across-seed variance as a sum of scalars, computed by a backward pass along the one stored run:
\[ \operatorname{Var} M(\theta_T) \approx \sum_{t=0}^{T-1} \frac{\eta^2}{P}\; b_{t+1}^{\top} \Sigma_t\, b_{t+1}, \qquad b_T = \nabla M(\theta_T), \qquad b_t = A_t^{\top} b_{t+1}. \tag{3} \]For count-based estimators the mean update is the gradient of a scalar potential, so the Jacobian is symmetric and each backward step is a Hessian-vector product; the measured asymmetry is 3e-10.
\[ g(\theta) = \nabla_\theta\, \mathbb{E}_x\!\left[\Lambda\big(p_x(\theta)\big)\right] \quad\Longrightarrow\quad J_t = J_t^{\top}. \tag{4} \]An exact finite-\(G\) covariance for the advantage estimators in use shows they differ only by a weight on prompt difficulty, and fixes the optimal group size at \(G^{*} = 1 + \sqrt{\tau_w / \tau_b}\).
One run was trained and frozen, and the forecast recorded, before the remaining seeds were trained. Over 192 settings spanning a 47x range of spreads the median absolute error is 1.11x, the worst case 1.58x, and the forecast lands inside the measured 95% interval in 162 of them. Across nine settings the log-log slope of divergence runs from −1.15 to +0.26, where a random walk needs +1.
On pretrained models the forecast is short: by 25x on Qwen2.5-0.5B under plain ascent and by 3x to 17x on Qwen2.5-7B under Adam, because runs end in groups rather than scattered around one. An earlier route via \(\sqrt{\mathrm{KL}}\), with a fitted exponent of +0.10 against a predicted +0.5, is marked withdrawn rather than deleted.
Not part of the repository. How the same problem reads through stochastic calculus, pricing theory and learning theory.
Near an optimum with Hessian \(H\), the deviation of a run from the mean run is an Ornstein-Uhlenbeck process, and its stationary covariance solves the continuous Lyapunov equation. Recursion (2) is the exact discrete version, with \(A = I - \eta H \approx e^{-\eta H}\). The fan at the top of this page is a discretized OU process; switching off the signal sets \(H = 0\) and leaves a Brownian motion, whose variance grows linearly: the coin-flip blow-up.
The variance of a stochastic integral is the integral of its squared integrand. Equation (3) is its discrete form: a sum over updates of the injected noise, projected on \(b_s\), how much the reported metric still depends on that update. Each term is computable along the one stored run.
The RLOO estimator subtracts a leave-one-out baseline: a control variate that leaves the mean gradient unchanged and shrinks \(\Sigma_t\) in recursion (2). The optimal group size \(G^{*}\) trades that variance reduction against the cost of extra rollouts.
The forecast is a protocol, not a training framework: any trainer that answers two questions about each stored update can be measured. Two backends ship, one exact for enumerable policies and one for pretrained models through MLX on Apple Silicon, using common random numbers so finite differences are directional derivatives rather than differences of noise. Every run leaves a manifest of configuration, commit and platform, and no number in the paper is transcribed by hand.
from caliper.forecast import run_backward class MyUpdate: # step_size, n_prompts as attributes def projected_variance(self, b): # Var over prompts of b . per-prompt gradient ... def jacobian_vector(self, b): # J b, one Hessian-vector product ... result = run_backward(metric_gradient, updates) result.std # forecast s.d. of the reported score across seeds result.interval(0.784) # a normal interval around the number you report
Stochastic approximation, adjoint methods, RLVR and RLOO, Hessian-vector products, prospective evaluation, PyTorch, MLX, Qwen2.5. 133 tests and continuous integration.
Spurious or structural? Low-rank confounding and partial identification of cross-asset price impact
The puzzle, simulated. Six stocks with no cross-impact at all and \(K\) common factors moving both prices and order flow. Each change draws 4,000 observations and runs \(\widehat\Lambda = \widehat\Sigma_{rq}\widehat\Sigma_{qq}^{-1}\) in your browser. The regression reports cross-impact that does not exist, and the singular values of its error show exactly \(K\) directions: Theorem 2, below, live.
Abstract. Cross-asset return-on-flow coefficients are routinely read as entries of a structural price-impact matrix. In a published study, adding one principal-component control flips the mean coefficient from +0.032 to −0.039 and the share of negative coefficients from 23.09% to 84.46%. I show that neither reading is identified, characterise exactly how, and price the consequence. Each step of the argument is one row of the derivation below.
Not part of the repository. How the same problem reads through stochastic calculus, pricing theory and learning theory.
Liquidating \(X\) shares with temporary impact \(\eta\) under a mean-variance objective with risk aversion \(\lambda\) gives this schedule. With many assets, \(\kappa\) becomes a matrix built from the covariance and the impact matrix, so the schedule depends on \(\Lambda\) exactly where identification fails. The figure below shows what a misestimated impact does to the plan.
In continuous time the regression slope is a ratio of quadratic covariations, which high-frequency data recovers exactly. But Proposition 3 says those second moments do not determine \(\Lambda\): the confounding gap \(\Gamma\Sigma_f\Delta_f^{\top}\Sigma_{qq}^{-1}\) survives any sampling frequency. Partial identification is a property of the model, not of the sample size.
In Kyle's model an informed trader and noise traders produce linear price impact. Correlated private information across assets is a structural reason \(\Lambda\) could be genuinely non-diagonal, and it is precisely the story the data cannot tell apart from a common factor.
Illustration. Almgren-Chriss holdings over the trading window, with unit volatility and impact. Overestimating impact makes the plan trade too slowly and carry more risk; underestimating it makes it trade too fast and pay more impact. The default error is the 54.23% from Figure 3, used only as an illustrative size.
The derivation was frozen before any simulation code existed (gate G0). The known-truth experiment ran on one master draw split into 100 immutable shards of 100,000 observations, each publishing only mergeable sufficient statistics: all 1,800 coefficient targets fall inside their simultaneous intervals, the maximum relative discrepancy is 0.00056 against a preregistered 0.001, and the replay reproduces the summary byte for byte (gate G1).
uv sync --locked --extra dev make check # ruff lint and format, strict mypy, 410 tests, smoke, drift make exhibits # regenerate every manuscript number; fails on drift make paper # build the preprint PDF
Econometrics, partial identification, factor models, market microstructure, parametric bootstrap, sharded simulation, strict typing, Python.
FLEX inventory in clearing data, and whether it predicts the visible market
The story. FLEX options are negotiated bilaterally, cleared centrally, and absent from the listed option chains most option research uses. The Options Clearing Corporation still publishes every cleared FLEX series, daily, for free. I built the pipeline that turns those reports into a panel, found two ways the obvious reading of the data is wrong, and then asked whether this hidden inventory predicts the visible market. It does not.
| Naive reading | What the data says |
|---|---|
| An off-grid strike marks a FLEX series | Over 486,895 cleared series that rule has precision 0.67, recall 0.34: two thirds of FLEX is struck on the grid. |
| A cleared series is a position | Counted leg by leg, a butterfly is credited with 49x its maximum value. |
| Four legs, four exposures | A four-leg SPY collar is counted about 3x over. |
| Strike notional measures exposure | A synthetic long call struck at $1.87 against a $765 spot is recorded as $7m when its exposure is $2.9bn. |
| File each report by the date it was fetched | OCC publishes overnight, so a capture taken during a session would enter one settlement twice; reports are filed by activity date. |
Scoring the strike rule \(\hat F\) against the report's ground truth \(F\): \(\;\text{precision} = |\hat F \cap F| / |\hat F| = 0.67,\ \ \text{recall} = |\hat F \cap F| / |F| = 0.34.\) I used that rule first, replaced it, and kept the superseded census public.
Live. Why legs must be grouped, with illustrative strikes: long one call at 90, short two at 100, long one at 110, valued at expiry. Each leg can be worth a great deal while the structure is never worth more than 10.
A dealer on the other side of a FLEX trade hedges in the listed market and carries the residual convexity. Inventory for day \(t\) is published overnight, so every predictor is measured at \(t\) or earlier and every outcome strictly after:
\[ y_{i,t+1} = \alpha_i + \delta_t + \beta\,\mathrm{flex}_{i,t} + \gamma^{\top} z_{i,t} + \varepsilon_{i,t+1}, \tag{1} \]with underlying and date fixed effects, errors clustered two ways, and controls for trailing realized volatility at 5 and 22 days, same-day absolute return and log dollar volume, over 58,014 underlying-days.
Four inventory measures, each estimated with equation (1) on next-day absolute returns.
The headline estimate is −0.00030 with t = −1.37. On an earlier, smaller panel it reached t = −1.95, the kind of number that gets written up as suggestive.
The placebo uses inventory dated after the outcome, so it cannot carry information. It lands on −0.00032, t = −1.37: whatever links inventory to volatility runs equally well backwards in time.
That is the signature of a common factor moving both, not of information flowing from one to the other.
A null is informative only in proportion to its power: \(\mathrm{MDE}_{80} = (z_{0.975} + z_{0.80})\,\widehat{\mathrm{SE}}(\hat\beta) \approx 2.80\,\widehat{\mathrm{SE}} \approx 0.0006\), against a typical daily absolute return near 0.02, so relative effects of 3 to 5% would have been detected.
Not part of the repository. How the same problem reads through stochastic calculus, pricing theory and learning theory.
A delta-hedged option earns its gamma times the gap between realized and implied variance. A dealer on the other side of a FLEX trade hedges \(\Delta = \Phi(d_1)\) and keeps the convexity; short gamma, it rebalances with the move and can amplify volatility, long gamma it dampens it. That hedging feedback is the channel this study tests, and the placebo-tested null bounds it at the daily, per-underlying level.
By Breeden and Litzenberger, a butterfly with wing width \(h\), scaled by \(1/h^2\), prices the risk-neutral density \(q\) of the underlying at expiry. Its value lives in the combination of legs, never in a single leg, which is the deeper reason a cleared series cannot be read as a position. The figure below converges as you narrow the wings.
Illustration. Black-Scholes call prices with \(S_0 = 100\), \(r = 0.02\), \(\sigma = 0.25\), \(T = 1\). Scaled butterflies across every strike converge to the lognormal density of \(S_T\) as the wings narrow.
uv venv && uv pip install -e '.[dev]' .venv/bin/pytest .venv/bin/python scripts/backfill_occ_flex.py --start 2025-07-23 --end 2026-08-17 .venv/bin/python scripts/run_predictive.py --start 2025-09-09 --end 2026-08-17
Data engineering, panel econometrics, fixed effects, two-way clustering, placebo tests, power analysis, options. 256 tests; code public, vendor captures kept in a private repository.
Simulation calibration of forecast comparison on correlated equity panels
Abstract. Short-horizon equity forecasting papers report a model that beats a benchmark. This one audits the apparatus instead: every standard test is run on panels where the truth is fixed by construction, and its measured size is set beside its nominal size. Three components fail badly. The corrected apparatus holds its size, and on Borsa Istanbul data it returns a negative result with an explicit statement of what it could and could not have found. Open a row to see why each test fails.
Stocks on the same session share market shocks, so for \(k\) units with average correlation \(\bar\rho\) the variance of a cross-sectional mean is inflated by \(\mathrm{VIF} = 1 + (k-1)\bar\rho\), and \(n_{\mathrm{eff}} = n/\mathrm{VIF}\). A test on rows overstates its statistic by \(\sqrt{\mathrm{VIF}}\):
\[ \alpha_{\mathrm{real}}(k) = 2\,\Phi\!\left(-\frac{z_{1-\alpha/2}}{\sqrt{1 + (k-1)\,\bar\rho}}\right). \]At \(\bar\rho = 0.5697\) the simulation tracks this closed form to within 0.0064 from 0.11 to 0.79.∎
Widening the panel makes it worse, because a session of \(k\) equicorrelated units carries only
\[ m(k) = \frac{k}{1 + (k-1)\,\bar\rho} \;\uparrow\; \frac{1}{\bar\rho} \]independent observations (Proposition 1). Past a few dozen names, breadth buys almost no precision; only more sessions help. On the committed panel, 480 rows over 120 sessions carry about 177 observations.
Holm controls the family-wise error rate only if each test in the family holds its own size. Fed row-level statistics, it inherits their inflation: a correction cannot repair the tests it is correcting.
Against a zero benchmark the models are nested, and the squared-error differential is
\[ \mathbb{E}\big[e_{1,t}^2 - e_{0,t}^2\big] = \mathbb{E}\big[\hat y_t^{\,2}\big] - 2\,\mathbb{E}\big[\hat y_t\, y_t\big], \]so a forecast with more variance is punished regardless of its covariance with the target. Clark-West subtracts the noise term, \(\hat f_t = e_{0,t}^2 - [e_{1,t}^2 - (\hat y_{0,t} - \hat y_{1,t})^2]\), and holds its size.
The deflated Sharpe ratio compares the best of \(N\) trials with the expected maximum of \(N\) skill-free ones,
\[ SR^{*}_0 = \sqrt{V[\widehat{SR}]}\,\Big[(1-\gamma)\,\Phi^{-1}\!\big(1 - \tfrac1N\big) + \gamma\,\Phi^{-1}\!\big(1 - \tfrac1{Ne}\big)\Big], \]but an expectation is not a critical value: a skill-free grid of 72 configurations clears it about half the time at every level of dependence. A joint stationary bootstrap gives a real 95th percentile, and the grid behaves like 3.77 independent trials, not 72.
Every loss differential is averaged across the stocks present on a date, the nested comparison uses Clark-West, and the search is resampled jointly. Measured size stays between 0.0385 and 0.0565 at nominal 0.05.
Each bar runs from the nominal 5% (black tick) to the measured size.
A test at 5% should reject a true null 5% of the time. On a single stock, Diebold-Mariano does. Keep scrolling to add stocks.
Four stocks correlated at 0.5697: the simulation measures 22.84%. The rows of a panel are not independent observations.
A hundred stocks: 79.13%. The curve is the closed form in the first row of the audit.
The corrected apparatus holds its size between 0.0385 and 0.0565.
Re-run the audit yourself. A Monte Carlo running in your browser: 120 sessions of \(k\) null loss differentials with \(\bar\rho = 0.5697\) per replication, tested on rows and on session averages. The row-level rate converges to the closed form, the session-level rate to 5%.
On a leakage-controlled panel of GARAN, ISCTR, KCHOL and THYAO with a target the backtest can trade, no model carries predictive content: White's Reality Check gives \(p = 0.9970\), Hansen's SPA \(p = 0.6891\). Inverting the test at size \(\alpha\) and power \(1-\beta\),
\[ \delta_{\min} = \big(t_{1-\alpha/2,\,n-1} + t_{1-\beta,\,n-1}\big)\cdot\mathrm{SE}(\bar d), \]bounds what it could have seen: an out-of-sample \(R^2\) of 0.113 in closed form and 0.2012 by simulation, against the 0.01 this literature treats as meaningful.
Not part of the repository. How the same problem reads through stochastic calculus, pricing theory and learning theory.
Sampling faster recovers \(\sigma\) exactly through quadratic variation, but the error in the drift depends only on calendar time. A return forecast is a statement about drift, so a short panel cannot resolve a plausible effect, however finely it is sampled: the same conclusion as the detectability bound, from first principles. The figure below runs it.
The best of \(N\) skill-free strategies looks good by extreme-value theory alone. The deflated Sharpe threshold (6) refines this expectation; correlation between trials shrinks the effective \(N\), which is why 72 configurations behave like 3.77.
Because \(\hat\mu\) carries almost all the error, so does the Sharpe ratio. This variance is the \(V_{\mathrm{ind}}\) behind the effective-trial count in Proposition 3.
Simulation. Three hundred five-year histories of a stock with 6% annual drift and 20% volatility, each estimated from data at the chosen frequency. The volatility estimates collapse onto the truth as sampling gets finer; the drift estimates do not move, because their error depends on the five years, not on the number of observations.
uv sync make reproduce-committed # replay the committed run, verify every artifact hash, offline uv run bist-predict calibrate # rebuild the calibration, about four minutes make verify-claims RUN_ID=20260810T230615Z-d346031-2a71b8
Monte Carlo calibration, Diebold-Mariano, Clark-West, Holm, White's Reality Check, Hansen's SPA, deflated Sharpe ratio, stationary bootstrap. Python with a Rust indicator library, 606 tests, an external review and my response in the repository.
Nested MSPE inference, purged walk-forward validation and multiple-testing control, with bootstrap bounds on net Sharpe ratios
Abstract. I test whether daily BTC, ETH and SOL returns can be forecast out of sample well enough to survive trading costs. Three assets, two horizons and six forecasters give 18 machine-learning settings against a martingale null. The headline depends entirely on the test, and the funnel below shows each one applied in turn: a faint and fragile statistical signal in BTC at one day, and no tradeable one anywhere.
A \(h\)-step target resolves at \(t + h\), so the \(h\) rows before each test block are purged and a further embargo of \(e\) bars is imposed; every fold satisfies \(\max(\mathcal{T}_k) + h + e < \min(\mathcal{V}_k)\). Scalers are fitted inside each fold, and gradient boosting purges its own early-stopping slice, which a test asserts by corrupting exactly those rows.
Against the recursively estimated mean \(\hat y^{\,b}\), Diebold-Mariano compares squared errors with a Newey-West variance, but the random walk is nested in every model, so estimation noise reads as evidence for the benchmark. Clark-West removes it:
\[ \hat f_t = \big(y_t - \hat y^{\,b}_t\big)^2 - \Big[\big(y_t - \hat y^{\,m}_t\big)^2 - \big(\hat y^{\,b}_t - \hat y^{\,m}_t\big)^2\Big], \qquad \mathrm{CW} = \frac{\bar f}{\sqrt{\hat V_f / n}}. \]Live. The target is pure noise and the larger model forecasts pure noise too, so the null is true by construction. Diebold-Mariano still convicts the model far above its nominal rate; Clark-West holds its size. This mirrors a test in the repository.
With \(m = 18\) ordered p-values, Holm rejects step-down while \(p_{(i)} \le \alpha/(m - i + 1)\), and Benjamini-Hochberg step-up at the largest \(i\) with \(p_{(i)} \le i\alpha/m\).
Not part of the repository. How the same problem reads through stochastic calculus, pricing theory and learning theory.
Under the null, log prices are martingales; under a risk-neutral measure every discounted price is one, which is the Girsanov change of measure from the notebook. A forecasting edge is a predictable piece of drift, and drift is the quantity Brownian motion hides best.
Ridge with infinite penalty is the zero forecast, the random walk itself; the elastic net shares that path, and gradient boosting \(F_m = F_{m-1} + \nu h_m\) with no trees is a constant. The benchmark sits inside every model family here, which is the nesting that biases Diebold-Mariano and the reason Clark-West is needed.
Brownian scaling, \(\operatorname{Var}(X_{t+q} - X_t) = q\sigma^2\), gives the Lo-MacKinlay variance-ratio test: a classic attack on the same null from the scaling side rather than the forecasting side.
make setup # install the package and dev extras make backtest # download data, run the study, regenerate reports and figures, about 15 s make test # 154 tests, including the no-lookahead and purge guarantees make check # ruff, black, mypy, 80% coverage floor: the CI gate on Python 3.12 and 3.13
Time-series cross-validation, Clark-West, Diebold-Mariano, Holm, Benjamini-Hochberg, stationary bootstrap, deflated Sharpe ratio, scikit-learn. 154 tests at 96% coverage; the report is generated from the results table, never hand-edited.
Eight theorems about random walks and Brownian motion, each simulated in your browser until it happens in front of you
Why this page exists. Quantitative finance runs on a small number of deep facts about randomness. Random walks become Brownian motion under the right scaling; fair games produce long, lopsided streaks; a fair process strays exactly as far as the law of the iterated logarithm allows; no stopping rule beats a martingale. In continuous time, Brownian paths have quadratic variation equal to elapsed time, so the chain rule gains a term; integrals depend on where you evaluate them; changing the measure reweights paths without moving them; and prices are expectations that solve partial differential equations. Each plate below is one of those facts, computed exactly enough to watch. The same processes run through the rest of the site: the line under the top bar is a Brownian path whose quadratic variation is your reading progress, and the rules between sections are Brownian bridges.
Part one. Discrete time
Rescale the first \(n\) steps of a random walk by \(1/\sqrt n\). As \(n\) grows, the law of the whole path converges to Wiener measure, and it does not matter what the steps were, only that they have mean zero and variance one. This is why binomial trees converge to Black-Scholes.
Play a fair game for a long time and record the fraction of time you are ahead. The most likely outcomes are almost always ahead or almost always behind; an even split is the least likely. It is why a long winning streak in a track record is weak evidence of skill.
The central limit theorem says \(S_n/\sqrt n\) is roughly normal at any fixed time. Khinchin's law says how far it strays along the whole path: out to \(\sqrt{2\log\log n}\), touched infinitely often and exceeded only finitely often. It is the boundary behind sequential testing.
A gambler starts with \(x\) and plays a fair coin until ruin at 0 or a target \(N\). Because \(S_n\) and \(S_n^2 - n\) are martingales and the stopping time is integrable, the probability of reaching the target and the expected length of the game follow without any computation of paths.
Part two. Continuous time
Cut a Brownian path at the dyadic times \(k2^{-n}\). The squared increments sum to elapsed time while the path's length diverges like \(2^{n/2}\). Every smooth function has zero quadratic variation, so this single number is what separates stochastic calculus from the ordinary kind, and why realized variance estimates volatility.
Because the increments do not vanish fast enough, a Riemann sum for \(\int W\,dW\) depends on where each interval is evaluated: the left endpoint gives Itô, the midpoint gives Stratonovich, and they differ by half the quadratic variation. Finance uses Itô because a strategy chooses its position before the price moves.
Weight each path by the likelihood ratio and \(W_t - \theta t\) becomes a Brownian motion: the same paths now carry drift \(\theta\). Risk-neutral pricing is this reweighting. The effective sample size shows the price: tilt far and a handful of paths carry all the weight, which is why importance sampling degenerates.
Under the risk-neutral measure, \(dS_t = rS_t\,dt + \sigma S_t\,dW_t\). By Feynman-Kac, the price of a claim on \(S_T\) is both an expectation over paths and the solution of a partial differential equation; for a call, the closed form below. Monte Carlo reaches it at the rate \(1/\sqrt n\).
Mehmet Demir Güven, Zürich, Switzerland. Contact: [email protected]. This is a personal, non-commercial website.
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