StatGuardia

Three disciplines,
one instrument.

StatGuardia operates at the intersection of statistical science, financial reasoning, and engineered software — built to hold each other accountable.

— 001

Return distribution · live ● live

n = 0 σ̂ = —

Discipline 001

Statistical Modeling

From first principles — probability, estimation, uncertainty quantification. We treat every parameter as a question and every assumption as a liability until it earns its place in the model.

Bayesian inference —
Hypothesis testing —
Time-series analysis —
Distribution fitting —
Causal reasoning —

“A good model explains what you already know before it predicts what you don’t.”

Modeling discipline starts long before a line of code. We spend the first hours of any engagement cataloguing what is assumed, what is known, and what is unknowable. Each category gets treated differently.

We default to the simplest model that holds up under adversarial scrutiny — then let the data argue us into additional structure if, and only if, it earns the complexity.

— 002

Monte Carlo · 35 paths ● live

5–95% median

Discipline 002

Financial Analysis

Pricing, risk, and return — examined through the language of probability rather than prediction. We measure what a signal is worth net of the noise surrounding it.

Portfolio analytics —
Risk decomposition —
Scenario simulation —
Volatility structure —
Valuation modeling —

“Every return is a noisy measurement of a phenomenon we cannot directly observe.”

We treat financial data as samples from a distribution, not as history. That shift changes everything downstream: how we measure performance, how we size positions, how we reason about tail events.

The goal is never to predict the market. The goal is to describe the shape of what markets do accurately enough that a human decision-maker can take an informed position within that shape.

— 003

Pipeline stream ● live

0 pkt/s

Discipline 003

Engineered Software

Models are only as credible as the systems that execute them. We build the pipelines, the validation harnesses, and the instrumentation that turns an idea into infrastructure.

Data pipelines —
Research platforms —
Backtesting frameworks —
Observability & audit —
Reproducible compute —

“Code is the only part of a model that survives contact with reality.”

We write software the way an engineer builds a bridge — with load calculations, inspection points, and documented failure modes. Every production pipeline ships with instrumentation that tells you, in real time, whether it is still operating within its validated envelope.

We believe in reproducibility above elegance. A result that cannot be regenerated on demand, from the same inputs, by a different engineer, is not a result — it is an anecdote.

A library
of first principles.

These are the classical tools of the trade — the shared vocabulary within which every StatGuardia engagement is framed. The art lies not in knowing them, but in knowing when each is the right one to reach for.

— Probability

Gaussian density

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

The bedrock: how probability mass organizes itself around a central tendency, scaled by uncertainty.

What it says

The normal distribution organizes mass symmetrically around a mean μ, with spread controlled by a single parameter σ. About 68% of the mass lives within one σ of μ, about 95% within two, and about 99.7% within three — the canonical 68/95/99.7 rule.

It is the most-invoked, most-misapplied distribution in all of quantitative work. Knowing when not to reach for it is as important as knowing the formula.

— Inference

Bayes' rule

$$P(\theta \mid D) = \frac{P(D \mid \theta)\, P(\theta)}{P(D)}$$

Belief, updated by evidence. The epistemology of the entire house.

What it says

The posterior probability of a hypothesis θ after seeing data D is the prior belief about θ, multiplied by how well θ explains D, normalized by how likely D was overall.

It is not really an equation — it is an instruction for how a rational mind should respond to evidence. Every statistical model is, at heart, a wrapper around this rule.

— Stochastic

Geometric Brownian motion

$$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$$

The continuous-time engine behind most modern pricing theory.

What it says

The asset price S evolves in continuous time with a drift term (μ S dt) and a random shock proportional to the price itself (σ S dW). The multiplicative form is what keeps prices positive and what makes log-returns tractable.

Nearly every closed-form pricing result in quantitative finance stands on this equation — as do all the caveats about what it quietly fails to capture.

— Performance

Sharpe ratio

$$S = \frac{\mathbb{E}[R_p - R_f]}{\sigma_p}$$

Return, denominated in the currency that actually matters — risk.

What it says

Excess return per unit of standard deviation. A scalar that lets you compare strategies with wildly different scales on the same axis: how much reward did you earn for each unit of risk you took on?

Powerful because it is simple; dangerous for exactly the same reason. Sharpe ratios computed on short windows or skewed distributions can mislead badly.

— Information

Shannon entropy

$$H(X) = -\sum_{x} p(x)\log p(x)$$

How much the world is allowed to surprise you.

What it says

The expected information content of a random variable — measured in bits when the log is base 2, nats when base e. Low entropy means predictable; high entropy means the world is allowed to surprise you.

Entropy shows up everywhere: decision trees, model compression, market microstructure, risk concentration. Whenever you need to quantify how much you don't know, reach here first.

— Derivatives

Black-Scholes call price

$$\begin{aligned} C &= S_0\,\Phi(d_1) - K e^{-rT}\,\Phi(d_2) \\[4pt] d_{1,2} &= \frac{\ln(S_0/K) + \left(r \pm \tfrac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}} \end{aligned}$$

A closed-form answer, and a cautionary tale about the assumptions one must swallow to get it.

What it says

Under a specific set of assumptions — continuous trading, constant volatility, log-normal returns, no arbitrage — the fair price of a European call is the expected discounted payoff, which collapses to this closed form.

The formula is less interesting than the cultural achievement it represents: the moment pricing became something a human could write down, critique, and build upon, rather than negotiate.

— Risk

Value-at-risk

$$\mathrm{VaR}_\alpha(L) = \inf\{\ell : P(L > \ell) \le 1-\alpha\}$$

A threshold, nothing more — useful when read with its fine print.

What it says

The smallest loss ℓ such that losses exceed ℓ with probability no greater than (1 − α). A summary statistic, not a risk measure in the modern sense — VaR tells you where the tail begins, but says nothing about how far it extends.

We use it, but always alongside complementary measures: Expected Shortfall, scenario analysis, stressed sensitivities. Never in isolation.

— Filter

Kalman update

$$\begin{aligned} \hat{x}_{k\mid k} &= \hat{x}_{k\mid k-1} + K_k\!\left(z_k - H_k\,\hat{x}_{k\mid k-1}\right) \\[6pt] K_k &= P_{k\mid k-1}\, H_k^{\top}\!\left(H_k P_{k\mid k-1} H_k^{\top} + R_k\right)^{-1} \end{aligned}$$

The mathematical formalization of changing your mind gracefully.

What it says

Given a prior estimate of hidden state x, a new noisy measurement z, and a model of how measurements relate to state (H), the optimal updated estimate is a weighted blend — where the weight K (the Kalman gain) depends on how noisy the measurement is versus how uncertain the prior was.

It is the mathematical formalization of a core epistemic virtue: trust evidence in proportion to its quality, and revise prior beliefs in proportion to how poorly they explained what you observed.