ArbitrageLab Documentation

The Distance Approach is a pairs trading strategy based on the work of Gatev et al. (2006). It identifies pairs of assets by calculating the Euclidean squared distance (SSD) between their normalized price time series.

Key Characteristics:

• No Cointegration Tests: Unlike mean-reversion approaches, it relies purely on distance metrics.
• Normalization: Prices are normalized using the formula: $P_{normalized} = \frac{P - min(P)}{max(P) - min(P)}$.
• Signal Generation: A trade is triggered when the difference between the normalized prices of a pair diverges by more than 2 standard deviations (calculated during the training period).
• Position Management:
  • Long/Short: If the portfolio value (difference in normalized prices) exceeds +2 standard deviations, a sell signal is generated (short the first element, long the second). If it falls below -2 standard deviations, a buy signal is generated (long the first element, short the second).
  • Exit: Positions are closed when the portfolio value crosses zero (the normalized prices cross) or when the trading period ends.

The Spread Modeling module implements research by Dr. Christian Dunis and colleagues focused on modeling and forecasting commodity spreads (e.g., Crack Spread, Soy Crush Spread, Corn Crush Spread).

Core Concepts

• Spread Trading: Trading the difference between two correlated assets. This approach aims to reduce exposure to information shocks and speculative bubbles, as movements in the two legs often offset each other.
• Modeling Goal: The module provides tools to model the spread accurately and, more importantly, to discriminate between large and small moves to optimize transaction costs.

Available Model Types

• Neural Networks: MLP (Multi-Layer Perceptron), RNN (Recurrent Neural Networks), and HONN (Higher Order Neural Networks).
• Benchmark Models: Fair Value Non-Linear Cointegration Models, ARMA, and MACD.
• Filters: Used to minimize transaction costs by selecting trades based on volatility or correlation. Includes Threshold Filters, Asymmetric Threshold Filters, Correlation Filters, and Time-Varying RiskMetrics volatility models.

The C-vine Copula strategy is a statistical arbitrage approach used to model multivariate dependencies among a cohort of stocks. It decomposes joint probability density into bivariate copula densities and marginal densities using a tree structure.

Key Concepts

• C-vine vs. R-vine: While R-vines are the most generic structure, they are computationally expensive and prone to overfitting. C-vines (Canonical vines) have a star-like structure that is more interpretable and efficient for modeling dependencies where one specific random variable (the target stock) is central to the group.
• Trading Logic: The strategy uses a "1-vs-the-rest" framework. It calculates conditional (cumulative) probabilities to determine if a target stock is underpriced or overpriced relative to the rest of the cohort, generating trading signals based on a mean-reversion bet.

Typical Workflow

1. Get data: Collect historical price/return data for the stock cohort.
2. Figure out the C-vine structure: Determine the tree structure for the dependencies.
3. Calculate probability density: Compute the joint density using the copula model.
4. Calculate the conditional probability: Derive the conditional probability to gauge mispricing.
5. Generate signals and trade: Execute trades based on the calculated mispricing signals.

A Copula is a statistical tool used to study the relationship among multiple random variables by focusing on the joint cumulative density of marginal quantiles, bypassing the idiosyncratic features of individual marginal distributions.

While standard bivariate copulas work well for two dimensions, they become rigid and lose detail in higher dimensions. Vine Copulas solve this by decomposing a high-dimensional probability density function (PDF) into a hierarchy of bivariate copulas and marginal densities.

The Decomposition Principle

The core idea is to decompose the joint PDF into conditional probabilities, which are then further decomposed into bivariate copulas. For example, a 3D joint density $f(x_1, x_2, x_3)$ can be decomposed as:

$f(x_1, x_2, x_3) = f_1(x_1) f_2(x_2) f_3(x_3) \times c_{2,3}(F_2(x_2), F_3(x_3)) c_{1,3}(F_1(x_1), F_3(x_3)) \times c_{1,2|3}(F_{1|3}(x_1 | x_3), F_{2|3}(x_2 | x_3))$

Visualizing the Structure

Vine copulas are visualized as tree structures where:

• Nodes represent either joint densities or marginal densities.
• Bivariate copulas connect the nodes.
• Leaves (at the top of the tree) represent the marginal densities of each random variable.

The Multivariate Cointegration strategy bets on the mean reversion of a spread formed by $N$ cointegrated assets.

Core Logic

1. Identify Cointegration: Use the Johansen test on log-price series to derive a cointegration vector $\mathbf{b}$.
2. Calculate the Spread: The spread is a linear combination of log-prices: $Y_t = \sum_{i=1}^N b^i \ln{P_t^i}$.
3. Trading Rule: Trade the assets based on the deviation of the recent sum of log-returns from zero. In practice, a finite lag $P$ is used to approximate the infinite sum of returns: $\sum_{p=1}^P Z_{t-p}$.
4. Dollar Neutrality: To ensure the portfolio is dollar-neutral, assets are partitioned into two sets based on the sign of their cointegration coefficient $b^i$:
   • Set $L$: Assets where $b^i \geq 0$
   • Set $S$: Assets where $b^i < 0$

The resulting portfolio maintains $C$ dollars in long positions and $C$ dollars in short positions, where $C$ is a positive scale factor.

To avoid the instability and convergence issues of generic maximization optimizers when fitting multiple parameters and weights, arbitragelab uses a two-step Expectation-Maximization (EM) algorithm adapted from [Cai, Wang 2014].

The Two-Step Process:

1. E-step: Iteratively calculates updated weights $w_k^{new}$ using the current parameters $\theta$ and weights $w$.
2. M-step: Uses the updated weights to find new parameters $\theta$ by maximizing the objective function $Q(\boldsymbol{\theta}, \mathbf{w})$ (often using truncated Newton or Newton-Raphson).

Key Characteristics:

• Oracle and Sparse: The algorithm has good asymptotic properties and uses a SCAD (smoothly clipped absolute deviation) penalty term to drive insignificant component weights toward zero.
• Performance Note: The CTGMixCop may take longer to converge due to the computational bottleneck of the Student-t copula component.
• Limitations: If copula components are very similar (e.g., Gaussian and Frank), the algorithm may struggle to distinguish weights accurately. The SCAD parameters $\gamma$ and $a$ are tuned via cross-validation in the package defaults.