Introduction to Vine Copula for Statistical Arbitrage

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VINE COPULA FOR STATISTICAL ARBITRAGE

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Possible Vine Copula Structures, picture from [Pham et al. (2018)]

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VINE COPULA FOR STATISTICAL ARBITRAGE

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Key Math Concepts

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Understand Vine Copula

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Trading Strategy

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Interesting Problems

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CONTENTS

VINE COPULA FOR STATISTICAL ARBITRAGE

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• Ph.D. Candidate in Applied Mathematics at University of Delaware
• Researcher at Hudson & Thames

LinkedIn: https://www.linkedin.com/in/hansen-pei-0949691b3/

Twitter: @HansenPei

ABOUT ME

Research Interests:

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• Stochastic Modeling
• Fokker-Planck and Broadwell Models
• Agent Based Modeling
• Numerical Methods for PDE
• Copula Modeling for Stats Arbitrage
• Applied Probability
• Stochastic Control

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Key Math Concepts

01.

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1. Get data from 2 random variables, say,

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• Map to their quantiles using marginal CDF and formulate

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Copula: Joint Cumulative Density on Quantiles

Review of Copula Concepts

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Decompose Joint Probability Density Via Copula

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Say we have the joint probability density via structure

Conditional (cumulative) density

• Key component for statistical arbitrage

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• Relative mispricing in a higher dimension

Decompose Joint Probability Density Via Copula

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Copula Density:

Joint Density:

Conditional Density:

Decompose Joint Probability Density Via Copula

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Understand Vine Copula

02.

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Let’s get slightly more complicated...

Vine Copula: An Example

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Let’s get slightly more complicated...

Vine Copula: An Example

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Let’s get slightly more complicated...

Be careful:

Bivar Copulas

Vine Copula: An Example

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Let’s get slightly more complicated...

Vine Copula: An Example

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Let’s get slightly more complicated...

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2, 3

1, 3

1,2 | 3

Vine Copula: An Example

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Let’s get slightly more complicated...

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2, 3

1, 3

1,2 | 3

Vine Copula: Decompose higher dim dependency by bivar copulas and graphical structure

Vine Copula: An Example

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Flexibility

Higher dim copulas are very rigid

Visual dependence structure

Tail risk is limited

Interpretability

Risk Control

Picture from [Killiches et al. (2016)]

Model Advantages

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R-Vine

(Regular)

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C-Vine

(Canonical)

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D-Vine

(Drawable)

Structure Types

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(1, 3, 2, 4)

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2, 4

3, 4

1, 2 | 4

2, 3 | 4

1, 3 | 2, 4

Structure has bijection map with an ordered tuple:

Center for each tree

C-Vine Notation

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1, 3

1, 2

1, 4

2, 3 | 1

2, 4 | 1

4, 3 | 2, 1

Structure has bijection map with an upper triangular matrix:

R-Vine Notation

1. Get data
2. Figure out the vine copula structure
3. Calculate point density
4. Calculate conditional probability
5. Generate signals

VINE COPULA FOR STATISTICAL ARBITRAGE

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Data

Marginal

CDF

Pseudo-

observation/

Quantile

Vine Copula Workflow

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• Get data
• Figure out the vine copula structure
• Calculate point density
• Calculate conditional probability
• Generate signals

VINE COPULA FOR STATISTICAL ARBITRAGE

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Complicated. Assume your computer can handle it for now.

Vine Copula Workflow

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• Get data
• Figure out the vine copula structure
• Calculate point density
• Calculate conditional probability
• Generate signals

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Calculate point density via vine structure:

Vine Copula Workflow

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• Get data
• Figure out the vine copula structure
• Calculate point density
• Calculate conditional probability
• Generate signals

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Vine Copula Workflow

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• Get data
• Figure out the vine copula structure
• Calculate point density
• Calculate conditional probability
• Generate signals

VINE COPULA FOR STATISTICAL ARBITRAGE

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Stock 1 Overpriced

Stock 1 Underpriced

Vine Copula Workflow

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Trading Strategy

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1. 4 stocks each cohort from top 20 stocks

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• C-Vine assumption

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• CMPI strategy (returns)

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• Bollinger Band

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• Against SPY index

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• Dollar neutral

VINE COPULA FOR STATISTICAL ARBITRAGE

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1. Pairwise Spearman’s rho
2. Generalized Spearman’s rho [Schmid and Schmidt (2007)]
3. Geometric distance to diagonal on Q-Q plot
4. Extremal approach [Mangold (2015)]

Note: This approach is based on [Stübinger et al. 2016]

Key Components

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• 4 stocks each cohort from top 20 stocks

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• C-Vine assumption

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• CMPI strategy (returns)

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• Bollinger Band

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• Against SPY index

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• Dollar neutral

VINE COPULA FOR STATISTICAL ARBITRAGE

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Note: This approach is based on [Stübinger et al. 2016]

Key Components

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• 4 stocks each cohort from top 20 stocks

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• C-Vine assumption

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• CMPI strategy (returns)

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• Bollinger Band

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• Against SPY index

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• Dollar neutral

VINE COPULA FOR STATISTICAL ARBITRAGE

Feb 2021

Note: This approach is based on [Stübinger et al. 2016]

Key Components

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• 4 stocks each cohort from top 20 stocks

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• C-Vine assumption

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• CMPI strategy (returns)

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• Bollinger Band

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• Against SPY index

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• Dollar neutral

VINE COPULA FOR STATISTICAL ARBITRAGE

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Note: This approach is based on [Stübinger et al. 2016]

Key Components

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• Long/short the target stock
• Hedge against SPY

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Vine Copula

t-distribution

Gaussian

Independent assumption

Picture from [Stübinger et al. 2016]

Performance

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Note: This approach is based on [Stübinger et al. 2016]

Finished:

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• Automatic C-Vine Fit

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• Generate positions for the target stock via Bollinger band

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• Translate positions as units against an index

Working on now:

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• Automatic R-Vine Fit

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• Stocks selection

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• Speed optimization

Functionalities In Our Module

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Exiting too early.

Performance seem only significant on stock groups.

Computation time.

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Possible Issues

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Interesting Problems

• Strategies for smaller cohorts

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• Stocks selection

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• Term structures

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• Fast computation for high dimensions

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• R-Vine fit

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• Optimal exit

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• Higher frequency data

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• Alternative data

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Q&A

References

• Killiches, M., Kraus, D. and Czado, C., 2016. Using model distances to investigate the simplifying assumption, goodness-of-fit and truncation levels for vine copulas. arXiv preprint arXiv:1610.08795.
• Pham, M.T., Vernieuwe, H., Baets, B.D. and Verhoest, N.E., 2018. A coupled stochastic rainfall–evapotranspiration model for hydrological impact analysis. Hydrology and Earth System Sciences, 22(2), pp.1263-1283.
• Dissmann, J., Brechmann, E.C., Czado, C. and Kurowicka, D., 2013. Selecting and estimating regular vine copulae and application to financial returns. Computational Statistics & Data Analysis, 59, pp.52-69.
• Stübinger, J., Mangold, B. and Krauss, C., 2018. Statistical arbitrage with vine copulas. Quantitative Finance, 18(11), pp.1831-1849.
• Czado, C., 2019. Analyzing dependent data with vine copulas. Lecture Notes in Statistics, Springer.
• Vine Copula Models - Lehrstuhl für Mathematische Statistik

VINE COPULA FOR STATISTICAL ARBITRAGE

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