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PCA for Implied Volatility Surfaces

Marco Avellaneda, Brian Healy, Andrew Papanicolaou and George Papanicolaou
The Journal of Financial Data Science Spring 2020, jfds.2020.1.032; DOI: https://doi.org/10.3905/jfds.2020.1.032
Marco Avellaneda
is a professor of mathematics at the Courant Institute of Mathematical Sciences at New York University in New York, NY
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  • For correspondence: avellaneda@courant.nyu.edu
Brian Healy
is a researcher at the Department of Financial Computing and Analytics at University College London in London, UK
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  • For correspondence: brian@decisionsci.net
Andrew Papanicolaou
Andrew Papanicolaou is an assistant professor at the Department of Finance and Risk Engineering at NYU Tandon School of Engineering in Brooklyn, NY
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  • For correspondence: ap1345@nyu.edu
George Papanicolaou
George Papanicolaou is a professor at the Department of Mathematics at Stanford University in Stanford, CA
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  • For correspondence: papanicolaou@stanford.edu
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Abstract

Principal component analysis (PCA) is a useful tool when trying to construct factor models from historical asset returns. For the implied volatilities of US equities, there is a PCA-based model with a principal eigenportfolio whose return time series lies close to that of an overarching market factor. The authors show that this market factor is the index resulting from the daily compounding of a weighted average of implied-volatility returns, with weights based on the options’ open interest and Vega. The authors also analyze the singular vectors derived from the tensor structure of the implied volatilities of S&P 500 constituents and find evidence indicating that some type of open interest- and Vega-weighted index should be one of at least two significant factors in this market.

TOPICS: Statistical methods, simulations, big data/machine learning

Key Findings

  • • Principal component analysis of a comprehensive dataset of implied volatility surfaces from options on US equities shows that their collective behavior is captured by just nine factors, whereas the effective spatial dimension of the residuals is closer to 500 than to the nominal dimension of 28,000, revealing the large redundancy in the data.

  • • Portfolios of implied volatility surface returns, weighed suitably by open interest and Vega, track the principal eigenportfolio associated with a market portfolio of options, in analogy to equity portfolios.

  • • Retention of the tensor structure in the eigenportfolio analysis improves the tracking between the open interest–Vega weighted (tensor) implied volatility surface returns portfolio and the (tensor) eigenportfolio, indicating that data structure matters.

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The Journal of Financial Data Science: 4 (2)
The Journal of Financial Data Science
Vol. 4, Issue 2
Spring 2022
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PCA for Implied Volatility Surfaces
Marco Avellaneda, Brian Healy, Andrew Papanicolaou, George Papanicolaou
The Journal of Financial Data Science Mar 2020, jfds.2020.1.032; DOI: 10.3905/jfds.2020.1.032

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PCA for Implied Volatility Surfaces
Marco Avellaneda, Brian Healy, Andrew Papanicolaou, George Papanicolaou
The Journal of Financial Data Science Mar 2020, jfds.2020.1.032; DOI: 10.3905/jfds.2020.1.032
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  • Article
    • Abstract
    • REVIEW OF LITERATURE
    • STRUCTURE AND RESULTS OF THE ARTICLE
    • MATRIX OF IMPLIED VOLATILITY RETURNS
    • PRINCIPAL EIGENPORTFOLIOS AND OI-WEIGHTED INDEXES
    • FACTORS AND EIGENPORTFOLIOS USING TENSORS
    • SUMMARY AND CONCLUSION
    • ADDITIONAL READING
    • APPENDIX A
    • APPENDIX B
    • ENDNOTES
    • REFERENCES
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