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The Geometry of the World of Currency Volatilities

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Abstract

Using empirical data and the properties they reveal, we develop a factor that captures changes of both currency implied correlation and volatilities. For this purpose, we apply the Guldin–Pappus theorem in Euclidean space for rotating triangles to construct a specific factor, which we define as gravity radius. This approach allows the construction of a portfolio index aggregating all currency pairwise trades. Our factor, which is a weighted sum of all gravity radius factors in a portfolio, exhibits characteristics that are similar to the well-known turbulence metric defined in the literature and has moderate correlation to the CBOE VIX index. This factor therefore can serve as a risk indicator. We argue that the changes in volatilities impact the gravity radius factor value considerably more than changes in correlations. Portfolio managers and risk managers can use the new metric to identify correlation and volatility changes that dynamically react to new information.

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Notes

  1. We use the terms “numeriare” and “base currency” interchangeably throughout this paper.

  2. Because we deal with Euclidean space, the second axiom is not shown explicitly. In Riemannian space with negative or positive curvature, the second axiom does not hold.

  3. For a further description of the Theorem of Pappus and Guldin, see Kern and Bland (1948), and Harris and Stocker (1998).

  4. Authors’ adaptation from Bulmer-Thomas (1984).

  5. Note that the perimeter of a triangle is the sum of the lengths of all edges.

  6. We are aware that other, simpler solutions of the problem might exist.

  7. Mathematically, rising correlations are associated with decreasing of the angle between \({\sigma }_{{fx}_{i}}^{2}\) and \({\sigma }_{{fx}_{j}}^{2}\); cos(90°) = 0, and cos (0°) = 1. Vidyamurthy (2004) calculated the angle between two vectors and provides a geometric interpretation of the distance measure.

  8. Bloomberg Ticker CVIX1I Index.

  9. We thank the anonymous reviewers for pointing this out.

  10. Konstantinov et al. (2020) showed that currencies exhibit large centrality scores in a directed asset network.

  11. See Gamov (1961) for a more detailed explanation of this issue.

References

  • Asness, C., Frazzini, A., Gormsen, N. J., & Pedersen, L. H. (2020). Betting against correlation: Testing theories of the low-risk effect. Journal of Financial Economics, 135(3), 629–652.

    Article  Google Scholar 

  • Black, F., Derman, E., & Toy, W. (1990). A one factor model of interest rates and its application to Treasury bond options. Financial Analysts Journal, 46(1), 33–39.

    Article  Google Scholar 

  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31, 307–327.

    Article  Google Scholar 

  • Bulmer-Thomas, I. (1984). “Guldin’s theorem—Or Papus’s? Isis, 75(2), 348–352.

    Article  Google Scholar 

  • Chen, S.-H. (2017). Carry trade strategies based on option-implied information: Evidence from a cross-section of funding currencies. Journal of International Money and Finance, 78, 1–20.

    Article  Google Scholar 

  • Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(2), 229–263.

    Article  Google Scholar 

  • Do, B., & Faff, R. (2010). Does simple pairs trading still work? Financial Analysts Journal, 66(4), 1–13.

    Article  Google Scholar 

  • Elliot, R. J., Van Der Hoek, J., & Malcolm, W. P. (2005). Pairs trading. Quantitative Finance, 5(3), 271–276.

    Article  Google Scholar 

  • Engelberg, J., Gao, P., & Jagannathan, R. (2009). An anatomy of pairs trading: The role of idiosyncratic news, common information and liquidity. In Third Singapore international conference in Finance.

  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U. K. inflation. Econometrica, 50, 987–1008.

    Article  Google Scholar 

  • Focardi, S. M., & Fabozzi, F. J. (2012). What’s wrong with today’s economics? The current crisis calls for an approach to economics rooted more on data than on rationality. Journal of Portfolio Management, 38(3), 104–119.

    Article  Google Scholar 

  • Focardi, S. M., Fabozzi, F. J., & Mazza, D. (2020). Quantum option pricing and quantum finance. Journal of Derivatives, 28(1), 79–98.

    Article  Google Scholar 

  • Gamov, G. (1961). One, two, three…infinity: Facts and speculations of science. Viking Press.

    Google Scholar 

  • Gatev, E., Goetzmann, W. N., & Rouwenhorst, K. G. (2006). Pairs trading: Performance of a relative-value arbitrage rule. Review of Financial Studies, 19(3), 797–827.

    Article  Google Scholar 

  • Geman, H., & Souveton, R. (1997). No arbitrage between economies and correlation risk management. Computational Economics, 10(2), 119–138.

    Article  Google Scholar 

  • Harris, J. W., & Stocker, H. (1998). Guldin’s rules. In Handbook of Mathematics and Computational Science. Springer.

  • Kern, W. F., & Bland, J. R. (1948). Theorem of Pappus. Chapter 40 in Solid Mensuration with Proofs, 2nd Edition (pp. 110–115) Wiley.

  • Kinlaw, W., & Turkington, D. (2014). Correlation surprises. Journal of Asset Management, 14(6), 385–399.

    Article  Google Scholar 

  • Konstantinov, G., Chorus, A., & Rebmann, J. (2020). A network and machine learning approach to factor, asset, and blended allocation. Journal of Portfolio Management, 46(6), 54–71.

    Article  Google Scholar 

  • Konstantinov, G., & Rebmann, J. (2019). From risk factors to networks: A case study on interconnectedness using currency funds. Journal of Financial Data Science, 1(3), 108–123.

    Article  Google Scholar 

  • Kritzman, M., & Li, Y. (2010). Skulls, financial turbulence, and risk management. Financial Analysts Journal, 66(5), 30–41.

    Article  Google Scholar 

  • Kritzman, M., Li, Y., Page, S., & Rigobon, R. (2011). Principal components as a measure of systemic risk. Journal of Portfolio Management, 37(4), 112–126.

    Article  Google Scholar 

  • Kritzman, M., Page, S., & Turkington, D. (2012). Regime shifts: Implications for dynamic strategies. Financial Analysts Journal, 68(3), 22–39.

    Article  Google Scholar 

  • Margrabe, W. (1993). Triangular equilibrium and arbitrage in the market for options to exchange two assets. Journal of Derivatives, 1(1), 60–69.

    Article  Google Scholar 

  • Meissner, G. (2016). Correlation trading strategies: Opportunities and limitations. Journal of Trading, 11(4), 14–32.

    Article  Google Scholar 

  • Menkhoff, L., Sarno, L., Schmeling, M., & Schrimpf, A. (2012). Carry trades and global foreign exchange volatility. Journal of Finance, 67(2), 681–718.

    Article  Google Scholar 

  • Moran, M. T. (2014). Thirty Volatility Indexes: Worldwide tools to gauge sentiment and diversify portfolios. Journal of Index Investing, 4(4), 69–87.

    Article  Google Scholar 

  • Neely, C. J., & Weller, P. A. (2013). Lessons from the evolution of foreign exchange trading strategies. Journal of Banking & Finance, 37(10), 3783–3798.

    Article  Google Scholar 

  • Phoa, W. (2013). Portfolio concentration and the geometry of co-movement. Journal of Portfolio Management, 39(4), 142–151.

    Article  Google Scholar 

  • Pojarliev, M., & Levich, R. (2008). Do professional currency managers beat the benchmark? Financial Analysts Journal, 64(5), 18–32.

    Article  Google Scholar 

  • Raffinot, T. (2018). Hierarchical clustering-based asset allocation. Journal of Portfolio Management, 44(2), 89–99.

    Article  Google Scholar 

  • Stoyanov, S. V., Rachev, S. T., Racheva-Yotova, B., & Fabozzi, F. J. (2011). Fat-tailed models for risk estimation. Journal of Portfolio Management, 37(2), 107–117.

    Article  Google Scholar 

  • Taleb, N. N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley.

    Google Scholar 

  • Vidyamurthy, G. (2004). Pairs Trading: Quantitative Methods and Analysis. Wiley.

    Google Scholar 

  • Whaley, R. E. (2009). Understanding the VIX. Journal of Portfolio Management, 35(3), 98–105.

    Article  Google Scholar 

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Authors and Affiliations

  1. LBBW Asset and Wealth Management, Stuttgart, Germany

    Gueorgui S. Konstantinov

  2. EDHEC Business School, Nice, France

    Frank J. Fabozzi

Authors
  1. Gueorgui S. Konstantinov
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  2. Frank J. Fabozzi
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Correspondence to Frank J. Fabozzi.

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Konstantinov, G.S., Fabozzi, F.J. The Geometry of the World of Currency Volatilities. Comput Econ 60, 125–145 (2022). https://doi.org/10.1007/s10614-021-10140-7

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  • DOI: https://doi.org/10.1007/s10614-021-10140-7

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