# Estimating Forward-Looking Stock Correlations from Risk Factors

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- It is symmetric;
- (ii)
- All its elements lie within the $[-1,1]$ interval;
- (iii)
- It has unit diagonal;
- (iv)
- It is positive semi-definite (psd).

- (v)
- The matrix is ex ante free of arbitrage;
- (vi)
- The values have a realistic structure.

## 2. Discussion on Existing Models

- Equi-Correlations

- Local Equi-Correlations

- Adjusted Ex Post

- Skewness Approach

## 3. Solutions from Factor Structures

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Definition**

**1.**

**Definition**

**2.**

#### 3.1. Quantitative Approach: Computing Nearest Implied

#### 3.1.1. Formulating the Problem

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

#### 3.1.2. Numerical Method

#### 3.2. Economic Approach

## 4. Empirical Experiment

^{®}Core™ i5-8250U CPU with 1.60 GHz, using the statistical programming software R.

_{avg}. The factor solely relying on implied data resulted by far in the highest fn

_{avg}. However, similarity to the target matrix plays an subordinate role here. Similar to that of fn

_{avg}, a reduction pattern can also be seen for $\tilde{\alpha}$ among hybrid models. In Section 3.2, $\tilde{\alpha}$ was defined as the weight of the boundary and $(1-\tilde{\alpha})$ as the weight of ${X}_{\mathbb{P}}$ inside the risk-neutral factor correlations ${X}_{\mathbb{Q}}$. This means that the larger the number of risk factors, the less modification was required to match the observed implied market variance in this empirical test. Thus, a larger k is likely to explain the hidden (true) implied correlation matrix better. This time, comparing the purely forward looking measure to the hybrid estimations shows that when implied data were used, less modification was required. Generally, as $\tilde{\alpha}$${}_{\mathrm{avg}}$ was close to zero across all five economic models, it can be concluded that correlation risk premia enter modestly in the factor-structured implied correlation matrix framework.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Non-Gaussian Copula: Example of Variance-Gamma

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**Figure 1.**Visualization of the constraints for the two-asset/one-factor/one-market-constraint case. The inequality constraint $\Omega $ spans the gray box of mathematically feasible solutions, and the blue line defines the solutions which satisfy the market constraint. As can be seen, the market constraint actually consists of two convex curves. Hence, two orthogonal projections of X onto $g(X)=0$ exist, but only one (i.e., $\dot{X}$) has minimum distance to X.

**Table 1.**The evaluation of existing models with respect to the developed requirements for realistic implied correlation matrices. A check mark ✓ indicates that the model is compliant with the condition. If it does not guarantee to comply, then it is marked with a ✗. A realistic solution has to fulfill all constraints (i)–(vi).

Mathematical | Economical | |||||
---|---|---|---|---|---|---|

(i) | (ii) | (iii) | (iv) | (v) | (vi) | |

Symmetric | |C_{ij}| ≤ 1 | Unit Diag. | Pos. Semi-Def. | Arbitrage-Free | Hetero. Struct. | |

Model: | ||||||

Equi-Correlations | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ |

Local Equi-Correlations | ✓ | ✓ | ✓ | ✗ | ✓ | ✗ |

Adj. Ex Post of [9], $AEP1$ | ✓ | ✓ | ✓ | ✗ | ✓ | ✓ |

Adj. Ex Post of [29], $AEP2$ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ |

Skewness Approach | ✓ | ✗ | ✓ | ✓ | ✗ | ✓ |

**Table 2.**Summary statistics of computing nearest (quantitative approach, Panels A–B) and risk-factor structured (economic approach, Panel C) implied correlation matrices on monthly option data. As for the quantitative approach, it turns out that computations were carried out in a small amount of time, converging towards the optimal solution within few iterations. The nearest factor-structured matrix can thus be used either as a stand-alone estimate, or as a tool to repair positive semi-definiteness and invertibility. Using the economic approach, implied as well as common risk factors from models, such as CAPM or Fama–French, can be used to estimate the implied correlation matrix.

k | A | t_{avg} | t_{sd} | fn_{avg} | fn_{sd} | |v.tol|_{avg} | |v.tol|_{max} | iter_{avg} | iter_{sd} | Index | |
---|---|---|---|---|---|---|---|---|---|---|---|

Panel A: | hist. matrix | ||||||||||

SP100 | 1 | hist. | 0.051 | 0.024 | 159.9 | 172.8 | 5.1 × ${10}^{-11}$ | 1.5 × ${10}^{-8}$ | 3.037 | 1.491 | SP100 |

SP100 | 3 | hist. | 0.106 | 0.050 | 111.0 | 168.3 | 1.3 $\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 6.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 4.819 | 2.137 | SP100 |

SP100 | 5 | hist. | 0.140 | 0.077 | 102.7 | 166.9 | 1.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 9.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 4.990 | 1.977 | SP100 |

mean-reverting matrix | |||||||||||

SP100 | 1 | m.r. | 0.055 | 0.027 | 158.0 | 158.2 | 2.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 2.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 3.064 | 1.438 | SP100 |

SP100 | 3 | m.r. | 0.107 | 0.047 | 113.3 | 173.2 | 3.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 9.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 4.708 | 1.790 | SP100 |

SP100 | 5 | m.r. | 0.149 | 0.082 | 104.6 | 172.5 | 2.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 8.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 5.054 | 1.989 | SP100 |

Panel B: | |||||||||||

SP500 | 1 | hist. | 5.347 | 2.311 | 2847.9 | 4395.0 | 1.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 5.003 | 1.892 | SP500 |

Repaired ${C}_{\mathbb{Q}}^{AEP1}$ | 15 | ${\mathrm{m}.\mathrm{r}.}_{\mathbb{Q}}$ | 0.223 | 0.070 | 16.41 | 6.775 | 3.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 9.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 7.184 | 2.351 | SP100 |

SQP | 1 | hist. | 0.337 | 0.098 | 161.4 | 166.8 | 3.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 9.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 3.517 | 0.721 | SP100 |

Panel C: | $\tilde{\alpha}$${}_{\mathrm{avg}}$ | $\tilde{\alpha}$${}_{\mathrm{sd}}$ | |||||||||

CAPM | 1 | skew${}_{\mathbb{Q}}$ | 0.018 | 0.014 | 4925.6 | 4548.0 | 5.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ | 4.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ | 0.099 | 0.083 | SP500 |

CAPM | 1 | hist. | 0.017 | 0.018 | 5108.5 | 5858.9 | 3.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ | 4.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ | 0.138 | 0.133 | SP500 |

Fama-Fr.3 | 3 | hist. | 0.019 | 0.018 | 4376.6 | 4740.8 | 3.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ | 2.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ | 0.091 | 0.070 | SP500 |

FF3+Mom. | 4 | hist. | 0.019 | 0.019 | 4167.1 | 4381.3 | 3.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ | 3.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ | 0.082 | 0.060 | SP500 |

Fama-Fr.5 | 5 | hist. | 0.019 | 0.024 | 4125.3 | 4509.1 | 3.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-17}$ | 3.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ | 0.076 | 0.055 | SP500 |

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**MDPI and ACS Style**

Schadner, W.; Traut, J.
Estimating Forward-Looking Stock Correlations from Risk Factors. *Mathematics* **2022**, *10*, 1649.
https://doi.org/10.3390/math10101649

**AMA Style**

Schadner W, Traut J.
Estimating Forward-Looking Stock Correlations from Risk Factors. *Mathematics*. 2022; 10(10):1649.
https://doi.org/10.3390/math10101649

**Chicago/Turabian Style**

Schadner, Wolfgang, and Joshua Traut.
2022. "Estimating Forward-Looking Stock Correlations from Risk Factors" *Mathematics* 10, no. 10: 1649.
https://doi.org/10.3390/math10101649