# Mean-Reverting 4/2 Principal Components Model. Financial Applications

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- We studied in detail a multivariate mean-reverting 4/2 stochastic volatility model based on PCA, which is inspired in the general framework of Cheng et al. (2019). The SC in the new model is decomposed into constant eigenvectors that capture the correlation among assets and a diagonal eigenvector matrix whose entries are modeled by the 4/2 process.
- The PCA structure allows us to find a semi-closed-form c.f. for the vector of returns. It permits the extension to multidimensions of simple but accurate approximation approaches, first introduced in Escobar-Anel and Gong (2020) for one dimension, to find closed-form approximations to the c.f., which are proven to be accurate for realistic parameter settings.
- We use the estimation approaches developed in Escobar-Anel and Gong (2020) to estimate the parameters for special cases of the proposed model. Here, we use two pairs of bivariate time series capturing both the asset and its variance. Estimation of multidimensional processes is rare in the literature, and our work demonstrates that many, but not all, of the parameters are statistically significant, confirming stylized facts of commodity prices and volatility indexes such as stochastic correlation and spill-over effects.
- A risk management application, based on a constant proportion strategies portfolio, for example Merton (1975) and DeMiguel et al. (2009), demonstrates the accuracy of the approximation.

## 2. Model Definition

#### 2.1. General Model Setup

- ${b}_{j}=0$: This is a generalization of Escobar et al. (2010) to multivariate mean-reverting asset classes. If $n=1$, we get the model considered in Benth (2011).
- ${\rho}_{j}=0$: This case applies to the assets whose price series demonstrates an abnormal increase or decrease, but no leverage effect is observed for the assets of interest. The term “leverage effect" was first defined and studied in Black (1976). It describes the negative correlation between an asset’s volatility and its return.
- ${b}_{j}=0,{\rho}_{j}=0$: This case can be generated by either of the two previous cases. It applies better to assets that exhibit mild behavior in their price series; at the same time, no leverage effect is identified.

#### 2.1.1. Separable Spillover Effect

#### 2.1.2. Model with No Spillover Effects

#### 2.2. Properties of the Variance Vector

## 3. Characteristic Functions and Approximations

#### 3.1. Characteristic Function for Model with Spillover Effects

**Corollary**

**1.**

#### 3.2. Characteristic Function for Model with Separable Spillover Effects

**Corollary**

**2.**

#### 3.3. Approximation Principle and Results

**Midpoint:**$m=\frac{\underset{[t,T]}{min}\left(C\left(s\right)\right)+\underset{[t,T]}{max}\left(C\left(s\right)\right)}{2}$, $n=\frac{\underset{[t,T]}{min}\left(B\left(s\right)\right)+\underset{[t,T]}{max}\left(B\left(s\right)\right)}{2}$.**Average:**$m=\frac{1}{T-t}{\int}_{t}^{T}C\left(s\right)ds$, $n=\frac{1}{T-t}{\int}_{t}^{T}B\left(s\right)ds$.

**Corollary**

**3.**

- ${\mathbf{b}}_{\mathbf{j}}=\mathbf{0}$: Given ${b}_{j}=0$, $C\left(s\right)=0$ and $D\left(s\right)=0$, $s\in [t,T]$. If ${n}_{j}\ge -\frac{{\alpha}_{j}^{2}}{2{\xi}_{j}^{2}}$, then$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \mathbb{E}\left[exp\left\{{\int}_{t}^{T}{B}_{j}\left(s\right){\nu}_{j}\left(s\right)ds+{\rho}_{j}\frac{{g}_{j}\left(T\right){\nu}_{j}\left(T\right)}{{\xi}_{j}}\right\}\mid {\mathcal{F}}_{t}\right]\approx \mathbb{E}\left[exp\left\{{\rho}_{j}\frac{{g}_{j}\left(T\right){\nu}_{j}\left(T\right)}{{\xi}_{j}}-{n}_{j}{\int}_{t}^{T}{\nu}_{j}\left(s\right)ds\right\}\mid {\mathcal{F}}_{t}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\left(\frac{({B}_{j}{\xi}_{j}^{2}+\alpha )\left({e}^{\sqrt{{A}_{j}}(T-t)}-1\right)+\sqrt{{A}_{j}}\left({e}^{\sqrt{{A}_{j}}(T-t)}+1\right)}{2\sqrt{{A}_{j}}{e}^{\frac{\sqrt{{A}_{j}}+{\alpha}_{j}}{2}(T-t)}}\right)}^{-\frac{2{\alpha}_{j}{\theta}_{j}}{{\xi}_{j}^{2}}}{e}^{{\nu}_{j}\left(t\right)\left(\frac{({B}_{j}{\alpha}_{j}-2{n}_{j})\left({e}^{\sqrt{{A}_{j}}(T-t)}-1\right)-{B}_{j}\sqrt{{A}_{j}}\left({e}^{\sqrt{{A}_{j}}(T-t)}+1\right)}{({B}_{j}{\xi}_{j}^{2}+{\alpha}_{j})\left({e}^{\sqrt{{A}_{j}}(T-t)}-1\right)+\sqrt{{A}_{j}}\left({e}^{\sqrt{{A}_{j}}(T-t)}+1\right)}\right)},\hfill \\ \hfill {B}_{j}& =-\frac{{\rho}_{j}{g}_{j}\left(T\right)}{{\xi}_{j}},{A}_{j}={\alpha}_{j}^{2}+2{n}_{j}{\xi}_{j}^{2},\hfill \end{array}$$
- ${\mathbf{b}}_{\mathbf{j}}\ne \mathbf{0},{\mathsf{\ae}}_{\mathbf{j}}=\mathbf{0}$: Given ${b}_{j}=0,{\rho}_{j}\ne 0$ and $D\left(s\right)=0$, $s\in [t,T]$. If ${m}_{j}>-\frac{{(2{\alpha}_{j}{\theta}_{j}-{\xi}_{j}^{2})}^{2}}{8{\xi}_{j}^{2}}$, ${n}_{j}\ge -\frac{{\alpha}_{j}^{2}}{2{\xi}_{j}^{2}}$, then$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \mathbb{E}\left[exp\left\{{\int}_{t}^{T}{B}_{j}\left(s\right){\nu}_{j}\left(s\right)ds+{\int}_{t}^{T}{C}_{j}\left(s\right)\frac{1}{{\nu}_{j}\left(s\right)}ds\right\}\mid {\mathcal{F}}_{t}\right]\approx \mathbb{E}\left[exp\left\{-{n}_{j}{\int}_{t}^{T}{\nu}_{j}\left(s\right)ds-{m}_{j}{\int}_{t}^{T}\frac{1}{{\nu}_{j}\left(s\right)}ds\right\}\mid {\mathcal{F}}_{t}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\left(\frac{{\gamma}_{j}(T,{\nu}_{j}\left(t\right))}{2}\right)}^{{k}_{j}+1}{\nu}_{j}{\left(t\right)}^{-\frac{{\alpha}_{j}{\theta}_{j}}{{\xi}_{j}^{2}}}{K}_{j}{\left(T\right)}^{-\left(\frac{1}{2}+\frac{{k}_{j}}{2}+\frac{{\alpha}_{j}{\theta}_{j}}{{\xi}_{j}^{2}}\right)}{e}^{\frac{1}{{\xi}_{j}^{2}}\left({\theta}_{j}(T-t)-\sqrt{{H}_{j}}{\nu}_{j}\left(t\right)coth\left(\frac{\sqrt{{H}_{j}}(T-t)}{2}\right)+{\alpha}_{j}{\nu}_{j}\left(t\right)\right)}\frac{\Gamma \left(\frac{1}{2}+\frac{{k}_{j}}{2}+\frac{{\alpha}_{j}{\theta}_{j}}{{\xi}_{j}^{2}}\right)}{\Gamma ({k}_{j}+1)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times {}_{1}{F}_{1}\left(\frac{1}{2}+\frac{{k}_{j}}{2}+\frac{{\alpha}_{j}{\theta}_{j}}{{\xi}_{j}^{2}},{k}_{j}+1,\frac{{\gamma}_{j}{(T,{\nu}_{j}\left(t\right))}^{2}}{4{K}_{j}\left(T\right)}\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {k}_{j}=\frac{1}{{\xi}_{j}^{2}}\sqrt{{(2{\alpha}_{j}{\theta}_{j}-{\xi}_{j}^{2})}^{2}+8{m}_{j}{\xi}_{j}^{2}},{H}_{j}={\alpha}_{j}^{2}+2{n}_{j}{\xi}_{j}^{2},{\gamma}_{j}(T,{\nu}_{j}\left(t\right))=\frac{2\sqrt{{H}_{j}{\nu}_{j}\left(t\right)}}{{\xi}_{j}^{2}sinh\left(\frac{\sqrt{{H}_{j}}(T-t)}{2}\right)},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {K}_{j}\left(T\right)=\frac{1}{{\xi}_{j}^{2}}\left(\sqrt{{H}_{j}}{\nu}_{j}\left(t\right)coth\left(\frac{\sqrt{{H}_{j}}(T-t)}{2}\right)+{\alpha}_{j}\right).\hfill \end{array}$$

## 4. Estimation

#### 4.1. Data Description

#### 4.2. Estimation of Volatility Group Parameters

#### 4.2.1. Estimation of Matrix $\mathbf{A}$ and the Scaling Parameters S

#### 4.2.2. Estimation of Volatility Group

#### 4.3. Estimation of Drift Group

## 5. Application to Risk Measures

#### 5.1. Portfolio Setup

#### 5.2. The Density Function of the Portfolio $\mathsf{\Pi}\left(t\right)$

#### 5.2.1. Density Function via Convolution

#### 5.2.2. Density Function via Fourier Inversion

**Corollary**

**4.**

#### 5.2.3. Numerical Implementation of Selected Method

- Step 1: Simulate two CIR processes ${\nu}_{1}\left(t\right)$ and ${\nu}_{2}\left(t\right)$ and compute ${\mathsf{\Phi}}_{MR}$ for ${\tilde{M}}_{1}^{*}\left(t\right)$ and ${\tilde{M}}_{2}^{*}\left(t\right)$.
- Step 2: Invert the c.f.s obtained in Step 1 to obtain ${f}_{1}\left({m}_{1}\right|{\mathcal{F}}_{t})$ and ${f}_{2}\left({m}_{2}\right|{\mathcal{F}}_{t})$.
- Step 3: Numerically integrate the product of the conditional density of ${\tilde{M}}_{1}^{*}\left(T\right)$ and ${\tilde{M}}_{2}^{*}\left(T\right)$ for the conditional density function of $ln\left(\mathsf{\Pi}\right(T\left)\right)$.

#### 5.3. The VaR for a Portfolio of USO and GLD

**Case**: ${\pi}_{1}={\pi}_{2}=0.5$

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notes

1 | The estimation methodology for volatility group parameters (Section 4.2) can also be applied to the general model, as for drift group parameters (Section 4.3) some modifications are needed to account for the vector autoregressive structure coming from spillover effects. |

2 | Similar results were obtained for equally weighted assets case (${\pi}_{1}={\pi}_{2}=\frac{1}{3}$). |

## References

- Alexander, Carol. 2001. Orthogonal garch. In Mastering Risk. Financial Times–Prentice Hall: London 2: 21–38. [Google Scholar]
- Andersen, Leif B. G. 2007. Efficient Simulation of the Heston Stochastic Volatility Model; SSRN 946405. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=946405 (accessed on 23 July 2021).
- Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. Coherent measures of risk. Mathematical Finance 9: 203–28. [Google Scholar] [CrossRef]
- Benth, Fred Espen. 2011. The stochastic volatility model of barndorff-nielsen and shephard in commodity markets. Mathematical Finance 21: 595–625. [Google Scholar] [CrossRef]
- Black, Fischer. 1976. Studies of stock market volatility changes. Paper presented at 1976 American Statistical Association Bisiness and Economic Statistics Section, Washington, DC, USA; pp. 177–81. Available online: https://www.scirp.org/(S(i43dyn45teexjx455qlt3d2q))/reference/ReferencesPapers.aspx?ReferenceID=2030459 (accessed on 23 July 2021).
- Campell, John Y., Andrew W. Lo, and A. Craig MacKinlay. 1997. The Econometrics of Financial Markets. Princeton: Princeton University Press. [Google Scholar]
- Carr, Peter, and Dilip Madan. 1999. Option valuation using the fast fourier transform. Journal of Computational Finance 2: 61–73. [Google Scholar] [CrossRef] [Green Version]
- Cheng, Peng, and Olivier Scaillet. 2007. Linear-quadratic jump-diffusion modeling. Mathematical Finance 17: 575–98. [Google Scholar] [CrossRef] [Green Version]
- Cheng, Yuyang, Marcos Escobar-Anel, and Zhenxian Gong. 2019. Generalized mean-reverting 4/2 factor model. Journal of Risk and Financial Management 12: 159. [Google Scholar] [CrossRef] [Green Version]
- Christoffersen, Peter, Steven Heston, and Kris Jacobs. 2009. The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science 55: 1914–32. [Google Scholar] [CrossRef] [Green Version]
- Cui, Zhenyu, Justin Lars Kirkby, and Duy Nguyen. 2021. Efficient simulation of generalized sabr and stochastic local volatility models based on markov chain approximations. European Journal of Operational Research 290: 1046–62. [Google Scholar] [CrossRef]
- Da Fonseca, José, Martino Grasselli, and Claudio Tebaldi. 2007. Option pricing when correlations are stochastic: An analytical framework. Review of Derivatives Research 10: 151–80. [Google Scholar] [CrossRef]
- De Col, Alvise, Alessandro Gnoatto, and Martino Grasselli. 2013. Smiles all around: Fx joint calibration in a multi-heston model. Journal of Banking & Finance 37: 3799–818. [Google Scholar]
- DeMiguel, Victor, Lorenzo Garlappi, and Raman Uppal. 2009. Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? The Review of Financial Studies 22: 1915–53. [Google Scholar] [CrossRef] [Green Version]
- Escobar, Marcos. 2018. A stochastic volatility factor model of heston type. statistical properties and estimation. Stochastics 90: 172–99. [Google Scholar] [CrossRef]
- Escobar, Marcos, Barbara Götz, Luis Seco, and Rudi Zagst. 2010. Pricing a cdo on stochastically correlated underlyings. Quantitative Finance 10: 265–77. [Google Scholar] [CrossRef]
- Escobar, Marcos, and Christoph Gschnaidtner. 2018. A multivariate stochastic volatility model with applications in the foreign exchange market. Review of Derivatives Research 21: 1–43. [Google Scholar] [CrossRef]
- Escobar, Marcos, and Pablo Olivares. 2013. Pricing of mountain range derivatives under a principal component stochastic volatility model. Applied Stochastic Models in Business and Industry 29: 31–44. [Google Scholar] [CrossRef]
- Escobar-Anel, Marcos, and Zhenxian Gong. 2020. The mean-reverting 4/2 stochastic volatility model: Properties and financial applications. Applied Stochastic Models in Business and Industry 36: 836–856. [Google Scholar] [CrossRef]
- Escobar-Anel, Marcos, and Harold A. Moreno-Franco. 2019. Dynamic portfolio strategies under a fully correlated jump-diffusion process. Annals of Finance 15: 421–53. [Google Scholar] [CrossRef]
- Gouriéroux, Christian. 2006. Continuous time wishart process for stochastic risk. Econometric Reviews 25: 177–217. [Google Scholar] [CrossRef]
- Grasselli, Martino. 2016. The 4/2 stochastic volatility model: A unified approach for the heston and the 3/2 model. In Mathematical Finance. Hoboken: Wiley Online Library. [Google Scholar]
- Heston, Steven L. 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6: 327–43. [Google Scholar] [CrossRef] [Green Version]
- Kirkby, J. Lars, and Duy Nguyen. 2020. Efficient asian option pricing under regime switching jump diffusions and stochastic volatility models. Annals of Finance 16: 307–51. [Google Scholar] [CrossRef]
- Langetieg, Terence C. 1980. A multivariate model of the term structure. The Journal of Finance 35: 71–97. [Google Scholar]
- Larsen, Linda Sandris. 2010. Optimal investment strategies in an international economy with stochastic interest rates. International Review of Economics & Finance 19: 145–65. [Google Scholar]
- Lin, Wei, Shenghong Li, Xingguo Luo, and Shane Chern. 2017. Consistent pricing of vix and equity derivatives with the 4/2 stochastic volatility plus jumps model. Journal of Mathematical Analysis and Applications 447: 778–97. [Google Scholar] [CrossRef] [Green Version]
- Luo, Xingguo, and Jin E. Zhang. 2012. The term structure of vix. Journal of Futures Markets 32: 1092–123. [Google Scholar] [CrossRef]
- McNeil, Alexander, Rudiger Frey, and Paul Embrechts. 2005. Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton Series in Finance; Princeton: Princeton University Press. [Google Scholar]
- Merton, Robert C. 1975. Optimum consumption and portfolio rules in a continuous-time model. In Stochastic Optimization Models in Finance. Amsterdam: Elsevier, pp. 621–61. [Google Scholar]
- Muhle-Karbe, Johannes, Oliver Pfaffel, and Robert Stelzer. 2012. Option pricing in multivariate stochastic volatility models of ou type. SIAM Journal on Financial Mathematics 3: 66–94. [Google Scholar] [CrossRef] [Green Version]
- Platen, Eckhard. 1997. A non-linear stochastic volatility model. In Financial Mathematics Research Report No. FMRR005-97. Canberra: Center for Financial Mathematics, Australian National University. [Google Scholar]
- Stuart, Alan, Steven Arnold, J. Keith Ord, Anthony O’Hagan, and Jonathan Forster. 1994. Kendall’s Advanced Theory of Statistics. Hoboken: Wiley. [Google Scholar]
- Zhang, Jin E., and Yingzi Zhu. 2006. Vix futures. Journal of Futures Markets 26: 521–31. [Google Scholar] [CrossRef]

**Figure 7.**Case 1: Density and histogram for ${\tilde{M}}_{1}^{*}\left(t\right)$ and ${\tilde{M}}_{2}^{*}\left(t\right)$.

**Table 1.**Empirical results (

**a**) covariance matrix and long term average of squared volatility indexes; (

**b**) eigenvectors, eigenvalues and scaling factors

(a) | |||

VIX & VSTOXX | USO & GLD | SLV & GLD | |

$\widehat{\mathsf{\Sigma}}$ | $\left(\begin{array}{cc}0.0062& 0.0030\\ 0.0030& 0.0048\end{array}\right)$ | $\left(\begin{array}{cc}4.78\times {10}^{-4}& 4.62\times {10}^{-5}\\ 4.62\times {10}^{-5}& 1.255\times {10}^{-5}\end{array}\right)$ | $\left(\begin{array}{cc}0.0724& 0.0331\\ 0.0331& 0.0237\end{array}\right)$ |

$({\widehat{\mu}}_{1},{\widehat{\mu}}_{2})$ | (0.0034, 0.0028) | (6.01$\times {10}^{-4}$, 1.67$\times {10}^{-4}$) | (3.62$\times {10}^{-4}$, 1.18$\times {10}^{-4}$) |

(b) | |||

VIX & VSTOXX | USO & GLD | SLV & GLD | |

$\widehat{\mathbf{A}}$ | $\left(\begin{array}{cc}0.7825& -0.6226\\ 0.6226& 0.7825\end{array}\right)$ | $\left(\begin{array}{cc}0.9918& -0.1278\\ 0.1278& 0.9918\end{array}\right)$ | $\left(\begin{array}{cc}0.8925& -0.451\\ 0.451& 0.8925\end{array}\right)$ |

$\dot{\mathbf{A}}$ | $\left(\begin{array}{cc}2.725& -1.725\\ -1.725& 2.725\end{array}\right)$ | $\left(\begin{array}{cc}1.0169& -0.0169\\ -0.0169& 1.0169\end{array}\right)$ | $\left(\begin{array}{cc}1.3429& -0.3429\\ -0.3429& 1.3429\end{array}\right)$ |

(${\sigma}^{\left(1\right)},{\sigma}^{\left(2\right)}$) | (2.1372, 0.592) | (0.121, 0.0299) | (0.0891, 0.007) |

(${\widehat{s}}_{1},{\widehat{s}}_{2}$) | (1.8074, 1.7229) | (0.795, 0.7518) | (0.799, 0.8) |

Data: ${\mathit{V}}_{1}\left(\mathit{t}\right)$ | Data: ${\mathit{V}}_{2}\left(\mathit{t}\right)$ | |
---|---|---|

$\widehat{\mathbf{b}}$ | 3.11 $\times {10}^{-4}$ | $\sim 0$ |

Mean of $\widehat{\mathbf{b}}$ (s.e) | 4.511$\times {10}^{-4}$(1.1581$\times {10}^{-6}$) | 0(0) |

$\widehat{\alpha}$ | 42.1811 | 19.1624 |

Mean of $\widehat{\alpha}$ (s.e) | 41.6498(0.0695) | 19.5827(0.0552) |

$\widehat{\theta}$ | 0.0079 | 0.0027 |

Mean of $\widehat{\theta}$ (s.e) | 0.0076(4.8578$\times {10}^{-6}$) | 0.0027(8.0796$\times {10}^{-6}$) |

$\widehat{\xi}$ | 0.3436 | 0.3885 |

Mean of $\widehat{\xi}$ (s.e) | 0.3537(1.3937$\times {10}^{-4}$) | 0.3877(5.1103$\times {10}^{-4}$) |

Data: ${\mathit{V}}_{1}\left(\mathit{t}\right)$ | Data: ${\mathit{V}}_{2}\left(\mathit{t}\right)$ | |
---|---|---|

$\widehat{\mathbf{b}}$ | 1.6776$\times {10}^{-5}$ | $\sim 0$ |

Mean of $\widehat{\mathbf{b}}$ (s.e) | 7.5578$\times {10}^{-5}$(1.3892$\times {10}^{-6}$) | 0(0) |

$\widehat{\alpha}$ | 3.62 | 5.3597 |

Mean of $\widehat{\alpha}$ (s.e) | 5.9253(0.0465) | 5.6994(0.0186) |

$\widehat{\theta}$ | 8.9803$\times {10}^{-4}$ | 1.1859$\times {10}^{-4}$ |

Mean of $\widehat{\theta}$ (s.e) | 4.6621$\times {10}^{-4}$(7.4231$\times {10}^{-6}$) | 1.1779$\times {10}^{-4}$(2.4962$\times {10}^{-7}$) |

$\widehat{\xi}$ | 0.0271 | 0.0231 |

Mean of $\widehat{\xi}$ (s.e) | 0.02(2.4446$\times {10}^{-4}$) | 0.0231(5.3704$\times {10}^{-6}$) |

Data: ${\mathit{V}}_{1}\left(\mathit{t}\right)$ | Data: ${\mathit{V}}_{2}\left(\mathit{t}\right)$ | |
---|---|---|

$\widehat{\mathbf{b}}$ | 2.0968$\times {10}^{-5}$ | $\sim 0$ |

Mean of $\widehat{\mathbf{b}}$ (s.e) | 7.4123$\times {10}^{-5}$(2.85$\times {10}^{-7}$) | 0(0) |

$\widehat{\alpha}$ | 5.178 | 24.3083 |

Mean of $\widehat{\alpha}$ (s.e) | 7.7687(0.0664) | 24.8349(0.057) |

$\widehat{\theta}$ | 8.3026$\times {10}^{-4}$ | 3.8644$\times {10}^{-5}$ |

Mean of $\widehat{\theta}$ (s.e) | 4.6417$\times {10}^{-4}$(7.5518$\times {10}^{-6}$) | 3.8753$\times {10}^{-5}$(6.5274$\times {10}^{-8}$) |

$\widehat{\xi}$ | 0.0307 | 0.0343 |

Mean of $\widehat{\xi}$ (s.e) | 0.024(3.0304$\times {10}^{-4}$) | 0.0344(1.1187$\times {10}^{-5}$) |

Data | $\widehat{\tilde{{\mathit{L}}_{\mathit{i}}}}$ | p-Value | $\widehat{\tilde{{\mathit{c}}_{\mathit{i}}}}$ | p-Value | $\widehat{\tilde{{\mathit{\beta}}_{\mathit{i}}}}$ | p-Value | $\widehat{\mathit{\rho}}$ | p-Value |
---|---|---|---|---|---|---|---|---|

${M}_{1}\left(t\right)$&${V}_{1}\left(t\right)$ | 25.6041 | 0 | −114.6521 | 0.436 | 6.1077 | 0 | 0.5621 | 0 |

${M}_{2}\left(t\right)$&${V}_{2}\left(t\right)$ | 9.4459 | 0 | 73.6448 | 0.734 | 15.6266 | 0 | 0.00419 | 0 |

Data | $\widehat{\tilde{{\mathit{L}}_{\mathit{i}}}}$ | p-Value | $\widehat{\tilde{{\mathit{c}}_{\mathit{i}}}}$ | p-Value | $\widehat{\tilde{{\mathit{\beta}}_{\mathit{i}}}}$ | p-Value | $\widehat{\mathit{\rho}}$ | p-Value |
---|---|---|---|---|---|---|---|---|

${M}_{1}\left(t\right)$&${V}_{1}\left(t\right)$ | 0.8096 | 0.154 | −416.2006 | 0.214 | 0.214 | 0.134 | −0.3723 | 0 |

${M}_{2}\left(t\right)$&${V}_{2}\left(t\right)$ | 2.6418 | 0.07 | −646.7339 | 0.384 | 0.5701 | 0.079 | −0.00294 | 0 |

Data | $\widehat{\tilde{{\mathit{L}}_{\mathit{i}}}}$ | p-Value | $\widehat{\tilde{{\mathit{c}}_{\mathit{i}}}}$ | p-Value | $\widehat{\tilde{{\mathit{\beta}}_{\mathit{i}}}}$ | p-Value | $\widehat{\mathit{\rho}}$ | p-Value |
---|---|---|---|---|---|---|---|---|

${M}_{1}\left(t\right)$&${V}_{1}\left(t\right)$ | 4.5459 | 0.009 | 842.936 | 0.153 | 1.0183 | 0.008 | −0.2323 | 0 |

${M}_{2}\left(t\right)$&${V}_{2}\left(t\right)$ | −1.5092 | 0.23 | −1787.0546 | 0.16 | −0.5401 | 0.191 | 0.1228 | 0 |

Data | $\widehat{\mathbf{L}}$ | $\widehat{\mathbf{C}}$ | $\widehat{\mathbf{B}}$ |
---|---|---|---|

VIX&VSTOXX | (25.9172, 8.5495) | $\left(\begin{array}{cc}-89.4091& 46.0451\\ -71.1886& -57.3209\end{array}\right)$ | $\left(\begin{array}{cc}9.7978& -4.6378\\ -4.6378& 11.9365\end{array}\right)$ |

USO&GLD | (0.4653, 2.7236) | $\left(\begin{array}{cc}-412.2959& -82.6444\\ -53.1823& 641.9225\end{array}\right)$ | $\left(\begin{array}{cc}0.2198& -0.0451\\ -0.0451& 0.5643\end{array}\right)$ |

SLV&GLD | (4.7379, 0.7032) | $\left(\begin{array}{cc}752.7178& -805.8599\\ 380.2658& 1595.3\end{array}\right)$ | $\left(\begin{array}{cc}0.7013& 0.6273\\ 0.6273& -0.2231\end{array}\right)$ |

Data | ${\mathbf{B}}^{-1}\mathbf{L}$ | $\mathbb{E}\left[\mathit{H}\left({\mathbf{V}}_{\mathit{t}}\right)\right]$ | Estimated Mean-Reverting Level | Empirical Averages |
---|---|---|---|---|

VIX&VSTOXX | (2.9042, 3.0832) | (−0.0952, −0.0631) | (2.809, 3.0219) | (2.6497, 2.9139) |

USO&GLD | (3.1598, 5.0794) | (−0.961, −0.2547) | (2.244, 4.8247) | (3.1342, 4.8096) |

SLV&GLD | (2.7242, 4.5075) | (0.2215, 0.2158) | (2.9457, 4.7232) | (2.9468, 4.8645) |

Simulation | Density w/o Approximation | Approx. Density (M) | Approx. Density (A) | |
---|---|---|---|---|

$Va{R}_{0.05}$ | 2.7833 (0.004006) | 2.783 (0.00401) | 2.7832 (0.003988) | 2.783 (0.004008) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Escobar-Anel, M.; Gong, Z.
Mean-Reverting 4/2 Principal Components Model. Financial Applications. *Risks* **2021**, *9*, 141.
https://doi.org/10.3390/risks9080141

**AMA Style**

Escobar-Anel M, Gong Z.
Mean-Reverting 4/2 Principal Components Model. Financial Applications. *Risks*. 2021; 9(8):141.
https://doi.org/10.3390/risks9080141

**Chicago/Turabian Style**

Escobar-Anel, Marcos, and Zhenxian Gong.
2021. "Mean-Reverting 4/2 Principal Components Model. Financial Applications" *Risks* 9, no. 8: 141.
https://doi.org/10.3390/risks9080141