Realized higher-order comoments

Abstract

We propose a new realized third-order comoment and new realized fourth-order joint cumulants, which are standardized comoments. They are obtained from sub-period returns and lower-order comoments and satisfy A. Neuberger’s (Realized skewness. Rev. Financ. Stud., 2012, 25(11), 3423–3455) aggregation property. Different from other realized higher-order comoments obtained from sub-period returns only, those in this study reflect characteristics of the volatility of volatility as well as jump contributions. As a result, our realized kurtosis and coskewness can reflect well-known phenomena such as the positive autocorrelation of volatility or negative correlation between returns and covariances.

Keywords:

• Realized kurtosis
• Realized coskewness
• Aggregation property
• Joint cumulants
• Comoments

JEL Classification:

• G11
• G12
• G13

Acknowledgements

We are grateful to an anonymous referee, Hyoung-Goo Kang, Jangkoo Kang, Hwa-Sung Kim, Sun-Joong Yoon, the editors (Michael Dempster and Jim Gatheral), and the Publishing Editor (Sadie Thrift), for valuable and detailed comments. All errors are the authors’ responsibility.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Let us consider an order $k=4$ and a jump-only martingale process $S_j$ with returns $\mathrm{\Delta }{S}_j=S_j-S_{j-1}$. Then, ${\text{E}}_0\left[{\left(\sum _{j=1}^N\mathrm{\Delta }{S}_j\right)}^4\right]$ contains nonnegative terms ${\text{E}}_0\left[{\left(\mathrm{\Delta }{S}_i\right)}^2{\left(\mathrm{\Delta }{S}_j\right)}^2\right]$ with $i\ne j$, which cannot be captured by the right-hand side of equation (1).

2 Hereafter, when we describe a process $X$ only at $t_j$’s with $j=0,\dots ,N$, $t_j$ is denoted by $j$. In addition, $X_j-X_{j-1}$ and $X_{i,j}-X_{i,j-1}$ are denoted by $\mathrm{\Delta }{X}_j$ and $\mathrm{\Delta }{X}_{i,j}$, respectively, for a (possibly vector-valued) index $i$.

3 The first equality is derived from the martingale property of process S, and the second equality from the aggregation property of a function $g\left(\mathrm{\Delta }S,\mathrm{\Delta }V\right)=(\mathrm{\Delta }S{)}^3+3\mathrm{\Delta }S\mathrm{\Delta }V$.

4 Each element in the information set trivially satisfies the aggregation property by itself. Thus, we can obtain non-trivial solutions for the aggregation property only through (linear combinations of) multiplications of the elements in the information set. In other words, a realized (co)moment would require lower-order (co)moments. For example, as Section 2 shows, the second-order realized moment $\sum (\mathrm{\Delta }S{)}^2$ requires $\mathrm{\Delta }S$, and the third-order realized moment $\sum \left({\left(\mathrm{\Delta }S\right)}^3+3\mathrm{\Delta }S\mathrm{\Delta }V\right)$ requires $\mathrm{\Delta }S$ and $\mathrm{\Delta }V$.

5 The authors are deeply thankful to an anonymous referee for the proof of sufficiency of equation (12) in the first statement of this proposition. Moreover, the referee discovered the last statement. Note that equation (13) uses a finer information set than equation (2) does. Thus, it is a stronger condition than the aggregation property in equation (2).

6 In equation (15), the prefixes $c$ and $TM$ represent co- and third moment, respectively. Additionally, the superscript ‘real’ indicates that the estimator is a realized moment, and the subscripts $a$ and $b$ represent the powers of the variables; thus, $cT{M}_{a,b}\left(S_1,S_2\right)$ denotes ${\text{E}}_0\left[{\left(S_{1,T}-S_{1,0}\right)}^a{\left(S_{2,T}-S_{2,0}\right)}^b\right]$. Later, $TM$ is replaced by $FM$ for the fourth cumulant, which is linked to the fourth moment. Similarly, $cFM$ represents the fourth joint cumulant, which is linked to the fourth comoment. A more detailed description of $FM$ and $cFM$ is provided later.

7 The first equality is from the aggregation property and the identity $M_{k,l,N}=0$ for any $k$ and $l$. The second equality is from the martingale property for each $S_1$ and $S_2$.

8 The approximation is from $E_{j-1}\left[\mathrm{\Delta }{M}_{2,0,j}+{\left(\mathrm{\Delta }{S}_{1,j}\right)}^2\right]=0$.

9 Although nth-order joint cumulants are defined on n-random variables, this study focuses on joint cumulants between two random variables; thus, for example, the subscript 3,1 of the $cFM$ in equation (20) means joint cumulants among three $S_1$s and one $S_2$. In general, the third-order joint cumulant of $\left(X_1,X_2,X_3\right)$ and the fourth-order joint cumulant of $\left(X_1,X_2,X_3,X_4\right)$ are $E\left[X_1{X}_2{X}_3\right]$ and $E\left[X_1{X}_2{X}_3{X}_4\right]-E\left[X_1{X}_2\right]E\left[X_3{X}_4\right]-E\left[X_1{X}_3\right]E\left[X_2{X}_4\right]-E\left[X_1{X}_4\right]E\left[X_2{X}_3\right]$, respectively, when the expectation of each $X_i$ is zero. Therefore, the last line of equation (16) is the third-order joint cumulant as well as the third comoment, because each $S_i$ is a martingale.

10 An element of $m$ (as well as $\alpha$ and $M$) has $\left(k,l\right)$ as a subscript, as in equation (A2). However, for convenience, we interchangeably use a single-number subscript, which represents the order of the element among the components of the vector $m$ (as well as $\alpha$ and $M$).

11 Regardless of the value of $\alpha$, by taking ${\pi }_1$ close to zero and using a large $n$ in equation (A1), we can construct an arbitrary $m$.

12 ${\alpha }_{-i}$ represents $\alpha$ without the i^th element. For example, ${\alpha }_{-1}=\left({\alpha }_{1,1},{\alpha }_{0,2},\cdots ,{\alpha }_{0.3}\right)$. $m_{-i}$ is defined similarly.

13 Recall that $M_{k,l,t}=E_t\left[{\left(S_{1,T}-S_{1,t}\right)}^k{\left(S_{2,T}-S_{2,t}\right)}^l\right]$. Thus, we have $V_t=a^2{M}_{2,0,t}+2ab{M}_{1,1,t}+b^2{M}_{0,2,t}$.

Additional information

Funding

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2018S1A5A8027910).