Geometric and arithmetic realized comoments Open Access

The framework suggested by Andersen et al. (2003) produces low-frequency variance from high-frequency returns. This so-called realized variance is defined as a sum of squares of sub-periodical returns. Kraus and Litzenberger (1976) and Dittmar (2002) demonstrate the relationship between higher-order moments and expected returns, and the concept of the realized variance has been extended to realized higher-order moments. In many studies, including those of Amaya et al. (2015), Sim (2016), Kim (2016), Mei et al. (2017), Kinateder and Papavassiliou (2019), and Ahmed and Al Mafrachi (2021) [¹], a realized kth order moment is defined as a sum of kth orders of sub-periodical returns. However, according to Amaya et al. (2015) and Bae and Lee (2021), these conventional realized higher-order moments can reflect neither the volatility of volatility nor cross-period relation among sub-periodical returns and are, therefore, flawed. Several studies attempt to resolve these problems by providing unbiased realized moments, and such research is summarized in Table 1.

The revised realized moments are developed based on Neuberger's (2012) Aggregation Property, through which the author presents arithmetic and geometric realized third moments using changes in prices and implied variances [²]. Bae and Lee (2021) extend the arithmetic realized moments in two folds. One is the extension of moments to comoments, and the other is an extension of the order from three to four. Furthermore, Fukasawa and Matsushita (2021) provide arithmetic moments of general orders. However, to the best of the author's knowledge, geometric comoments, geometric moments above the third order and arithmetic comoments above the fourth order have not yet been developed.

The current study attempts to complete Table 1. Our first target is the geometric realized moments and comoments. Many financial studies use geometric returns (log-returns) because they have useful features such as time-additivity. Accordingly, Neuberger (2012) proposes geometric realized third moment. To find the missing geometric measures in the aforementioned table, we extend information set to include lower order moments because all the revised moments are obtained through the lower order moments. However, unlike the aforementioned studies, the current research demonstrates that (even generalized) implied variance and covariance do not yield realized third comoment, although they yield realized covariance. Moreover, we reveal that (even generalized) implied third moment does not yield realized fourth moments.

Our second target is the arithmetic realized comoments for general orders. We previously mentioned the usefulness of the log-returns, and as shown in Table 1, arithmetic realized comoments up to the fourth-order are developed. However, arithmetic returns are also as well-used as geometric returns, and financial studies require the estimation of higher-order comoments. For example, Rubinstein (1973) extends the traditional Capital Asset Pricing Model (CAPM)

with $λ=1σM2E[rM−rf]$ to

and Chung et al. (2006) and Hung (2008) demonstrate that comoments above the fourth order are priced. Accordingly, we attempt to identify the realized comoment above the fourth-order under the arithmetic sense. To do so, we extend Fukasawa and Matsushita's (2021) arithmetic realized cumulants [³]. While Neuberger (2012) and Bae and Lee (2021) attempt to obtain all functions satisfying the Aggregation Property given information set, Fukasawa and Matsushita (2021) present a rule among realized cumulants. Adopting their methodology, we obtain arithmetic realized joint cumulants through complete Bell polynomials of lower-order joint cumulants. Our realized measures are unbiased estimators and they can, therefore, overcome the drawbacks of conventional realized measures.

The rest of the paper is organized as follows. Neuberger's (2012) Aggregation Property is reviewed, and generalized geometric moments are defined in section 2. The non-existence of geometric higher order moments and comoments is demonstrated in section 3. Joint cumulants are explained and arithmetic realized joint cumulants outlined in section 4. Finally, concluding remarks are presented in section 5.

Consider a martingale process $St$ and a partition ${t0,t1,…,tN}$ on $[0,T]$ such that $0=t0≤t1≤t2≤…≤tN=T$⁠. Equation (1) holds for $k=2$⁠.

Owing to this relation, $∑j=1N(Stj−Stj−1)2$ is referred to as realized second moment or realized variance. However, Equation (1) does not hold for the higher-order (⁠$k≥3$⁠), which makes obtaining realized higher-order moments non-straightforward. To solve this problem, Neuberger (2012) proposes the aggregation property that generalizes Equation (1) as follows.

Let $X=(Xt,0≤t≤T)$ be an adapted vector-valued stochastic process defined on a filtration. A function $g$ on a vector-valued process X satisfies the AP (aggregation property) if

Owing to the law of the iterated expectations, when a function $g$ satisfies the AP, we have

In this regard, $∑j=1Ng(Xtj−Xtj−1)$ can be called a realized $E0[g(XT−X0)]$⁠.

To develop the realized moments of log returns, $Xt$ needs to contain log prices $st=lnSt$⁠, and additional arguments can contribute to constructing the abundant functions that satisfy the AP. For example, Neuberger (2012) uses $Δs$ and $ΔvN$ that are changes in log price $st$ and specific generalized variance $vtN$⁠, respectively. Furthermore, he demonstrates that $eΔs−1$⁠, $Δs$⁠, $ΔvN$⁠, $eΔs(ΔvN+2Δs)$⁠, and their linear combination satisfy the AP when the stock price $St$ is a martingale. Thus, the following form satisfies the AP.

Moreover, the martingale property yields $Et[gN(lnST−lnSt,vTN−vtN)]=Et[K(lnST−lnSt)]$ for $K(x)=6(xex−2ex+x+2)=x3+O(x4)$⁠. Thus, Neuberger (2012) refers to,

as a realized third moment of log return $lnST−lnS0$⁠. However, the study presents neither any realized comoments nor realized fourth moments. It may be resolved by additional information of their lower-order implied comoments of log returns. According to Neuberger (2012), implied variance contributes to constructing the realized third moment for both arithmetic and log returns. Similarly, Bae and Lee (2021) show that realized comoments for the arithmetic returns require lower-order moments and comoments. Thus, implied covariance and variances of log returns may contribute to the realized third comoment of log returns, and implied variance and third moment of log returns may contribute to the realized fourth moment of log returns.

To consider covariation, we use two martingale processes $S1,t$ and $S2,t$⁠, and their log values are $s1,t$ and $s2,t$⁠, respectively. For the variant functions satisfying the AP, we allow flexibility on the forms of implied comoments, and we define generalized comoments as follows [⁴].

We refer to $Et[fk,l(s1,T−s1,t,s2,T−s2,t)]$ as a generalized (⁠$k$⁠, $l$⁠)-comoment at time $t$ when $fk,l$ is an analytic function such that $fk,l(a,b)akbl→1$ as $(a,b)→(0,0)$⁠. For convenience, we call it a generalized (⁠$k+l$⁠)-moment and replace $fk,l$ with $fk+l$ if $k$ or $l$ is zero.

Equipped with the above log prices processes and implied comoments, we investigate higher-order realized comoments. Consider a partitioned vector process $x=(s1,s2,m)$⁠, where $m$ is a vector process of comoments. When a function $g$ satisfies

with a function $grk,l(⋅,⋅)$ such that

$E0[∑j=1Ng(xtj−xtj−1)]$ is close to ordinary comoment. Thus, a realized comoment is defined as follows.

For a partitioned vector process $x=(s1,s2,m)$ including a vector process $mt$⁠, let us call

a realized (k, l)-comoment if a function $g$ satisfies the AP and is decomposed as follows

where $ϕ$ is a function that satisfies $Et[ϕ(xT−xt)]=0$⁠, and $grk,l$ is a function such that that $grk,l(a,b)akbl→1$ as $(a,b)→(0,0)$⁠. For convenience, we refer to Equation (8) as a realized (⁠$k+l$⁠)- moment if $k$ or $l$ is zero.

Note that when a function $g$ satisfies the AP and has the decomposition in Equation (9), we have

Thus, it is close to the standard comoment when $grk,l(s1,T−s1,0,s2,T−s2,0)≈(s1,T−s1,0)k(s2,T−s2,0)l$⁠.

Based on Definition 2.2, we denote the generalized 2-moment for the asset $i∈{1,2}$ as $vi$ with its underlying function $f2(⋅)$⁠. In addition, let us denote the generalized comoment as $vc$ with its underlying function $f1,1(⋅,⋅)$⁠. We first investigate the function satisfying the AP given the information set $x$ that includes log prices (⁠$s1$⁠, $s2$⁠), variances (⁠$v1$⁠, $v2$⁠) and covariance (⁠$vc$⁠) as follows.

The proof is provided in  Appendix 1.

Proposition 3.1 uses the information of implied covariance $vc$ in addition to the variation of a single process in Neuberger (2012). It makes it possible to obtain new terms that satisfy the AP: the 10th term $(Δv1−2Δs1)(Δv2−2Δs2)$ with conditions (1), (2) and (3), the 11th term $eΔs1(2Δvc−Δv2+2Δs2)$ with condition (1), and 12th term $eΔs2(2Δvc−Δv1+2Δs1)$ with condition (2). These new terms are generalizations of $(Δv1−2Δs1)2$ and $eΔs1(Δv1+2Δs1)$ observed in Neuberger (2012) in that the new terms become these when we set $S2,t$ to be identical to $S1,t$⁠. The new terms may contribute to constructing new realized comoments, and Corollary 3.2 states the result.

The proof is in  Appendix 1.

According to Corollary 3.2, we could not obtain realized (2,1)-comoment even when we have all its lower-order moment and comoment. This result is in contrast to Neuberger (2012) and Bae and Lee (2021), who obtain the realized third moment under both the arithmetic and log return and realized third comoment under the arithmetic return through their lower-order moments. Instead, Corollary 3.2 shows that $Δvc$ with $Δv1$ or $Δv2$ produces the realized (1,1) comoment through

or

Now, let us investigate functions satisfying the AP when the information includes higher-order moments for the single security. They are log price (⁠$s$⁠), implied second moment $(m2)$ and implied third moment $(m3)$⁠, where the underlying function for the $k$th moment is denoted by $fk(⋅)$⁠.

The proof is provided in  Appendix 1.

Proposition 3.3 shows that three terms are satisfying the AP and containing $m3$⁠; the 4th, 5th and 6th terms in Equation (14). The AP of the 4th term $Δm3$ is trivial because it is a (non-transformed) given process. Except for the 4th term, $Δm3$ always appears with $Δm2$ and $a$ as $Δm2+aΔm3$ with specific forms of $f2(Δs)+af3(Δs)$⁠, which satisfies the condition of a generalized second moment. Proposition 3.3 is therefore equivalent to a result under information set $x=(s,m˜2)$ with a generalized second moment $m˜2=m2+am3$ that is obtained from $f˜2=f2+af3$⁠. It implies that the additional information of $m3$ to the information set does not produce any non-trivial function satisfying the AP. Related to this, Corollary 3.4 indicates that there is no realized fourth moment.

The proof is similar to that for Corollary 3.4.

According to section 3, there is some skepticism about the geometric realized higher-order comoments. However, as mentioned in section 1, financial studies state the importance of the higher-order comoments even above the fourth-order. Different from geometric comoments, arithmetic ones up to the fourth-order are available (recall Table 1). This section provides an investigation of the arithmetic comoments of general orders. Strictly speaking, our goal is to present realized joint cumulants. Because these are lesser-known, let us see their definitions.

The lth cumulant of a random variable $Y$ is defined by

The joint cumulant of random variables $Y1,Y2,…,Yl$ is defined by

Recent studies such as Khademalomoom et al. (2019), Ahmed and Al Mafrachi (2021) and Cui et al. (2022) deal with the first six moments. Accordingly, the first six cumulants $κl(Y)$ are described in the second column of Table 2. The cumulants are kinds of normalized moments because $κl(Y)=0$ for $l≥3$ when Y follows a normal distribution. Moreover, a cumulant is a joint cumulant of an identical random variable with itself. In other words,

Moreover, for any constant number a, we have

where $κl−k,k(YM,Y)=κ(YM,⋯,YM,Y,⋯,Y)$ with $l−k$$YM$s and $k$$Y$s, and $κl−k,k(YM,Y)$ is linked to the comoment $E[YMl−kYk]$⁠. For example, $κ0,1(YM,Y)$⁠, … and $κ5,1(YM,Y)$ are described in the third column of Table 2.

Fukasawa and Matsushita (2021) present the relationships between cumulants and the AP, and the result is summarized as follows.

[⁵] where $BL$ is the Lth complete Bell polynomial defined as

and $Xt=(St,Mt(2),Mt(3),…,Mt(L−1),0)$ with $Mt(l)=κlt(ST)$⁠. Equation (19) implies that

is an unbiased estimator of $κL0(ST)$⁠. Therefore, it can overcome drawbacks of conventional realized moments. Accordingly, the authors name Equation (21) the realized Lth cumulant. For illustration, the realized cumulants of orders 2–6 are presented in Table 3. As stated in Neuberger (2012), Amaya et al. (2015), and Bae and Lee (2021), when l is not two, each summand requires additional terms more than $(ΔStj)l$⁠. For example, $ΔMtj(2)ΔStj$ can reflect leverage effect when l = 3, and $ΔMtj(2)(ΔStj)2$ can reflect volatility structure when l = 4.

By extending Fukasawa and Matsushita (2021), we provide realized joint cumulants in Proposition 4.2.

Proof is provided in  Appendix 2.

Based on Proposition 4.2, we can measure the relationship between $S1$ and $S2$ well through $cML−1,1real(S1,S2)$⁠. Because of Equation (23), it is an unbiased estimator of $κL−1,10(S1,T,S2,T)$⁠. Therefore, it can overcome drawbacks of conventional measures. For illustration, detailed forms of $cMl−1,1real(S1,S2)$ up to order six are presented in Table 4. Like the result of Table 3, it shows that the fifth joint cumulant $cM4,1real(S1,S2)$ requires more than $∑j=1N(ΔS1,tj)4ΔS2,tj$⁠. For example, it additionally requires $∑j=1NΔMtj(4,0)ΔS2,tj$⁠, which is related to covariation between the second asset return and the kurtosis of the first asset return. Similarly, $cM5,1real(S1,S2)$ requires more than $∑j=1N(ΔS1,tj)5ΔS2,tj$⁠.