Geometric and arithmetic realized comoments

Abstract

The author investigates realized comoments that overcome the drawback of conventional ones and derive the following findings. First, the author proves that (even generalized) geometric implied lower-order comoments yield neither geometric realized third comoment nor fourth moment. This is in contrast to previous studies that produce geometric realized third moment and arithmetic realized higher-order moments through lower-order implied moments. Second, arithmetic realized joint cumulants are obtained through complete Bell polynomials of lower-order joint cumulants. This study’s realized measures are unbiased estimators and they can, therefore, overcome the drawbacks of conventional realized measures.

Citation

Bae, K. (2022), "Geometric and arithmetic realized comoments", Journal of Derivatives and Quantitative Studies: 선물연구, Vol. 30 No. 2, pp. 89-113. https://doi.org/10.1108/JDQS-11-2021-0028

1. Introduction

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)

E[ri]=rf+λE[(rM−E[rM])(ri−E[ri])]

with λ=1σM2E[rM−rf] to

E[ri]=rf+∑l=2∞λlE[(rM−E[rM])l−1(ri−E[ri])],

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.

2. Preliminary: aggregation property and generalized geometric realized moments

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.

(1)E0[(ST−S0)k]=E0[∑j=1N(Stj−Stj−1)k]

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.

Definition 2.1.

Aggregation property

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

(2)Er[g(Xu−Xr)]=Er[g(Xu−Xt)]+Er[g(Xt−Xr)],∀(r,t,u)0≤r≤t≤u≤T.

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

(3)E0[g(XT−X0)]=E0[∑j=1Ng(Xtj−Xtj−1)].

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.

(4)gN(Δs,ΔvN)=−12(eΔs−1)+6Δs−3ΔvN+3eΔs(ΔvN+2Δs)=3ΔvN(eΔs−1)+6(ΔseΔs−2eΔs+Δs+2)

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,

(5)∑j=1NgN(lnSj−lnSj−1,vtjN−vtj−1N)

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].

Definition 2.2.

(Implied) generalized (k, l)-comoment

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

(6)E0[∑j=1Ng(xtj−xtj−1)]=E0[grk,l(s1,T−s1,0,s2,T−s2,0)]

with a function grk,l(⋅,⋅) such that

(7)grk,l(s1,T−s1,0,s2,T−s2,0)≈(s1,T−s1,0)k(s2,T−s2,0)l,

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

Definition 2.3.

Realized (k,l)-comoment

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

(8)∑j=1Ng(xtj−xtj−1)

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

(9)g(xτ−xt)=ϕ(xτ−xt)+grk,l(s1,τ−s1,t,s2,τ−s2,t),

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

(10)E0[∑j=1Ng(xtj−xtj−1)]=E0[g(xT−x0)]=E0[grk,l(s1,T−s1,0,s2,T−s2,0)].

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.

3. Nonexistence of geometric realized higher-order comoments

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.

Proposition 3.1.

An analytic function g satisfies the AP on the vector valued process x=(s1,s2,v1,v2,vc) if and only if g is represented as follows:

(11) g(Δs1,Δs2,Δv1,Δv2,Δvc)=h1(eΔs1−1)+h2Δs1+h3(eΔs2−1)+h4Δs2+h5Δv1+h6Δv2+h7Δvc+h8(Δv1−2Δs1)2+h9(Δv2−2Δs2)2+h10(Δv1−2Δs1)(Δv2−2Δs2)+h11eΔs1(2Δvc−Δv2+2Δs2)+h12eΔs2(2Δvc−Δv1+2Δs1)+h13eΔs1(Δv1+2Δs1)+h14eΔs2(Δv2+2Δs2)

for some constants h1,⋯,h14, which satisfy one of the following five conditions:

1. h12=h13=h14=0, f1,1(Δs1,Δs2)=Δs2(eΔs1−1) and f2(Δs)=2(eΔs−Δs−1),

2. h11=h13=h14=0, f1,1(Δs1,Δs2)=Δs1(eΔs2−1) and f2(Δs)=2(eΔs−Δs−1),

3. h11=h12=h13=h14=0 and f2(Δs)=2(eΔs−Δs−1),

4. h8=h9=h10=h11=h12=0 and f2(Δs)=2(ΔseΔs−Δs+1),

5. h8=h9=h10=h11=h12=h13=h14=0.

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.

Corollary 3.2.

When the information set is given by x=(s1,s2,v1,v2,vc), there is not a realized (2,1)-comoment but a realized (1,1)-comoment.

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

(12)g(Δs1,Δs2,Δv2,Δvc)=−Δvc+12Δv2+12eΔs1(2Δvc−Δv2+2Δs2)−Δs2=(eΔs1−1)(Δvc−12Δv2)+(eΔs1−1)Δs2,

or

(13)g(Δs1,Δs2,Δv2,Δvc)=(eΔs2−1)(Δvc−12Δv1)+(eΔs2−1)Δs1.

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 kth moment is denoted by fk(⋅).

Proposition 3.3.

An analytic function g on a vector valued process x=(s,m2,m3) has the Aggregation Property on the vector valued process x if and only if g is represented as follows:

(14) g(Δs,Δm2,Δm3)=h1(eΔs−1)+h2Δs+h3Δm2+h4Δm3+h5(Δm2+aΔm3−2Δs)2+h6(Δm2+aΔm3+2Δs)eΔs

for some constants h1,⋯,h6 and a, which satisfy one of the following three conditions:

1. h5=h6=0.

2. h6=0 and f2(Δs)+af3(Δs)=2(eΔs−Δs−1) for the constant a.

3. h5=0 and f2(Δs)+af3(Δs)=2(ΔseΔs−eΔs+1) for the constant a.

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.

Corollary 3.4.

When the information set is given by x=(s,m2,m3), there is no realized 4-moment.

The proof is similar to that for Corollary 3.4.

4. Arithmetic realized joint cumulants

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.

Definition 4.1.

Cumulants and joint cumulants

The lth cumulant of a random variable Y is defined by

(15)κl(Y)=∂l∂ullnE[exp(uY)]|u=0.

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

(16)κ(Y1,Y2,…,Yl)=∂l∂u1∂u2⋯∂ullnE[exp(∑i=1luiYi)]|u1=⋯=ul=0.

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,

(17)κl(Y)=κ(Y1,…,Yl), for Y1=⋯=Yl=Y.

Moreover, for any constant number a, we have

(18)κl(YM+aY)=∑k=0lak(lk)κl−k,k(YM,Y),

where κl−k,k(YM,Y)=κ(YM,⋯,YM,Y,⋯,Y) with l−k YMs and k Ys, 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.

(19)E0[∑j=1NBL(Xtj−Xtj−1)]=E0[BL(XT−X0)]=κL0(ST), [5]

where BL is the Lth complete Bell polynomial defined as

(20)BL(y1,⋯,yL)=∂L∂uLexp(∑l=1Lull!yl)|u=0,

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

(21)∑j=1NBL(Xtj−Xtj−1)

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.

Proposition 4.2.

For martingale processes S1,t and S2,t, let us define cML−1,1real(S1,S2) as follows

(22) cML−1,1real(S1,S2)=∑j=1N∑k=1L−1(L−1k)BL−k(ΔMtj(1,0),⋯,ΔMtj(L−k,0))ΔMtj(k−1,1)

with ΔMtj(l−k,k)=Mtj(l−k,k)−Mtj−1(l−k,k) and Mt(l−k,k)=κl−k,kt(S1,T,S2,T). Then, we have

(23) E0[cML−1,1real(S1,S2)]=κL−1,10(S1,T,S2,T).

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.

5. Concluding remarks

Neuberger (2012), Bae and Lee (2021), and Fukasawa and Matsushita (2021) demonstrate that realized third geometric moments and realized arithmetic moments of any orders are obtained by combining their lower-order implied moments and comoments. Extending the information set is therefore a natural trial to yield the higher order moments and comoments. Unlike previous studies, we show that geometric lower-order implied comoments do not yield geometric realized fourth moment and third comoment but yield geometric realized covariance only. The main reason for the non-existence is that the extension of the geometric information set does not produce additional non-trivial terms; the productions are only transformations of Neuberger (2012). Although this approach does not yield a meaningful measure, presenting this result can prevent the same trial and error for other scholars.

Furthermore, we yield the arithmetic realized lth joint cumulants, which are linked to E[(S1,T−S1,0)l−1(S2,T−S2,0)]. Several financial theories apply them; for example, the extended CAPM includes E[(rM−E[rM])l−1(ri−E[ri])] for l≥2. Given the drawbacks of conventional realized comoments, we believe that empirical studies can use our measure in the future.

Depending on combinations of assets, there are other joint cumulants such as E[(rM−E[rM])l−2(ri−E[ri])2] or E[(rM−E[rM])l−3(ri−E[ri])3]. We do not investigate them because they currently seem irrelevant to financial studies. However, we may obtain them as proof of Proposition 4.2 when the financial studies require them.

Table 1

Revised realized moments and comoments

Order	Arithmetic realized moments	Geometric realized moments
Panel A. Realized moments
3	Neuberger (2012)	Neuberger (2012)
4	Bae and Lee (2021)	-
Above 4	Fukasawa and Matsushita (2021)	-
Panel B. Realized comoments
3	Bae and Lee (2021)	-
4	Bae and Lee (2021)	-
Above 4	-	-

Table 2

The first six cumulants

l	κl(Y)	κl−1,1(YM,Y)
1	E[Y]	E[Y]
2	E[Yˆ2]	E[YˆMYˆ]
3	E[Yˆ3]	E[YˆM2Yˆ]
4	E[Yˆ4]−3E[Yˆ2]2	E[YˆM3Yˆ]−3E[YˆM2]E[YˆMYˆ]
5	E[Yˆ5]−10E[Yˆ3]E[Yˆ2]	E[YˆM4Yˆ]−6E[YˆM2]E[YˆM2Yˆ]−4E[YˆM3]E[YˆMYˆ]
6	E[Yˆ6]−15E[Yˆ4]E[Yˆ2]−10E[Yˆ3]2+30E[Yˆ2]3	E[YˆM5Yˆ]−10E[YˆM3Yˆ]E[YˆM2]−5E[YˆM4]E[YˆMYˆ]−10E[YˆM3]E[YˆM2Yˆ]+30E[YˆM2]2E[YˆMYˆ]

Note(s): The second column presents cumulants of a random variable Y. The third column presents joint cumulants κl−1,1(YM,Y) for l = 1,2, …,6. Yˆ and YˆM in the row 3–7 are Y−E[Y] and YM−E[YM], respectively

Table 3

Examples of realized cumulants

l	Realized lth cumulant
2	∑j=1N(ΔStj)2
3	∑j=1N((ΔStj)3+3ΔMtj(2)ΔStj)
4	∑j=1N((ΔStj)4+6ΔMtj(2)(ΔStj)2+3(ΔMtj(2))2+4ΔMtj(3)ΔStj)
5	∑j=1N((ΔStj)5+10ΔMtj(2)(ΔStj)3+15(ΔMtj(2))2ΔStj+10ΔMtj(3)(ΔStj)2+10ΔMtj(3)ΔMtj(2)+5ΔMtj(4)ΔStj)
6	∑j=1N((ΔStj)6+15ΔMtj(2)(ΔStj)4+20ΔMtj(3)(ΔStj)3+45(ΔMtj(2))2(ΔStj)2+15(ΔMtj(2))3+60ΔMtj(3)ΔMtj(2)ΔStj+15ΔMtj(4)(ΔStj)2+10(ΔMtj(3))2+15ΔMtj(4)ΔMtj(2)+6ΔMtj(5)ΔStj)

Note(s): This table describes the realized lth cumulants in Bae and Lee (2021) and Fukasawa and Matsushita (2021). ΔMtj(l) in the row 2–6 are Mtj(l)−Mtj−1(l). Recall that St=Mt(1)

Table 4

The 2–6 realized joint cumulants

l	cMl−1,1real(S1,S2)
2	∑j=1NΔS1,tjΔS2,tj
3	∑j=1N(((ΔS1,tj)2+ΔMtj(2,0))ΔS2,tj+2ΔS1,tjΔMtj(1,1))
4	∑j=1N(((ΔS1,tj)3+3ΔMtj(2,0)ΔS1,tj+ ΔMtj(3,0))ΔS2,tj+3((ΔS1,tj)2+ΔMtj(2,0))ΔMtj(1,1)+3ΔS1,tjΔMtj(2,1))
5	∑j=1N(((ΔS1,tj)4+6ΔMtj(2,0)(ΔS1,tj)2+3(ΔMtj(2,0))2+4ΔMtj(3,0)ΔS1,tj+ΔMtj(4,0))ΔS2,tj+4((ΔS1,tj)3+3ΔMtj(2,0)ΔS1,tj+ ΔMtj(3,0))ΔMtj(1,1)+6((ΔS1,tj)2+ΔMtj(2,0))ΔMtj(2,1)+4ΔS1,tjΔMtj(3,1))
6	∑j=1N(((ΔS1,tj)5+10ΔMtj(2,0)(ΔS1,tj)3+10ΔMtj(3,0)(ΔS1,tj)2+15(ΔMtj(2,0))2ΔS1,tj+10ΔMtj(3,0)ΔMtj(2,0)+5ΔMtj(4,0)ΔS1,tj+ΔMtj(5,0))ΔS2,tj+5((ΔS1,tj)4+6ΔMtj(2,0)(ΔS1,tj)2+3(ΔMtj(2,0))2+4ΔMtj(3,0)ΔS1,tj+ΔMtj(4,0))ΔMtj(1,1)+10((ΔS1,tj)3+3ΔMtj(2,0)ΔS1,tj+ΔMtj(3,0))ΔMtj(2,1)+10((ΔS1,tj)2+ΔMtj(2,0))ΔMtj(3,1)+5ΔS1,tjΔMtj(4,1))

Note(s): This table describes detailed forms of the realized lth joint cumulants cMl−1,1real(S1,S2) up to order six. ΔMtj(l−k,k) in the row 2–6 are Mtj(l−k,k)−Mtj−1(l−k,k). Recall that S1,t=Mt(1,0) and S2,t=Mt(0,1)

Notes

1. They use the realized moments for various purposes. Amaya et al. (2015) and Sim (2016) show that realized third moments can explain stock returns. Kim (2016) investigates the forecasting power of implied moments about realized moments. Mei et al. (2017) show realized third and fourth moments are related to future volatility. Kinateder and Papavassiliou (2019) show that realized fourth moment can predict sovereign bond returns during a crisis. Ahmed and Al Mafrachi (2021) show that realized moments up to the fifth-order can explain cryptocurrency returns.

2. Implied moments can be obtained from options (Bakshi and Madan, 2000; Bakshi et al., 2003; Kang et al., 2009; Neuberger, 2012).

3. Cumulants are normalized moments. See section 4 for details.

4. The rest of this section is preliminary of section 3 that proves the non-existence of the geometric realized comoments. Therefore, readers that are only interested in the form of the realized comoments should move to section 4.

5. A left superscript t of κ means a time-t conditional one. For example, κLt(Y)=∂L∂uLlnEt[exp(uY)]|u=0.

6. The ten coefficients, b0,⋯,b5,b12,b14,b23,b25, and b26 are replaced with d5,⋯,d14. More precisely, (d5,d8,d12,d13) replace (b1,b4,b12,b25), (d6,d9,d11,d14) replace (b2,b3,b14,b26), d10 replaces b23, and d7 replaces b0, given b3 and b12.

The author is grateful for the 2021 financial assistance provided by The Research Foundation, Graduate School of Business, Chonnam National University, Republic of Korea. The author is also grateful to the anonymous referees, Jun-Kee Jeon, Hyoung-Goo Kang, Jangkoo Kang, Hwa-Sung Kim, Hyeng-Keun Koo, Soonhee Lee, Sun-Joong Yoon and the editors (Sol Kim and Eun Jung Lee) for valuable and detailed comments. All errors are the authors' responsibility.

Appendix 1 Proofs for Propositions and Corollaries in Section 3

Beginnings of the proofs for Propositions 3.1 and 3.3 are identical. We denote them as common property A as follows:

Common property A: A common necessary condition of g that satisfies the aggregation property.

Consider a vector-valued process {(lnS1(t),lnS2(t),M(t)):t=0,1,2}. In addition, let

(A1)(lnS1,lnS2,M):(0,0,m)→{(s1,1,s2,1,α)→(s1,1+η1,s2,1+η2,0→)Pr=π1(s1,2,s2,2,0→)→(s1,2,s2,2,0→)Pr=π2⋮⋮⋮(s1,n,s2,n,0→)→(s1,n,s2,n,0→)Pr=πnt: 0→    1    → 2

with ∑j=1nπj=1, ∑j=1nπj⁡exp(si,j)=1, E[exp(ηi)]=1, E[fk,l(η1,η2)]=αk,l, α=(α2,0,α1,1,α0,2,α0,3) and m=(m2,0,m1,1,m0,2,m0,3) where

(A2)mk,l=π1E[fk,l(s1,1+η1,s2,1+η2)]+∑j=2nπjfk,l(s1,j,s2,j),

and fk,l is a generalized moment function such that fk,l(0,0)=0, lim(a,b)→(0,0)fk,l(a,b)akbl=1, fk,l(a,b)=fl,k(b,a), and fk(a)=fk,0(a,b).

When the process satisfies the aggregation property, we have

(A3)E[g(s1,1+η1,s2,1+η2,−m)]=g(s1,1,s2,1,α−m)+E[g(η1,η2,−α)]

with g(0,⋯,0)=0. Differentiating Equation (A3) with respect to the (k−2)th term of m, we obtain

(A4)E[gk(s1,1+η1,s2,1+η2,−m)]=gk(s1,1,s2,1,α−m), for k=3,⋯,6,

where gk is a partial differentiation with respect to the kth term. By substituting (s1,1,s2,1)=(0,0) and m=α into Equation (A4), we obtain:

(A5)E[gk(η1,η2,−α)]=gk(0,0,0→), for k=3,⋯,6.

Then, by Lagrangian, we have

(A6)gk(s1,s2,M)=ak,0+Ak,1(M)(es1−1)+Ak,2(M)(es2−1)+Ak,3(M)(f2(s1)+M2,0)+Ak,4(M)(f1,1(s1,s2)+M1,1)+Ak,5(M)(f2(s2)+M0,2)+Ak,6(M)(f3(s1)+M3,0)

where ak,0 is a constant and Ak,1,⋯,Ak,6 are functions of M. If we substitute Equation (A6), (s1,1,s2,1)=(0,0) and m=α except for the (l−2)th term into Equation (A4), we obtain:

(A7)Ak,l(−m)(αl−2−ml−2)=Ak,l(0,⋯,0,αl−2−ml−2,0,⋯,0)(αl−2−ml−2).

Because π, si,j and α are arbitrary, Ak,3(M),⋯,Ak,6(M) are constants. Thus, Equation (A7) is can be rewritten with notations of ak,3,…,ak,6, as follows:

(A8)gk(s1,s2,M)=ak,0+Ak,1(M)(es1−1)+Ak,2(M)(es2−1)+ak,3(f2(s1)+M2,0)+ak,4(f1,1(s1,s2)+M1,1)+ak,5(f2(s2)+M0,2)+ak,6(f3(s1)+M3,0).

To investigate Ak,1(M) and Ak,2(M), let us substitute (A8) into (A4) and differentiate it with respect to ml. It yields

(A9)∂Ak,1(−m)∂ml=∂Ak,1(α−m)∂ml,∂Ak,2(−m)∂ml=∂Ak,2(α−m)∂ml for l=3,…,6.

Therefore, Ak,1(M) and Ak,2(M) are affine functions. Accordingly, (A8) is represented as follows:

(A10)gk(s1,s2,M)=ak,0+(bk,0+bk,1M2,0+bk,2M1,1+bk,3M0,2+bk,4M3,0)(es1−1)+(ck,0+ck,1M2,0+ck,2M1,1+ck,3M0,2+ck,4M3,0)(es2−1)+ak,3(f2(s1)+M2,0)+ak,4(f1,1(s1,s2)+M1,1)+ak,5(f2(s2)+M0,2)+ak,6(f3(s1)+M3,0).

Proof for Proposition 3.1

(Proof for the first statement)

We use the common property A with restricting the M=(V1,V2,Vc) with V1=M2,0, V2=M0,2 and Vc=M1,1. Similarly, we use notations m=(v1,v2,vc) and α=(α1,α2,αc). In addition, f and fc replace f2 and f1,1, respectively. By integrating (A10) with respect to V1, Vc and V2, we can obtain three different forms of g(s1,s2,V1,V2,Vc) as follows.

(A11)g(s1,s2,V1,V2,Vc)=a1,0V1+(b1,0V1+12b1,1V12+b1,2V1Vc+b1,3V1V2)(es1−1)       +(c1,0V1+12c1,1V12+c1,2V1Vc+c1,3V1V2)(es2−1)       +a1,3(f(s1)V1+12V12)+a1,4(fc(s1,s2)V1+V1Vc)       +a1,5(f(s2)V1+V1V2)+g1(s1,s2,V2,Vc).

(A12)g(s1,s2,V1,V2,Vc)=a2,0Vc+(b2,0Vc+b2,1V1Vc+12b2,2Vc2+b2,3VcV2)(es1−1)       +(c2,0Vc+c2,1V1Vc+12c2,2Vc2+c2,3VcV2)(es2−1)       +a2,3(f(s1)Vc+V1Vc)+a2,4(fc(s1,s2)Vc+12Vc2)       +a2,5(f(s2)Vc+VcV2)+g2(s1,s2,V1,Vc).

(A13)g(s1,s2,V1,V2,Vc)=a3,0V2+(b3,0V2+b3,1V1V2+b3,2VcV2+12b3,3V22)(es1−1)       +(c3,0V2+c3,1V1V2+c3,2VcV2+12c3,3V22)(es2−1)       +a3,3(f(s1)V2+V1V2)+a3,4(fc(s1,s2)V2+VcV2)       +a3,5(f(s2)V2+12V22)+g3(s1,s2,V1,V2).

with some functions g1, g2, and g3. By combining Equations (A11), (A12) and (A13), we obtain

(A14)g(s1,s2,V1,V2,Vc)=b0Vc+b1V1+b2V2+(es1−1)(b3Vc+b4V1+b5V2    +b6VcV1+b7VcV2+b8V1V2+b9Vc2+b10V12+b11V22)    +(es2−1)(b12Vc+b13V1+b14V2+b15VcV1    +b16VcV2+b17V1V2+b18Vc2+b19V12+b20V22)    +b21(f(s1)Vc+V1Vc+V1fc(s1,s2))    +b22(f(s2)Vc+V2Vc+V2fc(s1,s2))    +b23(f(s2)V1+V1V2+f(s1)V2)+b24(2fc(s1,s2)+Vc)Vc    +b25(2f(s1)+V1)V1+b26(2f(s2)+V2)V2+gs(s1,s2)

for some constants b1,⋯,b26 and a function gs such that gs(0,0)=0.

Based on (η1,η2), which are in Equation (A1), let us construct (η1p,η2p)={(η1,η2)Pr=p(0,0),Pr=1−p for a constant p in [0,1]. For i∈{1,2}, we have E[eηip]=1, E[f(ηip)]=αip and E[fc(η1p,η2p)]=αcp. Then, by substituting Equation (A14) into (A3) and (η1p,η2p) into (η1,η2), we obtain:

(A15)0=p(es11−1)(b3αc+b4α1+b5α2+b6(αcα1p−α1vc−αcv1)+b7(α2αcp−α2vc−αcv2)+b8(α1α2p−α2v1−α1v2)+b9(αc2p−2αcvc)+b10(α12p−2α1v1)+b11(α22p−2α2v2))+p(es21−1)(b12αc+b13α1+b14α2+b15(αcα1p−α1vc−αcv1)+b16(α2αcp−α2vc−αcv2)+b17(α1α2p−α2v1−α1v2)+b18(αc2p−2αcvc)+b19(α12p−2α1v1)+b20(α22p−2α2v2))+pb21((E[f(s11+η1)]−f(s11)−α1)vc+f(s11)αc+α1fc(s11,s21)+(E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc)v1)+pb22((E[f(s21+η2)]−f(s21)−α2)vc+f(s21)αc+α2fc(s11,s21)+(E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc)v2)+pb23((E[f(s21+η2)]−f(s21)−α2)v1+f(s21)α1+f(s11)α2+(E[f(s11+η1)]−f(s11)−α1)v2)+2pb24((E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc)vc+fc(s11,s21)αc)+2pb25((E[f(s11+η1)]−f(s11)−α1)v1+f(s11)α1)+2pb26((E[f(s21+η2)]−f(s21)−α2)v2+f(s21)α2)−pE[gs(s11+η1,s21+η2)]+pgs(s1,s2)+pE[gs(η1,η2)]

Because (A15) holds for arbitrary p, the coefficient of p2 should be zero.

(A16)0=(es11−1)(b6αcα1+b7α2αc+b8α1α2+b9αc2+b10α12+b11α22)   +(es21−1)(b15αcα1+b16α2αc+b17α1α2+b18αc2+b19α12+b20α22)

Furthermore, because s11 and s21 are arbitrary, we have:

(A17)b6αcα1+b7α2αc+b8α1α2+b9αc2+b10α12+b11α22=0

(A18)b15αcα1+b16α2αc+b17α1α2+b18αc2+b19α12+b20α22=0

Given α1 and α2, we can construct arbitrary αc. Therefore, Equation (A17) yields:

(A19)b9=b6α1+b7α2=b8α1α2+b10α12+b11α22=0

Adopting this logic to Equation (A18) instead of (A17) and to α1 or α2 instead of αc, we can obtain

(A20)b6=b7=⋯=b11=0 and b15=b16=⋯=b20=0

Additionally, because the coefficient of p in Equation (A15) is zero, we have:

(A21)0=(es11−1)(b3αc+b4α1+b5α2)+(es21−1)(b12αc+b13α1+b14α2) +b21((E[f(s11+η1)]−f(s11)−α1)vc+f(s11)αc+α1fc(s11,s21)+(E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc)v1) +b22((E[f(s21+η2)]−f(s21)−α2)vc+f(s21)αc+α2fc(s11,s21)+(E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc)v2) +b23((E[f(s21+η2)]−f(s21)−α2)v1+f(s21)α1+f(s11)α2+(E[f(s11+η1)]−f(s11)−α1)v2) +2b24((E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc)vc+fc(s11,s21)αc) +2b25((E[f(s11+η1)]−f(s11)−α1)v1+f(s11)α1) +2b26((E[f(s21+η2)]−f(s21)−α2)v2+f(s21)α2) −E[gs(s11+η1,s21+η2)]+gs(s1,s2)+E[gs(η1,η2)]

Because vc is arbitrary, the coefficient of vc is zero. Thus, we have

(A22)b21(E[f(s11+η1)]−f(s11)−α1)+b22(E[f(s21+η2)]−f(s21)−α2)+2b24(E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc)=0

Now, consider a random variable η3 with η3 =d η2 and E[fc(η1,η3)]≠E[fc(η1,η2)]. Then,

(A23)b21(E[f(s11+η1)]−f(s11)−α1)+b22(E[f(s21+η3)]−f(s21)−α2)+2b24(E[fc(s11+η1,s21+η3)]−fc(s11,s21)−E[fc(η1,η3)])=0

By subtracting Equation (A23) from Equation (A22), one can see that b24=0 or

(A24)E[fc(s11+η1,s21+η2)]=E[fc(s11+η1,s21+η3)]+E[fc(η1,η2)]−E[fc(η1,η3)].

When we substitute

(A25)(η1,η2,η3)={(ln(1+k),ln(1+k),ln(1−k))Pr=1/2(ln(1−k),ln(1−k),ln(1+k))Pr=1/2

into Equation (A24) and multiply both sides of the equation with 12k2, and take the limit with k→0, we get

(A26)∂2fc(x,y)∂x∂y=1

Hence,

(A27)fc(s11,s21)=s11s21+F1(s11)+F2(s21)

for some functions F1 and F2. Then, applying the condition of lim(x,y)→(0,0)fc(x,y)xy=1, we obtain fc(s11,s21)=s11s21. By substituting it into Equation (A22), we obtain the following equation:

(A28)b21(E[f(s11+η1)]−f(s11)−α1)+b22(E[f(s21+η2)]−f(s21)−α2)+2b24(s11E[η2]+s21E[η1])=0

When s11=0, Equation (A28) becomes b22(E[f(s21+η2)]−f(s21)−α2)+2b24s21E[η1]=0. Because η1 can be chosen independently on s21 and η2,

(A29)b24=0.

The logic between Equations (A22) and (A29) shows that multiplier of E[fc(s11+η1,s21+η2)]−fc(s11,s21)−αc is zero. For alternatives of Equation (A22), as the coefficients of v1 and v2 instead of vc in Equation (A21), the same logic yields

(A30)b21=b22=0.

Because the coefficients of v1 and v2 in Equation (A21) are 0, equations (A29) and (A30) implies:

(A31)b23(E[f(s21+η2)]−f(s21)−α2)+2b25(E[f(s11+η1)]−f(s11)−α1)=b23(E[f(s11+η1)]−f(s11)−α1)+2b26(E[f(s21+η2)]−f(s21)−α2)=0.

Substituting s11=0 or s21=0 into the (A31) yields:

(A32)b23(E[f(s21+η2)]−f(s21)−α2)=b25(E[f(s11+η1)]−f(s11)−α1)=b23(E[f(s11+η1)]−f(s11)−α1)=b26(E[f(s21+η2)]−f(s21)−α2)=0

Thus,

(A33)E[f(s+η)]−f(s)−E[f(η)]=0 or b23=b25=b26=0.

Here, according to Neuberger (2012), E[f(s+η)]−f(s)−E[f(η)]=0 is equivalent to

(A34)f(x)=2(ex−1−x).

In sum, combining (A21) with (A29), (A30) and (A33) yields

(A35)0=(es11−1)(b3αc+b4α1+b5α2)+(es21−1)(b12αc+b13α1+b14α2) +b23(f(s21)α1+f(s11)α2)+2b25f(s11)α1+2b26f(s21)α2 −E[gs(s11+η1,s21+η2)]+gs(s1,s2)+E[gs(η1,η2)]

To Equation (A35), multiplying 2/k2, substituting (η1,η2)={(ln(1+k),0)Pr=1/2(ln(1−k),0)Pr=1/2, and taking the limit yields:

(A36)0=2b4(es11−1)+2b13(es21−1)+4b23(es21−1−s21)+8b25(es11−1−s11)−∂2gs(s11,s21)∂s112+∂gs(s11,s21)∂s11+∂2gs(0,0)∂s112−∂gs(0,0)∂s11.

Similarly, when use (η1,η2)={(0,ln(1+k))Pr=1/2(0,ln(1−k))Pr=1/2, we can obtain

(A37)0=2b5(es11−1)+2b14(es21−1)+4b23(es11−1−s11)+8b26(es21−1−s21)−∂2gs(s11,s21)∂s212+∂gs(s11,s21)∂s21+∂2gs(0,0)∂s212−∂gs(0,0)∂s21.

Alternatively, let us multiply 12k2 to Equation (A35), substitute (A38) and (A39) into Equation (A35), and subtract the equation obtained by the former substitution from that obtained by the latter; then, by taking limits, we get (A40).

(A38)(η1,η2)={(ln(1+k),ln(1+k))Pr=1/2(ln(1−k),ln(1−k))Pr=1/2

(A39)(η1,η2)={(ln(1+k),ln(1−k))Pr=1/2(ln(1−k),ln(1+k))Pr=1/2

(A40)0=b3(es11−1)+b12(es21−1)−∂2gs(s11,s21)∂s11s21+∂2gs(0,0)∂s11s21

Then, the solutions of Equation (A40), (A36) and (A37) are given as

(A41)gs(x,y)=b3(ex−x)y+b12(ey−y)x+h1(x)+h2(y)+b27xy

(A42)gs(x,y)=2b4(exx−ex+x)−2b13(ey−1)x−4b23x(ey−y−1)+4b25(2exx−2ex+x2+4x)+exh3(y)+h4(y)+b28x

(A43)gs(x,y)=2b14(eyy−ey+y)−2b5(ex−1)y−4b23y(ex−x−1)+4b26(2eyy−2ey+y2+4y)+eyh5(x)+h6(x)+b29y

for some functions h1,⋯,h6 and constants bi. Therefore, gs(x,y) is a linear combination of exy, eyx, xy, exx, ex, x2, x, eyy, ey, y2, y and 1. The consistency in the coefficients of exy and eyx requires b5=−12b3−2b23 and b13=−12b12−2b23. Thus, by Equation (A14), (A20), (A29), (A30), (A34), (A41), (A42), (A43) and gs(0,0)=0, g and gs are given by

(A44)g(s1,s2,V1,V2,Vc)=b0Vc+b1V1+b2V2 +(b3Vc+b4V1−(12b3+2b23)V2)(es1−1) +(b12Vc−(12b12+2b23)V1+b14V2)(es2−1) +b23(2(es2−s2−1)V1+V1V2+2(es1−s1−1)V2) +b25(4(es1−s1−1)+V1)V1+b26(4(es2−s2−1)+V2)V2+gs(s1,s2),

(A45)gs(s1,s2)=d1(es1−1)+d2s1+d3(es2−1)+d4s2+4b23s1s2+4b25s12   +4b26s22+b3es1s2+b12es2s1+(2b4+8b25)es1s1+(2b14+8b26)es2s2.

(A44) and (A45) can be arranged as

(A46)gs1,s2,V1,V2,Vc=d1es1−1+d2s1+d3es2−1+d4s2+d5V1+d6V2+d7Vc  +d8V1−2s12+d9V2−2s22+d10V1−2s1V2−2s2  +d11es1(2Vc−V2+2s2)+d12es2(2Vc−V1+2s1)  +d13es1V1+2s1+d14es2V2+2s2

where d5=b1−b4+12b12−4b25, d6=b2+12b3−b14−4b26, d7=b0−b3−b12, d8=b25, d9=b26, d10=b23, d11=12b3, d12=12b12, d13=b4+4b25 and d14=b14+4b26 [6].

Substituting these into equation (A3) yields the following:

(A47)0=2d8(−v1−2s11+α1)(α1+2E[η1])+2d9(−v2−2s21+α2)(α2+2E[η2]) +d10((−v1−2s11+α1)(α2+2E[η2])+(−v2−2s21+α2)(α1+2E[η1])) +(es11−1)(d13(α1−2E[η1eη1])+d11(2αc−2E[η2eη1]−α2)) +(es21−1)(d14(α2−2E[η2eη2])+d12(2αc−2E[η1eη2]−α1))

Since coefficients of v1 and v2 are zero, E[f(η)]=E[−2η] or d8=d9=d10=0. In addition, because s11 and s21 are arbitrary,

(A48)0=d13(α1−2E[η1eη1])+d11(2αc−2E[η2eη1]−α2)

(A49)0=d14(α2−2E[η2eη2])+d12(2αc−2E[η1eη2]−α1)

Conditions of Equations (A47)-(A49) can be fulfilled with one of following five cases.

1. If d11 is not zero, for some constants k1 and k2, we have

(A50)fc(η1,η2)=η2eη1+12f(η2)+d132d11(2η1eη1−f(η1))+k1(eη1−1)+k2(eη2−1).

Then d132d11(2η1eη1−f(η1))+k1(eη1−1)=0 and 12f(η2)+k2(eη2−1)=−η2 because fc(x,y)xy→1 as (x,y)→(0,0). Accordingly, k2=−1 because f(x)x2→1 as x→0. This implies that k1=d13=0. Therefore, fc(η1,η2)=η2(eη1−1), f(η)=2(eη−η−1). Then, d12=d14=0.

1. Similarly, if d12 is not zero, fc(η1,η2)=η1(eη2−1), f(η)=2(eη−η−1) and d11=d13=d14=0

Alternatively, when d11=d12=0, Equations (A47)-(A49) implies there are three more conditions as follows:

1. d11=d12=d13=d14=0, f(η)=2(eη−η−1), with arbitrary function fc.

2. d8=d9=d10=d11=d12=0, f(η)=2(ηeη−eη+1), with arbitrary function fc.

3. d8=d9=d10=d11=d12=d13=d14=0, with arbitrary functions f and fc.

(Proof for the second statement)

For the proof of the sufficiency of Equation (6) for the AP, we show that the function g in Equation (6) satisfies the SAP (strong aggregation property): Et[g(Xu−Xr)]=Et[g(Xu−Xt)]+Et[g(Xt−Xr)], which is stronger condition than the AP of Equation (2). The SAP of the first seven terms in the equation is obvious. The 10th term is a generalization of the 8th and the 9th term and all these three terms do not vanish only if f(η)=2(eη−η−1). Thus, SAP of 10th term implies the SAP of 8th and 9th terms. For convenience, let

(A51)Gu,t=ϕ1,u,tϕ2,u,t

with

(A52)ϕi,u,t=(Vi,u−Vi,t−2(si,u−si,t)) for i∈{1,2}

and

(A53)Vi,t≡Et[2(esi,T−si.t−(si,T−si.t)−1)]=Et[−2(si,T−si.t)]=Et[Vi,u−2(si,u−si,t)] for i∈{1,2}.

Then, we have

(A54)Et[Gu,0]=Et[(ϕ1,u,t+ϕ1,t,0)(ϕ2,u,t+ϕ2,t,0)]=Et[ϕ1,u,tϕ2,u,t+ϕ1,t,0ϕ2,t,0]   =Et[Gu,t]+Gt,0.

Thus, we have the SAP of 8th and 9th terms as well as the SAP of 10th term.

Additionally, the SAP of the 11th term under the condition (1) implies that of the 12th term under the condition (2) and the 13th and 14th terms under the condition (4). Thus, we finish this proof by showing Equation (A55).

(A55)Et[Fu,0]=Et[Fu,t]+Ft,0, 0≤t≤u≤T,

where

(A56)Fu,t=Et[es1,u−s1.t(Vˆu−Vˆt+2(s2,u−s2.t))]

and

(A57)Vˆt≡2Et[(s2,T−s2,t)(es1,T−s1.t−1)−(es2,T−s2.t−(s2,T−s2.t)−1)].

Equation (A57) is represented as:

(A58)Vˆt=Et[2es1,T−s1.t(s2,T−s2,t)] =Et[2es1,T−s1.ues1,u−s1.t(s2,T−s2,u)]+Et[2es1,u−s1.t(s2,u−s2,t)] =Et[es1,u−s1.t(Vˆu+2(s2,u−s2,t))].

It implies

(A59)Et[Fu,t]=Et[es1,u−s1.t(Vˆu+2(s2,u−s2.t))−es1,u−s1.tVˆt]=0

and

(A60)Et[Fu,0]=Et[es1,u−s1.0(Vˆu−Vˆ0+2(s2,u−s2.0))]   =Et[es1,u−s1.tes1,t−s1.0(Vˆu+2(s2,u−s2.t)−Vˆ0+2(s2,t−s2.0))]   =Et[es1,t−s1.0(Vˆt−Vˆ0+2(s2,t−s2.0))]   =Ft,0.

Due to Equations (A59) and (A60), Equation (A55) holds.

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Proof for Corollary 3.2

If a function is a realized (k,1)-comoment element for k∈{1,2}, Equation (11) should be decomposed as

(A61)g(Δs1,Δs2,Δv1,Δv2,Δvc)=(eΔs1−1)ϕ1(Δv1,Δv2,Δvc)+(eΔs2−1)ϕ2(Δv1,Δv2,Δvc)+gr(Δs1,Δs2)

such that gr(Δs1,Δs2)=O((Δs1)kΔs2) because of the restrictions E[eΔs1]=1 and E[eΔs2]=1. (Δv1)2 cannot be a part of (eΔs1−1)ϕ1(Δv1,Δv2,Δvc) or (eΔs2−1)ϕ2(Δv1,Δv2,Δvc), as well as gr(Δs1,Δs2), because it is only in h8(Δv1−2Δs1)2; thus, we have h8=0. In a similar manner, by considering (Δv2)2 and Δv1Δv2, we have h9=h10=0. Accordingly, eΔs1Δs2 and eΔs2Δs1 are only cross-terms between Δs1 and Δs2. Therefore, none of condition (3), (4) or (5) can generate a realized (k,1)-comoment element for k∈{1,2}.

Under the condition (1) in Proposition 3.1, eΔs1Δs2 is the only cross-term between Δs1 and Δs2. In the remaining terms, we have h5=0, h6=h11, and h7=−2h11 to separate Δv1, Δvc, and Δv2 from gr(Δs1,Δs2). Then, the remaining term h1(eΔs1−1)+h2Δs1+h3(eΔs2−1)+h4Δs2+2h11eΔs1Δs2 is at most O(Δs1Δs2) as (Δs1,Δs2)→(0,0) when h1=h2=h3=0 and h4=−2h11. It cannot be of O(s12s2), and it is a realized (1,1) comoment when h4=−1.

In a similar manner, under condition (2), the function is at most is at most O(s1s2), and it is a realized comoment when h1=h3=h4=h6=h8=h9=h10=0, h2=h7=−1, h5=h12=1/2.

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Proof for Proposition 3.3

(Proof for the first statement)

We use the common property A by omitting all terms related to the second security. Thus, g is a function of s1, M2 and M3 where M2=M2,0 and M3=M3,0. By integrating (A10) with respect to M2 and M3, we can obtain two different forms of the function g:

(A62)g(s1,M2,M3)=a1,0M2+(b1,0M2+12b1,1M22+b1,4M3M2)(es1−1)+a1,3(M2f2(s1)+12M22)+a1,6M2(f3(s1)+M3)+g1(s1,M3)

and

(A63)g(s1,M2,M3)=a2,0M3+(b2,0M3+b2,1M2M3+12b2,4M32)(es1−1)+a2,3M3(f2(s1)+M2)+a2,6(f3(s1)M3+12M32)+g2(s1,M2)

with some functions g1 and g2. By combining (A62) and (A63), we obtain

(A64)g(s1,M2,M3)=a1M2+a2M3      +(a3M2+a4M3+a5M22+a6M2M3+a7M32)(es1−1)      +a8(M22+2M2f2(s1))+a9(M32+2f3(s1)M3)      +2a10(M2M3+f2(s1)M3+f3(s1)M2)+gs(s1)

for some constants a1,⋯,a10 and a function gs such that gs(0)=0. Using the η1 in Equation (A1), let us construct ηp={η1Pr=p0Pr=1−p for a constant p in [0,1]. Then, by substituting Equation (A64) into (A3) and replacing η with ηp, we obtain:

(A65)0=p((es1,1−1)(a3α2+a4α3−2a5m2α2+a6(−m2α3−m3α2)−2a7m3α3)           +2a8((f2(s1,1+η1)−f2(s1,1)−α2)m2+α2f2(s1,1))      +2a9((f3(s1,1+η1)−f3(s1,1)−α3)m3+α3f3(s1,1))      +2a10((f2(s1,1+η1)−f2(s1,1)−α2)m3+α3f2(s1,1)+(f3(s1,1+η1)−f3(s1,1)−α3)m2+α2f3(s1,1))−E[gs(s1,1+η1)]+gs(s1,1)+E[gs(η1)])    +p2(es1,1−1)(a5α22+a6α2α3+a7α32)

Because we can set p arbitrary, coefficients of p and p2 are zero. Therefore, we have

(A66)a5α22+a6α2α3+a7α32=0

and

(A67)((es1,1−1)(a3α2+a4α3−2a5m2α2+a6(−m2α3−m3α2)−2a7m3α3)           +2a8((f2(s1,1+η1)−f2(s1,1)−α2)m2+α2f2(s1,1))      +2a9((f3(s1,1+η1)−f3(s1,1)−α3)m3+α3f3(s1,1))      +2a10((f2(s1,1+η1)−f2(s1,1)−α2)m3+α3f2(s1,1)+(f3(s1,1+η1)−f3(s1,1)−α3)m2+α2f3(s1,1))−E[gs(s1,1+η1)]+gs(s1,1)+E[gs(η1)])=0.

Equation (A66) implies that

(A68)a5=a6=a7=0

because η1 is an arbitrary random variable with E[eη1]=1. Also, in Equation (A67), coefficients of m2 and m3 are zero because we can set arbitrary values for them. Thus, we have:

(A69)0=a8(f2(s1,1+η1)−f2(s1,1)−α2)+a10(f3(s1,1+η1)−f3(s1,1)−α3)

(A70)0=a9(f3(s1,1+η1)−f3(s1,1)−α3)+a10(f2(s1,1+η1)−f2(s1,1)−α2)

There are three cases that satisfy both (A69) and (A70). We call them condition A.1 as follows:

Condition A.1

1. ∃f3 such that ∀(s1,η1), f3(s1,1+η1)−f3(s1,1)−α3=0 and a8=a10=0,

2. ∃(a,f2,f3) such that ∀(s1,1,η1), f2(s1,1+η1)−f2(s1,1)−α2+a(f3(s1,1+η1)−f3(s1,1)−α3)=0 with a10=a8a and a9=a2a8,

3. a8=a9=a10=0.

First, we check the condition in A.1.(1). Substituting η1={ln(1+k),Pr=0.5ln(1−k),Pr=0.5 into 2k2(f3(s1,1+η1)−f3(s1,1)−α3)=0, using limx→0f3(x)x3=1, and taking the limit for k→0 yield:

(A71)(f3)″(s1,1)−(f3)′(s1,1)=0

Thus, f3(s1,1)=b1es1,1+b2 for some constants b1 and b2. However, there are no b1 and b2 that makes limx→0f3(x)x3=1. Therefore, condition A.1 (1) is impossible.

Second, let us check condition A.1.(2). Substituting fa(x)=f2(x)+af3(x) and η1={ln(1+k),Pr=0.5ln(1−k),Pr=0.5 into f2(s1,1+η1)−f2(s1,1)−α2+a(f3(s1,1+η1)−f3(s1,1)−α3)=0, using limx→0f2(x)x2=1 and limx→0f3(x)x3=1, and taking the limit for k→0 yield:

(A72)(fa)″(s1,1)−(fa)′(s1,1)−2=0

Thus, we have fa(s1,1)=b1es1,1+b2−2s1,1 for some constants b1 and b2. Because of the conditions limx→0f2(x)x2=1 and limx→0f3(x)x3=1, f2 has the following form:

(A73)f2(s)=2(es−s−1)−af3(s).

Substituting (A68), (A73), and Condition A.1 (2) into (A67) yields:

(A74) E[gs(s1,1+η1)]−gs(s1,1)−E[gs(η1)]=(es1,1−1)(a3α2+a4α3)+4a8(α2+aα3)(es1,1−s1,1−1)

Equation (A74) with a8=0 satisfies (A67) with (A68) and condition A.1 (3). Therefore, Equation (A74) is a general equation for gs. Again, by letting η1={ln(1+k),Pr=0.5ln(1−k),Pr=0.5 and taking the limit, we can obtain a differential equation:

(A75)(gs)′−(gs)″+2a3(es−1)+8a8(es−s−1)=const.

Using gs(0)=0, gs is represented as follows:

(A76)gs(s)=a9s+a10(es−1)+4a8s2+(8a8+2a3)ses

with additional constants a9 and a10. Substituting it into (A64) yields:

(A77)g(s,M2,M3)=a1M2+a2M3+(a3M2+a4M3)(es−1)+a8(M2+aM3−2s)2+4a8(M2+aM3)(es−1)+a9s+a10(es−1)+(8a8+2a3)ses

or

(A78)g(s,M2,M3)=d1M2+d2M3+d3M3es+d4(M2+aM3−2s)2+d5(M2+aM3+2s)es+d6s+d7(es−1)

where d1=a1−a3, d2=a2−a4, d3=a4−aa3, d4=a8, d5=a3+4a8, d6=d9 and d7=d10. Then, substituting these into (A3) yields

(A79)d4(4s1,1+2(m2+am3−α2−aα3))(E[2η1]+α2+aα3)+(es1,1−1)(d5(E[2η1eη1]−α2−aα3)−d3α3)=0

Because s1 is arbitrary, we have the following cases.

Condition A.2

1. d3=d4=d5=0

2. d3=d5=0 and E[2η1]+α2+aα3=0

3. d4=0 and E[2η1eη1]−α2−hα3 with h=a+d3/d5.

Recall that E[eη1−1]=0, αk=E[fk(η1)] and f2(η1)+af3(η1)η12→1 for η1→0. Therefore, when condition A.2 (2) holds, we obtain f2(Δs)+af3(Δs)=2(eΔs−Δs−1). Next, condition A.2 (3) is equivalent to d3=d4=0 with

(A80)E[2η1eη1]−α2−aα3=0,

which implies that

(A81)f2(Δs)+af3(Δs)=2(ΔseΔs−eΔs+1).

Rearranging the above equations yields the equation and the condition of Proposition 3.1. This implies that Equation (14) is a candidate for a function with the aggregation property.

(Proof for the second statement)

Similar to Proposition 3.3, it is enough to show the SAP of Equation (14) holds. The SAP of the first four terms in the equation is obvious. Proofs for the 5th and 6th terms in this proposition are similar to those of the 10th and 11th terms in Proposition 3.1, respectively.

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Appendix 2 Proof for Proposition 4.2

Let us set St=S1,t+aS2,t and Xt=(Mt(1),…,Mt(L−1),0) with Mtl=κltST. Then, according to Fukasawa and Matsushita (2021),

(A82)E0∑j=1NBLXtj−Xtj−1=E0BLXT−X0,0=κL0ST.

According to Equation (18), Mt(l) is decomposed to ∑k=0lak(lk)Mt(l−k,k) for Mt(l−k,k)=κl−k,kt(S1,T,S2,T). In other words, the right hand side of Equation (A82) can be represented as

(A83)∑k=0Lak(Lk)κL−k,k0(S1,T,S2,T).

For convenience, we denote the summand of the left hand side of Equation (A82) as BL(ΔX). By the definition of BL, we can arrange it as follows.

(A84)BL(ΔX)=∂L∂uL(exp(∑i=1L−1ΔM(i)uii!))|u=0=∂L∂uL(exp(∑i=1L−1∑j=0iaj(ij)ΔM(i−j,j)uii!))|u=0=∂L∂uL(exp(∑i0=1L−1ΔM(i0,0)ui0i0!+a∑i1=1L−1i1ΔM(i1−1,1)ui1i1!+O(a2)))|u=0=∂L∂uL(exp(∑i0=1L−1ΔM(i0,0)ui0i0!))|u=0+a∑i1=1L−1(Li1)i1ΔM(i1−1,1)∂L−i1∂uL−i1(exp(∑i0=1L−1ΔM(i0,0)ui0i0!))|u=0+O(a2)=BL(ΔM(1,0),⋯,ΔM(L−1,0),0)+aL∑i1=1L−1(L−1i1−1)ΔM(i1−1,1)BL−i1(ΔM(1,0),⋯,ΔM(L−i1,0))+O(a2)

Because a is arbitrary, the coefficient of a of the left hand side of Equation (A82) is equal to the coefficient of a of the right hand side of Equation (A82). Therefore, by Equations (A82), (A83) and (A84), we have

(A85)κL−1,10(S1,T,S2,T)=E0[∑j=1N∑i1=1L−1(L−1i1−1)ΔM(i1−1,1)BL−i1(ΔM(1,0),ΔM(2,0),⋯,ΔM(L−i1,0))].

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