Investment time horizon and multifractality of stock price process

Abstract

We construct a multifractal random walk for a stock trades model with inverse power law interaction. Consider a stock in a stock market and introduce a discrete time model for pair trades [(first trade $\rightarrow$ second (reverse) trade] transacted by various types of traders. The type of trader is characterized by the investment time horizon defined as a time difference of the pair trades. We assume that probability distributions of the investment time horizons are given by inverse power law interactions with different exponents depending on the types of traders, and define a discrete time log-volatility process of the stock. Using the method of abstract polymer expansion developed in the study of mathematical physics we obtain a continuous type log-volatility process as a scale limit, and define a log-return process from time 0 as a stochastic integral with respect to a Brownian motion. We then find a condition of the continuous cascade equation which derives a multifractality in the log-return process. Finally, we construct a multifractal random walk using the martingale convergence theorem.

Appendices

Appendix 1: Proof of Theorem 1

The key point of the proof of Theorem 1 is a application of Kotecky–Preiss theory of abstract polymer expansion developed in the mathematical physics (Kotecky and Preiss 1986; Brickmont et al. 1985).

To describe Kotecky–Peiss theory, we extend $\Omega _n$ to more general configuration space $\mathcal{A}_n$ of polymers defined by,

$$\begin{aligned} \mathcal{A}_n=\left\{ A: \mathcal{P}_n \rightarrow \mathbf{N} \cup \{0\}; |A|=\sum _{ {\mathbf {p}}\in \mathcal{P}_n} A({\mathbf {p}}) <\infty \; \right\} \end{aligned}$$

where $\mathcal{P}_n$ be a set of all polymers.

Each $A \in \mathcal{A}_n$ is regarded as a configuration of finite number of polymers permitting intersections and multiplicities.

We define functionals $\varphi (A) , \alpha (A)$ for $A \in \mathcal{A}_n$ by

$$\begin{aligned}&\varphi (A)=\prod _{{\mathbf {p}}\in \text {supp}A} \varphi ({\mathbf {p}})^{A({\mathbf {p}})}\\&\alpha (A) ={\left\{ \begin{array}{ll} 1 &{}\text { if }\; A!=1 \,\text { and }\; b({\mathbf {p}})\cap b(\mathbf{q})=\emptyset \;\text { for\,all }\, {\mathbf {p}},\mathbf{q} \in \text {supp} \,{ A} \; ({\mathbf {p}}\not = \mathbf{q})\\ 0 &{}\text {otherwise} \end{array}\right. } \\&\chi _n(A)= {\left\{ \begin{array}{ll} 1 &{}\text { if }\; b({\mathbf {p}}) \cap (0, Tn^\alpha ] \not = \emptyset\, \text { \,for\,all } \,{\mathbf {p}}\in \text {supp}\,A \\ 0 &{}\text {otherwise} \end{array}\right. } \end{aligned}$$

where supp $A=\{{\mathbf {p}}\in \mathcal{P}_n; A({\mathbf {p}})>0 \}$ and $A!=\prod _{{\mathbf {p}}\in \text {supp}A} A({\mathbf {p}})!.$

Then the probability distribution (6) is rewritten as

$$\begin{aligned} P_n(A)= \frac{1}{Z_n} \varphi (A) \chi _n(A) \alpha (A). \end{aligned}$$

(14)

To state an important property of abstract polymer expansion we introduce a notion of cluster. $A \in \mathcal{A}_n$ is called a “cluster”, if $b({\mathbf {p}}) \subset S_j ({\mathbf {p}}\in \text {supp}A)$ for some $j \in \{1,\ldots ,n\}$ and there exists a chain of polymers $\{{\mathbf {p}}_{i_1}, {\mathbf {p}}_{i_2},\ldots , {\mathbf {p}}_{i_m}\} \subset \text {supp} A$ satisfying

$$\begin{aligned} {\mathbf {p}}_{i_1}={\mathbf {p}}_i , {\mathbf {p}}_{i_m}={\mathbf {p}}_\ell\, \text { and }\; b({\mathbf {p}}_{i_k}) \cap b({\mathbf {p}}_{i_{k+1}})\not = \emptyset \; (k=1,\ldots , m-1) , \end{aligned}$$

for any pair of distinct polymers $({\mathbf {p}}_i , {\mathbf {p}}_\ell )$ of supp A. In other words, a cluster is considered to be a chain of connected polymers. Remark that any cluster is contained in some $S_j$.

Kotecky-Preiss Theory (Kotecky and Preiss 1986) Let $D \subset \mathbf{C}$ be a complex domain. If a complex-valued functional $\varPhi ({\mathbf {p}},\xi )$ defined on $\mathcal{P}_n \times D$ satisfies

$$\begin{aligned} \mathop{\sum}\limits _{{\begin{array}{c} {{\mathbf {p}}\in \mathcal{P}_n}\\ { b({\mathbf {p}}) \cap b(\mathbf{q}) \not = \emptyset } \end{array}}} e^{a({\mathbf {p}})+\text {d}({\mathbf {p}})} |\varPhi ({\mathbf {p}},\xi )| \le a(\mathbf{q}) \quad ( \mathbf{q} \in \mathcal{P}_n,\xi \in D) \end{aligned}$$

(15)

for some nonnegative real-valued functionals $a({\mathbf {p}})$ and $d({\mathbf {p}})$ defined on $\mathcal{P}_n$ , then there exists a translation invariant functional $\alpha ^T(\cdot )$ on $\mathcal{A}_n$ and satisfies

$$\begin{aligned}&{\mathrm{(i)}} \quad \sum _{A \in \mathcal{A}_n} \phi (A,\xi ) \alpha (A) = \exp \left\{ \sum _{A \in \mathcal{A}_n} \phi (A,\xi ) \frac{\alpha ^T(A)}{A!} \right\} \\&{\mathrm{(ii)}}\quad \mathop{\sum\limits} _{{\begin{array}{c} {A \in \mathcal{A}_n}\\ { b(A) \cap b(\mathbf{q}) \not = \emptyset } \end{array}}} \left| \phi (A,\xi ) \frac{\alpha ^T(A)}{A!} \right| \; e^{\text {d}(A)}\;\le \; a(\mathbf{q}) \quad (\mathbf{q} \in \mathcal{P}_n) \end{aligned}$$

where $b(A)= \bigcup _{{\mathbf {p}}\in \text {supp}A} b({\mathbf {p}}), \phi (A,\xi )=\prod _{{\mathbf {p}}\in \mathcal{P}_n} \phi ({\mathbf {p}},\xi )^{A({\mathbf {p}})}$ and $\text {d}(A)=\sum _{{\mathbf {p}}\in \mathcal{P}_n} \text {d}({\mathbf {p}}) A({\mathbf {p}}).$ Furthermore, $\alpha ^T(A)=0$ unless A is a cluster.

We introduce a functional $Y_n({\mathbf {p}})$ on $\mathcal{P}_n$ and a functional $Y_(A)$ on $\mathcal{A}_n$ defined by

$$\begin{aligned} Y_n({\mathbf {p}})=\frac{1}{c(n)} \sum _{k=1}^m z_k \sum _{\ell \in I_j(t_k)} Z_{(j,\ell )}({\mathbf {p}}) \;( {\mathbf {p}}\subset S_j), \; \text {and}\; Y_n(A)=\sum _{{\mathbf {p}}\in \text {supp}A} Y_n({\mathbf {p}}) A({\mathbf {p}}), \end{aligned}$$

First, we remark that $|Y_n({\mathbf {p}})| \le \frac{M}{c(n)},$ where $M= 2H \cdot {\displaystyle \sum _{k=1}^m |z_k|}$, and define a complex domain D by $D=\{ \xi \in \mathbf{C}; | \text {Re} \xi | \le \frac{c(n)}{M} \}$. If $\xi \in D$,then $|e^{\xi Y_n({\mathbf {p}})}| \le e$ for any ${\mathbf {p}}$.

In this model, we define $\phi ({\mathbf {p}},\xi ), \phi (A,\xi ), a({\mathbf {p}})$ and $d({\mathbf {p}})$ by

$$\begin{aligned} \phi ({\mathbf {p}},\xi )&= e^{\xi Y_n(\phi )} \cdot \chi _n({\mathbf {p}}) \varphi ({\mathbf {p}}) \quad \phi (A,\xi )=\prod _{ {\mathbf {p}}\in \text {supp}A} \phi ({\mathbf {p}},\xi )^{A({\mathbf {p}})} \\ a({\mathbf {p}})&=C \text {d}_n^\varepsilon |{\mathbf {p}}|\quad \text {d}({\mathbf {p}})=-(1-\varepsilon ) \log \text {d}_n \end{aligned}$$

where $\varepsilon >0$ is arbitrarily chosen constant, $C={\displaystyle \frac{4c_1 e^2}{1-\beta (r)} }$ and ${d}_n=n^{-\zeta }$.

Using the estimate

$$\begin{aligned} \sum _{{\mathbf {p}}; 0 \in b({\mathbf {p}})} \varphi ({\mathbf {p}}) < \frac{4c_1}{1-\beta (r)} \cdot \frac{1}{n^\zeta } \end{aligned}$$

we obtain the inequality (15) for our model.

Applying Kotecky–Preiss Theory to our model, we have

$$\begin{aligned}&\varphi _{t_1,\ldots ,t_m}^{(n,r)} (z_1,\ldots ,z_m) \nonumber \\&\quad = \exp \left\{ \sum _A \Bigl (e^{ {\displaystyle \frac{1}{c(n)} \sum _{k=1}^m iz_k W_r^{(n)}( n^\alpha t_k, A)}}-1 \Bigr ) \varphi (A) \chi _n(A) \frac{\alpha ^T(A)}{A!}\right\} \end{aligned}$$

(16)

Setting $g(n)=\sqrt{n}$ and using Taylor’s expansion, the property that $\alpha ^T(A)=0$ unless A is a cluster, and arguments developed in Kuroda et al. (2011), we obtain

$$\begin{aligned}&\log \varphi _{t_1,\ldots ,t_m}^{(n,r)}(z_1,\ldots ,z_m) \\&\quad = -\frac{1}{2} \sum _{k=1}^m z_k^2 \cdot \Bigg \{ \frac{1}{n} \sum _{j=rn}^{n-1} \frac{f_1 \mu \left( \frac{j}{n}\right) }{1-\beta \left( \frac{j}{n}\right) } \cdot T \left( \frac{j}{n}\right) a +f_2 T \Bigg \} \cdot \frac{n^{1+\alpha -\zeta }}{c(n)^2} \\&\quad \quad -\frac{1}{2} \mathop{\sum\limits} _{{\begin{array}{c} {1\le k_1,k_2\le m}\\ { k_1 \not = k_2} \end{array}}} z_{k_1}z_{k_2} \Bigg \{ \frac{1}{n} \sum _{j=( (\frac{t}{T})^{\frac{1}{\alpha }}\wedge r)n}^{n-1} \frac{f_1 \mu \left( \frac{j}{n}\right) }{1-\beta \left( \frac{j}{n}\right) } \left( T\left( \frac{j}{n}\right) a -t(k_1,k_2) \right) \\&\quad \quad +f_2 (T -t(k_1,k_2)) \Bigg \} \frac{n^{1+\alpha -\zeta }}{c(n)^2} \end{aligned}$$

If we put $c(n)=n^{\frac{1}{2} (1+\alpha -\zeta )}$, then the proof of Theorem 1 is obtained.

Appendix 2: Proof of Proposition 4

Since $\{w_r(t)\}$ is a Gaussian process, $E[M_r(0,t)^q]$ is described as

$$\begin{aligned} E[M_r(0,t)^q]&=e^{-\frac{1}{2} T(f_2-f_1 \log r)q} \int _0^t \text {d}u_1 \cdots \int _0^t \text {d}u_q E[e^{w_r(u_1)+\cdots +w_r(u_q)}] \\&= e^{-\frac{1}{2} T (f_2-f_1 \log r)q} \int _0^t \text {d}u_1\cdots \int _0^t \text {d}u_q \\&\quad \times \exp \left\{ \frac{1}{2} \sum _{k=1}^q V(w_r(u_k))+ \frac{1}{2}\mathop{\sum}\limits _{{\begin{array}{c} {1\le k, \ell \le q}\\ {k \not =\ell } \end{array}}} \text {Cov}(w_r(u_k), w_r(u_\ell )) \right\} \\&:= I_1+I_2 \end{aligned}$$

where

$$\begin{aligned} I_1&=\int \int _{\begin{array}{c} {0\le u_1,\cdots ,u_q \le t}\\ { \left(\frac{|u_k-u_\ell |}{T}\right)^{\frac{1}{\alpha }} >r \;{\text {for all}} \;k\not =\ell } \end{array}} \exp \left\{ \frac{1}{2} \mathop{\sum}\limits _{{\begin{array}{c} {1\le k, \ell \le q}\\ {k \not =\ell } \end{array}}} \text {Cov}(w_r(u_k), w_r(u_\ell )) \right\} \text {d}u_1 \cdots \text {d}u_q \\ I_2&= \int \int _{\begin{array}{c} {0\le u_1,\ldots ,u_q \le t}\\ { \left(\frac{|u_k-u_\ell |}{T}\right)^{\frac{1}{\alpha }} \le r \;{\text {for some}} \;k\not =\ell } \end{array}} \exp \left\{ \frac{1}{2} \mathop{\sum}\limits _{{\begin{array}{c} {1\le k, \ell \le q}\\ {k \not =\ell } \end{array}}} \text {Cov}(w_r(u_k), w_r(u_\ell )) \right\} \text {d}u_1 \cdots \text {d}u_q \end{aligned}$$

First, we consider the limit of the main term $I_1$. Put $\lambda _0=\,{\displaystyle \frac{f_1T}{2\alpha }}$ and assume that $\lambda _0 q<1$. When ${ \displaystyle ( \frac{|u_k-u_\ell |}{T})^{\frac{1}{\alpha }}>r}$ for all $k \not = \ell$, the covariance of $w_r(u_k)$ and $w_r(u_\ell )$ is given by

$$\begin{aligned} \text {Cov}(w_r(u_k),w_r(u_\ell ))= -\frac{f_1T}{\alpha } \log \frac{|u_k-u_\ell |}{T}, \end{aligned}$$

so that

$$\begin{aligned}&I_1= \int \int _{\begin{array}{c} {0\le u_1,\cdots ,u_q \le t}\\ { (\frac{|u_k-u_\ell |}{T})^{\frac{1}{\alpha }} >r \;\text {for all} \;k\not =\ell } \end{array}} \mathop{\prod}\limits _{{\begin{array}{c} {1\le k,\ell \le q}\\ {k \not = \ell } \end{array}}} \frac{T^{\lambda _0}}{|u_k-u_\ell |^{\lambda _0}}\, \text {d}u_1 \cdots \text {d}u_q\\&\quad \uparrow C_1 \int _0^t \text {d}u_1 \cdots \int _0^t \text {d}u_q \prod _{1\le k<\ell \le q} \frac{1}{|u_k-u_\ell |^{2\lambda _0}}\quad ( r \downarrow 0) \\&\quad = C_1 t^{-\lambda _0 q^2 +(1+\lambda _0) q} \int _0^1 \text {d}u_1 \cdots \int _0^1 \text {d}u_q \prod _{1\le k<\ell \le q} \frac{1}{|u_k-u_\ell |^{2\lambda _0}} \end{aligned}$$

where $C_1=T^{\lambda _0(q^2-q)}$

The integral

$$\begin{aligned} I= \int _0^1 \text {d}u_1 \cdots \int _0^1 \text {d}u_q \prod _{1\le k<\ell \le q} \frac{1}{|u_k-u_\ell |^{2\lambda _0}} \end{aligned}$$

is called Selberg Integral and $I<\infty$ if $\lambda _0 q <1$.

Next, we consider an estimate for the remainder term $I_2$. Remark that the covariance of $w_r(u_k)$ and $w_r(u_\ell )$ satisfies an estimate

$$\begin{aligned} \text {Cov}(w_r(u_k),w_r(u_\ell )) \le -\frac{f_1T }{\alpha }\log \frac{|u_k-u_\ell |}{T} +f_2T \end{aligned}$$

(17)

in both cases $(\frac{|u_k-u_\ell |}{T})^{\frac{1}{\alpha }} \le r$ and $(\frac{|u_k-u_\ell |}{T})^{\frac{1}{\alpha }} > r$. In $I_2$ there exists a pair $(u_k,u_\ell )$ satisfying $(\frac{|u_k-u_\ell |}{T})^{\frac{1}{\alpha }} > r$. Without loosing generality we assume that $(\frac{|u_1-u_2|}{T})^{\frac{1}{\alpha }} > r$, so that

$$\begin{aligned} \exp \Big\{ \frac{1}{2} \text {Cov}(w_r(u_1),w_r(u_2))\Big\}= e^{-\lambda _0 \log \frac{|u_1-u_2|}{T} } \end{aligned}$$

Using this equality and (17) and putting

$$\begin{aligned} w_0=u_1-u_2, w_1=u_2, \ldots , w_{q-1}=u_q, \end{aligned}$$

we have

$$\begin{aligned}&I_2 \le \int _{-Tr^\alpha }^{Tr^\alpha } \text {d}w_0 \; e^{-\lambda _0 \log \frac{|w_0|}{T}} \cdot \int _0^t \text {d}w_1 \cdots \int _0^t \text {d}w_{q-1} \mathop{\prod}\limits _{{\begin{array}{c} {1\le k,\ell \le q-1}\\ { k \not = \ell } \end{array}}} e^{-\lambda _0 \log \frac{|w_k-w_\ell |}{T} +f_2T} \\&\quad = 2T^{\lambda _0} ( e^{f_2T} T^{\lambda _0})^{((q-1)^2-(q-1))} \int _0^{Tr^\alpha } \text {d}w_0 \, w_0^{-\lambda _0} \cdot \int _0^t\text {d}w_1 \cdots \int _0^t \text {d}w_{q-1} \\&\qquad \mathop{\prod}\limits _{{\begin{array}{c} {1\le k,\ell \le q-1}\\ { k \not = \ell } \end{array}}} \frac{1}{|w_k-w_\ell |^{\lambda _0}} \\&\quad =C_2 \cdot r^{\alpha (1-\lambda _0)} \cdot t^{-\lambda _0 (q-1)^2+(1+\lambda _0)(q-1)} \cdot \int _0^1 \text {d}v_1 \cdots \int _0^1 \text {d}v_{q-1} \\&\qquad \prod _{1\le k,\ell <\le q-1} \frac{1}{|v_k-v_\ell |^{2\lambda _0}} \end{aligned}$$

where $C_2 =2T^{\lambda _0} ( e^{f_2T} T^{\lambda _0})^{((q-1)^2-(q-1))} \cdot \frac{1}{1-\lambda _0}$.

Since $\lambda _0 q<1$, the integral in the right-hand side of the above formula is finite. Hence, $I_2 \rightarrow 0$ as $r \downarrow 0$.

Putting these results on $I_1$ and $I_2$ together we have the proof of Proposition 4.