n?u=RePEc:arx:papers:1611

**arXiv**:**1611.08088v1** [q-fin.ST] 24 Nov 2016

Multiple Time Series Ising Model for Financial

Market Simulations

Tetsuya Takaishi

Hiroshima University of Economics, Hiroshima 731-0192, JAPAN

E-mail: tt-taka@hue.ac.jp

Abstract. In this paper we propose an Ising model which simulates multiple financial

time series. Our model introduces the interaction which couples to spins of other systems.

Simulations from our model show that time series exhibit the volatility clustering that is often

observed in the real financial markets. Furthermore we also find non-zero cross correlations

between the volatilities from our model. Thus our model can simulate stock markets where

volatilities of stocks are mutually correlated.

1. Introduction

Thefinancialmarketsareconsideredtobecomplex systemswheremanyagents areinteractingat

differentlevels andactingrationally orinsomecases irrationally. Suchfinancialmarkets produce

a rich structure on time variation of various financial assets and the pronounced properties of

asset returns has been classified as the stylized facts, e.g. see [1]. The most prominent property

in the stylized facts is that asset returns show fat-tailed distributions that can not be explained

by the standard random work model. A possible explanation for the fat-tailed distributions is

that the return distributions are viewed as a finite-variance mixture of normal distributions,

suggested by Clark[2]. In this view asset returns follow a Gaussian random process with a timevaryingvolatility.

This view, usingrealized volatility[3, 4] constructed from high-frequency data,

hasbeentestedanditisfoundthattheassetreturnsareconsistent withthisview[5,6,7,8,9,10].

Bornholdt proposed an Ising model designed to simulate financial market as a minimalistic

agent based model[11]. The Bornholdt model successfully exhibits several stylized facts such

as fat-tailed return distributions and volatility clustering[11, 12, 13, 14]. Variants of the model

have also been proposed and they exhibit exponential fat-tail distributions[15] or asymmetric

volatility[16]. The view of finite-variance mixture of normal distributions has also been tested

for the Bornholdt models and it is shown that returns simulated from the models are consistent

with the view of the finite-variance mixture of normal distributions[17, 18].

So far the models studied only dealt with a financial market where a single asset is traded.

In the real financial markets various financial assets are traded and they are correlated each

other. Measuring correlations between assets is important to investigate stability of financial

markets and many studies have been conducted to reveal properties of correlations in financial

markets, e.g. [19, 20, 21, 22, 23]. In this study we propose an Ising model that extends the

single Bornholdt model to a multiple time series model. Then we perform simulations of our

model and show that the model can exhibit correlations between the return volatilities.

2. Multiple Time Series Ising Model

Let us consider a financial market where K stocks are traded and assume that each stock is

traded by N = L × L agents on a square lattice. Each agent has a spin s i and it takes two

states s i = ±1 corresponding to ”buy” (+1) or ”sell” (-1), where i stands for the ith agent. The

decision of agents are made probabilistically according to a local field. In our model the local

field h (k)

i

(t) at time t is given by

h (k)

i

(t) = ∑ Js (k)

j

(t)−αs (k)

i

(t)|M (k) (t)|+

〈i,j〉

K∑

γ jk M (j) (t), (1)

where 〈i,j〉 stands for a summation over the nearest neighbor pairs, k denotes the kth stock and

M (k) (t) is the magnetization that shows an imbalance between ”buy” and ”sell” states, given by

M (k) (t) = ∑ N

l=1 s(k) l

(t). J is the nearest neighbor coupling and in this study we set J = 1. The

third term on the right hand side of eq.(1) describes the interaction with other stocks that is not

present in the Bornholdt model. More precisely this interaction couples to the magnetization of

other stocks and introduce an effect of imitating the states of other stocks. Themagnitude of the

interaction is given by the interaction parameters that form a matrix γ lm having zero diagonal

elements, i.e. γ ll = 0. As in the Bornholdt model the states of spins are updated according to

the following probability.

s (k)

i

(t+1) = +1 p = 1/(1+exp(−2βh (k)

i

(t))), (2)

s (k)

i

(t+1) = −1 1−p.

j=1

0.1

Stock 1

Stock 2

0.1

Stock 1

Stock 2

M(t)

0

M(t)

0

-0.1

-0.1

1e+05 1.2e+05 1.4e+05

t

1e+05 1.2e+05 1.4e+05

t

Figure 1. Time series of the magnetization for

stocks 1 and 2 at γ = 0.0.

Figure 2. Time series of the magnetization for

stocks 1 and 2 at γ = 0.15.

3. Simulations

In this study we perform simulations for K = 2, i.e. we simulate a stock market consisting of

two stocks. The simulations are done on 120×120 square lattices with the periodic boundary

condition. We set simulation parameters to(β,α,J) = (2.0,30,1). Hereweassumesymmetricγ,

i.e. γ 12 = γ 21 , which means that the stocks we consider give the same interaction effect to other

stocks each other. Here we make simulations for five values of γ, γ = (0.0,0.05,0.07,0.10,0.15).

The states of spins are updated randomly according to eq.(2). After discarding the first 10 4

updates as thermalization we collect data from 5×10 5 updates.

0.01

Time series 1

Return

0

-0.01

1e+05 1.2e+05 1.4e+05

0.01

Time series 2

Return

0

-0.01

1e+05 1.2e+05 1.4e+05

t

Figure 3. Time series of return for stocks 1 and 2 at γ = 0.0.

0.02

0.01

Time series 1

Return

0

-0.01

-0.02

1e+05 1.2e+05 1.4e+05

0.02

0.01

Time series 2

Return

0

-0.01

-0.02

1e+05 1.2e+05 1.4e+05

t

Figure 4. Time series of return for stocks 1 and 2 at γ = 0.15.

Fig.1 shows the time series of the magnetization for two stocks ( 1 and 2 ) at γ = 0.0.

Simulating at γ = 0.0 means that two stocks are independent. On the other hand Fig.2 shows

the time series of the magnetization at γ = 0.15 wherethe interaction between stocks are present

and we find that synchronization occurs between the magnetizations. For other γ > 0 we also

see similar synchronization between the magnetizations to a certain extent.

Such synchronization also occurs between the returns defined by Return(t) = (M(t + 1) −

M(t))/2 as in [13]. The time series of returns at γ = 0.0 and 0.15 are shown in Figs. 3 and

4 respectively. It is clearly seen that volatility clustering occurs in the return time series. At

Table 1. Cross correlation between the volatilities for various γ.

γ 0.0 0.05 0.07 0.10 0.15

cross correlation 8.2×10 −3 5.8×10 −2 8.4×10 −2 0.15 0.31

γ = 0.0, however, we see no synchronization between the volatility clusterings. On the other

hand some volatility clusterings synchronize at γ = 0.15.

To quantify the strength of the synchronization we measure cross correlations between

volatilities from the two stocks. The cross correlation between the lth and the mth volatilities

is given by 〈(v(l) (t)−〈v (l) (t)〉)(v (m) (t)−〈v (m) (t)〉)〉

σ (l) σ (m) , where v (l) and σ (l) stands for the volatility

and its standard deviation of the lth stock respectively. Here we define the volatility by the

absolute value of return. Table 1 shows the values of cross correlations for various γ and we find

that the cross correlation of volatility increases with γ.

4. Conclusion

WehaveproposedanIsingmodelwhichcansimulatemultiplefinancialtimeseriesandperformed

simulations of a stock market with two stocks. The interaction parameter γ in the model tunes

the strength of correlations between stocks. We calculated cross correlation between volatilities

of the two stocks and found that the cross correlation increases with γ. Therefore our model

serves to simulate stock markets where volatilities of stocks are mutually correlated. Since we

have demonstrated the model includingonly two stocks it may bedesirableto furtherinvestigate

the more realistic model that includes many stocks.

Acknowledgement

Numerical calculations in this work were carried out at the Yukawa Institute Computer Facility

and the facilities of the Institute of Statistical Mathematics. This work was supported by JSPS

KAKENHI Grant Number 25330047.

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