Micro-Foundation of ARCH Model
Takanobu Mizuta
SPARX Asset Management Co., Ltd.
Many macroeconomic study argued macroeconomic models should be aggregated by micro processes models(“micro-foundation”) and many micro-founded macroeconomic models were built. On the other hand, there aremany models for price variation of a risk asset, which is macro phenomena, however, there are few studies for
micro-foundation of such models. In this study we tried micro-foundation of an ARCH model using intelligence ofartificial market simulation studies. That is we tried to clarify which micro processes determine each coeﬃcient ofan ARCH model. Then, we showed that the dispersion of investors’ estimated prices is larger or the orders by thebuy-sell imbalance taking liquidity are more, the volatility is larger. And we showed that the ration of the normalinvestors taking liquidity to the noise traders providing liquidity is higher or the measure of risk aversion of thenormal investors is lower, the magnitude of volatility clustering is larger.

1.

Introduction

Many macroeconomic study argued macroeconomic models should be aggregated by micro processes models (“microfoundation”)∗1 , and many micro-founded macroeconomic
models were built. On the other hand, there are manymodels for price variation of a risk asset, which is macro
phenomena, such as ARCH model [Engle 82] and GARCH
model [Bollerslev 86], however, there are few studies for
micro-foundation of such models than macroeconomic models.
On the other hand, there are many previous study using
artificial market simulation models, a kind of multi-agent
models, which simulate macro processes such as investors
and order matching on a computer. Artificial market studies observe macro phenomena such as price variations as aresult of the modeled macro processes∗2 .
Not only academies but also financial regulators andstock exchanges are recently interested in multi-agent simulations such artificial market models to investigate regulations and rules of financial markets. Indeed, the Science article by Battiston et al. [Battiston 16] described that ‘sincethe 2008 crisis, there has been increasing interest in usingideas from complexity theory (using network models, multiagent models, and so on) to make sense of economic andfinancial markets’. Actually, some artificial market studies
recently contributed to discussion what financial regulations
and rules should be∗3 .
Artificial market models only modelize the micro processes and observe macro phenomena, therefore, artificial
market models are fully micro-founded models. So, artificial market models have been gaining intelligence micromacro interaction mechanisms such as what micro processes
Contact: mizutata@gmail.com
http://www.geocities.jp/mizuta ta/∗1∗2[Lucas 76] is very famous example.
[LeBaron 06, Chen 12, Cristelli 14, Mizuta 16a] are excellent
reviews.
∗3 for example, tick size [Mizuta 13], speed of order matching systems on financial exchanges [Mizuta 15a], short selling
regulation [Mizuta 16b] and usage rate of dark pools [Mizuta15b].

amplify price variations.
Then in this study we try micro-foundation of theARCH(1) model [Engle 82] using intelligence of artificial
market simulation studies. That is we try to clarify whichmicro processes determine each coeﬃcient of the ARCH(1)model.

2.

ModelAn ARCH(1) model [Engle 82] with an average return iszero is finally derived,r t = σt ϵ t2σt2 = a0 + a1 rt−1.

(1)

Where rt is a log return, ϵt is a stochastic variable obeying
a standard normal distribution, and a0 , a1 are constants. a0
indicates a magnitude of a volatility (a degree of variationof a market price series) and a1 indicates a magnitude ofa volatility clustering (a large volatility tend to maintain alarge for a while) [Mandelbrot 63]. When a1 = 0, the model
indicates the case without a volatility clustering. Therefore,
micro processes determining a0 are cause of an ever-presentvolatility, and micro processes determining a1 are cause ofan additional volatility by the volatility clustering.
Next, we build a macro model aggregated by micro process discovered by the artificial market studies, and we compare the model with Eq.(1).
There are many artificial market models adopting a pricing model which determines a price variation by buy-sellorder imbalance∗4 . Even though they adopt such simple
pricing model, it is well known that they can replicate statistics of price variations in real financial markets. For example, [Palmer 94] defined rt = η(Abt − Ast ) where rt , Abt andAst are a return from time t − 1 to t, an amount of buy andsell orders respectively. η is some constant, and [Palmer 94]
did not show that what micro processes in real markets determine η. [Mizuta 12] tried to clarify which micro processes
determine η and obtained,
rt = ρ

Abt − Ast.
Abt + Ast

∗4 For examples, [Arthur 91, Palmer 94, Arthur 97, Lux 99].

(2)

Where ρ is proportional to a dispersion of investors’ estimated prices. In short, the dispersion is larger, the volatilityis larger. We explain details how they obtain Eq.(2) in theAppendix.
At first, we discuss the case only with noise traders. Weassume,Abt = 12 S + 12 kSϵt ,(3)
Ast = 21 S − 12 kSϵt .

obtain,Ut ∝(7)

Where rte is a normal investors’ estimated prices, α is ameasure of risk aversion. Usually, normal investors estimate
larger return, the price varied larger. Therefore, we assume2
that rte is in proportion to rt−1
and,
Ut =

Where S and k are constants. The first term describes orders that noise traders always make and the second term
describes the buy-sell order imbalance. k describes theamount of order by the imbalance per the amount of everpresent orders. Therefore, k means how much liquidity theorders by the imbalance take, and k is larger, the liquidity
is lower.
To substitute Eq.(2) for Eq.(3) we obtain,
rt = ρkϵt .

rte
α2
rt−1.
α(8)

To substitute Eq.(2) for Eq.(6) and Eq.(8) we obtain,(l 2 )
rt = ρk 1 + rt−1ϵt .
α(9)4≪ 1. Comparing Eq.(1) with
Furthermore we assume rt−1
this we obtain,(4)

To compare Eq.(1) with this we obtain,l 2
σt2 = ρ2 k2 + 2ρ2 k2 rt−1,α(10)a0 = ρ2 k2 ,a1 = 2ρ2 k2 αl .

(11)

and,
σt2 = ρ2 k2 = a0 .

(5)This shows the case with a1 = 0. As I mentioned micro processes determining a0 are cause of an ever-present volatility, and micro processes determining a1 are cause of anadditional volatility by the volatility clustering. Therefore,
the noise traders only generate the ever-present volatility,and do no generate the volatility clustering. The degree ofvolatility is determined by the dispersion of investors’ estimated prices(ρ) and the orders by the imbalance taking
liquidity(k).
For examples, an asset having high dispersion of estimated prices such as a stock has larger volatility than an asset having low dispersion such as a high credit rating bond.
And a stock having low liquidity due to be few waiting orders has larger volatility than a stock having high liquidity
provided by many waiting orders.
Next, we discuss the case not only with the noise traders
but also with the normal investors who determine amount oforders using a utility function. The normal investors correspond to fundamental and/or technical strategy investors inreal financial markets. Normal investors determine optimal
holdings of shares on their own strategies, and make theirholdings by taking liquidity provided by the noise traders.
Here, we assume,Abt = 12 S + 12 kS(1 + lUt )ϵt ,Ast = 12 S − 12 kS(1 + lUt )ϵt .

a0 is exactly same with the case without the normal investors. As we mentioned a1 indicates a magnitude of thevolatility clustering, and the magnitude of the volatility
clustering are determined not only by ρ and k same as theever-present volatility(a0 ) but also by the ratio of existing
the normal investors(l) and the measure of risk aversion of
the normal investors(α).
For examples, the ration of the normal investors taking
liquidity to the noise traders providing liquidity is higher,
the magnitude of volatility clustering is larger. And, themeasure of risk aversion of the normal investors is lower,in short investors make risk-taking trades more, the magnitude of volatility clustering is larger. On the other hand, thenormal investors do not aﬀect the ever-present volatility.

3.

In this study we tried micro-foundation of the ARCH(1)model [Engle 82] using intelligence of artificial market simulation studies. That is we tried to clarify which micro processes determine each coeﬃcient of the ARCH(1) model.
Then we obtained,l 2
σt2 = ρ2 k2 + 2ρ2 k2 rt−1.
α(6)

(12)

The dispersion of investors’ estimated prices(ρ) is larger
or the orders by the buy-sell imbalance taking liquidity(k)is larger, the volatility is larger. The ration of the normal investors taking liquidity to the noise traders providing
liquidity(l) is higher or the measure of risk aversion of thenormal investors(α) is lower, the magnitude of volatility
clustering is larger.
There are two future works. One is an empirical study
validating our model. Another one is more detail discussionof our assumptions that are too strong assumptions for real
financial markets.

lUt is a term added due to the normal investors where l isa ratio of existing the normal investors and Ut is a utility
function determining amount of orders. l means a ratio ofinvestors taking liquidity to traders providing liquidity, andl is larger, the liquidity is lower. Here, we assume a constantabsolute risk aversion (CARA) model∗5 as Ut . Therefore we∗5Conclusion and Future Work

[Pratt 64, Arrow 65] proposed CARA model. [Izumi 96,Arthur 97, Izumi 99, Yagi 10, Chiarella 09, Gsell 09] and so on
used CARA model for investors to determine amount of orders. [Mizuta 12] showed that CARA model play an importantrole to replicate the volatility clustering.

2Distribution of buy orders

Distribution of sell orders
amount of buyorders (area):

number orders

amount of sellorders (area):

number orders

𝐴𝑏𝑡𝐴𝑡𝑠log price (𝑝𝑡 )𝑝𝑡−1

log price (𝑝𝑡 )𝑝𝑡−1

Fig. 1: Distributions of buy and sell orders.
𝑏𝑠𝑏integrated (𝐵𝑡 , 𝐵𝑡 )number of orders𝑠

integrated (𝐵𝑡 , 𝐵𝑡 )number of orders

𝐴𝑏𝑡 = 𝐴𝑡𝑠

𝐴𝑡𝑠

𝐴𝑏𝑡sellsellbuy

𝑝𝑡 = 𝑝𝑡−1log price (𝑝𝑡 )buy𝑝𝑡 < 𝑝𝑡−1 log price (𝑝𝑡 )Fig. 2: Determining a trade price by the call market. Note that the horizontal axis shows prices and the vertical axis shows
cumulative amount of orders.

Appendixorder distribution from the cheapest price to pt , and thenthey obtained,

Here, we introduce the derivation of Eq.(2) by [Mizuta12]. They assumed there are enough buying investors andselling investors. They assumed that investors’ buy and sellorder prices(log prices) pt at time t are a normal distribution which average is pt−1 and standard deviation is ρ́ asFig.1. The dispersion of order prices (ρ́) is generated by adispersion of investors’ estimated prices. The distributionsof buy and sell orders are same shape, however, amounts oforders that correspond to the areas of distributions are notequal. They defined the areas of buy and sell order distributions as Abt and Ast . They employed a call auction as a
pricing mechanism in which a trade price (market price) isdetermined at a balance of supply and demand. As Fig.2
shows, the market price is determined at an intersectionof the demand curve made by the buy orders distribution
and the supply curve made by the sell orders distribution∗6 .
The supply curve Bts (pt ) is a cumulative amount of orders
less than some price pt . Therefore Bts (pt ) is integrated sellAsBts (pt ) = t2[(1 + erf

pt − pt−1√2ρ́2)].

(13)

Where erf is the error function and defined as,
2erf(x) = √π∫ x2e−t dt.

(14)0The demand curve Btb (pt ) is integrated buy order distribution from pt to the highest price and then they obtained,
AbBtb (pt ) = Abt − t2

[1 + erf(pt − pt−1√2ρ́2)].

(15)

As the left side of Fig. 2 shows, in the case of Ast = Abt
the market price pt is determined as pt−1 , and the market
price is not changed. On the other hand, as the right sideof Fig. 2 shows, in the case of Ast ̸= Abt the market priceis not pt . In short, due to the imbalance of buy-sell orders∗6 Note that in Fig.2 the horizontal axis shows prices and thevertical axis shows cumulative amount of orders.

3the market price determined as a diﬀerent price from theaverage of investors’ estimated price(pt−1 ).
Here, from Bts (pt ) = Btb (pt ) they obtained pt
pt − pt−1 = rt =√2ρ́2 erf −1 (Z).

[Izumi 96] Izumi, K. and Okatsu, T.: An artificial market analysis ofexchange rate dynamics, Evolutionary Programming V, pp. 27–36(1996)[Izumi 99] Izumi, K. and Ueda, K.: Analysis of dealers’ processing
financial news based on an artificial market approach, Journal ofComputational Intelligence in Finance, Vol. 7, pp. 23–33 (1999)(16)[LeBaron 06] LeBaron, B.: Agent-based computational finance,
Handbook of computational economics, Vol. 2, pp. 1187–1233(2006)

Where rt is log return, erf −1 is the inverse function of theerror function andAb − AstZ ≡ tb,At + Ast[Lucas 76] Lucas, R. E.: Econometric policy evaluation: A critique,in Carnegie-Rochester conference series on public policy, Vol. 1,pp. 19–46, Elsevier (1976)(17)[Lux 99] Lux, T. and Marchesi, M.: Scaling and criticality in astochastic multi-agent model of a financial market, Nature, Vol.
397, No. February, pp. 498–500 (1999)

where Z is always satisfied,|Z| < 1.

[Mandelbrot 63] Mandelbrot, B.: The variation of certain speculative prices, The journal of business, Vol. 36, No. 4, pp. 394–419(1963)(18)

Here they assumed Z ≪ 1 and then they could Maclaurin
expand erf −1 (z) by the first order term (there is no secondorder term),√πrt =ρ́Z.
(19)
2√
Furthermore, they defined ρ = π/2ρ́ and restored Z, finally they obtained,
rt = ρAbt − Ast.
Abt + Ast

[Mizuta 12] Mizuta, T., Yagi, I., and Izumi, K.: Development of anEvaluation Method for Artificial Market Settings Considering a
Realistic Pricing Mechanism, Transactions of the Japanese Society for Artificial Intelligence, Vol. 27, No. 6, pp. 320–327 (2012),
http://doi.org/10.1527/tjsai.27.320[Mizuta 13] Mizuta, T., Hayakawa, S., Izumi, K., and Yoshimura, S.:
Investigation of Relationship between Tick Size and Trading Volume of Markets using Artificial Market Simulations, in JPX working paper, No. 2, Japan Excgange
Group (2013), http://www.jpx.co.jp/english/corporate/researchstudy/working-paper/(20)[Mizuta 15a] Mizuta, T., Noritake, Y., Hayakawa, S., andIzumi, K.:
Impacts of Speedup of Market System on
Price Formations using Artificial Market Simulations, inJPX working paper, No. 9, Japan Excgange Group (2015),
http://www.jpx.co.jp/english/corporate/research-study/workingpaper/

Note that the opinions contained herein are solely those of theauthors and do not necessarily reflect those of SPARX Asset Management Co., Ltd.

[Mizuta 15b] Mizuta, T., Kosugi, S., Kusumoto, T., Matsumoto, W.,and Izumi, K.: Eﬀects of Dark Pools on Financial Markets’ Eﬃciency and Price-Discovery Function: An Investigation by Multi-Agent Simulations, Evolutionary and Institutional
Economics Review, Vol. 12, No. 2, pp. 375–394 (2015),
http://dx.doi.org/10.1007/s40844-015-0020-3

This equation is exactly Eq.(2).

Disclaimer

Reference[Arrow 65] Arrow, K. J.: Aspects of the theory of risk-bearing, Yrjö
Jahnssonin Säätiö (1965)[Mizuta 16a] Mizuta, T.:
A Review of Recent Artificial
Market Simulation Studies for Financial Market Regulations And/Or Rules, SSRN Working Paper Series (2016),
http://ssrn.com/abstract=2710495[Arthur 91] Arthur, W., Durlauf, S., Lane, D., and Program, S. E.:
Money and Financial Markets, pp. 354–368, Blackwell, Cambridge (1991)[Mizuta 16b] Mizuta, T., Kosugi, S., Kusumoto, T., Matsumoto, W.,Izumi, K., Yagi, I., and Yoshimura, S.: Eﬀects of Price Regulations
and Dark Pools on Financial Market Stability: An Investigation
by Multiagent Simulations, Intelligent Systems in Accounting,
Finance and Management, Vol. 23, No. 1-2, pp. 97–120 (2016),
http://dx.doi.org/10.1002/isaf.1374[Arthur 97] Arthur, W., Durlauf, S., Lane, D., and Program, S. E.:Asset pricing under endogenous expectations in an articial stock
market, The economy as an evolving complex system II, pp. 15–44, Addison-Wesley Reading, MA (1997)[Battiston 16] Battiston, S., Farmer, J. D., Flache, A., Garlaschelli, D., Haldane, A. G., Heesterbeek, H., Hommes, C.,Jaeger, C., May, R., and Scheﬀer, M.: Complexity theory and financial regulation, Science, Vol. 351, No. 6275, pp. 818–819 (2016)[Palmer 94] Palmer, R., Brian Arthur, W., Holland, J., LeBaron, B.,
and Tayler, P.: Artificial economic life: a simple model of a stockmarket, Physica D: Nonlinear Phenomena, Vol. 75, No. 1-3, pp.
264–274 (1994)[Bollerslev 86] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity, Journal of econometrics, Vol. 31, No. 3,
pp. 307–327 (1986)[Pratt 64] Pratt, J. W.: Risk Aversion in the Small and in the Large,
Econometrica, Vol. 32, No. 1/2, pp. 122–136 (1964)[Yagi 10] Yagi, I., Mizuta, T., and Izumi, K.: A Study on
the Eﬀectiveness of Short-selling Regulation using Artificial
Markets,
EvolutionaryandInstitutionalEconomics Review, Vol. 7, No. 1, pp. 113–132 (2010),
http://link.springer.com/article/10.14441/eier.7.113[Chen 12] Chen, S.-H., Chang, C.-L., and Du, Y.-R.: Agent-based
economic models and econometrics, Knowledge Engineering Review, Vol. 27, No. 2, pp. 187–219 (2012)[Chiarella 09] Chiarella, C., Iori, G., and Perelló, J.: The impactof heterogeneous trading rules on the limit order book and order flows, Journal of Economic Dynamics and Control, Vol. 33,
No. 3, pp. 525–537 (2009)[Cristelli 14] Cristelli, M.: Complexity in Financial Markets, Modeling Psychological Behavior in Agent-Based Models and Order
Book Models, Springer (2014)[Engle 82] Engle, R. F.: Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,
Econometrica, Vol. 50, No. 4, pp. 987–1007 (1982)[Gsell 09] Gsell, M.: Assessing the impact of algorithmic trading onmarkets: a simulation approach (2009)4