Price Volatility Model in terms of Trader’s Conditional Expected Return

Shu Lin Zhang College of Economics and Business Administration

North China University of Technology

Shijingshan District, Beijing, China

cseec@sina.com

Shu Ping Wang College of Economics and Business Administration

North China University of Technology

Shijingshan District, Beijing, China

dwangshuping@163.com

Juan Juan Ding College of Economics and Business Administration

North China University of Technology

Shijingshan District, Beijing, China

kameding@263.net

Abstract—We proposed an expectation model to explain

the observed price movements with active trader’s conditional expected return and trading strategies. Existing volatility models can be considered as a kind of price expectation with corresponding expectation model and disturbances distribution. These results questioned those intentions to reproduce stylized facts with artificial series, and call for a closer look at the trading rule and trader’s price expectation implied in both agent-based models and econometric researches.

Keywords-price formation;expectation model;trading strategies;forecasting;time varying model

I. INTRODUCTION There has been a large interest in using agent-based

model to explore the origins of the stylized facts observed in asset markets. Much has achieved in artificial markets to reproduce time series that exhibits the stylized facts such as volatility clustering, heavy tails and absence of linear auto-correlations.

However, few agent-based models deals with actual markets, especially in commodity futures market .This might be due to the hardness in modeling actual trader’s behavior in real markets.

Furthermore, neglecting the existing volatility model in the price formation was indeed common in majority papers about artificial markets.

However, stylized facts might be the consequence of the way of trading or the market designs of price formation [1].

A potential solution to this problem is to incorporate the trader’s price expectation and their trading decisions into price formation in agent-based market design. This can be considered as an economic foundation of agent-based research.

The paper is organized as follows: Section 2 presents the major articles inspired us. In Section 3, we propose a simple model to explain the observed price movements with conditional expected return in terms of active trader’s trading decisions. Finally, Section 4 gives an empirical application of the proposed model.

II. LITERATURE REVIEW Due to the increasing availability of high frequency

market data, the empirical analysis of trading behavior and

trading processes has become a major theme in modern financial econometrics [2].

On the contrary, many agent-based articles still try to produce artificial time series to capture the stylized facts observed in asset returns.

Up to now, the mainstream method of modeling price fluctuations in agent-based models is excess demand model or market impact function [3, 4, 5].

For example, Day and Huang (1990) identified three types of market participants: alpha-investors, beta-investors, and the market maker. The price change P(t+1)–P(t) is determined by the total excess demand E (P (t)):

P (t + 1) = P (t) + cγ [EP (t)], (1)

where γ (0) =0 and c is an adjustment coefficient [6]. In general, the market impact function can be considered

as a nonlinear function of excess demand:

r(t+1)= g(z(t)),z(t)=Σφi (t), (2)

where z (t) is aggregate excess demand, g (z) is a market impact function, φi (t) denoting the demand of the agent i [7].

In fact, the excess demand and market impact function g (.) are unknown and g (.) is often chosen as a linear function.

A simple way of modeling price fluctuations is to decompose price movements into direction and size [8]:

r (t) = A (t) D (t) S (t) (3)

where A (t) take on only two values: 0, 1. A (t) =1 means there are active price movements.D(t) denotes the direction of the price movements. S (t) indicates the size of price movements. The non–zero price movement is characterized as r (t) = D (t) S (t).

Christoffersen and Diebold (2005) explored the links between volatility dynamics and directional market movements, provided a rigorous investigation of the sign forecast ability in returns even if returns were conditional mean independent [9].

On the other hand, almost every asset price series exhibits volatility clustering, which can be modeled by the

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volatility models such as General Autoregressive Conditional Heteroskedastistic (GARCH).

So, another useful way of modeling price formation based on volatility clustering is using popular GARCH models.

r(t)=β0+β1h (t)+ε(t),h(t)=α0+α1ε2(t-1)+α2 h (t) (4)

For the disadvantages of additive model, the multiplicative error model (MEM) was introduced by Engle [10]. MEM is a general framework to model positive-valued dynamic process. A multiplicative error model of price formation may take the form:

r (t) = μ tεt (5)

where It denotes the information set up to t, μt is a non-negative conditionally deterministic process given It−1, and εt is a unit mean, i.i.d. variate process defined on non-negative support [2].

However, for the standard MEM, by construction, we have E (rt |It-1) =μ t, Var (rt|It-1) = μ t σ2

t, which means a too strict restriction we must follow [2]:

E(rt |It-1)/Var(rt|It-1)=1/σ2t (6)

In general, there are three categories of econometric models on price forecasting: time series models exploiting the statistical properties of the data, financial models based on the relationship between spot and future prices; and structural models describing the relation between specific economic factors [11]. All of which can be used as a basis to model price formation.

The main problem with price formation is that agents have to learn to extrapolate market impact from past information. Furthermore, its target keeps changing as a result of these learning, leading to a moving target function problem [12].

Moreover, the moving target function does not change randomly, it changes based on the learning dynamics of the agents in the system.

Vidal and Durfee (2003) proposed to model such moving target system with follow four parameters: Change rate, learning rate, Retention rate, and Impact [13].

Another issue that arises when building market with learning agents is the choice of modeling level, where high level means participants behave strategically with regard to others' expectation [14].

III. PRICE FLUCTUATIONS WITH TRADER’S PRICE EXPECTATION AND THEIR TRADING DECISIONS

We are interested in building a time varying model to forecast commodity prices with learning agents.

The model of price formation developed here attempts to incorporate the trader’s price expectation and their expected return based on widely accepted volatility model.

A. Market Price Movements with Trader’s Price Expectation Market prices are ultimately determined by supply and

demand, which result from traders' trading decisions. To understand the observed price movements, it is

important to assess how traders perceive price movements and how they make their trading decision over time.

Denote

Pie(t)=Fi(P(t+1)|It)

ei(t+1)= P(t+1)-Pie(t)

P(t+1) = Pie(t)+ei(t+1)

Here,Fi(P (t+1) |It) indicates the private price forecasting function of trader i at present time t, ei(t+1) is the unknown relative forecasting bias of trader i at future time t+1.

Clearly, current market price represents only present traders’ price expectation. For example, the trading price P(t+2) may not come from the same trader’s decision at t, so

P(t+2) = Pke(t+1)+ ek(t+2)

Furthermore, forecasting function Pke(t+1) may be

different from Pie(t+1), sometimes Pk

e(t+1) may even be different from Pk

e(t). By definition, ⊿P(t+1)= P(t+2)-P(t+1)

⊿P(t+1)= Pke(t+1)- Pi

e(t)+ ek(t+2)-ei(t+1)

Thus, the fluctuations of the market prices depends on two unobserved distinct components: the changes in the price forecasting functions and the changes in the forecasting bias.

In general, observed price movements are records of trader’s private forecasting function and their unknown errors at different times, and the log return may be considered as a nonlinear function as below:

r(t+2)=g(Pke(t+1), Pi

e(t), ek(t+1), ei(t+2)) (7)

For example, if all traders form their price expectation as a random walk, the relation between market price changes and trader’s price expectation can be directly inferred as below.

Suppose

Pie(t)= Fi(P(t+1)|It)=P(t)+ωi(t),

where ωi(t) is expected price changes of trader i. Denote

ei(t+1)= P(t+1)-Pie(t)

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Thus

P(t+1)=P(t)+ωi(t)+ ei(t+1)

ΔP(t)=P(t+1)-P(t)=ωi(t)+ ei(t+1)

Simillarly,we have

Pke(t+1)= P(t+1)+ωk(t+1)

ek(t+2)= P(t+2)-Pke(t+1)

By definition,

ΔP(t+1)=P(t+2)-P(t+1)

ΔP(t+1)=ΔP(t)+[ωk(t+1)-ωi(t)]+ [ek(t+2)-ei(t+1)] (8)

With such a simple price expectation model, observed market price movements can be decomposed into three distinct parts: past price movements, changes in trader’s expected price movements and changes in unexpected movements.

For so many possible models existing shown in Section 2, expectation may be different among traders even to the same observed price series P (t).

So, the problem is how Fi (P (t) |It-1) are formed and why they are specified so. The further question is how the impact of trader’s price expectation and learning on the market price formation process.

B. Market Price Movements with Trader’s Expected Return Equation (7) is a useful concept model to explore the

origins of price fluctuations .However, it is asymmetric to the two parties (i,k),and is too complicated for the others (j≠i,j≠k) because of its unknown private forecasts functions and their unknown errors. How can we get a simpler model to solve the problem ?

The main problem with such a price expectation model is that it requires a private price forecasting function ,and it is difficult to decide which model should be include and how to produces a parsimonious model to explain price fluctuations.

The basic idea here is to model the price changes in terms of active trader’s price expectation and their trading decisions.

Active trader’s investment decisions are those intensions can be characterized as to exploit short-term market opportunities such as trend following or momentum, so the trading prices pi(t)* are different from trader’s price expectations, otherwise no profit means no incentive to trade.

Suppose market prices came from active traders with their limit orders, and we further assume that the bid/ask prices are formed by the trader’s price forecasts or valuation pi

e(t) deducted by the required expected return rie(t).

In a other word, price forecasting function can be decomposed into bid/ask price multiply by its expected return ri

e(t). Denote

Pie(t)=Pi(t)ri

e(t) (9)

Define

ei(t+1)= r(t+1)/ rie(t)

Therefore,

r(t+1) = rie(t) ei(t+1) (10)

Given the behavior assumption (9) and the price expectation ri

e(t), we can get a simple time varying model: the future price movements in (7) can be reduced to conditional expected return with its unknown unexpected bias.

Clearly, there is no unique solution in such decomposition. There may be other price expectation models to the same market price movements for different traders.

r (t+1) = rke(t) ek(t+1)

Thus, the remained problem with such parsimonious price expectation models is how expected return rk

e(t) is formed and how to deal with the unexpected bias.

For the widely accepted volatility models in market returns , we proposed popular GARCH models as a possible way to model expected return rk

e(t). For example, if the conditional expected return is

assumed to be a linear function of risk premium, which can be formed as:

rie(t)=ρi+δiσ2(t)e

Here ρi denotes the trader’s expected price movement, δiσ2(t)e is the volatility compensation ,σ2(t)e indicates the expected volatility, which can be estimated from any kind of GARCH models or stochastic volatility models.

Therefore, for the trader i,

r(t+1)= μi (t) ei(t+1) , μi (t)=ρi+δiσ2(t)e (11)

Combining all of these definitions and assumptions, we proposed a simpler expectation model that incorporate GARCH-M model to explain the observed price movements.

By definition, conditional expected return and its components (ρi,δi) required for trader i are always non negative.

The proposed model can be extended to other volatility models by specify corresponding expectation model with special disturbances distribution.

However, it is clear that actual distribution of the bias here is unknown and can not be specified as those in standard volatility models.

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In such expectation models, there are no need to estimate the parameters in the expected volatility model, it is easy to implement the expectation model, but the problems remained are model choosing and how the expectation bias are corrected.

Such volatility models are sensitive to the assumptions on trader’s price expectation and expected return model, information set and learning scheme [15,16,17].

Furthermore, for the given expected return model μ(t) = ρ+δσ2(t)e ,market implied expected return can also be estimated as r (t) = μ(t) ε(t), ε(t) is a unit mean, i.i.d. variate process. This is a traditional GARCH-M model.

IV. EMPIRICAL RESULTS The trading and decision processes are dependent on the

models used. The proposed expectation models can be implemented and explored by agent-based simulations.

To examine the performance of this methodology, we present a simple empirical example, producing daily sign forecasts of returns on the Dow Jones Industrial Average index from October 2, 1928 through July 23, 2009. Historical Prices Data are getting from yahoo (http://uk.finance.yahoo.com/q/hp?s=%5EDJI).

For simplicity, the initial forecasting model used is a random walk. The forecasting model is estimated recursively in a daily expanding data set.

The forecasting bias generated by the agents is shown in Fig 1.

The trading rule is an example of Conts (2005) threshold behavior: as a result of the unknown bias, without sufficient incentive, agents remains inactive ,but if the daily return expected is above the certain threshold, agents will act.

-.2

-.1

.0

.1

.2

.3

.4

5000 10000 15000

BIAS

Figure 1. Dayly forecasting bias

The corresponding error ratio generated by the trading rule is given in table I.

The empirical results are promising: the forecasting bias exhibits the stylized facts such as volatility clustering, heavy tails. Clearly, the error ratio is governed by the threshold rule.

The example illustrates our methods, provides preliminary evidence to further exploration.

However, much has to be done to have a better model to forecast prices in such expectation approach, especially with learning agents.

TABLE I. ERROR RATIO UNDER DIFFERENT THRESHOLD

ex-ante expectation vs.

ex-post

threshold

0.01 0.02 0.03 0.04 0.05

UU 0.2653 0.3070 0.3553 0.3928 0.3880

UD 0.2394 0.2335 0.2452 0.2428 0.2537

DD 0.2355 0.25 0.2075 0.1928 0.2089

DU 0.2596 0.2094 0.1918 0.1714 0.1492 Here U indicts price moves upwards, and D means price moves

downwards .UU represents both the expectation and realization are upwards, etc.

V. CONCLUSION This article deals with the critical role of trader’s price

expectation and trading strategies on the price formation process for agent-based models design.

We proposed a simple price expectation model to explain the observed price movements.The future price movements can be reduced to trader’s conditional expected return with its unexpected bias.

This model differs to the previous agent-based models in that it incorporates trader’s subjective price expectation as the critical explanatory variables.

On the other hand, existing volatility models can be considered as a kind of price expectation with corresponding expectation model and disturbances distribution. This allows for the combination between econometric model and agent-based models.

Consequently, we might be able to understand the complex dynamics of market price fluctuations from agent’s expectation and learning.

These results questioned those intentions to generate stylized facts with artificial series, and call for a closer look at the trading rule and trader’s price expectation implied in both agent-based models and econometric researches.

ACKNOWLEDGMENT This project was supported by the grant for grand

research project on Computational Finance with Oil Derivatives from the North China University of Technology and the Youth Project of Humanities and Sociology (#08JC790004) from the Ministry of Education.

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