At period 0, each agent is endowed with one unit of risky asset and some amount of cash which are randomly drawn from uniform distribution on the interval \( \left[0,\overline{\mathrm{C}}\right] \). We define the wealth of each agent at the initial period as
$$ {\mathrm{w}}_{\mathrm{i}0}={\mathrm{c}}_{\mathrm{i}0}+{\mathrm{p}}_0{\mathrm{a}}_{\mathrm{i}0} $$
where wi0, ci0, ai0, p0 is the wealth, amount of cash, and amount of assets for agent I and asset price at initial period, respectively.
Heterogeneity of agents
In the literature which studies the asset markets, two types of agents are considered: fundamentalists and chartists. Fundamentalists are agents who make trading decisions based on estimates of the fundamental value of an asset. Unlike fundamentalists, chartists use past price trends as a basis for decisions. Now, if an agent knows both types of information, namely the fundamental value of the asset and the history of price change rate, it is reasonable to assume that one will use both information for anticipating the future asset price. Therefore, in this model, we do not distinguish between fundamentalists and chartists. Instead, we assume that all agents know both types of information for the asset price and thus use both to predict the future price of an asset. Yet the weight for each type of information is different for all agents according to their beliefs. In sum, the prediction of the asset price for the next period is determined by current fundamental value of the asset and the history of the asset price change.
To consider the fundamental value of the asset into our model, it is necessary to define the fundamental value of the asset. In many previous papers which deal with fundamentalists in the asset market, fundamental values are assumed to be a constant (Chiarella and Iori 2002) or random walk process (Chiarella et al. 2009). In this paper, we assume that the fundamental value of the asset follows the random walk process with zero drift and volatility σ2 to reflect the economic states, which are not constant. At time t, agents have the information for the fundamental value of the asset, \( {\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}} \) which determined by following equation and they use this information to predict the future price of the asset. Finally, we can define the fundamental value of the asset price at time t as following.
$$ {\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}}={\mathrm{p}}_{\mathrm{t}-1}^{\mathrm{f}}+{\upvarepsilon}_{\mathrm{t}}\;\mathrm{where}\kern0.32em {\upvarepsilon}_{\mathrm{t}}\sim \mathrm{N}\left[0,{\upsigma}^2\right] $$
Regarding the information on the history of the price change, the level of the current asset price and the price change rate may be considered. This means the level of the asset price for the next period is determined by the level of the current price asset and the trend. Here, γ is the weight for the price change rate. The information on the history of the price change which is available at time t can be expressed as following.
$$ {\mathrm{p}}_{\mathrm{t}}^{\mathrm{c}}={\mathrm{p}}_{\mathrm{t}}+\upgamma \left(\frac{{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}}{{\mathrm{p}}_{\mathrm{t}-1}}\right) $$
Finally, we have the prediction equation of the asset price for the next period, \( {\overline{\mathrm{p}}}_{\mathrm{it}+1} \). This equation represents that agents anticipate the future asset price by using both types of information; fundamental value, \( {\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}} \) and the price change rate, \( {\mathrm{p}}_{\mathrm{t}}^{\mathrm{c}} \). Here, αi means the weight of the fundamental value between two types of information, which is different for all agents.
$$ {\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}={\upalpha}_{\mathrm{i}}{\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}}+\left(1-{\upalpha}_{\mathrm{i}}\right){\mathrm{p}}_{\mathrm{t}}^{\mathrm{c}} $$
$$ ={\upalpha}_{\mathrm{i}}\left({\mathrm{p}}_{\mathrm{t}-1}^{\mathrm{f}}+{\upvarepsilon}_{\mathrm{t}}\right)+\left(1-{\upalpha}_{\mathrm{i}}\right)\left[{\mathrm{p}}_{\mathrm{t}}+\upgamma \left(\frac{{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}}{{\mathrm{p}}_{\mathrm{t}-1}}\right)\right] $$
where 0 < αi < 1, γ is constant for all i and t.
The trading rule is simple. Because agents examined in this paper pursue capital gains, they want to buy an amount of asset only if they expect that the price will rise above the current level. i.e., they can expect capital gains through asset trading. Otherwise, they choose to sell some of the assets they hold to prevent capital loss.
The demand function for agents reflects this trading strategy. If agents expect the capital gains, \( {\overline{\mathrm{p}}}_{\mathrm{it}+1}-{\mathrm{p}}_{\mathrm{t}}>0 \), they choose to buy an amounts of the asset with a fixed fraction g of cash. With the same logic, if agents anticipate the capital loss, \( {\overline{\mathrm{p}}}_{\mathrm{it}+1}-{\mathrm{p}}_{\mathrm{t}}<0, \) they decide to sell g portions of assets they currently hold. In this model, because we impose constraint on borrowing and short-selling, agents can only trade assets when they have the cash for buying or assets for selling (Harras and Sornette 2011). One additional feature we need to notice is that when agents trade assets according to their prediction rules, they use only g fractions of their cash or asset. During the simulation we set g = 0.1. This reflects that agents hold both a risk free asset (cash) and risky assets in their portfolio, not confining these to one type of financial asset. The demand function could be summarized as following.
$$ {\mathrm{D}}_{\mathrm{i}\mathrm{t}}=\left\{\begin{array}{c}\hfill \left({\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}-{\mathrm{p}}_{\mathrm{t}}\right)\mathrm{g}{\mathrm{c}}_{\mathrm{i}\mathrm{t}}=\left[{\upalpha}_{\mathrm{i}}\left({\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}}-{\mathrm{p}}_{\mathrm{t}}\right)+\left(1-{\upalpha}_{\mathrm{i}}\right)\upgamma \left(\frac{{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}}{{\mathrm{p}}_{\mathrm{t}-1}}\right)\right]{\mathrm{gc}}_{\mathrm{i}\mathrm{t}},\mathrm{if}\kern0.20em {\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}\ge {\mathrm{p}}_{\mathrm{t}}\hfill \\ {}\hfill \left({\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}-{\mathrm{p}}_{\mathrm{t}}\right)\mathrm{g}{\mathrm{a}}_{\mathrm{i}\mathrm{t}}=\left[{\upalpha}_{\mathrm{i}}\left({\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}}-{\mathrm{p}}_{\mathrm{t}}\right)+\left(1-{\upalpha}_{\mathrm{i}}\right)\upgamma \left(\frac{{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}}{{\mathrm{p}}_{\mathrm{t}-1}}\right)\right]{\mathrm{ga}}_{\mathrm{i}\mathrm{t}},\mathrm{if}\kern0.20em {\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}<{\mathrm{p}}_{\mathrm{t}}\hfill \end{array}\right. $$
The existence of threshold: tipping point
In this section, we consider the tipping point. Many previous studies, which deal with the financial markets, have pointed out that the existence of chartists and their strategy (technical trading) play a significant role in making volatility in asset markets (Joshi et al. 1998; Chiarella et al. 2009). That is to say, since they are speculators who seek excessive capital gains, they choose to buy additional assets if they believe that the price will rise even if the price is already high enough. During this process, the asset price reaches a very high level. At this point, there is a possibility that some agents might change their trading strategies from buying to selling due to a concern about a collapse of the asset price bubble if the price level increases above the level that agents believe to be the maximum. That means agents have expectations about the ‘tipping point’ for the asset price, and this level can vary among agents. To reflect this idea, we impose certain thresholds for the price level that agents believe to be the maximum. We model that this level, \( {\tilde{\upomega}}_{\mathrm{i}} \), is randomly drawn from uniform distribution on the interval \( \left[\underline{\Omega},\overline{\Omega}\right] \) and is different for all agents.
The trading rule for considering the tipping point is similar to that of when we did not consider it. If the current level of the asset price is lower than the expectations for the tipping point and the price is expected to increase, then agents decide to buy some amount of assets. On the other hand, if agents expect the future price to fall or the price level is high enough compared with the level they believe to be the maximum, they will choose to sell an amount of assets to prevent capital loss. The trading rule and demand function when we consider the expectations about the ‘tipping point’ of agents are the following.
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1)
If \( {\mathrm{p}}_{\mathrm{t}}<{\tilde{\upomega}}_{\mathrm{i}} \)
$$ {\mathrm{D}}_{\mathrm{i}\mathrm{t}}=\left\{\begin{array}{c}\hfill \left({\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}-{\mathrm{p}}_{\mathrm{t}}\right)\mathrm{g}{\mathrm{c}}_{\mathrm{i}\mathrm{t}}=\left[{\upalpha}_{\mathrm{i}}\left({\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}}-{\mathrm{p}}_{\mathrm{t}}\right)+\left(1-{\upalpha}_{\mathrm{i}}\right)\upgamma \left(\frac{{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}}{{\mathrm{p}}_{\mathrm{t}-1}}\right)\right]{\mathrm{gc}}_{\mathrm{i}\mathrm{t}},\mathrm{if}\;{\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}\ge {\mathrm{p}}_{\mathrm{t}}\ \hfill \\ {}\hfill \left({\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}-{\mathrm{p}}_{\mathrm{t}}\right){\mathrm{ga}}_{\mathrm{i}\mathrm{t}}=\left[{\upalpha}_{\mathrm{i}}\left({\mathrm{p}}_{\mathrm{t}}^{\mathrm{f}}-{\mathrm{p}}_{\mathrm{t}}\right)+\left(1-{\upalpha}_{\mathrm{i}}\right)\upgamma \left(\frac{{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}}{{\mathrm{p}}_{\mathrm{t}-1}}\right)\right]{\mathrm{ga}}_{\mathrm{i}\mathrm{t}},\mathrm{if}\;{\overline{\mathrm{p}}}_{\mathrm{i}\mathrm{t}+1}<{\mathrm{p}}_{\mathrm{t}}\ \hfill \end{array}\right. $$
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2)
If \( {\mathrm{p}}_{\mathrm{t}}<{\tilde{\upomega}}_{\mathrm{i}} \),
$$ {\mathrm{D}}_{\mathrm{it}}=-{\mathrm{ga}}_{\mathrm{it}} $$
Price determination
Once all agents make decisions, the returns and new asset price are determined. Returns are determined by the excess demand and asset price is determined by the asset price of the last period and the return.
$$ {\mathrm{r}}_{\mathrm{t}}=\frac{1}{\uplambda \mathrm{N}}{\displaystyle \sum_{\mathrm{i}=1}^{\mathrm{N}}}{\mathrm{D}}_{\mathrm{i}\mathrm{t}} $$
$$ {\mathrm{p}}_{\mathrm{t}}={\mathrm{p}}_{\mathrm{t}-1} \exp \left({\mathrm{r}}_{\mathrm{t}}\right) $$
where r(t) is the return at time t, and λ is the relative impact of the excess demand upon the price (Harras and Sornette 2011).
Cash and asset update
Finally, cash and asset are updated as a result of trading. New cash amounts are determined by the cost for purchasing new assets subtracted from the amount of cash held by agents. The new asset amounts are determined by the amount of asset agents held in the last period and the demand for asset in this period.
$$ {\mathrm{c}}_{\mathrm{it}+1}={\mathrm{c}}_{\mathrm{it}}-{\mathrm{p}}_{\mathrm{t}}{\mathrm{D}}_{\mathrm{it}} $$
$$ {\mathrm{a}}_{\mathrm{it}+1}={\mathrm{a}}_{\mathrm{it}}+{\mathrm{D}}_{\mathrm{it}} $$