A heterogeneous agent model of asset price dynamics with two time delays

See Hommes (2006) and LeBaron (2006) for overviews of early contributions, ter Ellen and Verschoor (2017) for review of empirical literature on heterogeneous beliefs and asset price dynamics and Dieci and He (2018) for survey of the current state-of-art of heterogeneous agents model.

The qualitative results to be obtained below are not affected by the choices of these values.

Chiarella (1992) gives two possible theoretical explanations of this shape.

We will have the same results even if \(\omega <0\) since conjugate is also a solution.

In particular, we take \(m=1/2\) and \(v=5,\) which lead to \(\tau _{0}(1/2,5)\simeq 0.628\). If m is increased to 0.62 as in Dibeh’s example, \(\tau _{0}(0.62,5)\simeq 1.5304.\)

Equation (6) implies that the price change is caused by the weighted demand of chartists and fundamentalists. The demand of fundamentalist is traditional, while the demand of chartists has a special form of asset price expectation that is a weighted average of the past two prices, \(p(t-\tau _{1})\) and \(p(t-\tau _{2})\). There are various expectation formations used in practice. In Chiarella (1992) and his successor He and Li (2012), the expectation is taken to be an exponentially decaying average of all the past prices,where \(1/\tau \) is the decay rate of the weights of the past prices. It is well known that in very rough sense, \(\bar{p}(t)\simeq p(t-\tau )\) as continuously distributed time delay can converge to a fixed delay under certain circumstance. Thus, the parameter \(\tau \,\)can be viewed as the time delay. Differentiating the above expression with respect to t presents the first-order differential equationwhere \(1/\tau \) can be viewed as the speed with which chartists adjust their expectations. In our model, the price trend is given byThe similarity between the roles of \(\sigma \) and \(1/\tau \) is clear. If \( \sigma =1/\tau ,\) then \(\eta =\tau -1\) implying that the delay \(\tau \) may be the weight of the past prices in our model.

If \(\sigma =1,\) then the two-delay model is reduced to the one-delay model. If \(\sigma =0\), then the average price cannot be defined.

Notice that the first equation of (12) has infinitely many solutions. Correspondingly, the general expression for the partition curve is.

Notice that the aspect ratio is appropriately adjusted.

In this example, \(\omega _{s}\simeq 1.414\) and \(\omega _{e}\simeq 2.449\). Since \(\tau _{2}^{-}(\omega ,0)\le 0\) for \(\omega \in [\omega _{s},2],\) we have \((\tau _{1}^{+}(\omega _{s},0),\ \tau _{2}^{-}(\omega _{s},0))\simeq (0.870,\ -1.351)\) and \((\tau _{1}^{+}(2,0),\ \tau _{2}^{-}(2,0))\simeq (0, 0.785).\)

This value is obtained by substituting \(\nu =5\) and \(m=2/5\) into \(\tau _{0}(m,\nu )\) given in Theorem 1 since the two-delay model is reduced to the one-delay model discussed in Sect. 1 when \(\sigma =1.\)

If the trend is formed by \(s=k[p(t)-p(t-\tau _{2})],\) then the threshold value is determined bySubstituting \(k=1/2\) that is obtained when \(\sigma =1/2\) yields this value.