Complexity and model comparison in agent based modeling of financial markets

Abstract

Agent based models of financial markets follow different approaches and might be categorized according to major building blocks used. Such building blocks include agent design, agent evolution and the price finding mechanism. The performance of agent based models in matching key features of real market processes depends on how these building blocks are selected and combined. For model comparison, both measures of model fit and model complexity are required. Some suggestions are made on how to measure complexity of agent based models. An application for the foreign exchange market illustrates the potential of this approach.

Appendices

Appendix 1: Approximate entropy algorithm

Let X(1), X(2), ..., X(N) be the time-series to be analyzed. Given the selected block length m (also referred to as pattern length or window size), a new series of vectors $V(1), \ldots , V(N-m+1)$ in $\mathcal {R}^m$ can be constructed, where $V(i) = X(i), \ldots , X(i+m-1)$.

Let the distance between two such vectors be $|V(i),V(j)| = \max (X(i+k) - X(j+k))$, for $0 \le k \le m-1$, i.e. the maximum difference in the scalar components. Given the tolerance window r (also referred to as threshold distance), a similarity function f can be defined as:

$$\begin{aligned} f(i,j,m,r) = {\left\{ \begin{array}{ll} 1 &{} |V(i),V(j)| \le r\\ 0 &{} \text {otherwiese} \end{array}\right. } \end{aligned}$$

(22)

The frequency when a vector V of size m is similar with another vector of the same size is denoted by:

$$\begin{aligned} C_i^m(r) = \frac{\sum _{j=1}^{N-m+1} f(i,j,m,r)}{N-m+1} \end{aligned}$$

(23)

Next, we define:

$$\begin{aligned} \varPhi ^m(r) = \frac{\sum \nolimits _{i=1}^{N-m+1} \ln C_i^m(r)}{N-m+1} \end{aligned}$$

(24)

Finally, the approximate entropy is:

$$\begin{aligned} ApEn(m,r) = \varPhi ^m(r) - \varPhi ^{m+1}(r) \end{aligned}$$

(25)

Related to the effective implementation of the algorithm, the original and basic version was introduced in Pincus et al. (1991), while an improved version is described in Manis (2008).

Appendix 2: Higuchi Fractal dimension algorithm

Let X(1), X(2), ..., X(N) be the finite set of N observations to be analyzed. A number of k new series $X^m_k$ can be constructed as follows:

$$\begin{aligned} X^m_k = {X(m), X(m+k), X(m+2\,k), \ldots , X\left( m+\left[ \frac{N-m}{k}\right] \,k\right) }, \end{aligned}$$

(26)

with $m = 1,2,\ldots ,k$ representing the initial values, k indicating the interval time (delay) and $[\cdot ]$ denoting the Gauss notation, i.e. the integer part.

For each of the previous $X^m_k$ series, the length of the curve can be computed as follows:

$$\begin{aligned} L_m(k) = \frac{1}{k}\,\left( \sum \limits _{i=1}^{[(N-m/k)]} \left| X(m+i\,k) - X(m+(i-1)\,k)\right| \right) \,\frac{N-1}{\left[ \frac{N-m}{k}\right] \,k}, \end{aligned}$$

(27)

where the last factor is a normalization factor.

The average length for each scale $k = 1,2,\ldots ,k_{max}$, L(k), is computed as the mean of $L_m(k)$ where $m = 1,2,\ldots ,k$ and is proportional to $k^{-D}$, i.e. the average lengths follow a power law. D is considered the fractal dimension computed by Higuchi’s algorithm and a least squares estimate for D is given by the slope of the line that fits the pairs ($\ln (L(k))$, $\ln (1/k)$).

Appendix 3: Search intervals for the calibration procedures

See Tables 7, 8 and 9.

Table 7 Search intervals and precision for the calibration of individual-based ($K_{\mathtt {ag}}$) and cluster-based ($K_{\mathtt {cl}}$) Kirman models

Table 8 Search intervals and precision for the calibration of the Lux model

Table 9 Search intervals and precision for the calibration of the Farmer–Joshi model