Model Calibration

In order to understand this model’s properties and its advantages, it is necessary to analyze how the model can reflect real data conditions. Thus, there is an issue of calibration of the model and applicability of the model for the description of a real market situation.

The percolation model allows us to simulate price change distribution in a one­time step as a hypothetical situation of interaction of agents for a certain time interval. The task of calibration is to select values of parameters and receive the

image076

Fig. 4 Algorithm of reverse engineering calibration

model’s empirical distribution, which is similar to the real-world market distribution in terms of some pre-selected measures Wiesinger et al. (2010).

This is the method of reverse engineering in the context of financial time-series. With its parameters and strategies, it optimizes the similarity between the actual data and simulated data.

An algorithm of the reverse engineering calibration of our model is presented in Fig. 4. We have already noticed that values of a, pbuy have very strong influence on the empirical distribution of A. Because of this, we will find the values of a, pbuy which will give the required similarity.

At the first stage, we do the processing of real market data. We consider hourly log returns of RTS index (leading Russian stock index) during the period of January 1st, 2008 to December 31st, 2009. This period could be characterized as an instable stage in the financial market. Thus, at calibration we are expecting a condition of infinite cluster occurrence, which most precisely characterizes a crisis situation in the market.

The next stage is optimization. There is the minimization of distance between a real sample of price changes and the model sample of A, as a result of the Monte Carlo simulation. The algorithm changes values of required parameters a, pbuy and generates a new percolation model as a result. We have a new sample of model price changes as a result of this iteration step.

The part of calculation the distance between modeling and the fact sheet assumes using various measures of distance between two probability distributions. In this research we decide to use Kullback-Leibler divergence. This is non-symmetric measure of the difference between two probability distributions and used for discrete and for continuous random variables. It defined to be:

dkl (p, q) = X)x/(x) ln qXy, (3)

where p(x), q(x)—the probability density of the corresponding discrete random variables X, Y. The main point of Kullback-Leibler divergence is that it base on information theory and reflect the difference between entropies of two distributions. It means that we try to minimize difference between indeterminacies of two samples of information. There are some other properties of Kullback-Leibler divergence:

• non-symmetric

• always nonnegative

• non-parametric.

In case of this research it’s possible to use divergence without information about form of distributions (Shengqiao 2012).

The optimization task was realized, using genetic algorithm. We minimize of Kullback-Leibler divergence with DEoptim R package, which is a global optimization algorithm from class of genetic algorithms, which uses biology – inspired principles. The main argument for this choice is the possibility to work with discontinuous and nondifferentiable functions, because we haven’t got enough information about function we have to minimize (Ardia et al. 2012).

Results of calibration are empirical distribution of modeling price change with parameters a = 0,02 and pbuy = 0,31. The results of empirical function are presented in Fig. 5.

The small value of parameter a = 0,02 is interpreted as a short time interval when market was observed. That’s why we can conclude, that high frequency traders are presented in this market. The value of probability to buy pbuy = 0,31, which can

image077

Fig. 5 Empirical distribution function of model and actual price changes

be interpreted as a small asymmetry between demand and supply and the most of agents prefer to sell in the market because of the critical crisis situation.

Подпись: Conclusion In this paper percolation model was used to describe agent’s cooperation in the financial market. The results of Monte Carlo simulation allow analyzing model price changes distribution and concluding about price change distribution and model parameters dependence. This dependence suggests the possibility of model calibration. Using optimization procedures help to find model parameters values which describe real market pricing process. The result shows that presented model generally comply with real-market data.

References

Ardia, D., Muller, K., Peterson, B., & Ulrich, J. (2012). Global optimization by differential revolution. Available via Internet: http://cran. r-project. org/web/packages/DEoptim.

Bouchaud, J.-P. (2002). An introduction to statistical finance. PhysicaA: Statistical Mechanics and Its Applications, 313, 238-251.

Chang, I., Stauffer, D., & Pandey, R. B. (2002). Asymmetries, correlations and fat tails in percolation market model. International Journal of Theoretical and Applied Finance, 5(6), 585-597.

Gould, H., Tobochnik, J. & Christian W. (1990). An introduction to computer simulation methods: Applications to Physical Systems Third Edition, Addison-Wesley, 2006, p. 796

Li, S. (2012). Fast nearest neighbor search algorithms and applications. Available via Internet: http://cran. r – proj ect. org/web/packages/FNN.

Sornette, D., Stauffer, D., & Takayasu, H. (1999). Market fluctuation II: Multiplicative and percolation models, size effects and prediction (Vo1, p. 30). Available via Internet: arXiv:cond – math/9909439.

Stauffer, D. (2001). Percolation models of financial market dynamics. Advances in Complex Systems, 4(1), 19-27.

Stauffer, D., & Sornette, D. (1990). Self-organized percolation model for stock market fluctuation. Physica A: Statistical Mechanics and Its Applications, 271, 496-506.

Wiesinger, J., Sornette, D., & Satinover, J. (2010). Reverse engineering financial mar­ket with majority and minority games using genetic algorithm. Available via Internet: arXiv:1002.2171v1.

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