Quality Analysis of the Models

Stylized facts are a good test for the identification of model quality, but another

important aspect is parity of basic market characteristics:

1. Returns. It is a well-known fact that simple Brownian motion does not allow the generation of heavy tails of distribution. The ZI model can generate fat tails, but the MF and Daniels models (in our case) can generate more heavy tails than in reality. It is interesting that MFWC generated returns, but without heavy tails (Fig. 10).

2. Distribution of spread. Farmer et al. (2005, 2006) in their research concentrated on spread. The spread of our model is not like the empirical one, but with heavy tails in their distribution (Fig. 11).

3. Cancellation time. The order cancellation process plays an important role in asset pricing, so it is important that its lifetime has heavy tails. The order cancellation process in the MF model shows complicated behavior, which is conditional on different market characteristics (just this process leads to a fat tail in an order’s life) (Fig. 12).

image046

Fig. 10 Distribution of minute returns of analyzing models

image047

image048

Fig. 11 Spread distribution of analyzing models

 

image049

lifetime of order

——– Empirical ————- Daniels…………. MF————— Upgrade

Fig. 12 Order lifetime distribution of analyzing models

 

Conclusion

We construct and estimate the parameters of two well-known models: Daniels and Mike-Farmer. During the process of the estimation of parameters, we find that distributions of price and probability of cancellation are conditional on the number of orders in the order book being quite different from the MF model. It is important that this model is very sensitive to small details in realization and small bugs in the code. Parameters being not carefully estimated can lead to a significant worsening of model results. We have tried to upgrade the model for our data, including an additional parameter for the

 

(continued)

 

Подпись: order cancellation process and fitting prices using two power-law distributions with t-Student’s center. The upgrade model for our sample shows the best results. It is important that the model represents only the microstructure of the market of Aeroflot stocks in January and cannot be spread to other instruments.

References

Achard, S., & Coeurjolly, J.-F. (2010). Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise. Statistics Surveys, 4, 117-147.

Arbuzov, V., & Frolova, M. (2012). Market liquidity measurement and econometric modeling. Market risk and financial markets modeling. Heidelberg: Springer.

Bouchaud, J.-P., Gefen, Y., Potters, M., & Wyart, M. (2004). Fluctuations and response in financial markets: the subtle nature of ‘random’ price changes. Quantitative Finance, 4(2), 176-190.

Chakraborti, A., Toke, I., Patriarca, M., & Abergel, F. (2011). Econophysics review: II. Agent – based models. Quantitative Finance, 11(7), 1013-1041.

Daniels, M. G., Farmer, J. D., Gillemot, L., Iori, G., & Smith, E. (2003). Quantitative model of price diffusion and market friction based on trading as a mechanistic random process. Physical Review Letters, 90(10), 108102.

Farmer, J. D., Gillemot, L., Iori, G., Krishnamurthy, S., Smith, D. E., & Daniels, M. G. (2006). A random order placement model of price formation in the continuous double auction. The economy as an evolving complex system III (pp. 133-173). New York: Oxford University Press.

Farmer, J. D., Patelli, P., & Zovko, 1.1. (2005). The predictive power of zero intelligence in financial markets. Proceedings of the National Academy of Sciences of the United States of America, 102, 2254-2259.

Gu, G.-F., & Zhou, W.-X. (2009). On the probability distribution of stock returns in the Mike – Farmer model. European Physical Journal B, 67(4), 585-592.

He, L.-Y., & Wen, X.-C. (2013) Statistical Revisit to the Mike-Farmer Model: can this model cap­ture the stylized facts in real world markets? Fractals, 21(2), 1-8. http://www. worldscientific. com/doi/abs/10.1142/S0218348X13500084

Lillo, F., & Farmer, J. D. (2004). The long memory of the efficient market. Studies in nonlinear dynamics & econometrics, 8(3), 1-33.

Lillo, F., Mike, S., & Farmer, J. D. (2005). Theory for long-memory of supply and demand. Physical Review E, 7106, 287-297.

Mike, S., & Farmer, J. D. (2008). An empirical behavioral model of liquidity and volatility. Journal of Economic Dynamics and Control, 32, 200-234.

R Core Team (2013) R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.

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