Transcript Document
A Financial Market Model G. Yang1, K. Wang1, B. Zhao1, L. Zheng1, T. Y. Fan1, C. H. Wang1 W. Wang1, L. Zhao1, Y. Chen2 and J. P. Huang1 1Department 2Department of Physics, Fudan University, Shanghai 200433, China of Systems Innovation, Graduate School of Engineering, University of Tokyo, Tokyo 113-8656, Japan Regulation of the trade Introduction For a long time, people are trying to predict the trend of price changes. But even Isaac Newton exclaimed that he could calculate celestial motion, but could not calculate human madness. We present here a financial market model using the technique of agent-based modeling (ABM). In the model, we mimic the human decision-making process of buying or selling a certain stock. We find that the trends of the simulated stock price are similar to what we see in the real markets. In particular, we also use the Bouchard-Sornette formula to do the European option pricing. With the adjusted parameters , the option prices based on the simulated data are well fitted with the ones on SH index (2000.1.2-2009.10.27). European option pricing The time series 0 … T-1 T T+1 T+2 Agent i Time T Market Situation: PT Time T+1 Market Situation: PT+1 His highly-scored strategy ST gives the prediction: +1 His highly-scored strategy ST+1 gives the prediction: -1 A buy order T A sell order A sell order 2) T+2 Note that if his predictions of the two periods (T to T+1 and T+1 to T+2) are the same, he will give a sell and a buy order at T+1. The total effect is: he will not give an order. Assuming a proportionality between price change and excess demand: ln xT 1 ln xT k ln Nb ln Ns we can get the time series of the price. Structure of the model For the European option, under several assumptions, we can use the BouchardSornette formula to do the option pricing: A buy order T+1 Assume that each agent can only have one unit stock 1) Call option && Put option: If you purchase a call option, you will have the right to choose whether to buy a given assets at a striking price X at the expiring date T. And if you buy a put option, you will have the right to sell the underlying assets. Simulated results N=1000, S=6, P=32, k=0.09 n 1 Vc (H[ x[ j] X ] Vp (H[ X x[ j]] j 0 n 1 j 0 n[ R x R n[ Rmin jR] ( x[ j ] X 0 )) n 2 x R n[ Rmin jR] ( X x[ j ] 0 )) n 2 j R ] The item min n in the formula can be calculated using the history price series of the real markets. It represents the occurrence frequency of de-trended return Rmin jR . Here we set the parameters of the financial market model as N=1000，P=32，S=6， k=0.09 and simulate for 2000 time steps. Then we use the simulated stock price series to do the option pricing (see the red dots in the figures below). To be contrast, we also use SH index (2000.1.2-2009.10.27) to do the pricing (the black dots) . Initial spot value x0=500, expiring date T=21. A strategy Conclusions +1 stands for the prediction of the stock price to rise and -1 is the contrary opinion. The number of +1 denoted as L. L represents the optimistic degree of the strategy. 1) 2) Each agent has S strategies. The integer L of each strategy is randomly chosen from 0, 1, 2, …, P. At each time, an agent uses his highly –scored strategy. The strategies which give a right prediction will be added 1 unit score. 1) 2) Fig.(a), (b), (c) are obtained by the input of the different values of the random function's original seed. Order flow is defined as the total number of sell orders and buy orders at each time. • We design a financial market model. Inputting different original seeds, we can simulate various trends of price changes. • We also use the simulated data to do the option pricing. It shows that with the parameters properly set, the results are well fitted with the ones using SH index. • In this work, we don’t take into account several nontrivial properties when designing the model, such as herd which can cause fat tails and extreme volatility. The methodology of the combination of option pricing and this model is not perfect, either. We will concentrate on these problems in the following work.