Using Agent Belief to Model Stock Returns

Transcription

1 Using Agent Belief to Model Stock Returns America Holloway Department of Computer Science University of California, Irvine, Irvine, CA Introduction It is clear that movements in stock prices are correlated with events in the larger world such as natural disasters, news, consumer attitudes, unemployment rates, etc. However, modeling even a few of these factors and their complex (largely hidden) interactions is infeasible. As a result, many choose to model stock returns as a purely stochastic process looking only to previous data to drive the model. The drawback to this approach is that seemingly spontaneous, drastic changes in stock volatility occur that are difficult to predict given only the past. In this paper, we propose a hybrid approach to modeling stock returns based on the assumption that the stock price is independent of external factors given every agent s subjective belief of the worth of the stock. An agent can be an institution, mutual fund, insider or any other entity that can buy or sell stock. Every agent weighs the external factors and previous stock performance and decides how much a single share of stock should be worth. The stock price is then a function of these hidden agent beliefs - independent of any external influences. To model an agent s latent belief of the stock price, we must also model how agents influence one another and how beliefs evolve over time. In particular, we want to capture the tendency of investors to flock together. There are many models from various disciplines that simulate this type of behavior: particle-motion in physics, flocking in computer graphics, rumor cascades in social networks, and herding models in finance. We consider a herding model and combine it with a stochastic time series model for a new approach to modeling stock returns. 2 Modeling Agent Belief We use a herding model to simulate a collection of agent s and their changing attitudes toward a particular stock. Let N be the total number of agents. At each time step, we summarize the beliefs of the agents in the network by the number of buyers or optimists. We denote this quantity as B t. Given B t we can compute B t+ as B t+ = B t X t + Y t where X t is the number of agents who were buyers at time t but became sellers at time t +, and Y t is the number of agents who were sellers at time t but became buyers at time t +. X t and Y t are sampled from Binomial distributions. X t Binom(B t, p) Y t Binom(N B t, q) p = α + λd N B t N q = α + λd B t N ()

2 The binomial parameters p and q are taken from the herding model as described in []. λ controls the degree of herding in the model, i.e. the degree to which an agent is influenced by the beliefs of its neighbors. α controls the degree to which an agent acts independently, and D is the average number of neighbors. The more buyers (sellers) there are, the greater the probability is of becoming a buyer (seller). We can now model a change in stock price using the following update equation, P t+ = P t + β(b t+ B t ) + ɛ t The price of the stock follows a very intuitive rule: If the number of buyers increases at time t + then the stock price increases, and if the number of buyers decreases at time t + then the stock price decreases. By subtracting the price at time t, we produce a model of stock returns. P t+ = P t + β(b t+ B t ) + ɛ t P t+ P t = β(b t+ B t ) + ɛ t (2) R t+ = β(b t+ B t ) + ɛ t For stock returns, large volatility tends to produce even larger volatility (a phenomenon known as volatility clustering) [2]. This means the error at time t, ɛ t, should depend upon the error at previous time steps. ARCH is a family of models that are used for time series that show such a dependence in error. Let ɛ t = u t ht where u t is a standard normal and h t is given by h t = α + α ɛ 2 t (3) Then equation 2 defines an ARCH() model whose mean is determined by the change in the number of buyers of the stock. 3 Estimating Properties of the Social Network The herding model described above requires knowing N, the total number of agents, and D, the average number of neighbors. One option is to learn these parameters from the data. However N can take on any value. It could be one million as easily as it could be ten; the only difference would be a corresponding change in the scaling parameter β. Furthermore, D and λ always occur together in the update equations. Given D we can learn λ. However if they are both unknown we run into another identifiability issue. To resolve this we use external data to determine N and D beforehand. For a given stock, we compile a list of major stock holders. This information is freely available and can be found on sites such as Yahoo! Finance. For each pair of major stock holders, we submit the names to Google and record the total number of hits returned. This gives us an estimate of the strength of the relationship between every pair of major holders. Since it was uncommon for an institution or mutual fund to be mentioned with an insider (a major stock holder who is employed by the same company) we removed insiders from the list. Each score was then normalized by the largest number of hits returned. This produced a fully connected, undirected, weighted graph. We discarded any edge with weight less than a set threshold (.5 for all experiments). N is set to be the total number of nodes, and D is the average node degree. 4 Learning the Posterior Distribution Let θ denote the vector of model parameters and R :t denote the returns from time to time t. We use a particle filter to approximate the posterior distribution p(θ t R :t ). Given a collection of L particles from the distribution p(θ t R :t ) we generate a new set of L particles by sampling from the prior p(θ t+ θ t ). The sampling order is described below. 2

3 3 AAPL Train Test Split ( min) x 4 Fig.. 3-minute return data for Apple Stock from Sample the set of global variables {β, λ, α, α, α } from a normal distribution whose mean is the previous parameter values and variance is. 2. Compute B t+ by sampling X t and Y t according to the equations given in. 3. Compute h t+ according to equation 3 using the updated values of α and α. 4. Update the particle weight according to w t+ = w t p(r t+ B t+, h t+ ). We resample when the effective number of particles falls below a threshold τ =.8L. The effective number of particles is computed as follows, Ñ = L i= w2 i 5 Implementation We consider three different models. The first model, ARCH, is an ARCH() model with fixed zero mean. We use this as a baseline. The second model, ARCH+SN, is an ARCH() model whose mean is estimated using the herding model described above. The third model, ARCH+SN β is the same model but β is fixed to be.5. The value.5 was chosen because it was the mean value of beta from the ARCH+SN model. We generate L = 2 particles for each model. N is set to 4 and D is set to 7. We compute returns from 3-minute intra-day prices of Apple stock (AAPL) from March 2, 997 to October 25, 22. Figure shows the 3-minute return data. The dotted line is the training-testing split. 8% of the data was used for training and the remaining 2% for testing. 3

4 6 Results At each time step, we model the distribution of returns as a mixture of L = 2 Gaussians. Each Gaussian is a particle whose mixing proportion is given by its respective weight. The log-likelihood of the test set is then given as log p(r :T ) = T i= ( L ) log w ij p(r i θ ij ) where θ ij is the jth particle at time i and its weight is given by w ij. Table shows the performance of all three models. j= ARCH ARCH+SN β ARCH+SN Table. Log-likelihood on Test Data The ARCH model performs significantly better than the other two models. In Figures[2-4], a kernel density estimate of the 2 particles has been shown for t = (the beginning of the test period) and t = ( time steps later). The ARCH model consistently has a smaller variance. The ARCH+SN has the largest variance. It is interesting to note that the ARCH+SN model is the only model that exhibits a significant heavy-tail and (for t=) is skewed. The ARCH+SN β model resembles the ARCH model except for its wider variance. Undoubtedly, the ARCH model benefits by having a smaller variance since the test set shows very small volatility. 7 Further Works Unfortunately, it seems a zero-mean model is the best. However, there are still a lot of experiments I want to try. Empirically, the distribution of returns (marginalized over time) has heavy-tails and is skewed [2]. The ARCH+SN model has the most potential to approximate such a distribution. One thing I want to try is restricting β to be greater than since this played a major role in the large variance seen in the ARCH+SN model. The number of buyers at time t is fairly constant (occasionally deviating from N) but β moves quite a bit. Also, I think the latter two models may perform better on a more volatile data set. Given the current market conditions, it would be interesting to run these models on recent stock returns. I did not talk about the continuous flocking model that our group first tried [3], but I would like to experiment more with that model. The discrete nature of the herding models is unnatural for this application whereas the flocking model has a very intuitive extension to the stock market. Lastly, it is quite time consuming to estimate N and D. Automating this process will let me run these models on different stocks. References. S. Alfarano, M. Milakovic, Should network structure matter in agent-based finance? (April 27). 2. R. Cont, Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance (2) T. VICSEK, A. CZIROK, E. BENJACOB, I. COHEN, O. SHOCHET, Novel type of phase-transition in a system of self-driven particles, Physical Review Letters 75 (995) For each time period, all 2 particles had equal weights. 4

5 !'% ()*+ ",-./.4 ARCH KDE (t=)!'&.2! "'#.8 "'$.6 "'%.4 "'&.2 "!!"!#!$!%!& " & % $ #!" Fig. 2. Kernel Density Estimates for ARCH.8 ARCH+SN KDE. ARCH+SN KDE (t=) Fig. 3. Kernel Density Estimates for ARCH+SN ARCH+SN! KDE (t=) ARCH+SN! KDE Fig. 4. Kernel Density Estimates for ARCH+SN β 5