Finance
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[edit] Introduction
The goal is to develop a micro-foundations model of financial markets that lends itself to econometric analysis. Financial markets[1] display many characteristics known in complex systems[2]. The interaction of a large number of optimizing agents in a recursive environment is commonly regarded as a main component of complexity. The limited ability of analytical models to explain the stylized facts from the empirical literature have confounded many scientists. This may be partly caused by multilevel dependence or emergence. To deal with the complexity of markets, SFI[3] has championed multi agent market simulation[4].
The SFI multi-agent simulation has made great contributions by modeling multi-agent environment. Some simulations showed fat tails and clustered volatility. Some have suggested that the results of the simulations changed with larger number of agents. Analytical models with heterogeneous agents, bounded rationality, with a focus on quantitative analysis are rare and much needed. Results from simulations should help whenever assuming functional form is necessary.
[edit] The Problem
Attempting to model individual agents using differential equations will be even harder than trying to move the movement of particles. In physics, they have correctly chosen a measure theoretic approach to model the interaction of a large number of particles. Fortunately, this modeling approach lends itself to data analysis. Before proceeding, we should attempt to describe the system we aim to model. This will keep the modeling honest and claims about results humble. Financial markets often have:
- Heterogeneous agents of several types
- Multi-dimensional dependence
- Inter-market spillovers
As you see, there is interdependence and hierarchy among a large number of nodes. All these acting together may cause emergence as it usually does with large complex systems
[edit] Project
- Define a financial market mathematically and use a directional graph that shows the interactions on all levels: inter-asset, inter-temporal, across individuals, and among markets.
[edit] Heterogeneity of Agents
In both theoretical and simulation communities, three types of agents have been recognized:
- Rational agents
- Followers or trend following
- Noise traders
[edit] Project
- Analyze the effects of a market with the three types together using evolutionary game theory.(Completed 2008)
[edit] Multi-Dimensional Dependence and Data Properties
The iid assumption is often assumed to use the CLT. However the independence assumption is violated in several dimensions:
- Inter-temporal dependence of the same assets
- Dependence across difference financial assets
- Dependence across individuals
Today's markets are very much connected. If someone takes a position in a foreign asset, they may decide to hedge in the foreign exchange market. Then, not only are assets correlated in the same market, but correlated across markets.
[edit] Project
- Create a table that shows the conditions for normality and determine the effects of multidimensional dependence on the limiting distribution of the data. (Completed 2007)
- Determine the effects of inter-market spillover on the properties of the data.
[edit] Microfoundations
Generally consider that the price in the current period is a function of the excess demand in the last period which is in turn a function of that period's price. How does the possible intransitivity of preferences amongst assets affect the continuity of utility function and how does that affect excess demand?
[edit] Modeling Aggregate Excess Demand
Micro-theory suggests that deviation from equilibrium price where quantity supplied must equal quantity demand. Otherwise, there will be excess demand and the price will correct until it reaches its equilibrium. So aggregate excess demand e(p(t)) is the sum of ei(p(t)). Micro theory[7] tells us that assuming nonnegative, continuous, and uniformly bounded above expectations, and non negative initial endowments, then individual excess demand has the following properties:
- Homogeneity of degree zero [8]
- Single valuedness and continuity
- Individual net demands are bounded from below
Aggregate excess demand is the sum of individual excess demands and therefore has the following properties:
- Homogeneity of degree zero
- Boundedness from below
- Discontinuity at zero
In that sense, we know that excess demand at time t is a function of the price at time t. Therefore, price at time t+1 is a function of price at time t. This implies a dynamical set of p(t+1)=F(p(t)) , but this ignores shifts in demand and supply. Conversely, this shows that the underlying dynamic F(p(t)) may change.
[edit] Project
- How does non-linear supply and demand affect aggregate excess demand?
- Treating individual excess demand as random variables, how would micro theory help you with the modeling? (Hint: [])
- Show how a shift would influence p(t+1)=F(p(t)). (Hint:[9]
[edit] Things that Cause shifts in market demand:
- Increase in market participants
- Change in price expectations
- Change in the price of compliments
- Change in the price of substitutes
- Change in income
- Change in herding
- Change in average order size
[edit] Things that Change Supply
- Subsidies
- Change in expectations
[edit] An Integrated Model
Notice that theory suggests the following:
- Price is a function of excess demand[10]
- Excess demand is bounded from above and below [11]
Question: The dynamic trajectories to the equilibrium prices may have a different distribution, what kind would it be?[12][13]
[edit] Exercises
Show that price models from micro foundations can be extended to the ones used in empirical literature. What difference would it make if the data was simply differenced?
[edit] Project
The shifts in the supply and demand change the manifold one in which the prices series lives. Certain shifts will create certain shifts, and therefore, there is hope in characterizing the behavior of the series itself.
[edit] Applied Financial Modeling
One of the main goals of the analysis above is to provide a unified framework under which we can validate the use of econometric tools [14] that are used by professionals on Wall Street. Time series econometric [15] analysis of financial markets is not without its advantages, but the ad hoc nature of its application in practice and lack of theoretical foundation weakens its credibility. Brock[16] has a review on the shortcomings of our statistical battery to provide consistent tests of interest. However, financial econometrics is a field of its own. Tsay's text gives a standard treatment[17]. For a good brief intro to the standard methods see[18]. There is also these lecture notes[19]
[edit] Data Frequency
The frequency of the data used should be the Nyquist frequency of the continuous signal one is trying to model. This is clear from the Shanon sampling theorem, but very few people in practice pay any attention to it. This is due to the fact that frequency of data is often taken as a given rather than chosen.
[edit] Modeling
Micro-foundation suggests the modeling of a time varying system. In large markets, this system has to be reduced in dimension.
[edit] Forecasting
The averaging of several models is known to yield better forecasts. The challenge is that the forecasts are then a mixture of densities.
[edit] Trading Signal
Forecasts for returns should provide a trading signal. There should be an optimal trade given the trading signal.
Project: Derive the optimal trade given the signal
[edit] Portfolio Optimization
A multi-period portfolio construction problem takes the form of a stochastic control problem where the control variable is the portfolio weights and the state is future returns.
[edit] Project
Prove or disprove that the greedy algorithm is applicable to the dynamic portfolio optimization problem
[edit] References
The contents of this site are original and any relevant materials I have included a link for. If you have any questions, please direct them to aldoctor@gmail.com
