Simulating Security Markets in Dynamic and Equilibrium Modes

Abstract

An asynchronous discrete-time model run in “dynamic mode” can model the effects on market prices of changes in strategies, leverage, and regulations, or the effects of different return estimation procedures and different trading rules. Run in “equilibrium mode,” it can be used to arrive at equilibrium expected returns.

Asynchronous discrete-time simulations, in which the time intervals between events are irregular, can be used to examine the mechanisms behind price movements and can thus be used to test the effects of changes in investors’ strategies, modifications in overall leverage, and switches in regulatory regimes. They can also be used to solve for equilibrium expected returns without requiring the kinds of unrealistic assumptions that some analytical models require. In this study, we used our asynchronous discrete-time simulation in (1) the dynamic analysis (DA) mode to investigate how changing input parameters changes simulated market behavior and (2) the capital market equilibrium (CME) mode to demonstrate how one can obtain estimates of equilibrium expected returns that are consistent with the aggregate holdings of heterogeneous investors subject to realistic constraints.

In the DA mode, ideal portfolio weights are determined by simulated portfolio analysts who use inputs from simulated statisticians and investors’ risk-aversion parameters and portfolio constraints. Prices and volume arise endogenously as simulated traders seek to complete the trades required to move investors’ current portfolios toward their ideal portfolios. Examining how prices and volumes react to changes in the initial random seeds that determine individual investor initial wealth and cash flows, we found that although idiosyncratic differences exist in security prices and volumes, markets overall are largely insensitive to such changes. We also examined the effects of changes in the proportions of entities that use various methods of estimating expected returns and changes in trading rules.

We found that simulated markets are sensitive to changes in both return estimation procedures and trading rules. For example, if all investors use return estimates based on securities’ historical returns and variances (which leads to “momentum” investing because the investors buy as prices increase and sell as prices decline), security prices are destabilized. Even when we varied the number of periods (days, months, quarters) used to estimate returns, we still found destabilization in the presence of investors who use historical returns. We introduced another type of investor, one who estimates returns on the basis of fixed future dollar returns per share (which leads to “value” investing: buying as prices decline and selling as prices rise). We found that when the ratio of momentum investors to value investors is large, security prices still tend to “explode”; but when the ratio is low, prices do not become destabilized.

Our simulations also showed that traders must be subject to some kind of anchoring rules designed to approximate real-world traders’ sense of how much to pay or ask for a security on the basis of not only current prices but also recent past prices. Absent such rules, security prices tend to either explode or implode.

In the CME mode, a simulation can find market-clearing expected returns. Securities’ expected returns are determined by an iterative adjustment procedure, rather than by the use of simulated statisticians, as is the case in the DA mode. In response to expected return changes, investors change their portfolios in such a way that the aggregate of all investors’ portfolios converges toward target market portfolio weights. In the CME mode, the simulator is able to solve for expected returns in markets with real-world constraints, such as restrictions on borrowing and/or short selling.