Levy, Solomon and Levy's Microscopic Simulation of Financial Markets points us towards the future of financial economics." Harry M. Markowitz, Nobel Laureate in Economics

microscopic element = individual investor

interaction=buying / selling of stocks / bonds

Discrete time

investment options:

Riskless: bond; fixed price

return rate r: investing W dollars at time t yields W r at time t+1

Risky: stock / index (SP) / market portfolio

price p(t) determined by investors (as described later )

Returns on stock

Capital gain / loss

If an investor i holds N i stock shares

a change P t+1 - P t

=> change N(P t - P t-1 ) in his wealth

Dividends Dt per share at time t

Overall rate of return on stock in period t:

H t = (P t - P t-1 + Dt )/ P t-1

Investors divide their money between the two investment options in the optimal way which maximizes their expected utility E{U[W]} = < ln W >.

To compute the expected future W, they assume that each of the last k returns H j ; j= t, t-1, …., t-k+1

Has an equal probability of 1/k to reoccur in the next time period t.

INCOME GAIN

N t (i) D t in dividends

and (W t (i)- N t (i) P t ) r

in interest

W t (i)- N t (i) P t is the money held in bonds as W t (i) is the total wealth and N t (i) P t is the wealth held in

stocks

Thus before the trade at time t the wealth of investor i is

W t (i) + N t (i) D t + (W t (i)- N t (i) P t ) r

Demand Function for stocks

We derive the aggregate demand function for various hypothetical prices Ph and based on it we find Ph = Pt

the equilibrium price at time t

Suppose that at the trade at time t the price of the stock is set at a hypothetical price Ph How many shares will investor i want to hold at this price?

First let us observe that immediately after the trade the wealth of investor i will change by the amount

N t (i) ( Ph - Pt ) due to capital gain or loss

Note that there is capital gain or loss only on the N t (i) shares held before the trade and not on shares bought or sold at the time t trade

Thus if the hypothetical price is Ph the hypothetical

wealth of investor i after the t trade Ph will be

Wh (i) = W t (i) + N t (i) D t

+ ( W t (i) - N t (i) P t ) r

+ N t (i) ( Ph - Pt )

The investor has to decide at time t how to invest this wealth He/she will attempt to maximize his/her expected utility at the next period time t

As explained before the expost distribution of returns is employed as an estimate for the exante distribution If investor i invests at time t a proportion X(i) of his/her

wealth in the stock his/her expected utility at time t will be given by

t-k+1

E{U[X(i)]} = 1/k ln[W ]

j=t

W= (1-X(i)) Wh (i) (1+r) +X(i) Wh (i)(1+ Hj ) bonds contribution stocks contribution

The investor chooses the investment proportion Xh (i) that maximizes his/her expected utility

E{U[X (i)]} / X(i) |X (i)= Xh (i) =0

The amount of wealth that investor i will hold in stocks at

the hypothetical price Ph is given by Xh (i) Wh (i)

Therefore the number of shares that investor i will want to hold at the hypothetical price Ph will be

Nh(i, Ph )= Xh (i) Wh (i) / Ph

This constitutes the personal demand curve of investor i

Summing the personal demand functions of all investors we obtain the following collective demand function

Nh(Ph )= i Nh(i, Ph )

Market Clearance

As the number of shares in the market denoted by N is assumed to be fixed the collective demand function Nh(Ph ) = N determines the equilibrium price Ph

Thus the equilibrium price of the stock at time t denoted by Pt will be Ph

History Update

The new stock price Pt+1 and dividend Dt+1 give us a

new return on the stock

H t = (P t+1 - P t + Dt+1 )/ P t

We update the stocks history by including this most

recent return and eliminating the oldest return H t-k+1

from the history

This completes one time cycle

By repeating this cycle we simulate the evolution of the stock market through time.

Include bounded rationality: Xh*(i)= Xh(i)+ (i)