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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)