Levy, Solomon and Levy's Microscopic Simulation of Financial Markets points us towards the future of

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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
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microscopic element individual investor
interactionbuying / 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 t1 Risky stock / index
(SP) / market portfolio price
p(t) determined by investors
(as described later )
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Returns on stock Capital gain / loss
If an investor i holds N i stock
shares a change P t1 - P t
gt 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
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Investors divide their money between the two
investment options in the optimal way which
maximizes their expected utility EUW lt ln W
gt. To compute the expected future W, they assume
that each of the last k returns H j j t, t-1,
., t-k1
Has an equal probability of 1/k to reoccur in the
next time period t.
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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
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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
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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 )
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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-k1 EUX(i) 1/k ?
lnW jt W (1-X(i)) Wh (i)
(1r) X(i) Wh (i)(1 Hj )
bonds contribution stocks
contribution
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The investor chooses the investment proportion
Xh (i) that maximizes his/her expected utility
dEUX (i) / dX(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
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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 ) Si Nh(i,
Ph )
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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
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History Update The new stock price Pt1 and
dividend Dt1 give us a new return on the stock
H t (P t1 - P t Dt1 )/ P t We update the
stocks history by including this most recent
return and eliminating the oldest return H t-k1
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)
e (i)
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