Complex evolutionary systems in
behavioural finance
Application: financial instruments and market stability
Florian Wagener
CeNDEF
University of Amsterdam
Dynamics in Games and Economics
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Overview
Brock, Hommes & Wagener (2009)
Extension of the Brock & Hommes (1998) heterogeneous agents
asset pricing model by adding hedging instruments (Arrow securities)
Main Question
Is it true that giving traders more trading options will improve the
market outcome?
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BH asset pricing model with Arrow securities
Stochastics
In the next period (“tomorrow”) one of S possible states of the world
occurs with probability αs
Investment possibilities
• Bonds, totally elastically supplied, price 1, return R
0
s
• Risky asset, total supply ζ 0 , price pt0 , return pt+1
+ yt+1
• n Arrow securities, total supply 0, price ptj , j = 1, .., n, return

1
δjs =
0
if
s = j,
otherwise
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Profits
0
Trader of “type h” expects price pht+1
of the risky asset;
1
0
excess return of portfolio (zt , zt , · · · , ztn ) is
s
0
s
πht+1
= (pht+1
+ yt+1
− Rpt0 )zt0 +
X
(δjs − Rptj )ztj
j
Demands and prices written as vectors
zt = (zt0 , z̃t ) = (zt0 , zt1 , · · · , ztn )
pt = (pt0 , p̃t ) = (pt0 , pt1 , · · · , ptn )
Expected excess returns in state s
Bsht
=
0
s
pht+1
+ yt+1
− Rpt0
δ s − R p̃t
!
,
Bht =
X
Bsht αs
s
Excess return of portfolio zt expected by trader type h
πht+1 = hBht , zt i
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Demands
Traders maximise expected risk-corrected profits
Uht (z) = Eht πht+1 −
a
1
Var πht+1 = hBht , zi − hz, Vn zi,
2
2
where
• a: coefficient of risk aversion
• Vn : (dividend) risk structure
s
s
Vn = a Cov(yt+1
, δ s ) = a Cov(yt+1
, δ1s , · · · , δns )
Key assumption: risk structure is common knowledge
Demands
zht = Vn−1 Bht
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Benchmark: homogeneous market
Single rational trader
• Outside supply of risky assets: ζ 0
• No outside supply of Arrow securities
• Total outside supply of assets ζ = (ζ 0 , 0, · · · , 0)
Market equilibrium at steady state prices p∗
ζ = Vn−1 B
implies
B=
s
(1 − R)p∗0 + Eyt+1
α − R p̃∗
!
= Vn ζ,
determines homogeneous benchmark prices p∗ = (p∗0 , p
~∗ )
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Heterogeneous agents market equilibrium
• nht : market fraction of traders of type h
• xt = (xt0 , x̃t ) = pt − p∗ : deviation from benchmark price
• Expectations of type h on the price deviation of the risky asset
0
0
− p∗0 = fht (xt−1 , · · · )
= pht+1
xht+1
Demands in deviation from the homogeneous demands ζ
zht (xt ) = ζ +
Vn−1
Market clearing ζ =
xt0 =
1X
nht fht ,
R
fht − Rxt0
− R x̃t
P
h
!
nht zht (xt ) implies
x̃t = 0.
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Reinforcement learning
Agents flock to strategies having good fitness scores
• Uht−1 is the fitness of type h after trading round t − 1
• The intensity of choice β is inversely related to noise when
observing Uht−1
• Discrete choice model
nht =
X eβUht−1
Zt
,
X
nht = 1
h
We take as fitness measure the average realised risk-adjusted profits
1
hzht , Vn zht i
2
0
1
= − e0 , Vn−1 e0 (xt+1
− fht )2 + Ct
2
Uht+1 = hBt , zht i −
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Risk measure
σn2 : measure of risk in presence of n Arrow securities
1
= a e0 , Vn−1 e0
2
σn
2
2
σ02 > σ12 > · · · > σS−2
> σS−1
=0
Theorem:
0
zht =
ζ
0
!


1
 2
+ (fht − Rxt0 )  σn 
w̃n
Less (percieved) risk means more demand ⇒ leveraging
Price dynamics
X
Rxt0 =
−
e
X
β
2
σn
−
e
β
2
σn
0
(xt−1
−fht−2 )2
fht
0
(xt−1
−fht−2 )2
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Example
Two-type model: near-fundamentalists versus trend chasers
f1t = 1, f2t = xt−1 + g(xt−1 − xt−2 )
1
x
SN2
12
SN1
Hopf
0
0
5
10
15
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Welfare
Welfare: population average of realised utility
X
Wt =
nht−1 Uht
With prediction errors εht = xt0 − fht−1 of type h, welfare can be written
as
Wt =
1 2 2 0 2
0
a σ0 (ζ ) + (xt0 − Rxt−1
)aζ 0
2
|
{z
}
|
{z
}
risk premium
−
n0,t−1 C
| {z }
irrationality bias
perfect foresight costs
−
1 Var εht
2a σ 2
| {z n }
variance of prediction errors
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Example II
Average welfare W̄ = T −1
X
Wt
Hopf
1
SN1
SN2
welfare
x
-0.1
12
SN1
-0.2
Hopf
0
0
5
10
15
Σn 2
SN2
-0.3
-0.4
-0.5
0
5
10
15
20
25
30
Σn 2
Theorem: limσn →0 W̄ = −∞
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Adding a perfect foresight type
• Demand of perfect foresight type has to clear the market at the
previously predicted price
• First round prediction should be such that local rational bubbles
are avoided
• Perfect foresight induces an additional cost C
Market clearing equation:
X
Rxt = n0t xt+1 +
nht fh (xt−1 , · · · )
h
Fitness (no prediction error)
U0t = −C
Typically C 1
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Singular perturbation theory
Introduce ε = e−βC , rewrite dynamics
X
X
εxt+1 = Rxt
e−βUht −
e−βUht fh (xt−1 , · · · , xt−L )
h
h
Here 0 < ε 1: singular perturbation (increases order)
Reduction to centre-stable manifold at stability loss
• rules out rational bubbles
• “correct” limit behaviour as C → ∞
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Effects of perfect foresight traders
If all boundedly rational types in the market are biased in the
fundamental equilibrium, then, if Arrow securities are added to the
market
• all boundedly rational types are driven out
• the price is driven to the fundamental
• welfare stabilises at forecasting costs
If some types are nonbiased in the fundamental equilibrium (trend
chasers, naive expectations), then
• nonbiased types are not driven out
• prices may remain volatile
• welfare may still decrease towards minus infinity
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Summary
In markets with heterogeneous, adaptively learning traders, adding
hedging instruments
• decreases risk
• amplifies effect of forecast errors
• destabilises rather than stabilises the market
• increases volatility
• decreases welfare
Adding perfect foresight traders can counteract the last three
conclusions in some cases
But:
No, adding trading options is not automatically a good idea
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