Strategic Traders and Liquidity Crashesalexanderrem.weebly.com/uploads/7/2/5/6/72566533/game... ·...

Introduction Liquidity Shocks Loss Limits Conclusion and Extensions

Strategic Traders and Liquidity Crashes

Alexander Remorov

6.254 Final Project

December 7, 2013

Remorov Strategic Traders and Liquidity Crashes 1 / 21


Introduction

Most of the time markets functioning well

Prices don’t move much

Supply and demand for stock relatively balanced

What happens if everyone wants to sell?



CREG Stock, Nov. 17, 2013

3.75

3.8

3.85

3.9

3.95

4

4.05

4.1

4.15

4.2

4.25

Pri

ce

0

20000

40000

60000

80000

Vo

lum

e

Data obtained from Google Finance



Liquidity Crashes

What happens if almost everyone wants to sell?

Much lower price needed for someone to buy the stock

Selling even a small position causes a large price decline

This is caused an illiquid market

Everyone selling at the same time causes a liquidity crash



Strategic Traders

Can view traders as playing a simultaneous game

Need to decide on positions in the stock throughout the period

Could “wait it out” since will price likely return to previous level

However may be forced to sell when price is low:

Due to an exogenous liquidity shock: Bernardo and Welch (2004)

Loss limits: Morris and Shin (2004)



Maket Participants

Short-term strategic investors

Risk-neutral; don’t have price impact by their own trades

May be forced to sell stock

External liquidity shock: Bernardo and Welch (2004)

Loss limit: Morris and Shin (2004)

Market-makers/long-term investors

Risk-averse

Buy asset from short-term investors

Set lower price if receive more sell orders



Bernardo and Welch Model

Three dates:

t = 0: investors (possibly) trade

t = 1: liquidity shock with prob. s, affected investors must trade

t = 2: liquidation; investors are paid

Investor behavior:

α: fraction of investors selling at time 01− α: fraction selling at time 1 (assume correlated shocks)

Prices

p0(α) = v − cα: price at time 0p1(α) = v − c(1 + α): price at time 1v : expected value at time 2



Strategic investor’s problem

Decide on probability α to sell now

Payoff to sell one share now: p0(α)

Payoff to wait until date 2:

u =

{p1(α), if experiences shock

v , if no shock

Therefore will sell now if and only if:

p0(α) ≥ s × p1(α) + (1− s)× v



Deriving Equilibrium

Define F (α) to be expected benefit to sell now:

F (α) = p0(α)− s × p1(α)− (1− s)× v

= c(s(1 + α)− α)

s ≤ 12 : mixed NE α∗ = s

1−s

s > 12 : pure NE α∗ = 1

If probability of shock is high enough, everyone will sell right awaybefore the shock even occurs

Drawback: model too discrete; all investors acting the same



Morris and Shin Model

Two dates:

t = 0: investors (possibly) trade, some get fired

t = 1: liquidation; investors are paid

Investor behavior:

α: number of investors selling at time 0

loss limit qi ; if breached, investor is fired

distribution: qi = θ + ωi , ωiIID∼ Unif [−ε, ε]

Prices

Time 0 orders executed at an uncertain price v − cU, U ∼ Unif [0, α]

Price at the end of time 0 is v − cα

v : expected liquidation value at time 1



Strategic investor’s problem

Investor has stop limit qi . It is breached if α̂i investors sell, where:

qi = v − cα̂i

Thus, if decides to hold the stock, expected payoff is:

u(α) =

{v , if α ≤ α̂i

0, if α > α̂i

If decides to sell, then payoff is equal to expected sell price if loss limitnot breached, and zero otherwise

w(α) =

{v − 1

2cα, if α ≤ α̂iα̂iα (v − 1

2cα̂i ), if α > α̂i



Deriving Equilibrium

Will look for threshold strategies:

(v , qi ) 7→

{sell, if qi > q∗(v)

hold, if qi ≤ q∗(v)

In equilibrium, the fraction α of traders selling, conditional on your limitbeing q∗, is uniform [0, 1]

Equilibirum condition: ∫ 1

0(u(α)− w(α))dα = 0

Substituting formulas, becomes:

v − qi = c exp[ qi − v

2(v + qi )

]Remorov Strategic Traders and Liquidity Crashes 12 / 21


Optimal Loss Limit Threshold

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Loss

Lim

it q

Price Sensitivity c

Optimal Loss Limit Threshold, v = 100

Loss Limit Threshold

Lowest Price



Discussion and Modification

Desired effect: trader sells as if most of other traders sell

When c is small, act as if everyone else is selling

Unrealistic! End up selling even if the price is 20% above limit...



New Equilibrium

Introduce transaction costs τ if successfully sell stock

Furthermore if limit is breached, payoff is R > 0

Old equilibrium condition:

v − qi = c exp[ qi − v

2(v + qi )

]New condition:

v − qi = c exp[ (qi − v)(1− τ)− 4vτ

2(v + qi )(1− τ)− 4R

]



Modified Model Results

80

82

84

86

88

90

92

94

96

98

100

0 2 4 6 8 10 12 14 16 18 20

Loss

Lim

it q

Price Sensitivity c


Lowest Price

Tau = 0 %, R = 0




80

82

84

86

88

90

92

94

96

98

100

0 2 4 6 8 10 12 14 16 18 20

Loss

Lim

it q

Price Sensitivity c


Lowest Price

Tau = 0 %, R = 0

Tau = 5 %, R = 20




80

82

84

86

88

90

92

94

96

98

100

0 2 4 6 8 10 12 14 16 18 20

Loss

Lim

it q

Price Sensitivity c


Lowest Price

Tau = 0 %, R = 0

Tau = 5 %, R = 20

Tau = 5 %, R = 40




80

82

84

86

88

90

92

94

96

98

100

0 2 4 6 8 10 12 14 16 18 20

Loss

Lim

it q

Price Sensitivity c


Lowest Price

Tau = 0 %, R = 0

Tau = 5 %, R = 20

Tau = 5 %, R = 40

Tau = 20%, R = 40



Conclusion

Nice framework for modeling liquidity crashes

In equilibrium investors sell in fear that other investors may sell

Due to a potential external shock

Due to a loss limit

When introduce transaction costs as well as positive payoffs if limitbreached, results become more realistic



Further extensions

Want a multi-period model; things get more complicated

Possible extension: three dates, two types of strategic investors:

First type can sell at time 0 or 1

Second type can sell only at time 1

Then we get an “optimal” price drop for two periods – more realistic

Consider possibility of investors buying at the low price

Example - Brunnermeier and Pedersen (2005)

Large investor forced to sell – others sell with him, then buy at theresulting very low price


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