POPULATION RISK ON FINANCIAL MARKETS Norges Bank Conference on the Microstructure of Equity and Currency Markets Jeremy Large All Souls College, University of Oxford 10 September 2005 First Version : January 2004

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POPULATION RISK ON FINANCIAL MARKETSNorges Bank
Conference on the Microstructure of Equity and
Currency MarketsJeremy LargeAll Souls
College, University of Oxford10 September 2005
First Version January 2004
2
OUTLINE

• In a simplification of Goettler, Parlour and
  Rajan (2004)
• Remove asymmetric information
• Introduce incomplete information about background
  parameters i.e.
• Number of traders ( ?N )
• Distribution and average of trader reservation
  values ( Fß )
• The paper calls this Population Risk
• Interested in cases where traders take a long
  (infinite) time to tighten their priors on
  ?N and Fß
• Point out a payoff concavity that arises from
  time-preferences
• ? Mean-preserving-spreads in prior on ?N and Fß
  deter limit orders
• But payoffs to market orders are (of course)
  unaffected
• To balance supply and demand for liquidity (limit
  and market orders)
• depths fall or spreads widen

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CONCAVITY IN LIMIT ORDER PAYOFF
Expected utility from limit bid at price P
V - P
?N
0
Quiet Market
Busy Market
In this paper, call the private motive to trade V
(not ß)
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EXAMPLE ? 5050 BETWEEN QUIET MARKET AND BUSY
MARKET
Expected utility from limit bid at price P
V - P
Amount by which uncertainty makes trader discount
limit order utility
Market sale intensity
0
Quiet Market
Busy Market
Average Market
Concavity implies aversion to uncertainty
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AGENDA

• Relationship to Literature
• Model
• Equilibrium Without Population Risk
• Equilibrium With Population Risk
• Conclusion

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CLOSEST LITERATURE

• Recent dynamic models of the limit order book
• Foucault, Kadan, Kandel (2003)
• Goettler, Parlour and Rajan (2003, 2004)
• Rosu (2004)
• Hollifield, Miller and Sandas (2003)
• Also a connection to Frailty effects in the
  credit risk literature

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CLOSEST LITERATURE COMMENT

• Recent dynamic models of the limit order book
• Foucault, Kadan, Kandel (2003)
• Goettler, Parlour and Rajan (2003, 2004)
• Rosu (2004)
• Hollifield, Miller and Sandas (2003)
• Typically these models, since depths are
  stationary,
• ergodic intensity of un-cancelled bids ergodic
  intensity of market sales, and
• ergodic intensity of un-cancelled asks ergodic
  intensity of market purchases
• Otherwise depths would explode or implode
  arbitrarily in finite time.

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CLOSEST LITERATURE COMMENT

• Recent dynamic models of the limit order book
• Foucault, Kadan, Kandel (2003)
• Goettler, Parlour and Rajan (2003, 2004)
• Rosu (2004)
• Hollifield, Miller and Sandas (2003)
• Typically these models, since depths are
  stationary,
• ergodic intensity of un-cancelled bids ergodic
  intensity of market sales, and
• ergodic intensity of un-cancelled asks ergodic
  intensity of market purchases
• In GPR (2004) there is (essentially) no
  cancellation without replacement therefore 1
• 50 of traders ultimately submit market orders
• 50 of traders ultimately submit limit orders
• And with buy-sell symmetry, 25 ultimately
  submit each type of order (bids, asks, buys,
  sells)

1 this also requires constant order size
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CLOSEST LITERATURE COMMENT

• Recent dynamic models of the limit order book
• Foucault, Kadan, Kandel (2003)
• Goettler, Parlour and Rajan (2003, 2004)
• Rosu (2004)
• Hollifield, Miller and Sandas (2003)
• Typically these models, since depths are
  stationary,
• ergodic intensity of un-cancelled bids ergodic
  intensity of market sales, and
• ergodic intensity of un-cancelled asks ergodic
  intensity of market purchases
• In GPR (2004) there is (essentially) no
  cancellation without replacement therefore 1
• 50 of traders ultimately submit market orders
• 50 of traders ultimately submit limit orders
• And with buy-sell symmetry, 25 ultimately
  submit each type of order (bids, asks, buys,
  sells)

I will think of this as two market clearing
conditions Supply and demand for liquidity
equate both at the bid, and at the ask
1 this also requires constant order size
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LITERATURE (2)

• The model also borrows substantially from Parlour
  (1998)
• Have in mind larger depths than GPR (2004), where
  
• average depth 3-4 units
• Contrast the extreme case, CBOT 10 yr treasury
  bond futures LOB
• average depth 25-30 units at each best quote
  alone
• bid-ask spread is fixed at one price tick
  (trades on a penny)
• Consequences (contestable)
• Can concentrate on strategies at the best quotes
• Picking-off risk is mitigated because traders can
  see own depths fall in good time, and cancel if
  they want
• I set picking-off risk to zero (constant common
  value)
• Useful to assume that prices are fixed bid at B,
  and ask at A gt B

(Average depth / average trade size) is in the
range of 25 - 30
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TRACTABLILITY SINGLETON STATE SPACE

• Previous papers have, realistically, had complex
  state spaces and transition densities
• Quoted prices and depths change with order
  submissions
• Essential for understanding dynamics
• But, few papers have yielded tractable equilibria
  (Rosu 2004)

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TRACTABLILITY SINGLETON STATE SPACE

• Previous papers have, realistically, had complex
  state spaces and transition densities
• Quoted prices and depths change with order
  submissions
• Essential for understanding dynamics
• But, few papers have yielded tractable equilibria
  (Rosu 2004)
• Here I look at the tractable simplification where
  the state space is a singleton
• But, retain the aforementioned market-clearing
  feature of richer models
• This just means constant market depths, L
• I capture the dynamic trade-off between
  submitting market and limit orders
• But I capture no order book dynamics at all
• Somewhat agnostic about trading mechanism (could
  be a specialist)

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AGENDA

• Relationship to Literature
• Model
• Equilibrium Without Population Risk
• Equilibrium With Population Risk
• Conclusion

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TWO PERIODS

• Time One Continuous Trading
• Time runs continuously
• Identical traders enter the market à la Poisson
  and choose between market orders and limit orders

• Time Zero Market Clearing
• Market-clearing condition is satisfied ex ante by
  setting prices and depths
• Agnostic about the agency or mechanism that
  imposes market clearing

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CONTINUOUS TRADING
Continuous Trading

• Time runs continuously from zero onwards
  indefinitely
• Only two prices, A (ask) and B (bid) (B lt A)
• Constant bid and ask depth L
• Identical traders arrive at market with intensity
  ?N N I
• Risk-neutral traders first draw reservation
  value, V, from a smooth, symmetric single-peaked
  CDF F(.). F is common knowledge.
• There is a trading deadline which stops the
  market
• Arrives randomly as the first event of a Poisson
  process of parameter ?.
• Models the motive to trade sooner rather than
  later

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TRADERS DO ONE OF FOUR ACTIONS ON ARRIVAL AT
MARKETOrder one unit at a time condition on
A, B, L
Continuous Trading
Bid Queue
Ask Queue
Market Orders
Market purchase
Market sale
L
L
B
A
Limit Orders
Limit bid
Limit ask
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IMMEDIATE VERSUS DELAYED EXECUTIONE.g. buyers
(analogous payoffs for sellers)
Continuous Trading
Bid Queue
Ask Queue
Take the Ask get ( V A ) now
L
L
B
A
Join the back of the Bid queue get ( V B )
when the ( L 1 ) th seller hits the bid queue
But get nothing (and lose nothing) if deadline
is passed
Traders can cancel and resubmit their choice of
order at any time but wont
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STATIONARY MARKOV-PERFECT EQUILIBRIUMTraders
condition action on A, B, L
Continuous Trading
The aggregate of many independent Poisson
processes is Poisson
Traders choose between market orders and limit
orders
Order arrival intensities determined in aggregate
µs µb
? s ? b
All traders need to know about others strategies
are the aggregate arrival rates
Rational anticipations
µs µs
µb µb
Traders anticipate order arrival intensities
?b ?b
?s ?s
µs µb
? s ? b
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MARKET CLEARINGA, B, AND L ARE SET IN ADVANCE
Market Clearing
N F
L
L
V
m
B
A
Markets for liquidity
CONDITION At both markets for liquidity, in any
given period, the expected number of limit
orders the expected number of market orders
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AGENDA

• Relationship to Literature
• Model
• Equilibrium Without Population Risk
• Equilibrium With Population Risk
• Conclusion

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BUYERS COST OF EXECUTION VERSUS IDEAL
BENCHMARKBenchmark trade immediately at ( V
B )
Continuous Trading
A limit bid incurs a cost of delayed execution
A market purchase incurs a cost of immediate
execution
( V B ) P deadline before trade ( A B
)
Cost

• Increasing in V
• Increasing in ?
• Tends to ( V B ) as ? tends to ?

• Insensitive to V
• Insensitive to ?

V
?
Therefore, high V sufficient for buyer to prefer
market orders to limit orders
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MARGINAL PRICE-TAKING BUYER Indifferent between
market purchases and limit bids
Continuous Trading
A limit bid incurs a cost of delayed execution
A market purchase incurs a cost of immediate
execution
( V B ) P deadline before trade
( A B )

• Increasing in V
• Increasing in ?
• Tends to ( V B ) as ? tends to ?

• Insensitive to V
• Insensitive to ?

V
?
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MARGINAL PRICE-TAKING BUYERIndifferent between
market purchases and limit bids
Continuous Trading
A limit bid incurs a cost of delayed execution
A market purchase incurs a cost of immediate
execution
( V B ) P deadline before trade
( A B )

• Increasing in V
• Increasing in ?
• Tends to ( V B ) as ? tends to ?

• Insensitive to V
• Insensitive to ?

V
?
Some Maths
So you get a cut-off, Vb, above which all buyers
submit market orders
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THREE CUT-OFFS DEFINING FOUR AREAS
Continuous Trading
F
Limit Bids
PDF of F
Limit Asks
Market Purchases
Market Sales
V
V
Vs
Vb
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MARKET CLEARING CONDITION IN EXPECTATIONPrices
are therefore set competitively in this model
Market Clearing
F
Limit Bids
PDF of F
Limit Asks
Market Purchases
Market Sales
Shaded areas are equal Unshaded areas are equal
V
V
Vs
Vb
µs ?b
µb ?s
This condition fixes A and B to surround the
median trader, V
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SYMMETRIC EQUILIBRIUM
Equilibrium
N F
L
L
V
m
B
A
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SYMMETRIC EQUILIBRIUM (II)
Equilibrium
sellers
buyers
Inter-quartile range
N F
LO
LO
L
L
MO
MO
V
m
F-1(1/4)
B
A
F-1(3/4)
Market Clearing Spread
Market clearing spread increasing in trader
dispersion and in mean wait for execution
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SYMMETRIC EQUILIBRIUM (III) Equilibrium spread
and depth are positively related
Equilibrium
sellers
buyers
Inter-quartile range
N F
LO
LO
L
L
MO
MO
V
m
F-1(1/4)
F-1(3/4)
B
A
Market Clearing Spread
I dont say much about which combination of L
and (A-B) obtains
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AGENDA

• Relationship to Literature
• Model
• Equilibrium Without Population Risk
• Equilibrium With Population Risk
• Conclusion

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THE PAYOFF TO A LIMIT ORDER IS CONCAVEL 25
Uncertainty recap
Expected utility from limit bid
V B
0.8 (V B)
Calibrated with HMSS (2004) and GPR (2004)
small convexity
inflection point
µs (normalized)
0
Truth
Expected time to top of queue is less than twice
the expected time to deadline
N.B. To the human eye, away from the inflection
point the calibrated schedule is invariant with L
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CONCAVITY IMPLIES AVERSION TO UNCERTAINTY 5050
between low-demand and high-demand world
Uncertainty recap
Expected utility from limit bid
V B
0.8 (V B)
Amount by which uncertainty makes traders
discount limit order utility
µs (normalized)
Low demand world
High demand world
Market order utility is unaffected by this
uncertainty. So, to clear the market, spreads
must widen in uncertain conditions.
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POPULATION RISK IS ABOUT F AND N not about my
V
1/3 prob
Overlay uncertainty about N (vertical stretch)
1/3 prob
1/3 prob
V
Anyone here submits market sell orders But
how many are they? The buyer is unsure about
µs , the rate of arrival of market sell orders
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THERE IS A SINGLETON FOR WHICH THE UNCERTAINTY IS
A MEAN-PRESERVING SPREADCompare spreads With
and Without Population Risk
V
Call the singleton (in green) F
For symmetric equilibrium, traders priors over
the states of the market, F? ? ??, are
symmetric about F.
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SYMMETRIC EQULIBRIUM WITH POPULATION RISK
Equilibrium
Inter-quartile range
1/3 prob
1/3 prob
LO
LO
1/3 prob
MO
MO
V
m
F-1(1/4)
F-1(3/4)
Market Clearing Spread
So, under Population Risk spreads widen or
depths fall
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CONCLUSION

• Traders are deterred from supplying liquidity by
  unresolved uncertainty about the population of
  traders they are up against.
• But, consuming liquidity is unproblematic.
• Depths fall or spreads widen to compensate
  liquidity suppliers.
• Glossed over
• V is slightly informative about which population
  you are in
• Complication but does not change results (use a
  monotonicity condition)
• Market Clearing is ex ante, but in many contexts
  ex post market clearing is more plausible
• Im working on this (making L variable)