Optimal trading strategies Optimal trading strategies with limit - - PowerPoint PPT Presentation

SLIDE 1

Optimal trading strategies Optimal trading strategies with limit orders: with limit orders:

a a stochastic stochastic programming programming approach approach

Rossella Rossella Agliardi Agliardi University University of

• f Bologna (Italy)

Bologna (Italy) Chemnitz Chemnitz, 27 , 27-

• 29 March 2019

29 March 2019

SLIDE 2

Trading in Trading in limit limit order

• rder books

books

• Consider a trader who has a block of shares to sell (or buy) in

Consider a trader who has a block of shares to sell (or buy) in an illiquid market. an illiquid market.

• The trader's problem is to design an optimal order execution

The trader's problem is to design an optimal order execution strategy such that he/she can unwind a portfolio position strategy such that he/she can unwind a portfolio position

• within a fixed time constraint

within a fixed time constraint

• subject to the optimization of certain criteria

subject to the optimization of certain criteria

(Cost minimization? Mean (Cost minimization? Mean-

• variance optimization? Maximize expected

variance optimization? Maximize expected utility? etc.) utility? etc.)

SLIDE 3

Limit orders Limit orders

• Limit orders are ex ante commitment to trade. They specify the

Limit orders are ex ante commitment to trade. They specify the security, the amount and a (limit) price. security, the amount and a (limit) price. They are order to trade a certain amount of a security at a They are order to trade a certain amount of a security at a given given price (or a better one). price (or a better one).

• Traders post their supply/demand in the form of limit orders to

Traders post their supply/demand in the form of limit orders to an electronic trading system. an electronic trading system.

• Accumulation of posted limit orders is known as the limit

Accumulation of posted limit orders is known as the limit

• rder book.
• rder book.
• Orders leave the order book either they get executed or

Orders leave the order book either they get executed or cancelled. cancelled.

• An active order at time t is an order that has been submitted at

An active order at time t is an order that has been submitted at t′≤t t′≤t, but has not been matched or cancelled by time t. , but has not been matched or cancelled by time t.

• A LOB is the set of all active orders in the market at time t.

A LOB is the set of all active orders in the market at time t.

SLIDE 4

Example of an Order Book

Sell Order Buy Order

Price 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 10 10 30 20 10 20 10 10

Best Ask Best Bid Spread

SLIDE 5

Market Market orders

• rders vs

vs limit limit orders

• rders

◯ ◯ Market order Market order A buy/sell order that is to be traded A buy/sell order that is to be traded immediately at the best available price. Always executed immediately at the best available price. Always executed if there is sufficient quantity available ⇒ if there is sufficient quantity available ⇒ PRICE RISK PRICE RISK ◯ ◯ Limit order Limit order A buy/sell order that is to be executed at A buy/sell order that is to be executed at its specified limit or at a better price. its specified limit or at a better price. If a trader files a limit order to buy, the limit order sets If a trader files a limit order to buy, the limit order sets a a maximum price that will be paid. In case of a limit order maximum price that will be paid. In case of a limit order to sell the limit sets the minimum price that will be to sell the limit sets the minimum price that will be accepted. accepted. Executed only if the limit price is reached ⇒ Executed only if the limit price is reached ⇒ EXECUTION RISK EXECUTION RISK

SLIDE 6

A key A key difference difference

• Market orders

Market orders have virtually zero execution have virtually zero execution-

• risk. Uncertainty of
• risk. Uncertainty of

the execution price, price impact (especially of large orders). the execution price, price impact (especially of large orders). A trader’s main concern is to avoid a high execution cost (e A trader’s main concern is to avoid a high execution cost (e.g. by .g. by splitting a large orders into several small orders). splitting a large orders into several small orders).

• Limit orders

Limit orders incur non incur non-

• execution (or partial execution) risk. Orders

execution (or partial execution) risk. Orders are sorted in price are sorted in price-

• time priority: higher bids and lower asks have

time priority: higher bids and lower asks have higher priority. If a buy (sell) limit price is too low (high), higher priority. If a buy (sell) limit price is too low (high), the order the order will not trade. will not trade.

• Buy (sell) limit orders with high (low) prices are

Buy (sell) limit orders with high (low) prices are aggressively aggressively priced priced. .

• The limit price of a `marketable’ limit buy (sell) order is at o

The limit price of a `marketable’ limit buy (sell) order is at or above r above (below) the best ask (bid). They are the most aggressive limit (below) the best ask (bid). They are the most aggressive limit

• rders.
• rders.

SLIDE 7

Optimal Optimal trade trade execution execution

• Statement of the optimal scheduling problem:

Statement of the optimal scheduling problem: Given a model for the evolution of the stock price, Given a model for the evolution of the stock price, find an optimal strategy for trading shares find an optimal strategy for trading shares. .

• Mathematical tool:

Mathematical tool: Stochastic dynamic programming Stochastic dynamic programming

• Market orders: trade

Market orders: trade-

• off between price risk and trade
• ff between price risk and trade

cost cost

• Limit orders: trade

Limit orders: trade-

• off between limit price and
• ff between limit price and

execution risk execution risk

SLIDE 8

Literature Literature

• Market

Market orders

• rders

D.

• D. Bertsimas

Bertsimas & A. Lo (1998), R. & A. Lo (1998), R. Almgren Almgren & N. & N. Chriss Chriss (1999, 2000), R. (1999, 2000), R. Almgren Almgren (2001, 2003), A. (2001, 2003), A. Alfonsi Alfonsi, , A.Fruth A.Fruth, A. , A. Schied Schied (2010), J. (2010), J. Gatheral Gatheral & A. & A. Schied Schied (2011), (2011), J.Gatheral J.Gatheral, A. , A. Schied Schied, A. , A. Slynko Slynko (2011), A. (2011), A. Obizhaeva Obizhaeva & J. & J. Wang Wang (2005, 2013),T. (2005, 2013),T. Schöneborn Schöneborn (2008), R. (2008), R. Agliardi Agliardi & & R.

• R. Gençay

Gençay (2014), (2014),………… …………. .

• Limit

Limit orders

• rders

Bayraktar Bayraktar & & Ludkovski Ludkovski (2011), (2011), O.

• O. Guéant

Guéant, C. , C. Lehalle Lehalle, J. , J. Fernandez Fernandez-

• Tapia

Tapia (2012), R. (2012), R. Cont Cont & A. & A. Kukanov Kukanov (2012), O. (2012), O. Guéant Guéant, C. , C. Lehalle Lehalle (2015), F. (2015), F. Guilbaud Guilbaud & H. & H. Pham Pham (2013), …..R. (2013), …..R. Agliardi Agliardi (2016), R. (2016), R. Agliardi Agliardi & R. & R. Gençay Gençay (2017) (2017)

Market and Market and Limit Limit orders

• rders

Huitema Huitema (2011) (2011)

SLIDE 9

Our Our problem problem

• A trader has a sell order to liquidate before T.

A trader has a sell order to liquidate before T.

• She submits sell

She submits sell limit orders limit orders by specifying a by specifying a limit limit price price and an and an order size

• rder size at any point in time.

at any point in time.

• Quotes and

Quotes and sizes sizes are chosen in order to optimize the are chosen in order to optimize the trader’s profit, while keeping control of her risk trader’s profit, while keeping control of her risk inventory and of the cost of getting rid of the inventory and of the cost of getting rid of the remaining shares within T. remaining shares within T.

• The use of limit orders exposes the trader to non

The use of limit orders exposes the trader to non-

• execution risk.

execution risk.

• Traders are allowed to control for order size, and not

Traders are allowed to control for order size, and not just for the posted prices. just for the posted prices.

SLIDE 10

Nomenclature Nomenclature for for sell sell orders

• rders

(L. Harris, Trading & (L. Harris, Trading & Exchanges Exchanges (2003), Oxford (2003), Oxford U.P. U.P.) )

‘ ‘marketable marketable’ ’ below below the best the best bid bid ‘ ‘marketable marketable’ ’ at the best at the best bid bid ‘ ‘in the market’ in the market’ between between the best the best bid bid and and the best the best ask ask ‘ ‘at the market’ at the market’ at the best at the best ask ask ‘ ‘behind behind the market’ the market’ above above the best the best ask ask The sell The sell order

• rder is

is …. …. LIMIT PRICE PLACEMENT LIMIT PRICE PLACEMENT

SLIDE 11

Model Model setup setup

• An agent has X shares to sell and trades through a limit

An agent has X shares to sell and trades through a limit

• rder book by submitting limit orders.
• rder book by submitting limit orders.
• (

(x xn

n,

,δ δn

n)

) order

• rder submitted

submitted at

at time time t tn

n , n=1,…,N,

, n=1,…,N, where where

x xn

n =

= order

• rder size

size δ δn

n =

= distance of the limit price from the reference price distance of the limit price from the reference price

• The

The reference reference price price follows follows a martingale a martingale

(e.g. (e.g. S St

t = S

= S0

0 +

+ σ σW Wt

t)

)

• Λ

Λ(x, (x,δ δ) ) =

= probability

probability that that an an order

• rder (x,

(x,δ δ) ) gets gets executed executed

An An exponential exponential shape shape A. A.exp( exp(-

• k

kx x-

• h

hδ δ) ) is is assumed assumed for for Λ Λ

SLIDE 12

The The probability probability of

• f limit

limit order

• rder

execution… execution…. .

• decreases with the order size and the distance of

decreases with the order size and the distance of the limit price from the reference price (Harris and the limit price from the reference price (Harris and Hasbrouck (1996)) Hasbrouck (1996))

• Exponential dependence of the fill rate on

Exponential dependence of the fill rate on δ δ ( (A A.exp( .exp(-

• h

hδ δ)) )) is is supported by the empirical literature ( supported by the empirical literature (Lo, Lo, MacKinlay MacKinlay, Zhang, 2002 , Zhang, 2002) ) and used in the theoretical literature ( and used in the theoretical literature (Gueant Gueant et al., 2012) et al., 2012)

• The parameters modeling the execution intensity are increasing i

The parameters modeling the execution intensity are increasing in n the the LOB’s LOB’s depth, because depth, because ‘when a book is thicker, limit orders take ‘when a book is thicker, limit orders take longer to execute’ ( longer to execute’ (Goettler Goettler, Parlour, , Parlour, Rajan Rajan (2004)) (2004))

SLIDE 13

The The optimization

• ptimization problem

problem

, ,... ,

1 1

su p E

N N

x x

− − δ

δ

[

n n N n n

x Λ

∑

− =

δ

1

(

• )

( v ar ) 2 ( 2

2 1 2 1 n n n n n n n n

S x X x X ∆ Λ + Λ −

− −

γ )- ) ) 2 ( ( 2

1 2 1 2 1 2 2 − − − − −

Λ − +

N N N N N

X x x X l ] (* ) w h ere

n n n n

I x X X − =

− 1

, n = ,… ,N

• 1

, an d X X =

− 1

.

Expected value of terminal wealth Risk of stock price fluctuation x trader’s risk aversion Cost of terminal liquidation (e.g.

through market orders)

SLIDE 14

A A mathematical mathematical trick… trick…

The risk of stock price fluctuation is modeled throughout the variance of

N

t

S X

+

) (

1

N n

t t n N n n

S S I x −

∑

− =

which can be rewritten as:

) (

1

1

n n

t t N n n

S S X −

+

∑

− =

=

n N n n S

X ∆

∑

− = 1

The law of total variance and a recursive argument give:

E [ ) ( var

1 n n N n n

S X ∆

∑

− =

]

SLIDE 15

Stochastic Stochastic dynamic dynamic programming programming

) ( = X JN (**) ) (

1 X

JN− =

1 1,

sup

− − N N

x δ [ 1 1 1 − − −

Λ

N N N

x δ

• 2

1 −

+

N

V γ l (

2

X -

1 2 1 1 1

2

− − − −

Λ + Λ

N N N N

x X x )] ) (X Jn =

n n

x δ ,

sup[

n n n

x Λ δ

• 2

n

V γ ) 2 (

2 2 n n n n

x X x X Λ + Λ − ) + ) ( [

1 n n n n

I x X J E −

+

]] n = 0,… ,N

• 2

Here Vn = varn(∆Sn).

SLIDE 16

Solution Solution

The solution ) , (

* * n n

x δ , n = 0,… , N - 1, of (**) is such that:

* 1 − N

x =2/(2k+

1

~

− N

v ) with

1

~

− N

v =

1 − N

v + h l and

j

v =

j

hV γ for any j;

* n

x (X), n=0,… ,N-2 is a solution to the following equation: (

n

E ) 1-k *

n

x -(

1 2

~

− − =

+

∑

N N n j j

v v ) h xn

*

+∑

− + =

        Λ + − Λ − Λ

1 1 * * * * * * * *

) ( )' ( ) ( ) (

N n j j j n n j j j j

X x x x X x X x = 0 where ) (

* X j

Λ = )) ( ), ( (

* *

X X x

j j

δ Λ ;

* n

δ (X) = h 1+(

1 2

~

− − =

+

∑

N N n j j

v v ) h X xn 2 2

* −

+∑

− + =

        − Λ − Λ

1 1 * * * * * *

) ( ) (

N n j n n j j j j

hx x X x X x , n = 0,… , N-1

SLIDE 17

Proof Proof details details ….. …..

JnX;xn,nden

• tesxnnn 

V

n

2 X2 2xnXn xn 2n EnJn1  XxnIn,n  0,,N1.

T h esolu tionxn

,n ,n  0,,N2,arefou

n dbyrecu rsion .Ifon eassu m esth atth eresu lth

• ldsforxn1

 ,n1  ,th

enon ecancom pu teth atJn1

 Xis



jn1 N 1 xj

j X

h



jn N 2vj 

vN

1X

2

2h .T

h erefore EnJn1

 XxnIn  jn1 N 1 xj

j Xxn nxj j X1 n

h



jn N 2vj 

vN

1X22xnXnxn

2 n

2h

. Plu g g in gth isex pressionin toJnXon eg etsth efollow in gex pression : JnX  xnn 

jn N 2vj 

vN

1xn

22xnX

2h



jn1 N 1 xj

j Xxnxj j X

h

n 

jn N 2vj 

vN

1X

2

2h 



jn1 N 1 xj

j X

h

. T h enxn

,n areobtain

edfromnJn  0an dxnJn  0.

SLIDE 18

Numerical Numerical example example (N=2)

(N=2)

h l

*

x

*

δ

* 1

x

* 1

δ (E)

* 1

δ (NE)

0.3 0.01 158.7 1.74 133.3

• 4.42
• 6.01

0.2 0.01 212.9 0.75 200

• 1.88
• 4.01

0.1 0.01 409 5.63 399.9 6.09 1.99 0.1 0.02 385.4 2.49 333.3 1.04

• 6.68

X=100, h/k=50, γ=0.1, σ=0.01

SLIDE 19

Numerical Numerical example example (N=3)

(N=3)

h

l

*

x

*

δ

* 1

x (E)

* 1

δ (E)

* 1

x (N)

* 1

δ (N)

* 2

x

* 2

δ (EE)

* 2

δ (EN)

* 2

δ (NE)

* 2

δ (NN)

0.6 0.02 139.5 1.16 89.3 1.78 119.7 0.58 83.13 3.08 1.29 1.51

• 1.52

0.6 0.01 130.8 1.24 15.9 1.88 118 0.94 110.7 2.68 1.53 1.77 0.21 1 0.02 151.1 0.99 54.7 0.89 149.9 0.98 49.9 1.62 0.52 0.88

• 2.52

1 0.01 101.1 0.47 69.9 0.70 76.9 0.23 66.4 1.04 0.34 0.54

• 0.68

1 0.005 90.6 0.63 80.6 0.87 81.2 0.54 79.7 1.06 0.65 0.84 0.18 2 0.01 84.1 0.36 40.7 0.10 84.1 0.41 33.2

• 0.09
• 0.50

0.10

• 1.35

X=200, h/k=100, γ=1, σ=0.01

SLIDE 20

We We find find that that …. ….

• The agent tries to benefit from the posted quote (

The agent tries to benefit from the posted quote (δ δ>0) >0) mainly at the inception or, at a later stage whenever mainly at the inception or, at a later stage whenever previous sell orders have been already filled. previous sell orders have been already filled.

• She places aggressive orders (

She places aggressive orders (δ δ <0) mainly when she is <0) mainly when she is closer to the end of the trading period and previous closer to the end of the trading period and previous

• rders have expired unfilled. (Interpretation: marketable
• rders have expired unfilled. (Interpretation: marketable
• rders)
• rders)
• The total size of limit orders is usually smaller for larger

The total size of limit orders is usually smaller for larger values of values of h h and and k k, that is, when the execution is more , that is, when the execution is more unlikely. unlikely.

• When the terminal liquidation at market price incurs a

When the terminal liquidation at market price incurs a lower cost she submits larger orders and sets less lower cost she submits larger orders and sets less aggressive limit prices aggressive limit prices – – or even behind the best price

• r even behind the best price -
• at the end of the trading period.

at the end of the trading period.

SLIDE 21

Risk Risk aversion aversion and and volatility volatility (N = 2)

(N = 2)

SLIDE 22

Risk Risk aversion aversion and and volatility volatility (N = 3)

(N = 3)

γ σ

*

x

*

δ

* 1

x (E)

* 1

δ (E)

* 1

x (N)

* 1

δ (N)

* 2

x

* 2

δ (EE)

* 2

δ (EN)

* 2

δ (NE)

* 2

δ (NN)

1 0.01 101.1 0.47 69.9 0.70 76.9 0.23 66.4 1.04 0.34 0.54

• 0.68

1 0.05 109.9 0.10 61.4 0.53 75.9

• 0.09

61.5 1.03 0.26 0.23

• 1.12

1 0.1 112.5 -1.12 48.3

• 0.11

124.4

• 0.16

50 0.70

• 0.27

0.09

• 2.50

10 0.01 103.2 0.30 66.5 0.62 75.7 0.10 64.5 1.02 0.29 0.42

• 0.85

50 0.01 121.5 -0.12 54.6 0.44 83.1

• 0.30

57.1 1.07 0.25 0.01

• 1.57

SLIDE 23

We We find find that that …. ….

• negative quotes may appear even in the

negative quotes may appear even in the first stages whenever first stages whenever

• the price volatility is high

the price volatility is high

• risk aversion is strong

risk aversion is strong because price risk is then an important because price risk is then an important consideration. consideration.

SLIDE 24

LOB’s LOB’s depth depth

h/k ℓ

*

x

*

δ

* 1

x(E)

* 1

δ (E)

* 1

x(N)

* 1

δ (N)

* 2

x

* 2

δ (EE)

* 2

δ (EN)

* 2

δ (NE)

* 2

δ (NN)

100 0.01 101.1 0.47 69.9 0.70 76.9 0.23 66.4 1.04 0.34 0.54

• 0.68

200 0.005 160.1 0.84 134.8 1.31 139.1 0.68 132.5 1.82 1.13 1.22 0.32 50 0.02 83.4 0.73 40.8 0.21 83.1 0.82 33.3

• 0.19
• 1.01

0.21

• 2.69

SLIDE 25

LOB’s LOB’s depth depth … …

• In a thicker book, orders of larger size can be submitted

In a thicker book, orders of larger size can be submitted and generally with more favorable quotes. and generally with more favorable quotes.

( (Goettler Goettler, , Parlour Parlour and and Rajan Rajan, 2005: increased depth beyond the , 2005: increased depth beyond the best price results in a lower frequency of aggressive limit orde best price results in a lower frequency of aggressive limit orders and rs and a higher frequency of limit orders placed on that side of the bo a higher frequency of limit orders placed on that side of the book

• k

behind the quotes) behind the quotes)

• In a very thin book: limit orders are set far behind the

In a very thin book: limit orders are set far behind the best price in the early trades, very aggressive orders at best price in the early trades, very aggressive orders at the close of the trading period. the close of the trading period.

• Conclusion: trader’s urgency plays a role in the chosen

Conclusion: trader’s urgency plays a role in the chosen strategy. strategy.

SLIDE 26

Hedging Hedging through through limit limit orders

• rders
• G(t,S

G(t,St

t) = option value

) = option value

• G

Gk

k := := G(t

G(tk

k,S

,St

tk

k)

)

• R

Rk

k =

= hedger hedger's 's wealth wealth at at time time t tk

k

• R

Rk

k = R₀+G

= R₀+Gk+

k+∑

∑j=0,…k

j=0,…k-

• 1

1 |

|x xj

j|

|δ δj

j+x

+xj

j(

(S Sk

k-

• S

Sj

j)

)I Ij

j

• Aim

Aim: : to to hedge hedge the derivative position the derivative position buying buying/ /selling selling shares shares by by submitting submitting limit limit

• rders
• rders

SLIDE 27

Optimization Optimization problem problem

SLIDE 28

A A mathematical mathematical trick trick

The following Lemma is useful: The following Lemma is useful:

for any k₁<k₂≤N and denoting for any k₁<k₂≤N and denoting

Ck1,k2 : .covk1Gk2  Gk1,Sk2  Sk1, Vk1,k2 : .vark1Sk2  Sk1.

Ek1Ck2,N  Ck1,NCk1,k2 and Ek1Vk2,N  Vk1,NVk1,k2

we have

SLIDE 29

An An explicit explicit solution solution is is found……… found………

The order aggressiveness increases with the term The order aggressiveness increases with the term | |Cov

Covk

k(

(∆ ∆G, G, ∆ ∆S) S),

, +X

+X var vark

k(

(∆ ∆S) S)|

| The deeper in The deeper in-

• the

the-

• money is the call option or

money is the call option or the shorter its time to maturity, the more the shorter its time to maturity, the more aggressive is the limit order. aggressive is the limit order. It is beneficial to post limit orders away from the It is beneficial to post limit orders away from the market price when the hedger is not pressed by market price when the hedger is not pressed by the urgency to complete her strategy. the urgency to complete her strategy.

SLIDE 30

         ! !! !