The Checklist - 1. Risk drivers identification - High frequency
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The “Checklist’’ > Risk drivers identification> 1.8. High frequency High frequency risk drivers Goal : define risk drivers at the microstructure level, to model the trading P&L for optimal execution (Step 10) Financial instruments tradable at high frequency: • Stocks: most documented market in the literature • Futures: rich market (currencies, commodities, government futures, etc) • Options: market played by market makers; main venue CBOE [W] ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
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Transcript of The Checklist - 1. Risk drivers identification - High frequency
The “Checklist’’ > Risk drivers identification> 1.8. High frequency
High frequency risk drivers
Goal: define risk drivers at the microstructure level, to model the tradingP&L for optimal execution (Step 10)
Financial instruments tradable at high frequency:• Stocks: most documented market in the literature• Futures: rich market (currencies, commodities, government futures, etc)• Options: market played by market makers; main venue CBOE [W]
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequency
High frequency risk drivers
Goal: define risk drivers at the microstructure level, to model the tradingP&L for optimal execution (Step 10)
Events (trades, new quotes, cancellations) occur at discrete, random times
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequency
High frequency risk drivers
Goal: define risk drivers at the microstructure level, to model the tradingP&L for optimal execution (Step 10)
Events (trades, new quotes, cancellations) occur at discrete, random times
• Theoretical framework: Marked point processes jointly model{Tκ event times ("points")XTκ quantities , e.g. transaction prices/volumes ("marks")
• Practical two track approach. Model:
1. Tκ event times (tick time)2. Xκ tick evolution of the marks
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Market microstructure
• Financial instruments are traded at prices Pt ∈ γ N
• Financial instruments are traded by filling orders
equally spaced gridtick-size: γ
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Market microstructure
• Financial instruments are traded at prices Pt ∈ γ N
• Financial instruments are traded by filling orders
Limit orders{
to buy (∆hlb , plb)
to sell (∆hls , pls)Market orders
{to buy ∆hmb
to sell ∆hms
equally spaced gridtick-size: γ
quantity(holdings)
price quantity(holdings)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Market microstructure
• Financial instruments are traded at prices Pt ∈ γ N
• Financial instruments are traded by filling orders
Limit orders{
to buy (∆hlb , plb)
to sell (∆hls , pls)Market orders
{to buy ∆hmb
to sell ∆hms
equally spaced gridtick-size: γ
instantaneously filled at the "best" price
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Market microstructure
• Financial instruments are traded at prices Pt ∈ γ N
• Financial instruments are traded by filling orders
Limit orders{
to buy (∆hlb , plb)
to sell (∆hls , pls)Market orders
{to buy ∆hmb
to sell ∆hms
equally spaced gridtick-size: γ
Limit order book (LOB): set of all the outstanding limit orders
LOB t : pj → (Hbidj,t , H
askj,t ) (1.91)
price: pj ∈ γN bid volumeat price pj
ask volumeat price pj
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Limit order book: how it works
Simplified evolution of the LOB• Limit orders remain outstanding until they are executed or cancelled• Market orders are instantaneously filled at the best available price
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Co-moving values
P bidt P ask
t
best bid best ask
bid volumeHbidt
ask volumeHaskt
buy orders sell orders
Reference prices that gravitates around the "center" of the LOB:
• Mid-quotePmidt ≡ (P bid
t + P askt )/2 (1.92)
• Microprice
Pmict ≡ P bid
t Haskt + P ask
t Hbidt
Haskt +Hbid
t
(1.94)
• Last transaction price P lastt
takes on a continuum of values =⇒ better suited for statistical exploration
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Co-moving values
P bidt P ask
t
best bid best ask
bid volumeHbidt
ask volumeHaskt
buy orders sell orders
Reference prices that gravitates around the "center" of the LOB:
• Mid-quotePmidt ≡ (P bid
t + P askt )/2 (1.92)
• Microprice
Pmict ≡ P bid
t Haskt + P ask
t Hbidt
Haskt +Hbid
t
(1.94)
• Last transaction price P lastt
takes on a continuum of values =⇒ better suited for statistical exploration
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
High frequency risk drivers: microprice
Top plot: Realized time series of pbidt , paskt , pmict , plastt , hbid
t and haskt
for the 10 year US Treasury bond futures (data source: QuantitativeBrokers).
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Transaction variables
• Cumulative volume
where |∆QTκ | is the volume exchanged during the κ-th event
• Cumulative sign
where
• Cumulative signed volume
• Cumulative monetary amount transacted
Qt ≡∑
Tκ≤t|∆QTκ | (1.96)
Sgnt ≡∑
Tκ≤t∆SgnTκ (1.99)
∆SgnTκ ≡{
+1 (“buy”) if PTκ ≈ P askTκ
−1 (“sell”) if PTκ ≈ P bidTκ
(1.98)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Transaction variables
• Cumulative volume
where |∆QTκ | is the volume exchanged during the κ-th event
• Cumulative sign
where
• Cumulative signed volume
• Cumulative monetary amount transacted
Qt ≡∑
Tκ≤t|∆QTκ | (1.96)
Sgnt ≡∑
Tκ≤t∆SgnTκ (1.99)
∆SgnTκ ≡{
+1 (“buy”) if PTκ ≈ P askTκ
−1 (“sell”) if PTκ ≈ P bidTκ
(1.98)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Transaction variables
• Cumulative volume
where |∆QTκ | is the volume exchanged during the κ-th event
• Cumulative sign
where
• Cumulative signed volume
• Cumulative monetary amount transacted
Qt ≡∑
Tκ≤t|∆QTκ | (1.96)
Sgnt ≡∑
Tκ≤t∆SgnTκ (1.99)
∆SgnTκ ≡{
+1 (“buy”) if PTκ ≈ P askTκ
−1 (“sell”) if PTκ ≈ P bidTκ
(1.98)
Cumulative volume and cumulative sign
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Transaction variables
• Cumulative volume
where |∆QTκ | is the volume exchanged during the κ-th event
• Cumulative sign
where
• Cumulative signed volume
• Cumulative monetary amount transacted
Qt ≡∑
Tκ≤t|∆QTκ | (1.96)
Sgnt ≡∑
Tκ≤t∆SgnTκ (1.99)
∆SgnTκ ≡{
+1 (“buy”) if PTκ ≈ P askTκ
−1 (“sell”) if PTκ ≈ P bidTκ
(1.98)
SgnQt ≡∑
Tκ≤t∆SgnQTκ
, ∆SgnQTκ≡ ∆SgnTκ |∆QTκ | (1.100)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyMarket microstructure
Transaction variables
• Cumulative volume
where |∆QTκ | is the volume exchanged during the κ-th event
• Cumulative sign
where
• Cumulative signed volume
• Cumulative monetary amount transacted
Qt ≡∑
Tκ≤t|∆QTκ | (1.96)
Sgnt ≡∑
Tκ≤t∆SgnTκ (1.99)
∆SgnTκ ≡{
+1 (“buy”) if PTκ ≈ P askTκ
−1 (“sell”) if PTκ ≈ P bidTκ
(1.98)
SgnQt ≡∑
Tκ≤t∆SgnQTκ
, ∆SgnQTκ≡ ∆SgnTκ |∆QTκ | (1.100)
TradeMoneyt ≡∑Tκ≤t ∆TradeMoneyTκ , ∆TradeMoneyTκ ≡ PTκ |∆QTκ | (1.101)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Activity time
Calendar time or wall clock time: progresses even when the market is closed
Activity time: progresses according to the activity in the market
At ≡∑
Tκ≤t∆Aκ (1.102)
increment in the activity clock at the κ-th event (calendar time Tκ)
• Specifications: tick time, volume time, common activity time
• At ≈ random walk with drift =⇒ At risk driver
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Activity time
Calendar time or wall clock time: progresses even when the market is closed
Activity time: progresses according to the activity in the market
At ≡∑
Tκ≤t∆Aκ (1.102)
increment in the activity clock at the κ-th event (calendar time Tκ)
• Specifications: tick time, volume time, common activity time
• At ≈ random walk with drift =⇒ At risk driver
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Activity time
Calendar time or wall clock time: progresses even when the market is closed
Activity time: progresses according to the activity in the market
At ≡∑
Tκ≤t∆Aκ (1.102)
increment in the activity clock at the κ-th event (calendar time Tκ)
• Specifications: tick time, volume time, common activity time
• At ≈ random walk with drift =⇒ At risk driver
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Activity time
• Tick time advances by one unit whenever any event occurs
At = Kt ≡∑
Tκ≤t1 (1.103)
=⇒ At = Kt counts the number of events up to time t
• Volume time advances by the number of contracts exchanged at Tκ
At = Qt ≡∑
Tκ≤t|∆QTκ | (1.104)
=⇒ At = Qt is the cumulative volume (1.96)
∆Aκ
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Activity time
• Tick time advances by one unit whenever any event occurs
At = Kt ≡∑
Tκ≤t1 (1.103)
=⇒ At = Kt counts the number of events up to time t
• Volume time advances by the number of contracts exchanged at Tκ
At = Qt ≡∑
Tκ≤t|∆QTκ | (1.104)
=⇒ At = Qt is the cumulative volume (1.96)
∆Aκ
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Tick-time vs clock-time
Left bottom plot:• Realized time series at of the tick time for the 10-year Treasurybond futures contract (tick=new trade)
• The ticks are not uniformly distributed because of the randomnessof the event times Tκ
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Volume-time vs clock-time
Left bottom plot:Realized time series at of the volume time for the 10-year Treasury bondfutures contract = evolution of the cumulative volume or number of con-tracts qt
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyActivity time
Activity time
• Common activity time advances by the amount of money transactedin a reference multi-instrument portfolio or marketM at each tradetime Tk
At =∑
m∈MTradeMoneym,t ≡
∑m∈M
∑Tκ≤t
∆TradeMoneym,Tκ(1.105)
sum over all the instrumentsin the reference portfolio/market
∆Aκ
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyTime-changed variables
Time-changed variables
• Since At is increasing in t we can compute the inverse of AtAt ⇐⇒ Ta (1.106)
Ta = random clock time at which the amount of activity a occurred
• Time-changed risk driver: X̃a ≡ XTa (1.107)
• If we sample the process X̃a at equally spaced increments ∆a (bins)
. . . , X̃a−∆a, X̃a, X̃a+∆a, . . . (1.108)
we obtain approximately a random walk
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyTime-changed variables
Time-changed variables
• Since At is increasing in t we can compute the inverse of AtAt ⇐⇒ Ta (1.106)
Ta = random clock time at which the amount of activity a occurred
• Time-changed risk driver: X̃a ≡ XTa (1.107)
Tick-time vs clock-time evolution
Volume-time vs clock-time evolution
• If we sample the process X̃a at equally spaced increments ∆a (bins)
. . . , X̃a−∆a, X̃a, X̃a+∆a, . . . (1.108)
we obtain approximately a random walk
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyTime-changed variables
Time-changed variables
• Since At is increasing in t we can compute the inverse of AtAt ⇐⇒ Ta (1.106)
Ta = random clock time at which the amount of activity a occurred
• Time-changed risk driver: X̃a ≡ XTa (1.107)
• If we sample the process X̃a at equally spaced increments ∆a (bins)
. . . , X̃a−∆a, X̃a, X̃a+∆a, . . . (1.108)
we obtain approximately a random walk
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
The “Checklist’’ > Risk drivers identification> 1.8. High frequencyTime-changed variables
Time-changed variables
• Since At is increasing in t we can compute the inverse of AtAt ⇐⇒ Ta (1.106)
Ta = random clock time at which the amount of activity a occurred
• Time-changed risk driver: X̃a ≡ XTa (1.107)
• If we sample the process X̃a at equally spaced increments ∆a (bins)
. . . , X̃a−∆a, X̃a, X̃a+∆a, . . . (1.108)
we obtain approximately a random walk
⇓
High-frequency risk drivers ⊇ (At, X̃a) (1.111)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-20-2017 - Last update
