The "Checklist"> 10. Execution> Market impact modelingExogenous impact

Exogenous impact

Goal: describe the empirical effect on the market of everyone’s trading.

Key variable: cumulative volume Q (1.97) traded in the market.

The simplest model is

∆Pκ =∑l<κbκ−l (∆SgnQ l) + εκ (10.11)

price changesin tick time (1.103)∆Pκ ≡ Pκ − Pκ−1

traded volumeswith sign (1.110)

i.i.d. noise

where, empirically• l is large due to the autocorrelation of the order flow (2.81);• bκ−l > 0 for the closest lags.

Exogenous impact model fitted on MSFT equity

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Endogenous impact

Goal: describe the empirical effect on the market of our own trades.

Key variable: holdings h traded by self.

For endogenous market impact modeling and order scheduling the naturalactivity time is volume time (1.104).

Clock-time t Volume time q

Fair price Pt Pq ≡ Pt(q)Transaction price Pt

˜P q ≡ Pt(q)

Cumulative traded volume Qt Qq ≡ Qt(q) = q

Holdings in a financial instrument ht hq ≡ ht(q)

Table 10.4: Key variables in market impact modeling and order scheduling

Going forward, we will write for instance Pq instead of Pq.

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Permanent market impact model

The price follows the permanent market impact model

Pq = Pqstart +∫ qqstart

σdBs︸ ︷︷ ︸no-trade

+∫ qqstart

b (q − s) f(hs)ds︸ ︷︷ ︸permanent

(10.14)

(stochastic) (deterministic)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Permanent market impact model

Speed of trading: hs ≡ dhsds

.

Volume-weighted-average-price (VWAP) strategy

The price follows the permanent market impact model

Pq = Pqstart +∫ qqstart

σdBs︸ ︷︷ ︸no-trade

+∫ qqstart

b (q − s) f(hs)ds︸ ︷︷ ︸permanent

(10.14)

(stochastic) (deterministic)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Permanent market impact model

Speed of trading: hs ≡ dhsds

.

Volume-weighted-average-price (VWAP) strategy

The price follows the permanent market impact model

Pq = Pqstart +∫ qqstart

σdBs︸ ︷︷ ︸no-trade

+∫ qqstart

b (q − s) f(hs) ds︸ ︷︷ ︸permanent

(10.14)

(stochastic) (deterministic)

market impactdecay kernel

• non-negative function

(instantaneous) marketimpact function

• f(0) = 0

• f ′ > 0

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Temporary market impact or slippage model

Pq 6= Pq

actual price at whichtransactions occur

price implied by themarket impact model (10.14)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Temporary market impact or slippage model

Pq 6= Pq

actual price at whichtransactions occur

price implied by themarket impact model (10.14)

The industry converged to the temporary market impact or slippagemodel

Pq = Pq + g(hq) (10.15)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Temporary market impact or slippage model

Pq 6= Pq

actual price at whichtransactions occur

fair "mark-to-market"price

The industry converged to the temporary market impact or slippagemodel

Pq = Pq + g(hq) (10.15)

(deterministic)where• g(0) = 0

• g′ > 0

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Market impact model

Permanent impact : Pq = Pqstart +∫ qqstart

σdBs︸ ︷︷ ︸no-trade

+∫ qqstart

b (q − s) f(hs)ds︸ ︷︷ ︸permanent

(10.14)

+

Temporary impact: Pq = Pq + g(hq) (10.15)

⇓Pq = Pq + g(hq) (10.15)

Market impact model

Pq = Pqstart + σBq−qstart︸ ︷︷ ︸no-trade

+∫ qqstart

b (q − s) f(hs)ds︸ ︷︷ ︸permanent

+ g(hq)︸ ︷︷ ︸slippage

(10.16)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update

The "Checklist"> 10. Execution> Market impact modelingEndogenous impact

Market impact model

Permanent impact : Pq = Pqstart +∫ qqstart

σdBs︸ ︷︷ ︸no-trade

+∫ qqstart

b (q − s) f(hs)ds︸ ︷︷ ︸permanent

(10.14)

+

Temporary impact: Pq = Pq + g(hq) (10.15)

⇓Pq = Pq + g(hq) (10.15)

Market impact model

Pq = Pqstart + σBq−qstart︸ ︷︷ ︸no-trade

+∫ qqstart

b (q − s) f(hs)ds︸ ︷︷ ︸permanent

+ g(hq)︸ ︷︷ ︸slippage

(10.16)

Almgren-Chriss: f(u) ≡ γu, b(u) ≡ 1, g(u) ≡ η sgn(u) |u|α (10.17)

Obizhaeva-Wang: f(u) = γu, b(u) = e−ρu, g(u) ≡ 0 (10.21)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update