The “Checklist” > 6. Aggregation > 6.3 Liquidity P&L/risk

Liquidity P&L/risk

Goal: liquidate our portfolio htnow

Theoretical valuevh,tnow = h′vtnow + casht (6.16)

not achievable: negative market impact of liquidation

portfolio holdingsvalues single instruments

trading speed ht

Solution: implement market-impact execution model in clock time

1 Market impact model: Almgren-Chriss model• Linear permanent impact f(u) ≡ γu (10.17)• Power law temporary impact: g(u) ≡ ±η|u|α, α = 0.5 (10.17)

2 Quasi-optimal power execution strategy• Power policy

ht ≡ htnow + (t− tnowtend − tnow

)β∆hparent , for t ∈ [tnow , tend ]

• Boundary conditions: liquidation htend = 0 and ∆hparent = −htnow

(10.34)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 6. Aggregation > 6.3 Liquidity P&L/risk

Liquidity P&L/risk

Goal: liquidate our portfolio htnow

Theoretical valuevh,tnow = h′vtnow + casht (6.16)

not achievable: negative market impact of liquidation

portfolio holdingsvalues single instruments

trading speed ht

Solution: implement market-impact execution model in clock time

1 Market impact model: Almgren-Chriss model• Linear permanent impact f(u) ≡ γu (10.17)• Power law temporary impact: g(u) ≡ ±η|u|α, α = 0.5 (10.17)

2 Quasi-optimal power execution strategy• Power policy

ht ≡ htnow + (t− tnowtend − tnow

)β∆hparent , for t ∈ [tnow , tend ]

• Boundary conditions: liquidation htend = 0 and ∆hparent = −htnow

(10.34)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 6. Aggregation > 6.3 Liquidity P&L/risk

Liquidity P&L/risk

Goal: liquidate our portfolio htnow

Theoretical valuevh,tnow = h′vtnow + casht (6.16)

not achievable: negative market impact of liquidation

portfolio holdingsvalues single instruments

trading speed ht

Solution: implement market-impact execution model in clock time

1 Market impact model: Almgren-Chriss model• Linear permanent impact f(u) ≡ γu (10.17)• Power law temporary impact: g(u) ≡ ±η|u|α, α = 0.5 (10.17)

2 Quasi-optimal power execution strategy• Power policy

ht ≡ htnow + (t− tnowtend − tnow

)β∆hparent , for t ∈ [tnow , tend ]

• Boundary conditions: liquidation htend = 0 and ∆hparent = −htnow

(10.34)

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 6. Aggregation > 6.3 Liquidity P&L/risk

Liquidity P&L/risk

• Expectation of P&L (10.36)

E{Πh(·),tnow→tend } = − γ

2h2tnow︸ ︷︷ ︸

“permanent impact”

− ηξ√tend − tnow

|htnow |3/2︸ ︷︷ ︸

temporary impact

(6.50)

≡ β3/2/(β + (β − 1)/2) > 0

⇓

Adjusted value for liquidity risk

vh,tnow ≈∑n

(hn,tnow vbidn,tnow + E{Πhn(·),tnow→tend })1{hn,tnow>0}

+∑n

(hn,tnow vaskn,tnow + E{Πhn(·),tnow→thor })1{hn,tnow<0} (6.52)

+ cashtnow

• Variance P&L (10.37)

V{Πh(·),tnow→tend } = σ2(tend − tnow )h2tnow (1− 2

β + 1+

1

2β + 1) (6.51)

uncertainty

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 6. Aggregation > 6.3 Liquidity P&L/risk

Liquidity P&L/risk

• Expectation of P&L (10.36)

E{Πh(·),tnow→tend } = − γ

2h2tnow︸ ︷︷ ︸

“permanent impact”

− ηξ√tend − tnow

|htnow |3/2︸ ︷︷ ︸

temporary impact

(6.50)

≡ β3/2/(β + (β − 1)/2) > 0

⇓

Adjusted value for liquidity risk

vh,tnow ≈∑n

(hn,tnow vbidn,tnow + E{Πhn(·),tnow→tend })1{hn,tnow>0}

+∑n

(hn,tnow vaskn,tnow + E{Πhn(·),tnow→thor })1{hn,tnow<0} (6.52)

+ cashtnow

• Variance P&L (10.37)

V{Πh(·),tnow→tend } = σ2(tend − tnow )h2tnow (1− 2

β + 1+

1

2β + 1) (6.51)

uncertainty

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 6. Aggregation > 6.3 Liquidity P&L/risk

Liquidity P&L/risk

• Expectation of P&L (10.36)

E{Πh(·),tnow→tend } = − γ

2h2tnow︸ ︷︷ ︸

“permanent impact”

− ηξ√tend − tnow

|htnow |3/2︸ ︷︷ ︸

temporary impact

(6.50)

≡ β3/2/(β + (β − 1)/2) > 0

⇓

Adjusted value for liquidity risk

vh,tnow ≈∑n

(hn,tnow vbidn,tnow + E{Πhn(·),tnow→tend })1{hn,tnow>0}

+∑n

(hn,tnow vaskn,tnow + E{Πhn(·),tnow→thor })1{hn,tnow<0} (6.52)

+ cashtnow

• Variance P&L (10.37)

V{Πh(·),tnow→tend } = σ2(tend − tnow )h2tnow (1− 2

β + 1+

1

2β + 1) (6.51)

uncertainty

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update