The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketRisky investment

Risky investment

The risky instrument V riskyt follows a diffusive process

dV riskyt ≡ µ(t, V risky

t )dt+σ(t, V riskyt )dBt (9c.4)

drift volatility Brownian motion

Arithmetic Brownian motion

µ(t, v) ≡ µ, σ(t, v) ≡ σ (9c.4)⇒ dV riskyt = µdt+ σdBt (9c.5)

Geometric Brownian motion

µ(t, v) ≡ µv, σ(t, v) ≡ σv (9c.4)⇒ dV riskyt

V riskyt

= µdt+ σdBt (9c.8)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketLow-risk investment

Low-risk investment

The low-risk instrument V low−riskt also follows a diffusive process

dV low−riskt ≡ µlow−risk (t, V low−risk

t )dt+ σlow−risk (t, V low−riskt ) dBlow−risk

t (9c.15)

|σlow−risk (t, V low−iskt )| � |σ(t, V risky

t )|

σlow−risk (t, v) ≈ 0 and µlow−risk (t, v) = rrf v

⇓

V low−iskt ≡ vrft

dvrftvrft

= rrf dt ⇒ vrft = vrftstart errf (t−tstart ) (9c.18)

risk-free instrument

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Holdings and weights

The value of strategy at time t reads

V stratt ≡ Hrisky

t V riskyt +Hrf

t Vrft (9c.19)

holdings

m

weights

W riskyt ≡ Hrisky

t V riskyt

V stratt

, W rft ≡

Hrft V

rft

V stratt

(9c.20)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Self-financing condition

The self-financing condition reads

V stratt+∆t ≡ Hrisky

t V riskyt+∆t +Hrf

t Vrft+∆t = Hrisky

t+∆tVriskyt+∆t +Hrf

t+∆tVrft+∆t ≥ 0 (9c.21)

m

dV stratt − V strat

t rrf dt = Hriskyt (dV risky

t − V riskyt rrf dt) (9c.22)

m

dV stratt

V stratt

=W riskyt

dV riskyt

V riskyt

+ (1−W riskyt )rrf dt (9c.23)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The allocation policy reads

Hriskyt ≡ hrisky( It ) ⇔ W risky

t ≡ wrisky( It ) (9c.24)

information available at time t

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The allocation policy reads

Hriskyt ≡ hrisky( It ) ⇔ W risky

t ≡ wrisky( It ) (9c.24)

information available at time t

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

Deterministic holding strategySuppose that

• V riskyt ← arithmetic Brownian motion (9c.5)

• rrf ≡ 0

• Hriskyt ≡ hrisky(t)

ThenV stratt ∼ N (µhrisky (·), σ

2hrisky (·)) (9c.27)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

Deterministic holding strategySuppose that

• V riskyt ← arithmetic Brownian motion (9c.5)

• rrf ≡ 0

• Hriskyt ≡ hrisky(t)

ThenV stratt ∼ N (µhrisky (·), σ

2hrisky (·)) (9c.27)

≡ vstrattnow + µ∫ t

tnowhrisky(s)ds (9c.28)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

Deterministic holding strategySuppose that

• V riskyt ← arithmetic Brownian motion (9c.5)

• rrf ≡ 0

• Hriskyt ≡ hrisky(t)

ThenV stratt ∼ N (µhrisky (·), σ

2hrisky (·)) (9c.27)

≡ σ2∫ t

tnow(hrisky(s))2ds (9c.29)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

Deterministic weight strategySuppose that

• V riskyt ← geometric Brownian motion (9c.8)

• rrf > 0

• W riskyt ≡ wrisky(t)

ThenV stratt

vstrattnow

∼ LogN (µwrisky (·), σ2wrisky (·)) (9c.30)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

Deterministic weight strategySuppose that

• V riskyt ← geometric Brownian motion (9c.8)

• rrf > 0

• W riskyt ≡ wrisky(t)

ThenV stratt

vstrattnow

∼ LogN (µwrisky (·), σ2wrisky (·)) (9c.30)

≡ rrf (t− tnow )− 12σ2wrisky (·)

+(µ− rrf

) ∫ t

tnowwrisky(s)ds

(9c.31)

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

The “Checklist” > 9c. Dynamic allocation: time series strategies > The marketStrategies

Allocation policy

The initial budget and the policy determine the value of the strategy

{vstrattnow , wrisky(·)⇔ hrisky(·)} 7→ V strat

t , t ≥ tnow (9c.26)

Deterministic weight strategySuppose that

• V riskyt ← geometric Brownian motion (9c.8)

• rrf > 0

• W riskyt ≡ wrisky(t)

ThenV stratt

vstrattnow

∼ LogN (µwrisky (·), σ2wrisky (·)) (9c.30)

≡ σ2∫ t

tnow(wrisky(s))2ds (9c.32)

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