The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance
Constant proportion portfolio insurance
Step 1. Specify a target minimum “floor”
V floort = vfloor
tnowe∫ ttnow
Rrfs ds
, t ≥ tnow (9c.68)
risk-free rate
Step 2. Set the initial investment vstrattnow
> vfloortnow
Step 3. Compute the holdings
Hriskyt V risky
t ≡ mult× Cusht , mult ∈ [0, 1] (9c.71)
≡ V stratt −V floor
t excess cushion
⇓
dCushtCusht
= (1−mult)× dvrft
vrft
+ mult × dVriskyt
Vriskyt
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance
Constant proportion portfolio insurance
Step 1. Specify a target minimum “floor”
V floort = vfloor
tnowe∫ ttnow
Rrfs ds
, t ≥ tnow (9c.68)
risk-free rate
Step 2. Set the initial investment vstrattnow
> vfloortnow
Step 3. Compute the holdings
Hriskyt V risky
t ≡ mult× Cusht , mult ∈ [0, 1] (9c.71)
≡ V stratt −V floor
t excess cushion
⇓
dCushtCusht
= (1−mult)× dvrft
vrft
+ mult × dVriskyt
Vriskyt
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance
Constant proportion portfolio insurance
Step 1. Specify a target minimum “floor”
V floort = vfloor
tnowe∫ ttnow
Rrfs ds
, t ≥ tnow (9c.68)
risk-free rate
Step 2. Set the initial investment vstrattnow
> vfloortnow
Step 3. Compute the holdings
Hriskyt V risky
t ≡ mult× Cusht , mult ∈ [0, 1] (9c.71)
≡ V stratt −V floor
t excess cushion
⇓
dCushtCusht
= (1−mult)× dvrft
vrft
+ mult × dVriskyt
Vriskyt
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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsConstant proportion portfolio insurance
Geometric Brownian motion
• vriskytnow
= $100, µ = 0.1, σ = 0.4
• vrftnow
= $100, rrf = 0.02
• vstrattnow
= $10, 000, vfloortnow
= $980, mult = 5
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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control
Drawdown control
Step 1. Compute the high watermark, or running maximum
HWM t ≡ maxtstart≤s≤t
{V strats } (9c.75)
Step 2. Compute drawdown
DDt ≡ HWM t−V stratt (9c.76)
⇓ (V stratt > V floor
t )
DDt < HWM t−V floort (9c.77)
Step 3. Compute the holdings
Hriskyt V risky
t = mult×(V stratt −γ×HWM t), γ ∈ (0, 1) (9c.78)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control
Drawdown control
Step 1. Compute the high watermark, or running maximum
HWM t ≡ maxtstart≤s≤t
{V strats } (9c.75)
Step 2. Compute drawdown
DDt ≡ HWM t−V stratt (9c.76)
⇓ (V stratt > V floor
t )
DDt < HWM t−V floort (9c.77)
Step 3. Compute the holdings
Hriskyt V risky
t = mult×(V stratt −γ×HWM t), γ ∈ (0, 1) (9c.78)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control
Drawdown control
Step 1. Compute the high watermark, or running maximum
HWM t ≡ maxtstart≤s≤t
{V strats } (9c.75)
Step 2. Compute drawdown
DDt ≡ HWM t−V stratt (9c.76)
⇓ (V stratt > V floor
t )
DDt < HWM t−V floort (9c.77)
Step 3. Compute the holdings
Hriskyt V risky
t = mult×(V stratt −γ×HWM t), γ ∈ (0, 1) (9c.78)
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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsDrawdown control
Geometric Brownian motion
• vriskytnow
= $100, µ = 0.1, σ = 0.4
• vrftnow
= $100, rrf = 0.02
• vstrattnow
= $10, 000, γ = 0.7, mult = 2
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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy
Signal
Linear time invariant (LTI) filter
Lt =
∫ ttstart
b(t− s)dV riskys (9c.79)
⇓
dLt = L̃tdt+ b(0)dV riskyt (9c.81)
impulse response
≡∫ ttstart
b′(t− s)dV riskys
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy
Signal
Linear time invariant (LTI) filter
Lt =
∫ ttstart
b(t− s)dV riskys (9c.79)
⇓
dLt = L̃tdt+ b(0)dV riskyt (9c.81)
⇓ b(t− s) ∝ e−ln(2)τHL
(t−s)
Momentum
Smomt ≡ ln(2)
τHL
∫ ttstart
e− ln(2)τHL
(t−s)dV risky
s (9c.79)
⇓
dSmomt = ln(2)
τHL(−Smom
t dt+ dV riskyt ) (9c.81)
half-life
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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy
Strategy
Consider the strategy
hrisky(It) ≡ hrisky(Lt) (9c.84)
⇓
Πtnow→thor︸ ︷︷ ︸P&L
= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile
−∫ thor
tnow
(g′(Lt)L̃t +1
2g′′(Lt)(b(0)σ(t, V risky
t ))2dt︸ ︷︷ ︸trading impact
(9c.86)
g(x) ≡ 1
b(0)
∫ x
c
hrisky(l)dl (9c.87)
volatility
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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy
Strategy
Consider the strategy
hrisky(It) ≡ hrisky(Lt) (9c.84)
⇓
Πtnow→thor︸ ︷︷ ︸P&L
= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile
−∫ thor
tnow
(g′(Lt)L̃t +1
2g′′(Lt)(b(0)σ(t, V risky
t ))2dt︸ ︷︷ ︸trading impact
(9c.86)
signal = momentum of arithmetic
Brownian motion⇓Lt ≡ Smom
t , σ(t, V riskyt ) ≡ σ, hrisky(s) ≡ γs
Πtnow→thor = γ τHL2 ln(2)
((Smomthor
)2−(smomtnow
)2)+γ
∫ thor
tnow
((Smomt )2−σ
2
2ln(2)τHL
)dt (9c.87)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy
Strategy
Consider the strategy
hrisky(It) ≡ hrisky(Lt) (9c.84)
⇓
Πtnow→thor︸ ︷︷ ︸P&L
= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile
−∫ thor
tnow
(g′(Lt)L̃t +1
2g′′(Lt)(b(0)σ(t, V risky
t ))2dt︸ ︷︷ ︸trading impact
(9c.86)
signal = value of risky instrument ⇓ constant filter b(·) ≡ 1⇒ L̃t = 0
Πtnow→thor = g(V riskythor
)−g(vriskytnow
)− 12
∫ thor
tnow
g′′(V riskyt ) (σ(t, V risky
t ))2 dt (9c.88)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy
Strategy
Consider the strategy
hrisky(It) ≡ hrisky(Lt) (9c.84)
⇓
Πtnow→thor︸ ︷︷ ︸P&L
= g(Lthor )− g(ltnow )︸ ︷︷ ︸option profile
−∫ thor
tnow
(g′(Lt)L̃t +1
2g′′(Lt)(b(0)σ(t, V risky
t ))2dt︸ ︷︷ ︸trading impact
(9c.86)
Simple signal-induced strategy
Πtnow→thor = hrisky(vriskytnow
)×Πriskytnow→thor
+ γ × (Πriskytnow→thor
)2︸ ︷︷ ︸option profile
−γσ2
∫ thor
tnow
(V riskyt )2 dt︸ ︷︷ ︸
trading impact
signal = value reversal of risky
instrument under gBm⇓
{b(·) ≡ 1, σ(t, V risky
t ) ≡ σV riskyt
hrisky(V riskyt ) ≡ δ + 2γ(V risky
t − vtarget)
> 0 trend-following strategy< 0 mean-reverting strategy
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The “Checklist” > 9c. Dynamic allocation: time series strategies > Rolling horizon heuristicsSignal induced strategy
Simple signal-induced strategy
• V riskyt follows an Ornstein-Uhlenbeck process
• mean reverting half-life varies• vrisky
tnow= $100, µ = 0, σ = 0.01
• vstrattnow
= $10, 000, δ = 90, γ ∈ [−2, 2] varies
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