The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insurancePayoff design

Payoff design

The option-based portfolio insurance (OBPI) strategy attains thedesired payoff by suitably trading the risky and the low-risk instruments.

Step 1. Specify an arbitrary payoff at thorV stratthor ≡ g(V risky

thor) (9c.40)

Call option

g(v) ≡ max(v − kstrk , 0) (9c.41)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insurancePartial differential equation

Partial differential equation

Step 2. Solve the partial differential equation

∂gstrat

∂t+∂gstrat

∂vrrf v+

1

2

∂2gstrat

∂v2(σ(t, v))2 −gstratrrf = 0 (9c.42)

≥ 0 risk-free rate ≥ 0 volatility

with boundary condition

gstrat(thor , v) ≡ g(v), for all v (9c.43)

payoff function

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insurancePartial differential equation

Partial differential equation

Step 2. Solve the partial differential equation

∂gstrat

∂t+∂gstrat

∂vrrf v+

1

2

∂2gstrat

∂v2(σ(t, v))2−gstratrrf = 0 (9c.42)

Arithmetic Brownian motion

Suppose that• V risky

t ← arithmetic Brownian motion (9c.5)⇒ σ(t, v) ≡ σ• rrf ≡ 0

• g (v) ≡ max(v − kstrk , 0

)Then the solution reads

gstrat (t, v) = cBachelier (σ,v − kstrk√thor − t

, thor − t) (9c.46)

Bachelier function (16.29)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insurancePartial differential equation

Partial differential equation

Step 2. Solve the partial differential equation

∂gstrat

∂t+∂gstrat

∂vrrf v+

1

2

∂2gstrat

∂v2(σ(t, v))2−gstratrrf = 0 (9c.42)

Geometric Brownian motion

Suppose that• V risky

t ← geometric Brownian motion (9c.8)⇒ σ(t, v) ≡ σv• rrf > 0

• g (v) ≡ max(v − kstrk , 0

)Then the solution reads

gstrat (t, v) = cBS(v, rrf , σ,ln v − ln kstrk√

thor − t, thor − t) (9c.48)

Black-Merton-Scholes function (16.32) moneyness

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insuranceBudget

Budget

Step 3. Set the required budget

vstrattnow ≡ gstrat(tnow , v

riskytnow

) (9c.49)

Arithmetic Brownian motion

vstrattnow = cBachelier (σ,vriskytnow

− kstrk√thor − tnow

, thor − tnow ) (9c.50)

Geometric Brownian motion

vstrattnow = cBS(vriskytnow, rrf , σ,

ln vriskytnow− ln kstrk

√thor − tnow

, thor − tnow ) (9c.51)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insurancePolicy

Policy

Step 4. Solve the OBPI allocation policy (9c.24)

Delta hedging

Hriskyt ≡

∂gstrat (t,Vriskyt )

∂v(9c.52)

⇓

V stratt = gstrat(t, V risky

t ), t ≤ thor (9c.53)

Arithmetic Brownian motion

Hriskyt = Φ(

V riskyt − kstrk

σ√thor − t

) (9c.55)

Geometric Brownian motion

Hriskyt = Φ

(1

σ(lnV risky

t − ln kstrk√thor − t

+ (r +σ2

2)√thor − t)

)(9c.57)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insurancePolicy

Payoff of the OBPI

Geometric Brownian motion

• vriskytnow= $100, µ = 0.1, σ = 0.4

• vrftnow = $100, rrf = 0.02

• vstrattnow = $10, 000, kstrk = $100

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insuranceA unified approach

OBPI and maximum utility

sdf tnow→thor (v) ≡ 1

fV (v)

∂2vcalltnow (k)

∂k2|k=v (14b.32)

stochastic discount factor

density of V riskythor

call option value

Maximum utility: hrisky(·) ≡ argmaxh(·)∈C(E{utility(V stratthor )}) (9c.37)

mOBPI: gθ(v) ≡ utility ′−1( θ er

rf (thor−tnow )sdf (v)) (9c.58)

s.t. E{gθ(V riskythor

)sdf (V riskythor

)} = vstrattnow

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insuranceA unified approach

OBPI and maximum utility

Maximum utility: hrisky(·) ≡ argmaxh(·)∈C(E{utility(V stratthor )}) (9c.37)

mOBPI: gθ(v) ≡ utility ′−1(θer

rf (thor−tnow )sdf (v)) (9c.58)

Arithmetic Brownian motion• Stochastic discount factor

sdf (v) = ae− µ

σ2v (9c.60)

• Payoffgλ(v) = cλ +

µ

λσ2v (9c.61)

• PDE solutiongstrat(t, v) = cλ + µv/(λσ2) (9c.62)

• Strategyhriskyt =

µ

λσ2(9c.63)

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Option-based portfolio insuranceA unified approach

OBPI and maximum utility

Maximum utility: hrisky(·) ≡ argmaxh(·)∈C(E{utility(V stratthor )}) (9c.37)

mOBPI: gθ(v) ≡ utility ′−1(θer

rf (thor−tnow )sdf (v)) (9c.58)

Geometric Brownian motion• Stochastic discount factor

sdf (v) = av−µ−r

rf

σ2 (9c.64)

• Payoffln gλ(v) = cλ + 1

λµ−rrfσ2 ln v (9c.65)

• PDE solution

ln gstrat(t, v) = cλ + 1λµ−rrfσ2 ln v + bλ(t− tnow ) (9c.66)

• StrategyHrisky = 1

λµ−rrfσ2

g(t,Vriskyt )

Vriskyt

(9c.67)

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