Almost-sure hedging under permanent price impact Y.Zou Universit e - - PowerPoint PPT Presentation

Almost-sure hedging under permanent price impact

Y.Zou

Universit´ e Paris Dauphine

April 20, 2016

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 1 / 16

Outline

1 Introduction 2 Process dynamics and hedging problems

Impact rule and continuous trading dynamics Hedging problem: covered options

3 Super-replication under gamma constraint

Statement of pricing PDE Viscosity solution of the pricing PDE

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 2 / 16

Introduction

Objective:

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16

Introduction

Objective:

1 Consider a pricing model with impact and liquidity cost. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16

Introduction

Objective:

1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16

Introduction

Objective:

1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16

Introduction

Objective:

1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact.

Approach:

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16

Introduction

Objective:

1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact.

Approach:

1 Define a continuous time trading dynamics from a discrete time rule. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16

Introduction

Objective:

1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact.

Approach:

1 Define a continuous time trading dynamics from a discrete time rule. 2 Provide the PDE characterization: stochastic target tool. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16

Impact rule & Trading signal

Basic rule: an order δ moves the price by Xt− − → Xt = Xt− + δf(Xt−), and costs δXt− + 1 2δ2f(Xt−) = δ(Xt− + Xt 2 ).

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 4 / 16

Impact rule & Trading signal

Basic rule: an order δ moves the price by Xt− − → Xt = Xt− + δf(Xt−), and costs δXt− + 1 2δ2f(Xt−) = δ(Xt− + Xt 2 ). A trading signal is an Itˆ

• process of the form

Y = Y0 + · bsds + · asdWs.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 4 / 16

Discrete trading dynamics

Trade at times tn

i = iT/n the quantity δn tn

i = Ytn i − Ytn i−1. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 5 / 16

Discrete trading dynamics

Trade at times tn

i = iT/n the quantity δn tn

i = Ytn i − Ytn i−1.

The stock price evolves according to X = Xtn

i +

·

tn

i

σ(Xs)dWs between two trades.(can add a drift or be multivariate.)

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 5 / 16

Discrete trading dynamics

Trade at times tn

i = iT/n the quantity δn tn

i = Ytn i − Ytn i−1.

The stock price evolves according to X = Xtn

i +

·

tn

i

σ(Xs)dWs between two trades.(can add a drift or be multivariate.) Dynamics of the wealth and of the asset: discrete trading and pass to the limit.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 5 / 16

Continuous trading dynamics

Passing to the limit n → ∞, we have the following convergence in S2 Y = Y0 + · bsds + · asdWs X = X0 + · σ(Xs)dWs + · f(Xs)dYs + · (µ + asσf′)(Xs)ds V = V0 + · YsdXs + 1 2 · a2

sf(Xs)ds,

at a speed √n, where V = cash part + Y X = “portfolio value”.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 6 / 16

Super-replication under gamma constraint

Covered options: No initial/final market impact.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 7 / 16

Super-replication under gamma constraint

Covered options: No initial/final market impact. Super-replication: The minimum initial total wealth to cover the final payoff in almost sure sense.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 7 / 16

Super-replication under gamma constraint

Covered options: No initial/final market impact. Super-replication: The minimum initial total wealth to cover the final payoff in almost sure sense. Gamma constraint: Re-write the dynamics of Y as below: dY = a dW + b dt = γa(X) dX + µa,b

Y (X) dt

with γa = a/(σ + af). Then γa(X) is bounded by some function.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 7 / 16

Mathematical definition

Super-hedging price under gamma constraint: v(t, x) := inf{v = c + yx : (c, y) ∈ R2 s.t. Gγ(t, x, v, y) = ∅} where Gγ(t, x, v, y) is the set of (a, b) s.t. φ := (y, a, b) satisfies: V t,x,v,φ

T

≥ g(Xt,x,φ

T

), and γa(X) ≤ ¯ γ(X) with f¯ γ < 1 − ǫ, ǫ > 0.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 8 / 16

Informal derivation

If we follow the delta-hedging rule: V = v(·, X) , and Y = ∂xv(·, X)

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 9 / 16

Informal derivation

If we follow the delta-hedging rule: V = v(·, X) , and Y = ∂xv(·, X) Equating the dt terms on V = v(·, X) & dX terms on Y = ∂xv(·, X): 1 2a2f(X) = ∂tv(·, X) + 1 2(σ + af)2(X)∂2

xxv(·, X)

γa = ∂2

xxv(·, X)

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 9 / 16

Informal derivation

If we follow the delta-hedging rule: V = v(·, X) , and Y = ∂xv(·, X) Equating the dt terms on V = v(·, X) & dX terms on Y = ∂xv(·, X): 1 2a2f(X) = ∂tv(·, X) + 1 2(σ + af)2(X)∂2

xxv(·, X)

γa = ∂2

xxv(·, X)

By definition of γa and some calculate:

• − ∂tv − 1

2 σ2 (1 − f∂2

xxv)∂2 xxv

• (·, X) = 0

Gamma constraint insures the non-singularity of the PDE.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 9 / 16

Pricing PDE and Terminal condition

Pricing PDE: on [0, T) × R F[v¯

γ](t, x)

:= min

• − ∂tv¯

γ − 1

2 σ2 (1 − f∂2

xxv¯ γ)∂2 xxv¯ γ, ¯

γ − ∂2

xxv¯ γ

• =

Propagation of the gamma constraint to the boundary: v¯

γ(T−, ·) = ˆ

g on R where ˆ g is the smallest (viscosity) super-solution of min{ϕ − g, ¯ γ − ∂2

xxϕ} = 0

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 10 / 16

Super-solution property

Theorem (Geometric Dynamic Programming Principle)

If V0 > v(0, X0), then there exists (a, b, Y0) such that Vθ ≥ v(θ, Xθ) for any stopping time θ ∈ [0, T]. Result: A super-solution v¯

γ ≤ v¯ γ. cf. N.Touzi, M.Soner.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 11 / 16

Sub-solution property: shaken operator

Objective: Construct a solution ¯ v¯

γ ≥ v¯ γ.

Theorem

Define the shaken operator F ǫ[ϕ](t, x) := min

x′∈Bǫ(x) min

• −∂tϕ+

σ2(x′) 2(1 − f(x′)∂2

xxϕ), ¯

γ(x′)−∂2

xxϕ

• (t, x)

then ∀ǫ > 0, ∃¯ vǫ

¯ γ the unique continuous viscosity solution of

F ǫ[ϕ](t, x)✶[0,T) + [ϕ − (ˆ g + ǫ)]✶{T} = 0 Moreover, ¯ vǫ

¯ γ ≥ ˆ

g + ǫ/2 on [T − cǫ, T] × R.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 12 / 16

Sub-solution property: shaken operator

Objective: Construct a solution ¯ v¯

γ ≥ v¯ γ.

✶ ✶

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 13 / 16

Sub-solution property: shaken operator

Objective: Construct a solution ¯ v¯

γ ≥ v¯ γ.

By the stability of viscosity solution: ¯ vǫ

¯ γ → ¯

v¯

γ,

as ǫ → 0 where ¯ v¯

γ is the unique viscosity solution of

F[ϕ](t, x)✶[0,T) + (ϕ − ˆ g)✶{T} = 0

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 13 / 16

Sub-solution property: shaken operator

Objective: Construct a solution ¯ v¯

γ ≥ v¯ γ.

By the stability of viscosity solution: ¯ vǫ

¯ γ → ¯

v¯

γ,

as ǫ → 0 where ¯ v¯

γ is the unique viscosity solution of

F[ϕ](t, x)✶[0,T) + (ϕ − ˆ g)✶{T} = 0 Remain to prove: ¯ v¯

γ ≥ v¯ γ.

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 13 / 16

Sub-solution property: regularization

Theorem

For ψ ∈ C∞, define ψδ = δ−2ψ(·/δ). Then ¯ vǫ,δ

¯ γ

:= ¯ vǫ

¯ γ ⋆ ψδ is a

super-solution of F[¯ vǫ,δ

¯ γ ](t, x)✶[δ,T) + (¯

vǫ,δ

¯ γ − ˆ

g)✶{T} = 0 Then Y := ∂x¯ vǫ,δ

¯ γ (·, X) provides a super-hedging strategy, i.e. ¯

vǫ,δ

¯ γ

≥ v¯

γ.

Moreover, lim

ǫ,δ→0¯

vǫ,δ

¯ γ

= ¯ v¯

γ.

Conclusion: ¯ v¯

γ = lim ǫ,δ→0 ¯

vǫ,δ

¯ γ

≥ v¯

γ ≥ v¯ γ ≥ ¯

v¯

γ

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 14 / 16

Numerical res: Bachelier model with constant impact

Model: dXt = 0.2 dWt; Butterfly: g(x) = (x + 1)+ − 2x+ + (x − 1)+

• 2
• 1

1 2 0.0 0.2 0.4 0.6 0.8 x Value Function

• 2
• 1

1 2 0.00 0.05 0.10 0.15 x Price Difference

Figure 1: Left: Super-hedging price. Dashed line: λ = 0.5, ¯ γ = 1.75; solid line: λ = 0, ¯ γ = 1.75; dotted line: λ = 0, ¯ γ = +∞. Right: Difference with the price associated to λ = 0, ¯ γ = +∞. Dashed line: λ = 0.5, ¯ γ = 1.75; solid line: λ = 0, ¯ γ = 1.75 .

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 15 / 16

Thank you very much!

Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 16 / 16