SSVI a la Bergomi Stefano De Marco 1 , Claude Martini 2 1 Ecole - - PowerPoint PPT Presentation

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SSVI a la Bergomi Stefano De Marco 1 , Claude Martini 2 1 Ecole - - PowerPoint PPT Presentation

Nov 15, 2023 118 likes •802 views

Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi SSVI a la Bergomi Stefano De Marco 1 , Claude Martini 2 1 Ecole Polytechnique 2 Zeliade Systems Jim Gatheral 60th birthday conference 1 / 30 Outline

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

SSVI a la Bergomi

Stefano De Marco1, Claude Martini2

1 Ecole Polytechnique 2 Zeliade Systems

Jim Gatheral 60th birthday conference

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline:

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline: Jim’s → Zeliade (ZQF, Model Validation) → Banks, HFs, CCPs

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline: Jim’s → Zeliade (ZQF, Model Validation) → Banks, HFs, CCPs

◮ Double Lognormal Model

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline: Jim’s → Zeliade (ZQF, Model Validation) → Banks, HFs, CCPs

◮ Double Lognormal Model ◮ Variational Most likely Path

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline: Jim’s → Zeliade (ZQF, Model Validation) → Banks, HFs, CCPs

◮ Double Lognormal Model ◮ Variational Most likely Path ◮ SVI

2 / 30

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline: Jim’s → Zeliade (ZQF, Model Validation) → Banks, HFs, CCPs

◮ Double Lognormal Model ◮ Variational Most likely Path ◮ SVI ◮ SSVI

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline: Jim’s → Zeliade (ZQF, Model Validation) → Banks, HFs, CCPs

◮ Double Lognormal Model ◮ Variational Most likely Path ◮ SVI ◮ SSVI ◮ ..and more to come!

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Foreword To Jim

A nice pipeline: Jim’s → Zeliade (ZQF, Model Validation) → Banks, HFs, CCPs

◮ Double Lognormal Model ◮ Variational Most likely Path ◮ SVI ◮ SSVI ◮ ..and more to come!

So, Jim, on behalf of Zeliade I say: thank you!

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

SSVI a la Bergomi

Stefano De Marco1, Claude Martini2

1 Ecole Polytechnique 2 Zeliade Systems

Jim Gatheral 60th birthday conference

3 / 30

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

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1st ingredient: (e)SSVI

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral SVI

Formula for the implied total variance at a given maturity T: v(k) = a + b(ρ(k − m) +

  • (k − m)2 + σ2)

where: v = implied vol2T. k is the log forward moneyness.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral SVI

Formula for the implied total variance at a given maturity T: v(k) = a + b(ρ(k − m) +

  • (k − m)2 + σ2)

where: v = implied vol2T. k is the log forward moneyness. 5 parameters, calibration not so immediate (Zeliade Quasi Explicit whitepaper, 2009).

7 / 30

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral SVI

Formula for the implied total variance at a given maturity T: v(k) = a + b(ρ(k − m) +

  • (k − m)2 + σ2)

where: v = implied vol2T. k is the log forward moneyness. 5 parameters, calibration not so immediate (Zeliade Quasi Explicit whitepaper, 2009). No arbitrage conditions essentially unknown.

7 / 30

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral SVI

Formula for the implied total variance at a given maturity T: v(k) = a + b(ρ(k − m) +

  • (k − m)2 + σ2)

where: v = implied vol2T. k is the log forward moneyness. 5 parameters, calibration not so immediate (Zeliade Quasi Explicit whitepaper, 2009). No arbitrage conditions essentially unknown. Fits super well (the best 5 parameters model around?).

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral-Jacquier Surface SVI

Formula for the implied total variance for the whole surface: w(k, θt) = θt 2 (1 + ρϕ(θt)k +

  • (ϕ(θt)k + ρ)2 + ¯

ρ2) where: θ :ATM TV, ρ :(constant) spot vol correlation, ϕ : (function) curvature. k is the log forward moneyness.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral-Jacquier Surface SVI

Formula for the implied total variance for the whole surface: w(k, θt) = θt 2 (1 + ρϕ(θt)k +

  • (ϕ(θt)k + ρ)2 + ¯

ρ2) where: θ :ATM TV, ρ :(constant) spot vol correlation, ϕ : (function) curvature. k is the log forward moneyness. Typical shape for ϕ: power law, ϕ(θ) = ηθ−λ, 0 ≤ λ ≤ 1/2.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral-Jacquier Surface SVI

Formula for the implied total variance for the whole surface: w(k, θt) = θt 2 (1 + ρϕ(θt)k +

  • (ϕ(θt)k + ρ)2 + ¯

ρ2) where: θ :ATM TV, ρ :(constant) spot vol correlation, ϕ : (function) curvature. k is the log forward moneyness. Typical shape for ϕ: power law, ϕ(θ) = ηθ−λ, 0 ≤ λ ≤ 1/2. θ taken directly as a parameter: feature quite unique to SSVI. Unlike Bergomi Variance Swap curve parameterization.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Gatheral-Jacquier Surface SVI

Formula for the implied total variance for the whole surface: w(k, θt) = θt 2 (1 + ρϕ(θt)k +

  • (ϕ(θt)k + ρ)2 + ¯

ρ2) where: θ :ATM TV, ρ :(constant) spot vol correlation, ϕ : (function) curvature. k is the log forward moneyness. Typical shape for ϕ: power law, ϕ(θ) = ηθ−λ, 0 ≤ λ ≤ 1/2. θ taken directly as a parameter: feature quite unique to SSVI. Unlike Bergomi Variance Swap curve parameterization. (Historically, stems out of SVI. SSVI slices are a subfamily of SVI).

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

No arbitrage in SSVI

Proposition (GJ, SSVI paper, Theorems 4.1 and 4.2 )

There is no calendar spread and no butterfly arbitrage if ∂tθt ≥ 0 (2.1) 0 ≤ ∂θ(θϕ(θ)) ≤ 1 ρ2 (1 + ¯ ρ)ϕ(θ), ∀θ > 0 (2.2) θϕ(θ) ≤ min

  • 4

1 + |ρ|, 2

  • θ

1 + |ρ|

  • , ∀θ > 0

(2.3) where ¯ ρ =

  • 1 − ρ2.

Condition 2.3 implies that limθ→0 θϕ(θ) = 0.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

SSVI in practice

Usage: implied vol smoother, risk models Widely used on Equity (indexes, stocks), works very well Also on some FI and FX markets Easy to implement (calibration easier than SVI)

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

e(xtended) SSVI

(joint work with Sebas Hendriks) Idea: allows for time (θ) dependent correlation ρ in SSVI.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

e(xtended) SSVI

(joint work with Sebas Hendriks) Idea: allows for time (θ) dependent correlation ρ in SSVI. Motivation: correlation in the calibration of a joint slice SSVI model:

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

e(xtended) SSVI

eSSVI slices are SSVI slices: same no-butterfly arbitrage conditions. Question: investigate calendar-spread arbitrage.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

e(xtended) SSVI

eSSVI slices are SSVI slices: same no-butterfly arbitrage conditions. Question: investigate calendar-spread arbitrage. Starting point: look at 2 SSVI slices with different correlations ρ1, ρ2.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

w1 = θ1 2 (1 + ρ1ϕ1k +

  • ϕ2

1k2 + 2ρ1ϕ1k + 1)

w2 = θ2 2 (1 + ρ2ϕ2k +

  • ϕ2

2k2 + 2ρ2ϕ2k + 1)

(2.4)

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

w1 = θ1 2 (1 + ρ1ϕ1k +

  • ϕ2

1k2 + 2ρ1ϕ1k + 1)

w2 = θ2 2 (1 + ρ2ϕ2k +

  • ϕ2

2k2 + 2ρ2ϕ2k + 1)

(2.4) [ Haute Couture on parametric quadratic polynomials here]

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

w1 = θ1 2 (1 + ρ1ϕ1k +

  • ϕ2

1k2 + 2ρ1ϕ1k + 1)

w2 = θ2 2 (1 + ρ2ϕ2k +

  • ϕ2

2k2 + 2ρ2ϕ2k + 1)

(2.4) [ Haute Couture on parametric quadratic polynomials here]

Proposition (Sufficient conditions for no crossing)

The 2 smiles don’t cross if θ2 ≥ θ1 and ϕ2 ≤ ϕ1 θ2ϕ2 θ1ϕ1 ≥ max 1 + ρ1 1 + ρ2 , 1 − ρ1 1 − ρ2

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

eSSVI: continuous time, ρ(θ).

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

eSSVI: continuous time, ρ(θ). Passing to the limits h → 0 for 2 smiles at t, t + h yields

Proposition

Let γ := 1 ϕ d(θϕ) dθ , δ := θd(ρ) dθ Then there is no calendar spread arbitrage in eSSVI iff ∂tθt ≥ 0 and −γ ≤ δ + ργ ≤ γ and either:

  • 1. γ ≤ 1
  • 2. −√2γ − 1 ≤ δ + ργ ≤ √2γ − 1

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

eSSVI: continuous time, ρ(θ). Passing to the limits h → 0 for 2 smiles at t, t + h yields

Proposition

Let γ := 1 ϕ d(θϕ) dθ , δ := θd(ρ) dθ Then there is no calendar spread arbitrage in eSSVI iff ∂tθt ≥ 0 and −γ ≤ δ + ργ ≤ γ and either:

  • 1. γ ≤ 1
  • 2. −√2γ − 1 ≤ δ + ργ ≤ √2γ − 1

When δ = 0, we re-find Gatheral-Jacquier condition from 2 (which implies 1 in this case).

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

eSSVI: continuous time, ρ(θ). Passing to the limits h → 0 for 2 smiles at t, t + h yields

Proposition

Let γ := 1 ϕ d(θϕ) dθ , δ := θd(ρ) dθ Then there is no calendar spread arbitrage in eSSVI iff ∂tθt ≥ 0 and −γ ≤ δ + ργ ≤ γ and either:

  • 1. γ ≤ 1
  • 2. −√2γ − 1 ≤ δ + ργ ≤ √2γ − 1

When δ = 0, we re-find Gatheral-Jacquier condition from 2 (which implies 1 in this case). Can be proven rigorously directly, investigating ∂θw.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Representation formula for ρ(θ)

If we restrict to the case where 0 ≤ γ ≤ 1, we can get all possible ρ satisfying −γ ≤ δ + ργ ≤ γ by solving the ODE δ + ργ = γu where u is any function with values in [−1, 1].

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Representation formula for ρ(θ)

If we restrict to the case where 0 ≤ γ ≤ 1, we can get all possible ρ satisfying −γ ≤ δ + ργ ≤ γ by solving the ODE δ + ργ = γu where u is any function with values in [−1, 1]. This gives:

Proposition

Assume 0 ≤ γ ≤ 1 Then there is no calendar spread arbitrage in eSSVI iff ρ(θ) = 1 θϕ(θ) θ u(τ)d(τϕ(τ)) (2.5) for some u → [−1, 1]

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

2nd ingredient: Chriss-Morokoff-Gatheral-Fukasawa formula

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

VIX reminder

For a continuous model: lim E[

  • log S(k+1)h

Skh

2

] = E[−2 log(ST S0 )] and one has always the replication formula for the log contract: E[−2 log(ST FT )] = 2 FT P(K, T) K 2 dK + 2 ∞

FT

C(K, T) K 2 dK where we assume that there is no interest rate. Here C(K, T) (resp. P(K, T)) is the price of a Call (resp. Put) with strike K and time to maturity T. FT is the Forward at maturity T VIX: synthetic index with a discrete version of this formula (and fixed 30 days time to maturity) Notation: VIX 2(T) = E[−2 log( ST

FT )]/T

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Chriss-Morokoff-Gatheral-Fukasawa formula

In Jim’s Practitioner book, the following formula is obtained: E[−2 log(ST FT )] =

  • σ2(g2(z))e− z2

2

√ 2π dz (3.6) (we drop the T dependence in the RHS) where g2 is the inverse function of the transformation k → d2(k, σ(k)) where d2(k, σ) = − k

σ − σ 2 .

Fukasawa (2010) proved that under no butterfly arbitrage conditions d2(k., σ(.)) is indeed invertible and proved rigorously 3.6.

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General shape of σ(g2)

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

General shape of σ(g2)

Lemma (Fukasawa)

The inequality 2g2(z) ≤ z2 holds for all z ∈ R. There exists a unique z∗ > 0 such that 2g2(z∗) = (z∗)2. Moreover, we have σ(g2(z)) = z +

  • z2 − 2g2(z) below z∗

and σ(g2(z)) = z −

  • z2 − 2g2(z) above z∗. In particular, σ(g2(z∗)) = z∗.

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σ(g2) in SSVI

SSVI: σ2(g2(z)) = θ 2(1 + ρϕg2 +

  • (ϕg2 + ρ)2 + ¯

ρ2) so θ(1 + ρϕg2 +

  • (ϕg2 + ρ)2 + ¯

ρ2) = 4(z2 − g2 ± z

  • z2 − 2g2)

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

σ(g2) in SSVI

SSVI: σ2(g2(z)) = θ 2(1 + ρϕg2 +

  • (ϕg2 + ρ)2 + ¯

ρ2) so θ(1 + ρϕg2 +

  • (ϕg2 + ρ)2 + ¯

ρ2) = 4(z2 − g2 ± z

  • z2 − 2g2)

[EASY COMPUTATIONS HERE]

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

σ(g2) in SSVI

SSVI: σ2(g2(z)) = θ 2(1 + ρϕg2 +

  • (ϕg2 + ρ)2 + ¯

ρ2) so θ(1 + ρϕg2 +

  • (ϕg2 + ρ)2 + ¯

ρ2) = 4(z2 − g2 ± z

  • z2 − 2g2)

[EASY COMPUTATIONS HERE]

Setting v2 = σ(g2(z)) we get the quadratic equation: θ2(1 − ρ2)ϕ2 (2z − v2)2 4 = 4[v 2

2 − θ(1 + ρϕv2(2z − v2)

2 )] (3.7)

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Close formula for σ(g2) and the VIX in (e)SSVI

Let u := θϕ(θ) and set: a = 1 + ρu 2 − ¯ ρ2u2 16 b = −ρu + ¯ ρ2u2 4

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Close formula for σ(g2) and the VIX in (e)SSVI

Let u := θϕ(θ) and set: a = 1 + ρu 2 − ¯ ρ2u2 16 b = −ρu + ¯ ρ2u2 4

Lemma

We have a > 0, and σ(g2(z)) = −bz+

√ u2z2+4aθ 2a

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Close formula for σ(g2) and the VIX in (e)SSVI

Let u := θϕ(θ) and set: a = 1 + ρu 2 − ¯ ρ2u2 16 b = −ρu + ¯ ρ2u2 4

Lemma

We have a > 0, and σ(g2(z)) = −bz+

√ u2z2+4aθ 2a

Proposition (VIX in (e)SSVI)

T VIX 2(T) = (b2 + u2) + 4aθ 4a2

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Close formula for σ(g2) and the VIX in (e)SSVI

Let u := θϕ(θ) and set: a = 1 + ρu 2 − ¯ ρ2u2 16 b = −ρu + ¯ ρ2u2 4

Lemma

We have a > 0, and σ(g2(z)) = −bz+

√ u2z2+4aθ 2a

Proposition (VIX in (e)SSVI)

T VIX 2(T) = (b2 + u2) + 4aθ 4a2

Proof.

σ(g2(z))2 = (b2+u2)z2+4aθ−2bz

√ u2z2+4aθ 4a2

, integrate in z wrt Gauss kernel.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Close formula for σ(g2) and the VIX in (e)SSVI

Let u := θϕ(θ) and set: a = 1 + ρu 2 − ¯ ρ2u2 16 b = −ρu + ¯ ρ2u2 4

Lemma

We have a > 0, and σ(g2(z)) = −bz+

√ u2z2+4aθ 2a

Proposition (VIX in (e)SSVI)

T VIX 2(T) = (b2 + u2) + 4aθ 4a2

Proof.

σ(g2(z))2 = (b2+u2)z2+4aθ−2bz

√ u2z2+4aθ 4a2

, integrate in z wrt Gauss kernel.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Some properties

In the sequel: V (θ) := T VIX 2(T).

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Some properties

In the sequel: V (θ) := T VIX 2(T). Assume ρ ≤ 0 and no calendar-spread arbitrage. Then:

  • 1. θ → V (θ) is non-decreasing.
  • 2. V (θ) ≥ θ

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Conclusion: (e)SSVI a la Bergomi

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

What for?

  • 1. e.g. directly get the VIX term structure from a SSVI calibration on Vanillas.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

What for?

  • 1. e.g. directly get the VIX term structure from a SSVI calibration on Vanillas.
  • 2. also: take V = T VIX 2(T) as parameter instead of θT (a la Bergomi).

Why a la Bergomi?

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

What for?

  • 1. e.g. directly get the VIX term structure from a SSVI calibration on Vanillas.
  • 2. also: take V = T VIX 2(T) as parameter instead of θT (a la Bergomi).

Why a la Bergomi? T VIX 2(T) = T

0 ξ0(t)dt where ξ0 is the intial Forward Variance curve.

Key input of Bergomi approach.

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From V to θ

We need to invert the θ → V formula.

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From V to θ

We need to invert the θ → V formula. To do so, we assume some ϕ(θ) is chosen, e.g. the popular ϕ(θ) = η/ √ θ (sqrt SSVI).

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From V to θ

We need to invert the θ → V formula. To do so, we assume some ϕ(θ) is chosen, e.g. the popular ϕ(θ) = η/ √ θ (sqrt SSVI). Same principle for ϕ(θ) = ηθ−λ with λ = 1/2.

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θ(V ) formulas

Proposition (ATM implied total variance, uncorrelated sqrt SSVI)

Assume ϕ(θ) =

η √ θ and ρ = 0. Then:

θ = 8(1 −

  • 1 + η2

2 + η4 8 (V + 1 2)) + η2(V + 2)

η2(1 + η2(V −4)

16

)

Proposition (ATM implied total variance, uncorrelated sqrt SSVI, small parameter expansion)

Assume ϕ(θ) =

η √ θ and ρ = 0. Then at first order in η2:

θ = V (1 − η2(V + 4) 16 )

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

θ(V ) formulas,ctd

Proposition (Short term ATM implied total variance, sqrt SSVI)

Assume ϕ(θ) =

η √ θ. Then for small θ:

θ = V ( (1+ρ2)

4

η2 + 1) [1+ρη √ V ( (3+ρ2)

8

η2 + 1

2)

( (1+ρ2)

4

η2 + 1)

3 2 −η2V

(3ρ2+1) 16

+ η2 3(ρ4+6ρ2+1)

64

( (1+ρ2)

4

η2 + 1)2 +o(V )]

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

SSVI a la Bergomi

SSVI a la Bergomi: V , ρ, η, λ → SSVI(θ(V )) with one of these formulas.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

SSVI a la Bergomi

SSVI a la Bergomi: V , ρ, η, λ → SSVI(θ(V )) with one of these formulas. Same parameters as Bergomi type models. Vol and correlation are disantangled.

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Hurst exponent from short term skew in Rough Bergomi models

Identification of the short term skew:

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Hurst exponent from short term skew in Rough Bergomi models

Identification of the short term skew: (SSVI) √ T∂kσBS(k = 0) ≈ ρ/2 √ θϕ(θ) ∝ ρT 1/2−λ

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Hurst exponent from short term skew in Rough Bergomi models

Identification of the short term skew: (SSVI) √ T∂kσBS(k = 0) ≈ ρ/2 √ θϕ(θ) ∝ ρT 1/2−λ (Rough Bergomi: Fukasawa, 2015) ∝ ρT H

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Hurst exponent from short term skew in Rough Bergomi models

Identification of the short term skew: (SSVI) √ T∂kσBS(k = 0) ≈ ρ/2 √ θϕ(θ) ∝ ρT 1/2−λ (Rough Bergomi: Fukasawa, 2015) ∝ ρT H which yields λ = 1/2 − H . Temptative rough (e)SSVI

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Hurst exponent from short term skew in Rough Bergomi models

Identification of the short term skew: (SSVI) √ T∂kσBS(k = 0) ≈ ρ/2 √ θϕ(θ) ∝ ρT 1/2−λ (Rough Bergomi: Fukasawa, 2015) ∝ ρT H which yields λ = 1/2 − H . Temptative rough (e)SSVI

Thank you for your attention !

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Outline Reminder on SSVI Chriss-Morokoff-Gatheral-Fukasawa formula SSVI a la Bergomi

Thanks and joyeux anniversaire Jim !!!

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