A Regularity Structures for Rough Volatility

A regularity structures for rough volatility

Peter K. Friz

Barcelona, 9 June 2017

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 1 / 43 Black–Scholes (BS)

Given σ > 0 and scalar Brownian motion B,

dSt /St = σdBt .

Dynamics under pricing measure, zero rates, for option pricing. Call prices

+ C(S0, K , T ) := E(ST K ) − + σ2 = E S exp σB T K 0 T − 2 − 2 =: CBS S0, K ; σ T

Time-inhomogenous BS: σ σt determistic Z T 2 C(S0, K , T ) = CBS S0, K ; σ dt 0

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 2 / 43 Examples:

Dupire local volatility, 1-factor model, ∞ parameters

σt (ω) = σloc (St (ω), t)

Stochastic volatility

Make volatility q σ σt (ω) vt (ω) stochastic, adapted ≡ so that q dSt /St = σt (ω)dBt = vt (ω)dBt deﬁnes a martingale.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 3 / 43 Stochastic volatility

Make volatility q σ σt (ω) vt (ω) stochastic, adapted ≡ so that q dSt /St = σt (ω)dBt = vt (ω)dBt deﬁnes a martingale. Examples:

Dupire local volatility, 1-factor model, ∞ parameters

σt (ω) = σloc (St (ω), t)

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 3 / 43 More (classical) stochastic volatility

More examples: Heston, 2-factors: W , W¯ indep. BMs, 5 parameters dSt /St = √vdBt √v ρdWt + ρ¯dW¯ t ≡ Z t Z t vt = v0 + (a bv)dt + c√vdW 0 − 0

with correlation parameter ρ and ρ2 + ρ¯ 2 = 1 SABR, as above but 4 parameters

β dSt /S = σdBt σ ρdWt + ρ¯dW¯ t t ≡ 1 2 σt = σ exp(αWt α t) 0 − 2 with correlation parameter ρ and ρ2 + ρ¯ 2 = 1

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 4 / 43 Pricing under classical stochvol (selection)

Numerics: PDE pricing Monte Carlo pricing Strong / weak rates, Multi level MC (Kusuoka-Lyons-Victoir) Cubature methods Ninomia-Victoir splitting scheme

Analytics: Malliavin calculus, asymptotic expansions Large deviations, e.g. Freidlin-Wentzell Famously used to understand smile: SABR formula etc

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 5 / 43 Rough fractional stochastic volatility

Some history [ALV07], [Fuk11,15] .... Now much interest in

H vt W Wˆ t ≈ t ≡ a fractional Browniation motion (fBm) in the rough regime of Hurst paramter H < 1/2: worse than Brownian regularity Recall: Volterra (a.k.a. Riemann-Liouville ) fBm

Z t Z t H H 1/2 Wˆ t = K (t, s)dWs = √2H t s − dWs 0 0 | − | Popular “simple” form (also in [GJR14x], [BFG16])

σt := f (Wˆ t , t)

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 6 / 43 EFR volatility dynamics

In [EFR16], El Euch, Fukasawa and Rosenbaum show that stylized facts of modern market microstructure naturally give rise to fractional dynamics and leverage effects. Speciﬁcally, they construct a sequence of Hawkes processes suitably rescaled in time and space that converges in law to a rough volatility model of rough Heston form

dSt /St = √vdBt √v ρdWt + ρ¯dW¯ t ≡ Z t a bv Z t c√v vt = v0 + − dt + dW 0 (t s)1/2 H 0 (t s)1/2 H − − − − (As earlier, W , W¯ independent Brownians.)

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 7 / 43 Pricing under rough volatility

Numerics: PDE pricing NO! Monte Carlo pricing X [BFG16], [BLP15x], ... Strong / weak rates, Multilevel MC: Partial answers [NS16x], ... Cubature, Ninomia-Victoir? Even classical Wong-Zakai?

Analytics: Malliavin calculus, asymptotic expansions X [ALV07], [Fuk11,Fuk15], ... Large deviations: Partial answers [FZ17], ...

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 8 / 43 Moderate and large deviations

In recent joint work with S. Gerhold and A. Pinter we introduced [FGP17] moderate deviation analysis in option pricing. In [BFGHS17x] we extend this to simple rough volatility of the form σt = f (Wˆ ). In particular, we reﬁne Forde–Zhang rough large deviations [FZ17] while regaining analytical tracktability. To state the result, set σ0 := f (0) (spotvol) and also σ0 := f 0(0). Theorem (Rough vol skew, “moderate” regime)

1 H+β Let 0 < β < H < 1/2 and ﬁx y, z > 0 with y = z. Set yˆ := yt 2 − and similarly for z. Then, as t 0, 6 →

σimpl(yˆ , t) σimpl(zˆ, t) ρ σ0 H 1 − t − 2 . yˆ zˆ ∼ (H + 1/2)(H + 3/2) σ − 0 Extends CLT regime of [ALV07, Fuk11].

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 9 / 43 A SV call price formula

Any SV model of form dSt /St = √vdBt √v ρdWt + ρ¯dW¯ t ≡ vt adapted to (W )

allows to price calls as averages of Black–Scholes calls

Z T ρ2 Z T ρ¯ 2 Z T E CBS S0 exp ρ √vdW vdt , K , vdt 0 − 2 0 2 0 (Romano–Touzi proof: condition payoff with respect to W .) Even for “simple” rough volatility: √v = f (Wˆ )

Z T Question: efﬁcient simulation f (Wˆ )dW ? 0 R T Also need to simulate 0 vdt, but this one harmless.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 10 / 43 Intermezzo on approximating Brownian motion

In terms of (Znk ), i.i.d. standard Gaussians, can expand Bm

∞ 2n 1 − Wt = Z0(ω)t + ∑ ∑ Znk (ω)hnk (t), n=0 k=0 where hnk is the integrated (here: Haar) wavelet basis Hnk .

2 n/2 1 n/2 hnk(t) ✁❆ − − Hnk(t) 2 ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Ink ❆ ✲ Ink ✲ k k+1 t k k+1 t 2n 2n 2n 2n

(a) (b) 2n/2 −

ε n Truncation at level n: natural approximation W at scale ε = 2− .

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 11 / 43 Brownian approximations cont’d

Figure: Based on (a) Haar (b) Alpert-Rokhlin wavelets

1 n = 3 0.8 n = 10

0.6

0.4 (t)

B 0.2

−0.2 (a) −0.4 0 0.2 0.4 0.6 0.8 1 t 0.5 n = 3 n = 10 0 (t)

−0.5 B

−1

(b) −1.5 0 0.2 0.4 0.6 0.8 1 Peter K. Friz Rouch vol and regularityt structures Barcelona, 9 June 2017 12 / 43 Approximation theory: simple rough vol

We encountered the (Ito)ˆ integral

Z T f (Wˆ )dW 0 where Wˆ is rough, namely a fBm with, say, H 0.05 << 1/2 ≈ Z t H Wˆ t = K (t, s)dWs 0

Badly behaved even under classical Wong-Zakai approximations! Problem: ∞ Ito-Stratonovichˆ correction whenever H < 1/2, Z Z f (Wˆ ε)dW ε f (Wˆ ) dW → ◦ Z 1 = f (Wˆ )dW + [f (Wˆ ), W ] = +∞ 2 = Itoˆ but Stratonovich @ .... ⇒ X Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 13 / 43 Lessons from SPDE theory

Universal model for ﬂuctuations of interface growth: the Kardar–Parisi–Zhang (KPZ) equation

2 2 ∂t u = ∂ u + ∂x u + ξ x | | with space-time white noise ξ = ξ(x, t; ω). Fact: Ito-solutionˆ X (“Cole-Hopf”) but

with molliﬁed noise ξε = get: uε +∞ ⇒ → hence @ Stratonovich solution ...

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 14 / 43 What is going on?

Formally at least, with u 0 for simplicity, 0 ≡ 2 u = H ∂x u + ξ ∗ | | with space-time convolution ? and heat-kernel

1 x2 H(t, x) = exp 1 t>0 √4πt − 4t { } Naiv Picard iteration

2 u = H ? ξ + H ? ((H0 ? ξ) ) + ...

does not work because

2 ε 2 @(H0 ? ξ) , @ lim(H0 ? ξ ) ... ε 0 →

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 15 / 43 KPZ analysis cont’d

However, there exists a diverging sequence (Cε) such that

ε 2 2 lim(H0 ? ξ ) Cε (new object) =: (H0 ? ξ) . ∃ ε 0 − → → Rough path inspired idea (Hairer): Accept

2 H ? ξ, (H0 ? ξ) (and a few more)

as enhanced noise (”model”) upon which solution depends in pathwise robust fashion. This unlocks the seemingly ﬁxed relation

2 H ? ξ ξ (H0 ? ξ) . → → (NB: last term does not make analytical sense and in fact diverges upon approximation)

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 16 / 43 This can be traced back to the Milstein-scheme for SDEs (and then “rough paths”). Take

dY = f (Y )dW ,

with Y0 = 0 for simplicity, and consider Z Y f (0)W + ff 0(0) WdW ≈

The Hairer result

Theorem [Hairer] There exist diverging constants Cε such that u˜ ε (Ito-solution)ˆ → in terms of the renormalized approximating equation

ε 2 ε ε 2 ε ∂t u˜ = ∂ u˜ + ∂x u˜ C + ξ . x | | − ε 4 (Same approach works e.g. for Φ3 from QFT.)

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 17 / 43 The Hairer result

Theorem [Hairer] There exist diverging constants Cε such that u˜ ε (Ito-solution)ˆ → in terms of the renormalized approximating equation

ε 2 ε ε 2 ε ∂t u˜ = ∂ u˜ + ∂x u˜ C + ξ . x | | − ε 4 (Same approach works e.g. for Φ3 from QFT.) This can be traced back to the Milstein-scheme for SDEs (and then “rough paths”). Take

dY = f (Y )dW ,

with Y0 = 0 for simplicity, and consider Z Y f (0)W + ff 0(0) WdW ≈

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 17 / 43 SDEs vs. SPDEs

SDE case cont’d: unlock the seemingly ﬁxed relation Z W W˙ W W˙ , → → for there is a choice to be made (e.g. Itoˆ / Stratonovich) ! Take this one step further (rough path): accept R WdW W as new object, ≡

SDE theory = analysis based on (W , W)

Similarly

SPDE theory a` la Hairer = analysis based on (renormalized) enhanced noise.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 18 / 43 Inside Hairer’s theory

Motivation: The Taylor-expansion (at x) of a smooth function,

1 2 f (y) = f (x) + f 0(x)(y x) + f 00(x)(y x) + ... , − 2 − can be written as abstract polynomial (“jet”) at x,

F (x) := f (x) 1 + g(x)X + h(x)X 2 + ... (1)

[necessarily: g = f 0, h = f 00/2, ...] (2)

If we “realize” these abstract symbols again as honest monomials

k k Πx : X (. x) 7→ −

and extend Πx linearly, then we recover the full Taylor expansion:

1 2 Πx [F (x)](.) = f (x) + g(x)(. x) + h(x)(. x) + ... − 2 −

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 19 / 43 Inside Hairer’s theory cont’d

Hairer looks for solution of this form: at every space-time point a jet is attached, which in case of KPZ turns out to be of the form U(x, s) = u(x, s) 1 + + + v(x, s) X + 2 + v(x, s) . As before, every symbol is given concrete meaning by “realizing” it as honest function (or Schwartz distribution). In particular, ( H ? ξe, molliﬁed noise; or (3) 7→ H ? ξ noise and then, more interestingly,  H ? (H ? ξe)2, canonically enhanced molliﬁed noise; or  0 e 2 H ? [(H0 ? ξ ) Ce], renormalized or 7→  2 − ∼ H ? (H0 ? ξ) , renormalized enhanced noise (REM) (4) etc ... This realization map is called “model”.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 20 / 43 Inside Hairer’s theory cont’d

Hence, the model captures (in a pathwise fashion) your choice of noise. Writing Πx,s for the realization map for REM have e.g.

2 Πx,s[U(x, s)](.) = u(x, s) + H ? ξ ( ) + H ? (H0 ? ξ) ( + ... | ∗ | ∗) where ( ) indicates suitable centering at (x, s). Mind that U takes values in∗ a (ﬁnite) linear space spanned by (sufﬁciently many) symbols,

U(x, s) ..., 1, , , X, , , ... =: T ∈ h i Together with some regularity, (x, s) U(x, s) is an example of a modelled distribution. Also need to keep7→ track of regularity, better: degree, of each symbol (e.g. X 2 = 2, = 1/2 κ), collected in index set. Last not least, in order| | to compare| | jets− at different points (think (X δ1)3 = ...), use a group of linear maps on , called structure− group. Last not least, the reconstruction mapT uniquely maps modelled distributions to function / Schwartz distributions.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 21 / 43 Back to rough volatility

We can apply these ideas in the context of rough volatility. Recall the basic problem Z Z f (Wˆ ε)dW ε f (Wˆ ) dW → ◦ Z 1 = f (Wˆ )dW + [f (Wˆ ), W ] = +∞ 2 The (wavelet) expansion of white noise

N 2n 1 ˙ ε − W (t) = Z0(ω) + ∑ ∑ Znk (ω)Hnk (t) n=0 k=0 naturally induced fBm approximation, namely

Z t N 2n 1 ˆ ε ε ˆ − ˆ W (t) = K (t, s) dW (s) = Z01(t) + ∑ ∑ Znk (ω)Hnk (t) 0 n=0 k=0 where fˆ(t) = √2H R t t s H 1/2f (s)ds. 0 | − | − Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 22 / 43 Back to rough volatility

On scale ε 2 N , set ≡ − N ε √2H 2 N N 1+H 1/2 C (t)= t t2 2− − , (5) 1 + H 1/2| − b c | −

which, at least when H > 1/4, can be replaced (below) by diverging local average

√2H H 1/2 ε − =: C . (H + 1/2)(H + 3/2) ε

Theorem (BFGMS17) For any H 1/2 and nice function f , with strong rate H, ≤ Z T Z T Z T ˆ ε ε ε ˆ ε ˆ lim f (W )dW C (t)f 0(Wt )dt = f (W )dW ε 0 0 − 0 0 →

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 23 / 43 Rough idea of proof

Robust interpretation of Itoˆ solution (here: integral) ? = Identify correct enhancement of noise ⇒ Z t k k,1 ˆ ˆ Ws,t := Wr Ws dWr k = 0, 1, ..., ? s − Formally, for any partition P of [0, T ],

Z T Z t ˆ ˆ f Wr dWr = ∑ f Wr dWr 0 [s,t] P s ∈ ( ) 1 = f (k) Wˆ Wk,1 + ... ∑ ∑ k s s,t [s,t] P 0 krenormalization at this level.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 24 / 43 Rough path type considerations

Degree of regularity of building blocks? We can see a.s.

k,1 kH+1/2 ε Ws,t . t s − | − |

Terms with exponent > 1 may be dropped since Wk,1 = o(t s) s,t − = keep k < K := min j : jH + 1/2 > 1 ⇒ { } e.g. when H = 1/2, K = 2 and K 1/(2H) as H 0. ∼ → Expect (from rough paths theory) ( ) Z T ˆ 1 (k) ˆ k,1 f Wr dWr = lim f Ws Ws,t mesh(P) 0 ∑ ∑ k 0 [s,t] P 0 k

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 25 / 43 A regularity structure for rough volatility

Although previous heuristics are (hopefully) convincing, a direct proof would amount to (re)invent rough path in a non-geometric setting, with renormalization. Fortunately, Hairer’s theory already provides a framework to all this! A regularity structures is a triplet (T , A, G) with T spanned by symbols, a set of degrees A and a structure group G. In case of rough volatility,

D K 1 K 1E T = W˙ , W˙ Wˆ , ... , W˙ Wˆ − , 1, Wˆ ... , Wˆ − 1 1 1 A = , H , ... , (K 1)H , 0, H, ... , (K 1)H −2 − 2 − − 2 − Structure group ?

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 26 / 43 Structure group explicit

Take K = 3 for better visibility, D E T = W˙ , W˙ Wˆ , W˙ Wˆ 2, 1, Wˆ , Wˆ 2

Then G = Γh : h (R, +) with Γh Lin(T , T ) given by block-matrix{ ∈ } ∈  1 h h2   0 1 2h 0     0 0 1     1 h h2     0 0 1 2h  0 0 1

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 27 / 43 The Itoˆ enhanced “model” for rough vol

Unsurprisingly,

k k ΠsWˆ (r) := Wˆ r Wˆ s − k ˆ k ˙ ˆ ˆ ˙ d k,1 ΠsW W := W Ws W := Ws,t (in 0) · − dt D and also Γs,t = Γh with h = Wˆ s,t Deﬁnes a model in the sense of Hairer, call it ΠIto. Gaussian (since Wk,1 ﬁrst k Wiener-Itoˆ chaos’) s,t ∈

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 28 / 43 Reconstructing the integral

A modelled distribution of regularity γ is deﬁned by 1 (k) ˆ ˆ k ˙ t ∑ f Wt W W 7→ 0 k 1 , { } γ = KH 1/2 > 0 − The (unique) reconstruction is precisely the Schwartz derivative of Z f Wˆ dW ,

which in turn is recover by testing again indicator functions Best of all, continuous dependence of integrals as function of the model

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 29 / 43 Rough vol: Approximation and renormalization ...

Canonical model for ε-molliﬁed noise (divergent!) k Πε Wˆ k (r) := Wˆ ε Wˆ ε s r − s k ε ˆ k ˙ ˆ ε ˆ ε ˙ ε ΠsW W := W Ws W · − Renormalized model: deﬁne k Πˆ ε Wˆ k (r) := Wˆ ε Wˆ ε s r − s k k 1 ε ˆ k ˙ ˆ ε ˆ ε ˙ ε ˆ ε ˆ ε − Πˆ sW W := W Ws W Cεk W Ws · − − · −

Theorem: There exists a choice of Cε (necessarily divergent) such that Πˆ ε ΠIto as ε 0 → ↓ in probablity and model distance. (This implies the rough vol approximation result stated several slides ago.)

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 30 / 43 True, (so far) we mostly talked about a scalar Itoˆ integral. Of course, Euler / left-point approximation also works: with mesh P to zero, in every good sense, Z ˆ ˆ ∑ f (Ws)Ws,t f (W )dW [s,t] P → ∈ Attention: Euler Simulation of W not good for Wˆ = R (singular)dW , but can simulate (W , Wˆ ) directly via known covariance structure. So what did we achieve? Itoˆ integral also for illustration. Below a more complex situation. Good approximation theory for rough vol evolution. Rates! Flexible wavelet approximations possible. Often highly effective in combination with QMC (Sobol); Alpert-Rokhlin wavelets expected to go well with fBm. (Numerics in progress.) Robustness, also as analytical tool!

So what?

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 31 / 43 Itoˆ integral also for illustration. Below a more complex situation. Good approximation theory for rough vol evolution. Rates! Flexible wavelet approximations possible. Often highly effective in combination with QMC (Sobol); Alpert-Rokhlin wavelets expected to go well with fBm. (Numerics in progress.) Robustness, also as analytical tool!

So what?

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 31 / 43 Good approximation theory for rough vol evolution. Rates! Flexible wavelet approximations possible. Often highly effective in combination with QMC (Sobol); Alpert-Rokhlin wavelets expected to go well with fBm. (Numerics in progress.) Robustness, also as analytical tool!

So what?

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 31 / 43 Flexible wavelet approximations possible. Often highly effective in combination with QMC (Sobol); Alpert-Rokhlin wavelets expected to go well with fBm. (Numerics in progress.) Robustness, also as analytical tool!

So what?

True, (so far) we mostly talked about a scalar Itoˆ integral. Of course, Euler / left-point approximation also works: with mesh P to zero, in every good sense, Z ˆ ˆ ∑ f (Ws)Ws,t f (W )dW [s,t] P → ∈ Attention: Euler Simulation of W not good for Wˆ = R (singular)dW , but can simulate (W , Wˆ ) directly via known covariance structure. So what did we achieve? Itoˆ integral also for illustration. Below a more complex situation. Good approximation theory for rough vol evolution. Rates!

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 31 / 43 Robustness, also as analytical tool!

So what?

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 31 / 43 So what?

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 31 / 43 Large deviation principle (reminder)

A family of random variables X δ : δ > 0 satisﬁes a LDP iff

I(x) P X δ x exp . ≈ ∼ − δ2

Formal deﬁnition: with rate function I 0 and speed δ2, have ≥ inf I(x) + o(1) δ2 log P X δ A inf I(x) + o(1). x A¯ ≤ − ∈ ≤ x A◦ ∈ ∈ Contraction principle: basic fact of large deviation theory - stability under continuous maps. LDP for Y δ = ΦX δ with rate function

J(y) = inf I(x) : Φ(x) = y . { } Works well with rough paths / regularity structures!

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 32 / 43 Rough vol large deviations

Theorem For nice f , a LDP for

Z 1 δ ˆ X1 = f (δWt )δdWt 0

holds with speed δ2 and rate function

Z 1 Z t 1 2 H 1/2 J(x) = inf h L2 : x = f t s − h(s)ds h(t)dt 2k k 0 0 | − |

H 1/2 1 2H By scaling, this gives also short time LDP for t − Xt with speed t and same rate function. “Simple” rough vol large deviations [FZ17] as a consequence ...

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 33 / 43 EFR type rough vol dynamics

Until now we considered “simple” rough vol of form σt (ω) = f (Wˆ t ). Following Rosenbaum and coworkers, we consider

Z t Z t 2 g(vs) h(vs) σt vt = v0 + dWs + ds ≡ 0 t s 1/2 H 0 t s 1/2 H | − | − | − | − for general (but nice) coefﬁcient functions g, h. This is a Volterra stochastic differential equation / usual SDE theory. ∈ Theorem (BFGMS17) For any H (0, 1/2], this is a subcritical equation and has a unique Itoˆ solution. Naiv∈ approximations based on W ε diverge; but this is ﬁxed by renormalization: Z t ε Z t ε ε ε g(v˜s) ε (g C (.)h h0)(v˜s) v˜t = v0 + dWs + − ds 0 t s 1/2 H 0 t s 1/2 H | − | − | − | −

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 34 / 43 EFR type rough vol dynamics cont’d

Solution theory a` la Hairer identiﬁes limiting (Ito)ˆ solution as robust image of the enhanced noise. At least when H > 1/4 can replace renormalization function C ε(.) by a (diverging) constant Cε, so that

Z t ε Z t ε ε g(v˜s) ε (g Cεh h0)(v˜s) v˜t = v0 + dWs + − ds 0 t s 1/2 H 0 t s 1/2 H | − | − | − | − Immediate large deviations! But: rate function not explicit, more work along [BFGHS17] needed Everything works, though more involved, for H 1/4. ≤

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 35 / 43 All this is joint work with

Christian Bayer (WIAS Berlin) Paul Gassiat (Paris Dauphine) Jorg¨ Martin (HU Berlin) Benjamin Stemper (TU and WIAS Berlin)

Financial support: ERC GoG 683164, ArXiv-preprint to appear (soon) ...

Thank you for your attention!

References:

[ALV07] E. Alos, J. Leon, J. Vives. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Fin. Stoch. 2007. [BFG16] C. Bayer, P. Friz, J. Gatheral. Pricing under rough volatility. Quant. Fin. 2016. Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 36 / 43 [BFGHS17x] C. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper. Short-time near-the-money skew under rough vol. arXiv 2017. [BLP15x] M. Bennedsen, A. Lunde, M. Pakkanen. Hybrid scheme for Brownian semistationary processes. arXiv 2015. [FZ17] M. Forde, H. Zhang, Asymptotics for rough stochastic volatility models, SIAM J. Finan. Math. 2017. [FGP17] P. Friz, S. Gerhold, A. Pinter: Option Pricing in the Moderate Deviations Regime; Math. Fin., to appear [FH14] P. Friz, M. Hairer: A Course on Rough Paths. With an Introduction to Regularity Structures, Springer 2014 [Fuk11] M. Fukasawa. Asymptotic analysis for stochastic volatility: martingale expansion. Fin. Stoch. 2011. [Fuk17] Masaaki Fukasawa. Short-time at-the-money skew and rough fractional volatility. Quant. Fin. 2017. [GJR15x] Volatility is rough. Jim Gatheral, Thibault Jaisson, Mathieu Rosenbaum. arXiv 2015. [NS16x] A. Neuenkirch, T. Shalaiko; The Order Barrier for Strong Approximation of Rough Volatility Models. arXiv 2016.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 37 / 43 Appendix on regularity structures 1

Deﬁnition

A regularity structure T = (A, T , G) consists of the following elements: An index set A R such that 0 A, A is bounded from below, and A is locally ﬁnite.⊂ ∈ L A model space T , which is a graded vector space T = α A Tα, ∈ with each Tα a Banach space ; elements in Tα are said to have homogeneity (or degree) α. Furthermore T = 1 = R. Given 0 h i ∼ τ T , we will write τ for the norm of its component in T . ∈ k kα α A structure group G of (continuous) linear operators acting on T such that, for every Γ G, every α A, and every τ T , one has ∈ ∈ α ∈ α def M Γτα τα T<α = Tβ . (6) − ∈ β<α

Furthermore, Γ1 = 1 for every Γ G. ∈ Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 38 / 43 Appendix cont’d: rough path structure

Deﬁnition

Let α (1/3, 1/2]. The regularity structure for α-Holder¨ rough paths (over R∈e) is given by The set of possible homogeneities is given by A = α 1, 2α 1, 0, α . { − − } The model space T is given by e+e2+1+e T = Tα 1 T2α 1 T0 Tα = R with − ⊕ − ⊕ ⊕ ∼ 1 e T0 = 1 , Tα = W , ... , W , h i 1 e h ij i Tα 1 = W˙ , ... , W˙ , T2α 1 = W˙ : 1 i, j e . − h i − h ≤ ≤ i The group G Re, + acts on T via ∼ Γ 1 = 1 , Γ W i = W i + hi 1 , h h (7) i i ij ij i j ΓhW˙ = W˙ , ΓhW˙ = W˙ + h W˙ .

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 39 / 43 Appendix cont’d: models

Given a test function φ on Rd , we write φλ as a shorthand for x ≡ λ d 1 φ (y) = λ− φ λ− (y x) . x − Given an integer r > 0, we also denote by r the set of all functions d r B φ : R R such that φ with φ r 1 that are furthermore b b → ∈ C k kC ≤ d supported in the unit ball around the origin. We also write 0(R ) for the space of Schwartz distributions on Rd . D Deﬁnition

Given a regularity structure T and an integer d 1, a model d ≥ M = (Π, Γ) for T on R consists of maps

d d d d Π : R T , 0(R ) Γ : R R G → L D × → x Πx (x, y) Γxy 7→ 7→

such that Γxy Γyz = Γxz and Πx Γxy = Πy . We then say that Πx realizes an element of T as a Schwartz distribution. Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 40 / 43 Appendix cont’d: models

Deﬁnition (model, cont’d) Furthermore, write r for the smallest integer such that r > min A 0. d | | ≥ We then impose that for every compact set K R and every γ > 0, ⊂ there exists a constant C = C(K, γ) such that the bounds

λ α α β Πx τ (φ ) Cλ τ , Γxy τ C x y − τ , (8) x ≤ k kα k kβ ≤ | − | k kα

hold uniformly over φ r , (x, y) K, λ (0, 1], τ T with α γ ∈ B ∈ ∈ ∈ α ≤ and β < α.

Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 41 / 43 Appendix on regularity structures: modelled distributions

Deﬁnition (modelled distribution) Given a regularity structure T equipped with a model M = (Π, Γ) over d γ γ R , the space DM = DM(T ) is given by the set of functions d f : R T<γ such that, for every compact set K and every α < γ, there exists→ a constant C with

γ α f (x) Γxy f (y) C x y − (9) k − kα ≤ | − | uniformly over x, y K. Such functions f are called modelled ∈ distributions. For ﬁxed K, a semi-norm f M,γ;K is deﬁned as the k k γ smallest constant C in the bound (9). The space DM endowed with this family of seminorms is then a Frechet´ space.

Distance between models: the smallest constant C in the bound γ α f (x) Γxy f (y) f¯(x) + Γ¯ xy f¯(y) C x y − . k − − kα ≤ | − | Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 42 / 43 Appendix on regularity structures: reconstruction

The most fundamental result in the theory of regularity structures then states that given f D γ with γ > 0, there exists a unique Schwartz ∈d d distribution f on R such that, for every x R , f “looks like Πx f (x) near x”. MoreR precisely, one has ∈ R Theorem (Reconstruction)

Let M = (Π, Γ) be a model for a regularity structure T on Rd . Assume f D γ with γ > 0. Then, there exists a unique linear map ∈ M γ d = M : D 0(R ) R R M → D such that λ γ f Πx f (x) (φ ) λ , (10) R − x . uniformly over φ r and λ as before, and locally uniformly in x. ∈ B Without the positivity assumption on γ, everything remains valid but uniqueness of . R Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 43 / 43