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 s > 0 and scalar Brownian motion B, dSt /St = sdBt . Dynamics under pricing measure, zero rates, for option pricing. Call prices + C(S0, K , T ) := E(ST K ) − + s2 = E S exp sB T K 0 T − 2 − 2 =: CBS S0, K ; s T Time-inhomogenous BS: s st determistic Z T 2 C(S0, K , T ) = CBS S0, K ; s dt 0 Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 2 / 43 Examples: Dupire local volatility, 1-factor model, ¥ parameters st (w) = sloc (St (w), t) Stochastic volatility Make volatility q s st (w) vt (w) stochastic, adapted ≡ so that q dSt /St = st (w)dBt = vt (w)dBt defines a martingale. Peter K. Friz Rouch vol and regularity structures Barcelona, 9 June 2017 3 / 43 Stochastic volatility Make volatility q s st (w) vt (w) stochastic, adapted ≡ so that q dSt /St = st (w)dBt = vt (w)dBt defines a martingale. Examples: Dupire local volatility, 1-factor model, ¥ parameters st (w) = sloc (St (w), 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 = pvdBt pv rdWt + r¯dW¯ t ≡ Z t Z t vt = v0 + (a bv)dt + cpvdW 0 − 0 with correlation parameter r and r2 + r¯ 2 = 1 SABR, as above but 4 parameters b dSt /S = sdBt s rdWt + r¯dW¯ t t ≡ 1 2 st = s exp(aWt a t) 0 − 2 with correlation parameter r and r2 + r¯ 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 = p2H t s − dWs 0 0 j − j Popular “simple” form (also in [GJR14x], [BFG16]) st := 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 = pvdBt pv rdWt + r¯dW¯ t ≡ Z t a bv Z t cpv 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 st = f (Wˆ ). In particular, we refine Forde–Zhang rough large deviations [FZ17] while regaining analytical tracktability. To state the result, set s0 := f (0) (spotvol) and also s0 := f 0(0). Theorem (Rough vol skew, “moderate” regime) 1 H+b Let 0 < b < H < 1/2 and fix y, z > 0 with y = z. Set yˆ := yt 2 − and similarly for z. Then, as t 0, 6 ! simpl(yˆ , t) simpl(zˆ, t) r s0 H 1 − t − 2 . yˆ zˆ ∼ (H + 1/2)(H + 3/2) s − 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 = pvdBt pv rdWt + r¯dW¯ t ≡ vt adapted to (W ) allows to price calls as averages of Black–Scholes calls Z T r2 Z T r¯ 2 Z T E CBS S0 exp r pvdW vdt , K , vdt 0 − 2 0 2 0 (Romano–Touzi proof: condition payoff with respect to W .) Even for “simple” rough volatility: pv = 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(w)t + ∑ ∑ Znk (w)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 − # n Truncation at level n: natural approximation W at scale # = 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 −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 #)dW # 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 + x x j j with space-time white noise x = x(x, t; w). Fact: Ito-solutionˆ X (“Cole-Hopf”) but with mollified noise x# = get: u# +¥ ) ! 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 + x ∗ j j with space-time convolution ? and heat-kernel 1 x2 H(t, x) = exp 1 t>0 p4pt − 4t f g Naiv Picard iteration 2 u = H ? x + H ? ((H0 ? x) ) + ... does not work because 2 # 2 @(H0 ? x) , @ lim(H0 ? x ) ... # 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 (C#) such that # 2 2 lim(H0 ? x ) C# (new object) =: (H0 ? x) . 9 # 0 − ! ! Rough path inspired idea (Hairer): Accept 2 H ? x, (H0 ? x) (and a few more) as enhanced noise (”model”) upon which solution depends in pathwise robust fashion. This unlocks the seemingly fixed relation 2 H ? x x (H0 ? x) . ! ! (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 C# such that u˜ # (Ito-solution)ˆ ! in terms of the renormalized approximating equation # 2 # # 2 # ¶t u˜ = ¶ u˜ + ¶x u˜ C + x . x j j − # 4 (Same approach works e.g. for F3 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 C# such that u˜ # (Ito-solution)ˆ ! in terms of the renormalized approximating equation # 2 # # 2 # ¶t u˜ = ¶ u˜ + ¶x u˜ C + x . x j j − # 4 (Same approach works e.g. for F3 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.