Rough volatility models

Transcription

1 Mohrenstrasse Berlin Germany Tel October 18, 2018 Weierstrass Institute for Applied Analysis and Stochastics Rough volatility models Christian Bayer EMEA Quant Meeting 2018

2 Outline 1 Implied volatility modeling 2 The rough Bergomi model 3 Case studies 4 Further challenges and developments Rough volatility models October 18, 2018 Page 2 (28)

3 SPX implied volatility surface Rough volatility models October 18, 2018 Page 3 (28)

4 SPX ATM volatility skew Rough volatility models October 18, 2018 Page 4 (28)

5 Questions and goals ds t = v t S t dz t, dv t =... We look for time-homogeneous models. Term structure of ATM volatility skew (k = log(k/s t )) ψ(τ) = k σ BS (k, τ) 1/τ α, α [0.3, 0.5] k=0 Conventional stochastic volatility models produce ATM skews which are constant for τ 1 and of order 1/τ for τ 1. Hence, conventional stochastic volatility models cannot fit the full volatility surface. Do we need jumps? Rough volatility models October 18, 2018 Page 5 (28)

6 Some notations and assumptions Given a traded asset S t satisfying ds t = v t S t dz t Interest rate r = 0; model (and expectations) formulated under Q In this talk, S corresponds to the S & P 500 index (SPX). Realized variance w t,t = T t Log-strip formula: ( S t P(K) E t w t,t = 2 0 K 2 dk + S t v s ds, forward variance ξ t (u) = E t [v u ] ) C(K) K 2 dk Rough volatility models October 18, 2018 Page 6 (28)

7 Intraday realized variance Rough volatility models October 18, 2018 Page 7 (28)

8 Outline 1 Implied volatility modeling 2 The rough Bergomi model 3 Case studies 4 Further challenges and developments Rough volatility models October 18, 2018 Page 8 (28)

9 Bergomi model Recall ξ t (u) = E t v u ds t = ξ t (t)s t dz t, n t ξ t (u) = ξ 0 (u)e η i e κi(u s) dws i i=1 0 E(X) exp(x 1 2 E[ X 2 ]) for Gaussian r.v. X Market model In practice, n = 2 needed for good fit, contains seven parameters n η i ψ(τ) (1 1 ) e κ iτ κ i τ κ i τ i=1 Tempting to replace the exponential kernel by a power law kernel! Rough volatility models October 18, 2018 Page 9 (28)

10 Bergomi model Recall ξ t (u) = E t v u ds t = ξ t (t)s t dz t, n t ξ t (u) = ξ 0 (u)e η i e κi(u s) dws i i=1 0 E(X) exp(x 1 2 E[ X 2 ]) for Gaussian r.v. X Market model In practice, n = 2 needed for good fit, contains seven parameters n η i ψ(τ) (1 1 ) e κ iτ κ i τ κ i τ i=1 Tempting to replace the exponential kernel by a power law kernel! Rough volatility models October 18, 2018 Page 9 (28)

11 Rough Fractional Stochastic Volatility Gatheral, Jaisson, and Rosenbaum (2014) study time series of realized variance and find amazing fits of a stochastic volatility model based on log v u log v t = 2ν ( ) Wu H Wt H Mandelbrot Van Ness representation of fbm (with γ = 1/2 H) ( t Wt H dw P 0 [ s = C H (t s) γ + 1 (t s) γ 1 ] ) ( s) γ dws P 0 v u is not a Markov process. With W t P(u) = 2H u dws P t (u s) γ, we get Rough volatility models October 18, 2018 Page 10 (28)

12 Rough Fractional Stochastic Volatility Gatheral, Jaisson, and Rosenbaum (2014) study time series of realized variance and find amazing fits of a stochastic volatility model based on log v u log v t = 2ν ( ) Wu H Wt H Mandelbrot Van Ness representation of fbm (with γ = 1/2 H) ( u dw P t [ ] ) s log v u log v t = 2νC H (u s) γ + 1 (u s) γ 1 (t s) γ dws P v u is not a Markov process. With W t P(u) = 2H u dws P t (u s) γ, we get t Rough volatility models October 18, 2018 Page 10 (28)

13 Rough Fractional Stochastic Volatility Gatheral, Jaisson, and Rosenbaum (2014) study time series of realized variance and find amazing fits of a stochastic volatility model based on log v u log v t = 2ν ( ) Wu H Wt H Mandelbrot Van Ness representation of fbm (with γ = 1/2 H) ( u dw P t [ ] ) s log v u log v t = 2νC H (u s) γ + 1 (u s) γ 1 (t s) γ dws P t v u is not a Markov process. With W t P(u) = 2H u dws P t (u s) γ, we get v u = E P [v u F t ]E ( η W t P (u) ) Rough volatility models October 18, 2018 Page 10 (28)

14 Rough Fractional Stochastic Volatility Gatheral, Jaisson, and Rosenbaum (2014) study time series of realized variance and find amazing fits of a stochastic volatility model based on log v u log v t = 2ν ( ) Wu H Wt H Mandelbrot Van Ness representation of fbm (with γ = 1/2 H) ( u dw P t [ ] ) s log v u log v t = 2νC H (u s) γ + 1 (u s) γ 1 (t s) γ dws P t v u is not a Markov process. With W t P(u) = 2H u dws P t (u s) γ, we get v u = E Q [v u F t ]E ( η W Q t (u) ) Rough volatility models October 18, 2018 Page 10 (28)

15 Rough Fractional Stochastic Volatility Gatheral, Jaisson, and Rosenbaum (2014) study time series of realized variance and find amazing fits of a stochastic volatility model based on log v u log v t = 2ν ( ) Wu H Wt H Mandelbrot Van Ness representation of fbm (with γ = 1/2 H) ( u dw P t [ ] ) s log v u log v t = 2νC H (u s) γ + 1 (u s) γ 1 (t s) γ dws P t v u is not a Markov process. With W t P(u) = 2H u dws P t (u s) γ, we get v u = ξ t (u)e ( η W Q t (u) ) Rough volatility models October 18, 2018 Page 10 (28)

16 The Rough Bergomi model (under Q) ds t = v t S t dz t v t = ξ 0 (t)e ( η W t ) dw t dz t = ρdt, W t = 2H t dw s 0 (t s) γ, γ = 1/2 H W is a Volterra process (or Riemann-Liouville fbm ) Covariance: E [ W ] 2H u v W 1/2+H u = 1/2 + H v 1/2 H 2 F 1 (1, 1/2 H, 3/2 + H, u/v), u v, E [ W ] 2H ( v Z u = ρ v 1/2+H [v min(u, v)] 1/2+H) 1/2 + H ψ(τ) 1/τ γ Typical parameter values: H 0.05, η 2.5 Rough volatility models October 18, 2018 Page 11 (28)

17 The Rough Bergomi model (under Q) ds t = v t S t dz t v t = ξ 0 (t)e ( η W t ) dw t dz t = ρdt, W t = 2H t dw s 0 (t s) γ, γ = 1/2 H W is a Volterra process (or Riemann-Liouville fbm ) Covariance: E [ W ] 2H u v W 1/2+H u = 1/2 + H v 1/2 H 2 F 1 (1, 1/2 H, 3/2 + H, u/v), u v, E [ W ] 2H ( v Z u = ρ v 1/2+H [v min(u, v)] 1/2+H) 1/2 + H ψ(τ) 1/τ γ Typical parameter values: H 0.05, η 2.5 Rough volatility models October 18, 2018 Page 11 (28)

18 The Rough Bergomi model (under Q) ds t = v t S t dz t v t = ξ 0 (t)e ( η W t ) dw t dz t = ρdt, W t = 2H t dw s 0 (t s) γ, γ = 1/2 H W is a Volterra process (or Riemann-Liouville fbm ) Covariance: E [ W ] 2H u v W 1/2+H u = 1/2 + H v 1/2 H 2 F 1 (1, 1/2 H, 3/2 + H, u/v), u v, E [ W ] 2H ( v Z u = ρ v 1/2+H [v min(u, v)] 1/2+H) 1/2 + H ψ(τ) 1/τ γ Typical parameter values: H 0.05, η 2.5 Rough volatility models October 18, 2018 Page 11 (28)

19 The Rough Bergomi model (under Q) ds t = v t S t dz t v t = ξ 0 (t)e ( η W t ) dw t dz t = ρdt, W t = 2H t dw s 0 (t s) γ, γ = 1/2 H W is a Volterra process (or Riemann-Liouville fbm ) Covariance: E [ W ] 2H u v W 1/2+H u = 1/2 + H v 1/2 H 2 F 1 (1, 1/2 H, 3/2 + H, u/v), u v, E [ W ] 2H ( v Z u = ρ v 1/2+H [v min(u, v)] 1/2+H) 1/2 + H ψ(τ) 1/τ γ Typical parameter values: H 0.05, η 2.5 Rough volatility models October 18, 2018 Page 11 (28)

20 Outline 1 Implied volatility modeling 2 The rough Bergomi model 3 Case studies 4 Further challenges and developments Rough volatility models October 18, 2018 Page 12 (28)

21 02/04/2010; SPX Vol surface for H = 0.07, η = 1.9, ρ = 0.9 Rough volatility models October 18, 2018 Page 13 (28)

22 02/04/2010; SPX short maturity smile for H = 0.07, η = 1.9, ρ = 0.9 Rough volatility models October 18, 2018 Page 14 (28)

23 02/04/2010; SPX volatility skew for H = 0.07, η = 1.9, ρ = 0.9 Rough volatility models October 18, 2018 Page 15 (28)

24 02/04/2010; SPX ATM volatility for H = 0.07, η = 1.9, ρ = 0.9 Rough volatility models October 18, 2018 Page 16 (28)

25 Variance swap forecast Variance v is not a martingale, hence non-trivial forecast. Using a result in (Nuzman and Poor, 2000), we have E P [ ] cos(hπ) t log v t+ F t = H+1/2 log v s ds π (t s + )(t s) H+1/2 E P [v t+ F t ] = exp ( E P [ log v t+ F t ] + 2cν 2 2H) Use realized variance as proxy for v Problem: realized variance only available from opening to close, not from close to close Forecasts must be re-scaled by (time-varying) factor; hence should predict variance swap curve up to a factor Rough volatility models October 18, 2018 Page 17 (28)

26 Variance swap forecast Variance v is not a martingale, hence non-trivial forecast. Using a result in (Nuzman and Poor, 2000), we have E P [ ] cos(hπ) t log v t+ F t = H+1/2 log v s ds π (t s + )(t s) H+1/2 E P [v t+ F t ] = exp ( E P [ log v t+ F t ] + 2cν 2 2H) Use realized variance as proxy for v Problem: realized variance only available from opening to close, not from close to close Forecasts must be re-scaled by (time-varying) factor; hence should predict variance swap curve up to a factor Rough volatility models October 18, 2018 Page 17 (28)

27 Variance swap forecasts for the Lehman weekend Actual and predicted variance swap curves, 09/12/08 (red) and 09/15/08 (blue), after scaling. Rough volatility models October 18, 2018 Page 18 (28)

28 Rough voltility is ubiquitous in equity (Bennedsen, Lunde and Pakkanen 2017) compare timeseries data over 10 years of 2000 assets (US equities). They find overwhelming evidence of rough volatility! Figure: Estimates for α H 1/2 according to sector. Rough volatility models October 18, 2018 Page 19 (28)

29 Outline 1 Implied volatility modeling 2 The rough Bergomi model 3 Case studies 4 Further challenges and developments Rough volatility models October 18, 2018 Page 20 (28)

30 Mathematical challenges of rough volatility models Theory: Lack of general fractional stochastic calculus (for instance, no rough path framework for H 1/4) Difficult to generalize dynamics (needed to capture higher order effects) Difficult to analyze even very simple models such as rough Bergomi Computations: No Markov structure, hence no (tractable) pricing PDE or tree approximations Large deviations depend on truly infinite dimensional variational problems, making asymptotic analysis more difficult Simulation expensive but doable relying on the Gaussian structure Rough volatility models October 18, 2018 Page 21 (28)

31 Rough Heston model [Rosenbaum and El Euch, 2017,... ] ds t = v t S t dz t Fractional Riccati ODE 0 t v t = v (t s) α 1 λ(θ v s )ds Γ(α) t (t s) α 1 λν v s dw s Γ(α) E [ exp ( iu log(s t ) )] = exp (g 1 (u, t) + v 0 g 2 (u, t)), with g 1 (u, t) θλ t 0 h(u, s)ds, g 2 (u, t) I 1 α h(u, t), D α h(u, t) = 1 2 ( u2 iu) + λ(iuρν 1)h(u, t) + (λν)2 2 h2 (u, t), I 1 α h(u, 0) = 0. I r f (t) 1 Γ(r) t 0 (t s) r 1 f (s)ds, D r f (t) 1 d Γ(1 r) dt t 0 (t s) r f (s)ds. Rough volatility models October 18, 2018 Page 22 (28)

32 Microstructural foundation Assumption Market orders are indep. Hawkes processes N a/b, with intensities λ a/b t = µ + t 0 φ(t s)dn a/b s Market impact exists and has a non-vanishing transient component. The market is highly endogenous. Under some additional assumptions, we obtain a rough Heston type model as scaling limit of price changes obtained from the market orders. ([El Euch, Fukasawa, Rosenbaum, 2016], [Jusselin, Rosenbaum 2018]) Rough volatility models October 18, 2018 Page 23 (28)

33 Microstructural foundation Assumption Market orders are indep. Hawkes processes N a/b, with intensities λ a/b t = µ + t 0 φ(t s)dn a/b s Market impact exists and has a non-vanishing transient component. The market is highly endogenous. Under some additional assumptions, we obtain a rough Heston type model as scaling limit of price changes obtained from the market orders. ([El Euch, Fukasawa, Rosenbaum, 2016], [Jusselin, Rosenbaum 2018]) Rough volatility models October 18, 2018 Page 23 (28)

34 Microstructural foundation Assumption Market orders are indep. Hawkes processes N a/b, with intensities λ a/b t = µ + t 0 φ(t s)dn a/b s Market impact exists and has a non-vanishing transient component. The market is highly endogenous. Under some additional assumptions, we obtain a rough Heston type model as scaling limit of price changes obtained from the market orders. ([El Euch, Fukasawa, Rosenbaum, 2016], [Jusselin, Rosenbaum 2018]) Rough volatility models October 18, 2018 Page 23 (28)

35 Simulation Recall that (Ŵ, Z) is a Gaussian process. Hence, we can simulate samples on a grid 0 = t 0 < t 1 < < t N = T by Cholesky factorization of the covariance (exact, but cost O(N 2 ) per sample); Hybrid scheme by [Bennedsen, Lunde, Pakkanen, 2017] (inexact, but cost O(N log N)). Leads to Riemann approximation T 0 f ( t, Ŵ t ) dzt N 1 i=0 f ( t i, Ŵ ti ) ( Zti+1 Z ti ). Theorem (Neuenkirch and Shalaiko 16) The strong rate of convergence is H and no better. Rough volatility models October 18, 2018 Page 24 (28)

36 Simulation Recall that (Ŵ, Z) is a Gaussian process. Hence, we can simulate samples on a grid 0 = t 0 < t 1 < < t N = T by Cholesky factorization of the covariance (exact, but cost O(N 2 ) per sample); Hybrid scheme by [Bennedsen, Lunde, Pakkanen, 2017] (inexact, but cost O(N log N)). Leads to Riemann approximation T 0 f ( t, Ŵ t ) dzt N 1 i=0 f ( t i, Ŵ ti ) ( Zti+1 Z ti ). Theorem (Neuenkirch and Shalaiko 16) The strong rate of convergence is H and no better. Rough volatility models October 18, 2018 Page 24 (28)

37 Weak error of the Euler scheme in the rough Bergomi model ATM-call with ξ 0.04, H = 0.06, η = 2.5, ρ = 0.8, T = 0.8 error 5e 06 5e 05 5e 04 5e steps Rough volatility models October 18, 2018 Page 25 (28)

38 Weak error The weak rate of convergence seems unknown even for Y 1 0 f (s, Ŵ s )dw s. Standard methods for SDEs rely on PDE arguments. Using metrics for weak convergence such as Wasserstein distance seems difficult. Techniques based on Malliavin calculus work in principle. For Y as above, one can get weak rate 2H, but numerical experiments suggest much better rates. Partial result: for Euler approximation Y, f nice [ E Y 2 Y 2] C h. Rough volatility models October 18, 2018 Page 26 (28)

39 Weak error The weak rate of convergence seems unknown even for Y 1 0 f (s, Ŵ s )dw s. Standard methods for SDEs rely on PDE arguments. Using metrics for weak convergence such as Wasserstein distance seems difficult. Techniques based on Malliavin calculus work in principle. For Y as above, one can get weak rate 2H, but numerical experiments suggest much better rates. Partial result: for Euler approximation Y, f nice [ E Y 2 Y 2] C h. Rough volatility models October 18, 2018 Page 26 (28)

40 References Bayer, C., Friz, P., Gassiat, P., Martin, J., Stemper, B. A regularity structure for rough volatility, preprint, Bayer, C., Friz, P., Gatheral, J. Pricing under rough volatility, Quant. Fin., Bayer, C., Friz, P., Gulisashvili, A., Horvath, B., Stemper, B. Short time near-the-money skew in rough fractional volatility models, Quant. Fin., to appear. Bennedsen, M., Lunde, A., Pakkanen, M. Hybrid scheme for Brownian semistationary processes, Fin. Stoch., Bennedsen, M., Lunde, A., Pakkanen, M. Decoupling the shortand long-term behavior of stochastic volatility, preprint, Bergomi, L. Smile dynamics II, Risk, October Gatheral, J., Jaisson, T., Rosenbaum, M. Volatility is rough, Quant. Fin., El Euch, O., Rosenbaum, M. The characteristic function of the rough Heston model, Math. Fin., to appear. Rough volatility models October 18, 2018 Page 27 (28)

41 Fractional Brownian motion Definition Fractional Brownian motion is a continuous time Gaussian process B H (with Hurst index 0 < H < 1) with B0 H = 0, E[BH t ] = 0 and E[Bt H B H s ] = 1 ( t 2H + s 2H t s 2H). 2 B H with H = 1 2 is classical Brownian motion. Increments are neg. corr. for H < 1 2 and pos. corr. for H > 1 2. fbm with H = 0.1 (left) H = 1/2 (middle) and H = 0.9 (right) Rough volatility models October 18, 2018 Page 28 (28)

42 Fractional Brownian motion Definition Fractional Brownian motion is a continuous time Gaussian process B H (with Hurst index 0 < H < 1) with B0 H = 0, E[BH t ] = 0 and E[Bt H B H s ] = 1 ( t 2H + s 2H t s 2H). 2 B H with H = 1 2 is classical Brownian motion. Increments are neg. corr. for H < 1 2 and pos. corr. for H > 1 2. fbm with H = 0.1 (left) H = 1/2 (middle) and H = 0.9 (right) Rough volatility models October 18, 2018 Page 28 (28)