Rough volatility models - Weierstrass Institute · 2018. 12. 5. · 2 The rough Bergomi model 3...
Transcript of Rough volatility models - Weierstrass Institute · 2018. 12. 5. · 2 The rough Bergomi model 3...
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Weierstrass Institute forApplied Analysis and Stochastics
Rough volatility models
Christian BayerEMEA Quant Meeting 2018
Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.deOctober 18, 2018
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Outline
1 Implied volatility modeling
2 The rough Bergomi model
3 Case studies
4 Further challenges and developments
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SPX implied volatility surface
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SPX ATM volatility skew
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Questions and goals
dS t =√
vtS tdZt,
dvt = . . .
I We look for time-homogeneous models.
I Term structure of ATM volatility skew (k = log(K/S t))
ψ(τ) =
∣∣∣∣∣ ∂∂kσBS (k, τ)
∣∣∣∣∣k=0∼ 1/τα, α ∈ [0.3, 0.5]
I Conventional stochastic volatility models produce ATM skewswhich are constant for τ � 1 and of order 1/τ for τ � 1. Hence,conventional stochastic volatility models cannot fit the fullvolatility surface.
I Do we need jumps?
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Some notations and assumptions
I Given a traded asset S t satisfying
dS t =√
vtS tdZt
I Interest rate r = 0; model (and expectations) formulated under Q
I In this talk, S corresponds to the S & P 500 index (SPX).
I Realized variance wt,T =∫ T
t vsds, forward variance ξt(u) = Et[vu]
I Log-strip formula:
Etwt,T = −2(∫ S t
0
P(K)K2 dK +
∫ ∞
S t
C(K)K2 dK
)
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Intraday realized variance
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Outline
1 Implied volatility modeling
2 The rough Bergomi model
3 Case studies
4 Further challenges and developments
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Bergomi model
I Recall ξt(u) = Etvu
dS t =√ξt(t)S tdZt,
ξt(u) = ξ0(u)E
n∑i=1
ηi
∫ t
0e−κi(u−s)dW i
s
I E(X) B exp(X − 1
2 E[|X|2]) for Gaussian r.v. X
I Market model
I In practice, n = 2 needed for good fit, contains seven parameters
I ψ(τ) ∼n∑
i=1
ηi
κiτ
(1 −
1 − e−κiτ
κiτ
)I Tempting to replace the exponential kernel by a power law kernel!
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Bergomi model
I Recall ξt(u) = Etvu
dS t =√ξt(t)S tdZt,
ξt(u) = ξ0(u)E
n∑i=1
ηi
∫ t
0e−κi(u−s)dW i
s
I E(X) B exp(X − 1
2 E[|X|2]) for Gaussian r.v. X
I Market model
I In practice, n = 2 needed for good fit, contains seven parameters
I ψ(τ) ∼n∑
i=1
ηi
κiτ
(1 −
1 − e−κiτ
κiτ
)I Tempting to replace the exponential kernel by a power law kernel!
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Rough Fractional Stochastic Volatility
I Gatheral, Jaisson, and Rosenbaum (2014) study time series ofrealized variance and find amazing fits of a stochastic volatilitymodel based on
log vu − log vt = 2ν(WH
u −WHt
)I Mandelbrot – Van Ness representation of fBm (with γ = 1/2 − H)
WHt = CH
(∫ t
0
dWPs
(t − s)γ+
∫ 0
−∞
[1
(t − s)γ−
1(−s)γ
]dWP
s
)I vu is not a Markov process.I With WP
t (u) =√
2H∫ u
tdWP
s(u−s)γ , we get
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Rough Fractional Stochastic Volatility
I Gatheral, Jaisson, and Rosenbaum (2014) study time series ofrealized variance and find amazing fits of a stochastic volatilitymodel based on
log vu − log vt = 2ν(WH
u −WHt
)I Mandelbrot – Van Ness representation of fBm (with γ = 1/2 − H)
log vu−log vt = 2νCH
(∫ u
t
dWPs
(u − s)γ+
∫ t
−∞
[1
(u − s)γ−
1(t − s)γ
]dWP
s
)I vu is not a Markov process.I With WP
t (u) =√
2H∫ u
tdWP
s(u−s)γ , we get
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Rough Fractional Stochastic Volatility
I Gatheral, Jaisson, and Rosenbaum (2014) study time series ofrealized variance and find amazing fits of a stochastic volatilitymodel based on
log vu − log vt = 2ν(WH
u −WHt
)I Mandelbrot – Van Ness representation of fBm (with γ = 1/2 − H)
log vu−log vt = 2νCH
(∫ u
t
dWPs
(u − s)γ+
∫ t
−∞
[1
(u − s)γ−
1(t − s)γ
]dWP
s
)I vu is not a Markov process.I With WP
t (u) =√
2H∫ u
tdWP
s(u−s)γ , we get
vu = EP[vu|Ft]E(ηWP
t (u))
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Rough Fractional Stochastic Volatility
I Gatheral, Jaisson, and Rosenbaum (2014) study time series ofrealized variance and find amazing fits of a stochastic volatilitymodel based on
log vu − log vt = 2ν(WH
u −WHt
)I Mandelbrot – Van Ness representation of fBm (with γ = 1/2 − H)
log vu−log vt = 2νCH
(∫ u
t
dWPs
(u − s)γ+
∫ t
−∞
[1
(u − s)γ−
1(t − s)γ
]dWP
s
)I vu is not a Markov process.I With WP
t (u) =√
2H∫ u
tdWP
s(u−s)γ , we get
vu = EQ[vu|Ft]E(ηWQ
t (u))
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Rough Fractional Stochastic Volatility
I Gatheral, Jaisson, and Rosenbaum (2014) study time series ofrealized variance and find amazing fits of a stochastic volatilitymodel based on
log vu − log vt = 2ν(WH
u −WHt
)I Mandelbrot – Van Ness representation of fBm (with γ = 1/2 − H)
log vu−log vt = 2νCH
(∫ u
t
dWPs
(u − s)γ+
∫ t
−∞
[1
(u − s)γ−
1(t − s)γ
]dWP
s
)I vu is not a Markov process.I With WP
t (u) =√
2H∫ u
tdWP
s(u−s)γ , we get
vu = ξt(u)E(ηWQ
t (u))
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The Rough Bergomi model (under Q)
dS t =√
vtS tdZt
vt = ξ0(t)E(ηWt
)I dWtdZt = ρdt, Wt =
√2H
∫ t0
dWs(t−s)γ , γ = 1/2 − H
I W is a “Volterra” process (or “Riemann-Liouville fBm”)
I Covariance:
E[WvWu
]=
2H1/2 + H
u1/2+H
v1/2−H 2F1 (1, 1/2 − H, 3/2 + H, u/v) , u ≤ v,
E[WvZu
]= ρ
√2H
1/2 + H
(v1/2+H − [v −min(u, v)]1/2+H
)I ψ(τ) ∼ 1/τγ
I Typical parameter values: H ≈ 0.05, η ≈ 2.5
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The Rough Bergomi model (under Q)
dS t =√
vtS tdZt
vt = ξ0(t)E(ηWt
)I dWtdZt = ρdt, Wt =
√2H
∫ t0
dWs(t−s)γ , γ = 1/2 − H
I W is a “Volterra” process (or “Riemann-Liouville fBm”)
I Covariance:
E[WvWu
]=
2H1/2 + H
u1/2+H
v1/2−H 2F1 (1, 1/2 − H, 3/2 + H, u/v) , u ≤ v,
E[WvZu
]= ρ
√2H
1/2 + H
(v1/2+H − [v −min(u, v)]1/2+H
)I ψ(τ) ∼ 1/τγ
I Typical parameter values: H ≈ 0.05, η ≈ 2.5
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The Rough Bergomi model (under Q)
dS t =√
vtS tdZt
vt = ξ0(t)E(ηWt
)I dWtdZt = ρdt, Wt =
√2H
∫ t0
dWs(t−s)γ , γ = 1/2 − H
I W is a “Volterra” process (or “Riemann-Liouville fBm”)
I Covariance:
E[WvWu
]=
2H1/2 + H
u1/2+H
v1/2−H 2F1 (1, 1/2 − H, 3/2 + H, u/v) , u ≤ v,
E[WvZu
]= ρ
√2H
1/2 + H
(v1/2+H − [v −min(u, v)]1/2+H
)I ψ(τ) ∼ 1/τγ
I Typical parameter values: H ≈ 0.05, η ≈ 2.5
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The Rough Bergomi model (under Q)
dS t =√
vtS tdZt
vt = ξ0(t)E(ηWt
)I dWtdZt = ρdt, Wt =
√2H
∫ t0
dWs(t−s)γ , γ = 1/2 − H
I W is a “Volterra” process (or “Riemann-Liouville fBm”)
I Covariance:
E[WvWu
]=
2H1/2 + H
u1/2+H
v1/2−H 2F1 (1, 1/2 − H, 3/2 + H, u/v) , u ≤ v,
E[WvZu
]= ρ
√2H
1/2 + H
(v1/2+H − [v −min(u, v)]1/2+H
)I ψ(τ) ∼ 1/τγ
I Typical parameter values: H ≈ 0.05, η ≈ 2.5
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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)
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02/04/2010; SPX Vol surface for H = 0.07, η = 1.9, ρ = −0.9
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02/04/2010; SPX short maturity smile for H = 0.07, η = 1.9, ρ = −0.9
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02/04/2010; SPX volatility skew for H = 0.07, η = 1.9, ρ = −0.9
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02/04/2010; SPX ATM volatility for H = 0.07, η = 1.9, ρ = −0.9
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Variance swap forecast
I Variance v is not a martingale, hence non-trivial forecast.
I Using a result in (Nuzman and Poor, 2000), we have
EP [log vt+∆|Ft
]=
cos(Hπ)π
∆H+1/2∫ t
−∞
log vs
(t − s + ∆)(t − s)H+1/2 ds
EP [vt+∆|Ft] = exp(EP [
log vt+∆|Ft]+ 2cν2∆2H
)I Use realized variance as proxy for v
I Problem: realized variance only available from opening to close,not from close to close
I Forecasts must be re-scaled by (time-varying) factor; henceshould predict variance swap curve up to a factor
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Variance swap forecast
I Variance v is not a martingale, hence non-trivial forecast.
I Using a result in (Nuzman and Poor, 2000), we have
EP [log vt+∆|Ft
]=
cos(Hπ)π
∆H+1/2∫ t
−∞
log vs
(t − s + ∆)(t − s)H+1/2 ds
EP [vt+∆|Ft] = exp(EP [
log vt+∆|Ft]+ 2cν2∆2H
)I Use realized variance as proxy for v
I Problem: realized variance only available from opening to close,not from close to close
I Forecasts must be re-scaled by (time-varying) factor; henceshould predict variance swap curve up to a factor
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Variance swap forecasts for the Lehman weekend
Actual and predicted variance swap curves, 09/12/08 (red) and09/15/08 (blue), after scaling.
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Rough voltility is ubiquitous in equity
(Bennedsen, Lunde and Pakkanen 2017) compare timeseries dataover 10 years of 2000 assets (US equities). They find overwhelmingevidence of rough volatility!
Figure: Estimates for α B H − 1/2 according to sector.
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Outline
1 Implied volatility modeling
2 The rough Bergomi model
3 Case studies
4 Further challenges and developments
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Mathematical challenges of rough volatility models
Theory:I Lack of general fractional stochastic calculus (for instance, no
rough path framework for H ≤ 1/4)I Difficult to generalize dynamics (needed to capture higher order
effects)I Difficult to analyze even very simple models such as rough
Bergomi
Computations:I No Markov structure, hence no (tractable) pricing PDE or tree
approximationsI Large deviations depend on truly infinite dimensional variational
problems, making asymptotic analysis more difficultI Simulation expensive but doable relying on the Gaussian
structure
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Rough Heston model [Rosenbaum and El Euch, 2017, . . . ]
dS t =√
vtS tdZt
vt = v0 +1
Γ(α)
∫ t
0(t − s)α−1λ(θ − vs)ds
+1
Γ(α)
∫ t
0(t − s)α−1λν
√vsdWs
Fractional Riccati ODEE
[exp
(iu log(S t)
)]= exp (g1(u, t) + v0g2(u, t)), with
g1(u, t) B θλ
∫ t
0h(u, s)ds, g2(u, t) B I1−αh(u, t),
Dαh(u, t) =12
(−u2 − iu) + λ(iuρν − 1)h(u, t) +(λν)2
2h2(u, t), I1−αh(u, 0) = 0.
Ir f (t) B1
Γ(r)
∫ t
0(t−s)r−1 f (s)ds, Dr f (t) B
1Γ(1 − r)
ddt
∫ t
0(t−s)r f (s)ds.
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Microstructural foundation
Assumption
I Market orders are indep. Hawkes processes Na/b, with intensities
λa/bt = µ +
∫ t
0φ(t − s)dNa/b
s
I Market impact exists and has a non-vanishing transientcomponent.
I The market is highly endogenous.
Under some additional assumptions, we obtain a rough Heston typemodel as scaling limit of price changes obtained from the marketorders. ([El Euch, Fukasawa, Rosenbaum, 2016], [Jusselin,Rosenbaum 2018])
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Microstructural foundation
Assumption
I Market orders are indep. Hawkes processes Na/b, with intensities
λa/bt = µ +
∫ t
0φ(t − s)dNa/b
s
I Market impact exists and has a non-vanishing transientcomponent.
I The market is highly endogenous.
Under some additional assumptions, we obtain a rough Heston typemodel as scaling limit of price changes obtained from the marketorders. ([El Euch, Fukasawa, Rosenbaum, 2016], [Jusselin,Rosenbaum 2018])
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Microstructural foundation
Assumption
I Market orders are indep. Hawkes processes Na/b, with intensities
λa/bt = µ +
∫ t
0φ(t − s)dNa/b
s
I Market impact exists and has a non-vanishing transientcomponent.
I The market is highly endogenous.
Under some additional assumptions, we obtain a rough Heston typemodel as scaling limit of price changes obtained from the marketorders. ([El Euch, Fukasawa, Rosenbaum, 2016], [Jusselin,Rosenbaum 2018])
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Simulation
Recall that (W,Z) is a Gaussian process. Hence, we can simulatesamples on a grid 0 = t0 < t1 < · · · < tN = T by
I Cholesky factorization of the covariance (exact, but cost O(N2)per sample);
I Hybrid scheme by [Bennedsen, Lunde, Pakkanen, 2017](inexact, but cost O(N log N)).
Leads to Riemann approximation∫ T
0f(t, Wt
)dZt ≈
N−1∑i=0
f(ti, Wti
) (Zti+1 − Zti
).
Theorem (Neuenkirch and Shalaiko ’16)The strong rate of convergence is H — and no better.
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Simulation
Recall that (W,Z) is a Gaussian process. Hence, we can simulatesamples on a grid 0 = t0 < t1 < · · · < tN = T by
I Cholesky factorization of the covariance (exact, but cost O(N2)per sample);
I Hybrid scheme by [Bennedsen, Lunde, Pakkanen, 2017](inexact, but cost O(N log N)).
Leads to Riemann approximation∫ T
0f(t, Wt
)dZt ≈
N−1∑i=0
f(ti, Wti
) (Zti+1 − Zti
).
Theorem (Neuenkirch and Shalaiko ’16)The strong rate of convergence is H — and no better.
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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
2 5 10 20 50 100 200 500 2000
5e−
065e
−05
5e−
045e
−03
steps
erro
r
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Weak error
The weak rate of convergence seems unknown even for
Y ≡∫ 1
0f (s, Ws)dWs.
I Standard methods for SDEs rely on PDE arguments.I Using metrics for weak convergence such as Wasserstein
distance seems difficult.I Techniques based on Malliavin calculus work in principle.I For Y as above, one can get weak rate 2H, but numerical
experiments suggest much better rates.I Partial result: for Euler approximation Y, f “nice”
E[Y2 − Y
2]≤ C h.
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Weak error
The weak rate of convergence seems unknown even for
Y ≡∫ 1
0f (s, Ws)dWs.
I Standard methods for SDEs rely on PDE arguments.I Using metrics for weak convergence such as Wasserstein
distance seems difficult.I Techniques based on Malliavin calculus work in principle.I For Y as above, one can get weak rate 2H, but numerical
experiments suggest much better rates.I Partial result: for Euler approximation Y, f “nice”
E[Y2 − Y
2]≤ C h.
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References
Bayer, C., Friz, P., Gassiat, P., Martin, J., Stemper, B. A regularitystructure for rough volatility, preprint, 2017.
Bayer, C., Friz, P., Gatheral, J. Pricing under rough volatility,Quant. Fin., 2016.
Bayer, C., Friz, P., Gulisashvili, A., Horvath, B., Stemper, B. Shorttime near-the-money skew in rough fractional volatility models,Quant. Fin., to appear.
Bennedsen, M., Lunde, A., Pakkanen, M. Hybrid scheme forBrownian semistationary processes, Fin. Stoch., 2017.
Bennedsen, M., Lunde, A., Pakkanen, M. Decoupling the short-and long-term behavior of stochastic volatility, preprint, 2017.
Bergomi, L. Smile dynamics II, Risk, October 2005.
Gatheral, J., Jaisson, T., Rosenbaum, M. Volatility is rough,Quant. Fin., 2018.
El Euch, O., Rosenbaum, M. The characteristic function of therough Heston model, Math. Fin., to appear.
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Fractional Brownian motion
DefinitionFractional Brownian motion is a continuous time Gaussian processBH (with Hurst index 0 < H < 1) with BH
0 = 0, E[BHt ] = 0 and
E[BHt BH
s ] =12
(t2H + s2H − |t − s|2H
).
I BH with H = 12 is classical Brownian motion.
I Increments are neg. corr. for H < 12 and pos. corr. for H > 1
2 .
fBm with H = 0.1 (left) H = 1/2 (middle) and H = 0.9 (right)
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Fractional Brownian motion
DefinitionFractional Brownian motion is a continuous time Gaussian processBH (with Hurst index 0 < H < 1) with BH
0 = 0, E[BHt ] = 0 and
E[BHt BH
s ] =12
(t2H + s2H − |t − s|2H
).
I BH with H = 12 is classical Brownian motion.
I Increments are neg. corr. for H < 12 and pos. corr. for H > 1
2 .
fBm with H = 0.1 (left) H = 1/2 (middle) and H = 0.9 (right)
0.0 0.2 0.4 0.6 0.8 1.0
−1.
5−
0.5
0.5
1.5
0.0 0.2 0.4 0.6 0.8 1.0
−1.
0−
0.5
0.0
0.5
0.0 0.2 0.4 0.6 0.8 1.0−
0.5
−0.
3−
0.1
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