Empirical evidence
Efficient simulation methods
Concluding remarks
R OUGH V OLATILITY:
E MPIRICAL E VIDENCE AND
E FFICIENT S IMULATION M ETHODS
Mikko Pakkanen1,2
1 Department of Mathematics, Imperial College London, UK
2 CREATES, Aarhus University, Denmark
At the Frontiers of Quantitative Finance, ICMS, 27 June 2016
Joint work with Mikkel Bennedsen and Asger Lunde
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Empirical evidence
Efficient simulation methods
Concluding remarks
Empirical evidence
Efficient simulation methods
Concluding remarks
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Empirical evidence
Efficient simulation methods
Concluding remarks
Volatility is rough
In their thought-provoking paper, Gatheral, Jaisson, and
Rosenbaum (2014) suggested that volatility is rough.
Suppose that the “frictionless” price of an asset follows an Itô
semimartingale
dSt
= µt dt + σt dWt ,
St
t Ê 0.
Then the slogan “volatility is rough” refers to the idea that the
sample paths of the log (spot) volatility process
vt := log σt ,
t Ê 0,
would be rougher than standard Brownian sample paths.
3 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Volatility is rough — at first glance
Wt
−0.6 −0.2 0.2
Standard Brownian motion
0.0
0.2
0.4
0.6
0.8
1.0
Source: Oxford−Man Realized Library
−12
log(RVt)
−9 −7
−5
t
Daily realised variance of CAC 40 index
2008−01−01
2010−01−01
2012−01−01
2014−01−01
2016−01−01
t
−1.5
Zt
0.0 1.0
Fractional Brownian motion with Hurst index 0.15
0.0
0.2
0.4
0.6
0.8
1.0
t
4 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Volatility is rough — statistical evidence
Gatheral, Jaisson, and Rosenbaum (2014) found evidence of rough
volatility by analysing high-frequency price data on:
• DAX and Bund futures contracts,
• S&P 500 and NASDAQ equity indices.
They analysed statistically the roughness of volatility using volatility
proxies computed at daily frequency.
Our contribution (Bennedsen et al., 2016) is to carry out a similar
statistical analysis of the volatility of:
• E-mini S&P 500 futures contracts at intraday time scales,
• 29 individual US equities at daily frequency.
5 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Characterising roughness
Definition
Suppose that the log volatility v = (vt )tÊ0 is covariance stationary
with autocorrelation function (ACF)
ρ(h) := Corr(v0 , vh ),
h Ê 0.
We say that v is rough if
1 − ρ(h) ∼ c h2α+1 ,
h → 0,
for some c > 0 and α ∈ (−1/2, 0).
Lemma
If v is rough and Gaussian, then it has a modification with γ-Hölder
continuous sample paths for any γ ∈ (0, α + 1/2).
6 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Volatility is not actually observable. . .
When we try to assess the roughness of volatility empirically, we
face the problem that neither v nor σ is observable ex post.
However, by semimartingale theory, the realised variance
n
:=
RV[t,t+∆]
bn∆c
X¡
log St+i/n − log St+(i−1)/n
¢2
i=1
is a consistent estimator of the integrated variance
t+∆
Z
IV[t,t+∆] :=
t
σ2s ds
under infill asymptotics n → ∞.
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Empirical evidence
Efficient simulation methods
Concluding remarks
. . . but we might be able to find a proxy for it
This suggests that for large n and small ∆,
b 2t (n, ∆) :=
σ
1
1
1
RV n
≈ IV[t,t+∆] =
∆ [t,t+∆] ∆
∆
t+∆
Z
t
σ2s ds ≈ σ2t ,
and that the ACF of v,
ρ(h) for h = 0, ∆, 2∆, . . .
could be estimated by the empirical ACF of the log volatility proxy
b 20 (n, ∆), log σ
b 2∆ (n, ∆), log σ
b 22∆ (n, ∆), . . . ,
log σ
denoted ρbn∆ (h), h = 0, ∆, 2∆, . . ..
8 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Measuring roughness
We can estimate the roughness parameter α semiparametrically by
running a regression
¡
¢
log 1 − ρbn∆ (h) = a + b log(h) + εh ,
h = ∆, 2∆, . . . , m∆,
and using α̂ := (b̂OLS − 1)/2.
Inference (confidence intervals etc.) on α can be conducted using a
(fractional) bootstrap method (Bennedsen, 2016).
Ideally m should be large and ∆ small (less than a day, if possible).
n
But RV[t,t+∆]
was computed using bn∆c observations, so this means
that n should be large.
n
So we should compute RV[t,t+∆]
using “ultra” high-frequency data!
9 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Dealing with market microstructure noise
In practise we do not observe the “frictionless” price St of the asset,
but instead something that can be described as
St∗ = St + et ,
where et stems from market microstructure (MMS) noise.
The presence of MMS noise becomes apparent when we go to
“ultra” high frequencies (n → ∞).
Since the asymptotic properties of the realised variance are
impaired by MMS noise, we replace it with a robust estimator, the
realised kernel (Barndorff-Nielsen, Hansen, Lunde, and Shephard,
2008). Realised kernel
10 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Empirical study
We have studied the roughness of the realised volatility of the
following assets:
• E-mini S&P 500 futures contract (3 Jan 2005–31 Dec 2014),
• 29 major US equities (2 Jan 1997–31 Dec 2013).
The proxy for the spot volatility is constructed from realised kernels,
using:
• ∆ ∈ [10 sec, 1 day] for E-mini S&P 500,
• ∆ = 1 day for equities.
The realised kernels are based on 1-second calendar-time sampled
transaction prices.
We only use the data over the period when NYSE is open
(9:30am–4pm EST).
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Empirical evidence
Efficient simulation methods
Concluding remarks
Intraday volatility — an example
∆ = 2 min
1.0
Without diurnal adjustment
With diurnal adjustment
0.0
−1.0
−0.5
log(RK) (adjusted)
−4.0
−4.5
−5.0
−1.5
−5.5
−6.0
log(RK)
∆ = 2 min
0.5
−3.5
−3.0
Data: E-mini S&P 500, 3 Jan 2013.
10:00
11:00
12:00
13:00
Time (EST)
14:00
15:00
16:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
Time (EST)
12 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Roughness of volatility
Data: E-mini S&P 500, 2 Jan 2013–31 Dec 2014.
∆ = 2 minutes
-0.5
∆ = 10 minutes
-0.5
slope = 0.30
log(1 − ACFh )
log(1 − ACFh )
slope = 0.08
-0.6
-0.7
-0.8
-1.5
α̂ = −0.46
-0.9
-1
-6
-1
-5
-4
-3
-2
α̂ = −0.35
-1
-2
-4
-3
-2
log(h)
¡
log(trading days)
¢
¡
-1
0
1
log(h)
¢
log(trading days)
13 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Roughness of volatility — a “signature plot”
Data: E-mini S&P 500, 2 Jan 2013–31 Dec 2014.
−0.25
−0.3
α̂
−0.35
−0.4
−0.45
−0.5
0
200
400
600
800
1000
∆ (seconds)
1200
1400
1600
1800
Solid line: estimate; dashed line: 95% confidence interval.
14 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Day-to-day variation of roughness
Data: E-mini S&P 500, 2 Jan 2013–31 Dec 2014.
−0.2
∆ = 2 minutes
−0.1
Median = -0.46
∆ = 5 minutes
Median = -0.44
−0.2
−0.3
α̂
α̂
−0.3
−0.4
−0.4
−0.5
−0.6
2013
0
−0.5
2014
−0.6
2013
2015
∆ = 10 minutes
0.2
2014
2015
∆ = 15 minutes
Median = -0.37
0
−0.2
α̂
α̂
−0.2
−0.4
−0.4
−0.6
−0.6
Median = -0.41
−0.8
2013
2014
2015
−0.8
2013
2014
2015
15 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Is roughness time-varying?
Data: E-mini S&P 500; ∆ = 10 minutes, estimation of α̂ using 10-day rolling window.
−0.1
−0.15
← Lehman bankruptcy
α̂
S m oot h
M e d i an = - 0. 36
−0.2
← Greek debt crisis
← Flash Crash
α
−0.25
−0.3
−0.35
−0.4
−0.45
−0.5
2005
2006
2007
2008
2009
2010
T im e
2011
2012
2013
2014
2015
16 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Roughness of US equity volatility
∆ = 1 day
AA
JNJ
●
AIG
●
AXP
BA
●
MCD
BAC
●
CAT
●
T
●
●
●
VZ
●
●
WMT
●
XOM
−0.3
^
α
−0.2
−0.1
0.0
●
●
UTX
IBM
−0.4
●
●
SPY
●
INTC
−0.5
●
●
PG
●
GE
HD
●
●
MSFT
●
DD
GM
MRK
estimate
95% CI
●
CVX
DIS
MMM
●
●
●
KO
●
C
●
JPM
−0.5
●
●
−0.4
−0.3
^
α
−0.2
−0.1
0.0
17 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Empirical evidence
Efficient simulation methods
Concluding remarks
18 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Pricing under rough volatility
Inspired in part by the empirical results on rough volatility, Bayer,
Friz, and Gatheral (2016) proposed an option pricing model that
incorporates rough volatility dynamics.
In the so-called rough Bergomi model (Bayer, Friz, and Gatheral,
2016), the price of the underlying follows
tp
µZ
St := S0 exp
0
1
Vs dBs −
2
t
Z
0
¶
Vs ds ,
t ∈ [0, T ],
where
µ
¶
Z t
p
η2
Vt := ξ exp η 2α + 1 (t − s)α dWs − t 2α+1 ,
2
0
t ∈ [0, T ],
with S0 , ξ, η > 0, d〈B, W 〉t = ρ dt, and α ∈ (−1/2, 0).
19 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Capturing rough volatility
The spot variance process V is specified using the (Gaussian)
Riemann–Liouville process
t
Z
Yt :=
0
(t − s)α dWs ,
t ∈ [0, T ],
whose sample paths are γ-Hölder continuous for all γ ∈ (0, α + 1/2).
Bayer, Friz, and Gatheral (2016) observed that the implied volatility
smile k 7→ IV (k, T ) corresponding to the call option price
C(k, T ) := E[(ST − S0 ek )+ ]
fits nicely to empirical IV smiles when α ≈ −0.4, a value which is
compatible with the measured roughness of realised volatility.
20 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Pricing by Monte Carlo
So far, the only known method of pricing mere vanilla options in
this model is Monte Carlo simulation.
In principle this involves no complications:
1. The processes B and Y are jointly Gaussian and their
covariance structure can be expressed using special functions.
2. Thus, B and Y can be simulated exactly by sampling from a
high-dimensional Gaussian distribution.
3. Finally, S can be approximated by Riemann sums.
Unfortunately the second step can be quite costly computationally.
An algorithm based on the Cholesky decomposition has typically
complexity O (n3 ), where n is the number of time steps.
21 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Approximation by Riemann sums
The efficiency of the Monte Carlo procedure could be improved by
using an approximate scheme to simulate Y .
An obvious way to approximate Yt would be to use Riemann sums:
Yt =
btnc
X Z t− nk + n1
k
k=1 t− n
(t − s)α dWs ≈
µ
btnc
X
k=1
k
n
¶α ³
´
Wt− k + 1 − Wt− k .
n
n
n
• This corresponds to approximating the kernel function x 7→ xα
by a step function.
• The approximation misses the singularity of the kernel
function at zero.
• The first summands are the problematic ones, as x 7→ xα is
evaluated near zero therein.
22 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
The hybrid scheme
The idea behind the hybrid scheme (Bennedsen et al., 2015) is to
eschew approximation in the first κ Ê 1 summands and use exact
simulation.
For k = 1, . . . , κ, we use
Z
t− nk + n1
t− nk
(t − s)α dWs
in lieu of
µ ¶α ³
´
k
Wt− k + 1 − Wt− k .
n n
n
n
Collecting the new summands, we define
Y̌tn :=
Z t− k + 1
min{κ,bntc}
X
n n
k=1
t− nk
(t − s)α dWs .
23 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
The hybrid scheme
We adopt the remaining summands from the Riemann sum, but we
allow the point at which x 7→ xα is evaluated to be choosen freely
within each discretisation cell.
That is, we define
Ŷtn
:=
btnc
X
k=min{κ,btnc}+1
µ
bk
n
¶α ³
´
Wt− k + 1 − Wt− k ,
n
n
n
where b = {bk }∞
is a sequence that must satisfy
k=κ+1
bk ∈ [k − 1, k] \ {0},
k Ê κ + 1,
but otherwise can be chosen freely.
24 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Approximating the kernel function
n = 10
g(x) = xα
α = −0.4
n = 10
α = −0.4
κ=2
●
6
8
true value
approximation
4
g(x)
6
4
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0
●
2
2
●
0
g(x)
8
10
●
true value
approximation
10
g(x) = xα
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
x
0.6
0.8
1.0
x
Riemann sums vs. the hybrid scheme
25 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
The hybrid scheme
The hybrid scheme for Yt combines the two sums:
Yt ≈ Ytn := Y̌tn + X̂tn .
Remark
Define b0 := {k}∞
. Then in the case κ = 0 and b = b0 we recover
k=κ+1
the approximation by Riemann sums.
n
n
In practise the variates Y0n , Y1/n
, . . . , Ybntc/n
can be generated by:
• sampling bntc observations from a κ + 1 dimensional Gaussian
distribution,
• computing a discrete convolution using FFT.
The complexity of this procedure is O (n log n).
26 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Asymptotics of the mean square error
Theorem
For all t > 0,
E[|Yt − Ytn |2 ] ∼ J(α, κ, b) n−(2α+1) ,
where
J(α, κ, b) :=
∞ Z
X
k
k=κ+1 k−1
n → ∞,
2
(y α − bα
k ) dy < ∞.
• For any α, we can find b that minimises the asymptotic RMSE
p
J(α, κ, b). We denote the minimiser by b∗ .
27 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
asymptotic RMSE
102
100
10-2
κ=0
κ=1
κ=2
κ=3
10-4
-0.4
-0.2
0.0
α
0.2
0.4
reduction in asymptotic RMSE (%)
Asymptotic root mean square error
100
80
κ=0
κ=1
κ=2
κ=3
60
40
20
0
-0.4
-0.2
0.0
0.2
0.4
α
Solid line: b = b∗ ; dashed line: b = b0 .
p
p
J(α, κ, b) − J(α, 0, b0 )
reduction in asymptotic RMSE =
· 100%.
p
J(α, 0, b0 )
28 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Application — IV smile using the hybrid scheme
T = 0.041
0.8
T = 1
0.4
exact
κ=0
κ=1
κ=2
exact
κ=0
κ=1
κ=2
0.3
IV (k, T )
IV (k, T )
0.6
0.4
0.2
0
0.2
0.1
−0.4
−0.2
k
0
−0.5
0
0
k
0.5
Solid/patterned line: b = b∗ ; dashed line: b = b0 .
S(0)
ξ
η
α
ρ
1
0.2352
1.9
−0.43
−0.9
29 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Empirical evidence
Efficient simulation methods
Concluding remarks
30 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
Concluding remarks
• The evidence that volatility is rough extends to intraday time
scales and to individual equities.
• At short time scales volatility seems to be very rough — a sign
of jumps in volatility?
• A flexible and highly tractable framework for rough stochastic
volatility modelling and forecasting will be presented in the
upcoming paper Bennedsen et al. (2016).
• The rough Bergomi model can be simulated efficiently using
the hybrid scheme, without sacrificing accuracy in comparison
with the exact method.
• Open question: How error-prone is the approximation of S by
Riemann sums?
31 / 32
Empirical evidence
Efficient simulation methods
Concluding remarks
References
O. E. Barndorff-Nielsen, P. R. Hansen, A. Lunde, and N. Shephard (2008):
Designing realized kernels to measure the ex post variation of equity prices in
the presence of noise. Econometrica 76(8), 1481–1536.
C. Bayer, P. K. Friz, and J. Gatheral (2016): Pricing under rough volatility.
Quantitative Finance 16(6), 887–904.
M. Bennedsen (2016): Semiparametric bootstrap inference on the fractal
index of a time series: with application to the assessment of the fine structure
of stochastic volatility. In preparation.
M. Bennedsen, A. Lunde, and M. S. Pakkanen (2015): Hybrid scheme for
Brownian semistationary processes. Preprint,
http://arxiv.org/abs/1507.03004.
M. Bennedsen, A. Lunde, and M. S. Pakkanen (2016): Decoupling the shortand long-term behavior of stochastic volatility. In preparation.
J. Gatheral, T. Jaisson, and M. Rosenbaum (2014): Volatility is rough. Preprint,
http://arxiv.org/abs/1410.3394.
32 / 32
The realised kernel
The realised kernel (Barndorff-Nielsen, Hansen, Lunde, and
Shephard, 2008) is defined by
n
:= γn[t,t+∆] (0) + 2
RK[t,t+∆]
³j−1´
J
X
K
γn[t,t+∆] (j),
J
j=1
where
γn[t,t+∆] (j) =
bn∆c
X
i=|j|+1
¡
log S∗
t+ ni
− log S∗
t+ i−1
n
¢¡
log S∗
t+
i−|j|
n
− log S∗
t+
¢
i−|j|−1
n
is the j-th “autocovariation” of log S and K : [0, 1] → R is a weight
function (“kernel”) that satisfies K (0) = 1 and K (1) = 0.
For example, the cubic kernel K (x) = 1 − 3x2 + 2x3 can be used.
1/2
The realised kernel
Letting the bandwidth J = J(n) grow to ∞ as n → ∞ turns the
realised kernel into a consistent estimator of the integrated
R t+∆
variance IV[t,t+∆] = t σ2s under rather general assumptions.
The realised kernel satisfies also a central limit theorem.
Remark
In practise the bandwidth J is like a smoothing parameter and it
should be selected based on data. Implementing the whole
procedure requires a fair amount of work.
Back to MMS noise
2/2