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Trading in VIX Derivatives1
Presented at IAQF Thalesian Series
Andrew Papanicolaou
FRE DepartmentNYU TandonBrooklyn NY
April 25th 2017
1Joint work with Marco Avellaneda.1 / 85
Fear
Figure : The VIX is the market’s impulse response to fear.
2 / 85
The VIX Index
For t ≤ T let the options on SPX be
P(t,K ,T ) = e−r(T−t)Et(K − ST )+
C(t,K ,T ) = e−r(T−t)Et(ST − K )+ ,
with Et being risk-neutral expectation.
The VIX is
VIXt =
√√√√2erτ
τ
(∫ EtSt+τ
0
P(t,K , t + τ)dK
K 2+
∫ ∞EtSt+τ
C(t,K , t + τ)dK
K 2
)
with τ = 30 days.
3 / 85
Highlights of Talk
1. The VIX futures curve exhibits stationary behavior, with meanreversion toward a contango.
2. Which is a good model for capturing statistical dynamics ofVIX futures?
3. How do complacency and the unknown factor into astationary model?
4. Can we manage the negative roll yield in VIX ETNs?
5. Is the market showing too little concern?
4 / 85
The VIX Time Series
0
20
40
60
80
100VIX Daily Closing (1/2/2004 to 1/3/2017)
02−Ja
n−2004
30−D
ec−
2004
27−D
ec−
2005
22−D
ec−
2006
21−D
ec−
2007
18−D
ec−
2008
16−D
ec−
2009
14−D
ec−
2010
09−D
ec−
2011
10−D
ec−
2012
06−D
ec−
2013
04−D
ec−
2014
02−D
ec−
2015
29−N
ov−
2016
Figure : The VIX has measured market fear since 2004.
5 / 85
Statistical Properties of VIX & VX Futures
From the VIX time series...
1. time series mean, 17.16
2. time series mode, 12.64
3. augmented Dicky-Fuller stat: reject (no unit root)
Typically...
1. most days the VIX futures is in contango, with backwardationcoming when there’s fear.
2. backwardation mean reverts within a few weeks, which is fastcompared to interest rates or oil
6 / 85
VX Term Structure
Figure : A typical contango, Jan 23, 2017.
7 / 85
VX Term Structure
Figure : Backwardation for 5 months, Aug 17, 2011. Obviousdollar-neutral trade here.
8 / 85
VX Term Structure
Figure : Full on backwardation, Oct 16, 2008
9 / 85
VX Term Structure
Figure : April 11, 2017. Notice the “scoop”, which is typically howtrouble starts.... Perhaps elections in France and the possibility of aFrench exit are causing fear.
10 / 85
VX Term Structure
Figure : VIX weeklies included, April 11, 2017. Weeklies are not liquid.
11 / 85
The “Dull” or Most Likely State
Figure : April 10th 2017. The blue line connects the modes for each VXcontract.
12 / 85
Comparison with Other Stationary Curves
Other curves are stationary, but with different properties,I Oil: normal backwardation and storage theory
I normal for producers to hedge price (Keynes),I 100 years later, storage industry in Cushing OK,
cash-and-carry arb puts lower bound on contango.I inventories/storage lower vol.
I Gold: considering only (21m)3 in human hands, relativelycheap to store, relatively flat curve/low vol
I Rates: Lot’s of instruments (bonds, swaps, etc..), relativelycomplete market; hedgable
Deep contango and high vol in non-storables:
I electricity
I VIX futures
13 / 85
ContangoRollYield
Thisisthemostlikelycurve:Longposi)onsinshort-termVXlosefasterthanlong-term
T
VX
Figure : The most likely yield, and the red arrows illustrating the negativeroll yield for long positions at all maturities.
14 / 85
ANon-Sta*onaryCurve
T
VX
The“Scoop-Shaped”curvewillrevertbacktothecontango.
Figure : A non-stationary or transient state. The black arrows illustratethe positive roll yield for long positions at shorter maturities.
15 / 85
The “Dull” State of Fear
Figure : Sooner or later it becomes normal......
16 / 85
Bergomi’s Model [Bergomi, 2005, Bergomi, 2008]
Let T > 0 denote a future’s maturity,
Ft,T = EtVIXT ∀t ≤ T ,
where Et denotes a time-t conditional risk-neutral expectation.
Bergomi model arises naturally from the risk-neutral martingale,
dFt,TFt,T
=d∑
i=1
σi (T − t)dW it ,
where each σi (t) is a diffusion coefficient that tends toward zero ast →∞, and vector dWt is Brownian increments with correlations
dW it dW
jt = ρijdt .
17 / 85
Bergomi’s Model
The SDE for Ft,T is
Ft,T = Ft0,T exp
d∑i=1
∫ t
t0
σi (T − s)dW is − 1
2
d∑i ,j=1
∫ t
t0
ρijσ2ij(T − s)ds
,
where σ2ij = σiσj and t0 ≤ t is an initial time.
Different kernels:
I The exponential kernel σi (t) = σie−κi t with κi > 0 (Markov)
I Power law σi (t) = t−γ with γ > 0 (fractional Brownianmotion, non-Markov, [Gatheral et al., 2014])
18 / 85
Rolling ContractsDenote τ = T − t to have rolling futures contract,
V τt = Ft,t+τ ,
for which there is the following expression:
V τt = V t+τ
t0exp
(d∑
i=1
∫ t
t0
σi (τ + t − s)dW is
−12
d∑i ,j=1
∫ t
t0
ρijσ2ij(τ + t − s)ds
.
Take the volatility functions to be
σi (t) = σie−κi t
where κi > 0. We take the factor Xt to be a stationary OUprocess,
X it = σi
∫ t
−∞e−κi (t−s)dW i
s ,
and letting t0 tend toward −∞,.......... 19 / 85
Stationary State
.......... we obtain the stationary model for the futures curve,
V τt = V∞ exp
d∑i=1
e−κiτX it −
1
2
d∑i ,j=1
ρij σi σjκi+κj
e−(κi+κj )τ
.
In particular, evaluating at τ = 0 give
VIXt = exp
d∑i=1
X it −
1
2
d∑i ,j=1
ρij σi σjκi+κj
.
20 / 85
The “Dull” or Most Likely State
The mode for this model.....
mode(V τt ) = V∞ exp
−1
2
d∑i ,j=1
ρij σi σjκi+κj
e−(κi+κj )τ
.
This should be a contango....
21 / 85
The Dull State for a 2-Factor Model
0 2 4 6 8 10 1210
15
20
25
30
35
VIX Term Structure from 2−Factor Gaussain OU
maturity (in months)
VIX0 = 12.7028
VIX0 = 24.0596
VIX0 = 32.0561
Figure : 2 Factors, X 1t and X 2
t mean-reverting OU processes. 22 / 85
PCA and Model Selection
PCA from February 8th 2011 to December 15th 2016
I 8 rolling contracts (including the VIX)
I N = 1, 499 days
I each day the VIX and the VX future curve form row entry inN × 8 matrix.
Notation,Vij = ln(V
τjti )− ln(V τj ) ,
where
ln(V τj ) =1
N
∑i
ln(Vτjti ) ,
with i = 1, 2, 3, . . . ,N, and j = 0, 1, 2 . . . , 7 with τj = j 30365 .
23 / 85
PCA and Model SelectionThe singular value decomposition (SVD),
USψ′ = V ,
where
I U is an N × 8 matrix orthonormal columns,
I S is an 8× 8 diagonal matrix containing the singular values,and
I ψ is an 8× 8 orthonormal matrix whose columns are theprincipal components used to form any given futures curve.
In other words, for d ≤ 8 we have
ln(Vτjti ) = ln(V τj ) +
d∑`=1
ai`ψj` ,
where the coefficient matrix is a = US .24 / 85
PCA and Model Selection
0 1 2 3 4 5 6 713
14
15
16
17
18
19
20
VX Curve Reconstructed with 1 Components
0 1 2 3 4 5 6 715.5
16
16.5
17
17.5
18
18.5
VX Curve Reconstructed with 2 Components
0 1 2 3 4 5 6 715.5
16
16.5
17
17.5
18
18.5
VX Curve Reconstructed with 3 Components
Maturity (in months)0 1 2 3 4 5 6 7
15.5
16
16.5
17
17.5
18
18.5
VX Curve Reconstructed with 4 Components
Maturity (in months)
Figure : PCA reconstruction of VX curves for April 10th 2014.
25 / 85
PCA and Model Selection
0 1 2 3 4 5 6 7
25
30
35
40
45
50
VX Curve Reconstructed with 1 Components
0 1 2 3 4 5 6 7
25
30
35
40
45
50
VX Curve Reconstructed with 2 Components
Maturity (in months)0 1 2 3 4 5 6 7
25
30
35
40
45
50
VX Curve Reconstructed with 3 Components
Maturity (in months)0 1 2 3 4 5 6 7
25
30
35
40
45
50
VX Curve Reconstructed with 4 Components
Figure : PCA reconstruction of VX futures curve for August 8th 2011,US credit downgrade.
26 / 85
Mode of PCA Weights
-0.05 0 0.05 0.1
0
20
40
60
80
100
120
Histogram of 1st PCA Weight
-0.15 -0.1 -0.05 0 0.05 0.1
0
20
40
60
80
100
120
140
Histogram of 2nd PCA Weight
Figure : The histogram of weights for the 1st and 2nd principalcomponents, ai1 and ai2 respectively.
27 / 85
The Most Likely Curve via PCA
0 1 2 3 4 5 6 713
14
15
16
17
18
19
20
21
22
Maturity (in months)
Most Likely Curve
most likely curve
mean curve
Figure : Recall mean VIX higher than mode VIX. Here mean curve ishigher than mode curve,
mode(
ln(Vti ))≈ ln(V ) + mode
(ai1)ψ1 + mode
(ai2)ψ2
28 / 85
The Real-World or Statistical ModelThe stationary, risk-neutral process,
dX 1t = −κ1X
1t dt + σ1dW
1t ,
dX 2t = −κ2X
2t dt + σ2dW
2t .
The real-world or statistical dynamics of the bivariate OU processare
dX 1t = κp1(µ1 − X 1
t )dt + σ1dWp,1t ,
dX 2t = κp2(µ2 − X 2
t )dt + σ2dWp,2t ,
where
d
(W 1
t
W 2t
)=
(σ1 00 σ2
)−1(κp1µ1 − (κp1 − κ1)X 1
t
κp2µ2 − (κp2 − κ2)X 2t
)︸ ︷︷ ︸
market price of volatility risk
dt+d
(W p,1
t
W p,2t
).
29 / 85
Parameter Estimation
The data is the observed VIX and VX futures,
Yτji = ln(V
τjti )
with τj = j × 30 days for j = 0, 1, . . . , 7. Let the parameters of theOU process by denoted by θ,
θ = (V∞, κ1, κ2, σ1, σ2, ρ︸ ︷︷ ︸risk neutral
, κp1 , κp2 , µ1, µ2︸ ︷︷ ︸
real world
) .
Using the model
Yi = HθXti + G θ + ετji (observed) , (1)
Xti+1 = AθXti + µθ + ∆W pi+1 (latent) , (2)
where cov(ετji ) = R and cov(∆W p
i+1) = Qθ.
30 / 85
Estimated Parameters
Estimated θ
V∞ 21.2068κ1 0.6879κ2 23.7273σ1 1.3393σ2 1.8297ρ 0.2018
κp1 1.2938κp2 17.0911µ1 0.1904µ2 -0.0056
Table : Estimated VIX risk-neutral mode of 10.5068. For the statistical,the optimization has constraints to look for a mean and mode that areequal those of the VIX data; the estimated model’s statistical mode is12.6400 and it’s mean is 18.9844, compared to the mode and mean ofthe VIX time series of 12.6400 and 17.1639, respectively. The totalfitting error to the VX term structure is 8.50378.
31 / 85
Goodness of Fit
0 1 2 3 4 5 6−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Kalman Filter
t (in years)
X1
X2
0 1 2 3 4 5 6−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Least−Squares Estimator
t (in years)
X1
X2
Figure : Left: The Kalman filter X θi . Right: The least-square estimator
X θ,lsqi .
I Important to use Kalman filter; maintains our a prioripreference for OU dynamics X .
I Daily Least squares (LSQ) estimation can be used afterparameters estimated, but if daily LSQ used in iterativeparameter estimation algorithm then there is overfitting.
32 / 85
Goodness of Fit
0 1 2 3 4 5 610
20
30
40
50
Vt
τ
, τ = 0 days
0 1 2 3 4 5 610
20
30
40
50
Vt
τ
, τ = 30 days
0 1 2 3 4 5 610
20
30
40
Vt
τ
, τ = 60 days
0 1 2 3 4 5 610
20
30
40
Vt
τ
, τ = 90 days
0 1 2 3 4 5 610
20
30
40
Vt
τ
, τ = 120 days
t (in years)0 1 2 3 4 5 6
10
20
30
40
Vt
τ
, τ = 150 days
t (in years)
data
fit
33 / 85
Goodness of Fit
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
200
400
600
800
1000
1200
Innovations
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40
50
100
150
200
250
300
350
400
450
Residuals X θi+1− AθX θ
i− µθ
Figure : Left: The histograms of the innovations νi , which are normallydistributed under the null hypothesis of normally-distributed εi and∆W p. Right: The histograms of the residuals X θ
i+1 − AθX θi − µθ, which
would also be normally distributed under the null hypothesis.
34 / 85
Most Likely VX Curve Given High VIX
The outlier probability for V τt is equivalent to the probability of the
normal random variable exceeding some large M > 0,
Pp
(X 1t + X 2
t ≥ M)≤ exp
−1
2
(M − (µ11 + µ2
2))2(1, 1
)Σ
(11
) ,
where Pp denotes the statistical probability measure, and
Σ =
σ2
1
2κp1
ρσ1σ2
κp1+κp2
ρσ1σ2
κp1+κp2
σ22
2κp2
35 / 85
Most Likely VX Curve Given High VIX
Conditional on an exceedance over a threshold M, the most likelyvalue of (X 1
t ,X2t ) is found by maximizing the joint density subject
to a constraint. The density is
p(x1, x2) =1
2π|Σ|exp
(−1
2
(x1 − µ1, x2 − µ2
)Σ−1
(x1 − µ1
x2 − µ2
)),
and the Lagrangian optimization problem is
minx1,x2
1
2
(x1 − µ1, x2 − µ2
)Σ−1
(x1 − µ1
x2 − µ2
)− δ
(x1 + x2 −M
),
where δ ≥ 0 is a Lagrange multiplier.
36 / 85
Most Likely VX Curve Given High VIX
The solution is (X 1
X 2
)ml(τ,M)
=
(µ1
µ2
)+ δ∗Σ
(11
),
where δ∗ is the optimal Lagrange multiplier, which forM > µ1
1 + µ22 is
δ∗ =M − (µ1
1 + µ22)(
1, 1)
Σ
(11
) ;
obviously δ∗ = 0 if M ≤ µ11 + µ2
2.
37 / 85
Most Likely VX Curve Given High VIX
Maturity (in months)0 5 10 15 20 25
10
15
20
25
30
35
40
Most Likely Curve Given VIX High
most likely curve given X1+X
2 > 1.2
most likely curve
Figure : The most likely curve given X 1 + X 2 ≥ M.
38 / 85
Most Likely VX Curve Given High VIX
Maturity (in months)0 5 10 15 20 25
12
13
14
15
16
17
18
19
20
21
22
Most Likely Curve Given VIX High
most likely curve given X1+X
2 > 0.25
most likely curve
Figure : The most likely curve given X 1 + X 2 ≥ M.
39 / 85
Or the Curve if we know more...
Maturity (in months)0 2 4 6 8 10 12
12
13
14
15
16
17
18
19
20
High VIX with Dull State Expected in 4 Months
Non-Stationary Curvemost likely curve
Figure : Most likely given X 1 + X 2 ≥ M andX 1e−κ1τ4 + X 2e−κ2τ4 = µ1e−κ1τ4 + µ2e−κ2τ4 .
40 / 85
Summary to this Point
I Characterized stationarity of VX curve
I Looked at PCA and found 2 factors is sufficient
I Fit the Gaussian Bergomi model and found it captures someof features.
41 / 85
Complacency
42 / 85
Complacency
Of considerable concern right now.
James Mackintosh of WSJ writes:
“At the moment, the gap between realized and impliedvolatility is normal. Investors are prepared for some rise involatility, but from an abnormally low level.”
Typically, the spread VIX minus realized vol is positive to reflectpremium in options.
However, JM remarks that low VIX isn’t the whole story, as thepremium in SPX put options is more indicative of fear.
43 / 85
Complacency
Indeed, there a several causes and effects:
I stimulus and easy money every time the market goes down;younger traders only know the “Big Dip Era”
I The ETF craze, entire market “waxes and wanes” in unisonbecause of the migration to passive funds
I post-crisis corporate buy backsI Self-fullfilling prophecy:
I VIX will stay low, so short the contango, which in turn drivesdown VIX prices
I many funds shorting the long-dated VIX futures and collectingpremium
I The ZIV is an example of this short
44 / 85
Past VIX Events
Non-Complacent moments in the last 2 decades:
I the Russia crisis 8/98
I the dotcom collape 3/00
I market euphoria begins 1/06
I the credit crunch 8/07
I Lehman collapse 9/08
I Greece debt 5/10
I Eurozone debt/US downgrade 8/11
Brexit and Trump election didn’t have much effect.
The French election had some effect in the last 2 weeks.
45 / 85
We’re Overdue for an Event
I The forest servicepractices controlled burnsas a means to preventdisastrous wildfires.
I We’re overdue for avolatility event, the forestis littered with kindling..,,,
I Is there a volatility forestservice?
46 / 85
The VIX ETNs
0
5000
10000
01-M
ar-
2017
03-M
ar-
2016
06-M
ar-
2015
10-M
ar-
2014
12-M
ar-
2013
12-M
ar-
2012
15-M
ar-
2011
18-M
ar-
2010
VXX and VXZ Daily Closing (3/1/2010 to 3/1/2017)
0
200
400
VXX
VXZ
Figure : Negative roll yields for positions in front end or in the back end.Front end is more volatile. $1 million invested in the first VIX ETN in2009 would be $600 today. 47 / 85
Negative Roll Yield
-6
-4
-2
0
2
4
6
8
07-3
0-1
6
10-2
4-1
5
01-1
7-1
5
Contango Yield
04-1
2-1
4
06-2
8-1
3
09-0
8-1
2
11-2
2-1
1
02-0
8-1
1
V2-V1
V7-V4
Figure : Negative roll yields for positions in front end or in the back end.Front end is more volatile.
48 / 85
Pairs Trading for VIX ETNs
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Co-Integrated Time Series log(VXX)-β1-β
2log(VXY)
01-M
ar-
2017
03-M
ar-
2016
06-M
ar-
2015
10-M
ar-
2014
12-M
ar-
2013
12-M
ar-
2012
15-M
ar-
2011
18-M
ar-
2010
Figure : Engle-Granger rejects H0: ln(VXX ) and ln(VXY ) have noco-integration. Run regression ln(VXX ) = β1 + β2 ln(VXY ), thenDickey-Fuller test rejects unit root in residual.
49 / 85
Construction of ETNsETNs roll between 2 or more VX contracts
I let t be the calendar dateI let T1 and T2 be first and second expirations after t.I let r = T2 − T1. i.e. number business days in the period
between T1 and T2
I let θ be number business days between today and 1 monthfrom today.
Figure : Contracts Ft,T1 and Ft,T2 used in ETN with roll horizon θ.50 / 85
Construction of ETNsFor T1 ≤ t + θ ≤ T2 define
a(t) =T2 − (t + θ)
rand notice
I 0 ≤ a(t) ≤ 1I a(T1 − θ) = 1 and a(T2 − θ) = 0I linear in t.
Denote interpolation as
Vt = a(t)FT1t + (1− a(t))FT2
t .
The change in Vt is
dVt =(a(t)FT1
t − a(t)FT2t
)dt + a(t)dFT1
t + (1− a(t))dFT2t
=1
θ
(FT2t − FT1
t
)dt + a(t)dFT1
t + (1− a(t))dFT2t
51 / 85
Construction of ETNs
The value of the ETN in given by E given by the formula
dEt
Et=
a(t)dFT1t + (1− a(t))dFT2
t
a(t)FT1t + (1− a(t))FT2
t
+ rdt
where r is the interest rate.
This corresponds to the variation in a futures trading account inwhich an investor buys the front contract with 100% of his capitaland then gradually buys the second contract and sells the first(“rolls”) – buying calendar spreads on each date – so as to be fullyinvested in the ‘first contract after the next expiration date, and soon.
52 / 85
The Roll Yield
The continuous-time analogue of the evolution of VXX (or anycontinuously rolled ETF/ETN) is therefore:
dEt
Et=
dV θt
V θt
− ∂ ln(V θt )
∂θdt + rdt.
Note: The futures strategy implemented by the ETN issuermaintains the number of contracts constant in the roll, bytransferring a number of contracts equal to
Total number of contracts
number of business days between expirations
from one maturity to the next one, by buying that number ofcalendar spreads daily.
53 / 85
Losing to Contango
Since under a risk-neutral measure all self-financing strategiesshould return zero, we should have
Et
[dV θt
V θt
− ∂ ln(V θt )
∂θdt]
= rdt, (3)
where Et [∗] is conditional expectation.
I This statement does not hold under the empirical measure,
since we expect V θt to be mean-reverting and ∂ ln(V θt )
∂θ to bemostly positive,
I due to contango.
54 / 85
Simulated ETNs
0 0.5 1 1.5 2 2.5 30
100
200
ETNs and VIX
ET
N V
alu
e
1 Month ETN
7 Month ETN
VIX
0 0.5 1 1.5 2 2.5 30
50
100
VIX
Figure : VXX and VXY simulated from fitted 2-factor model.
55 / 85
Simulated ETNs
0 0.5 1 1.5 2 2.5 3−100
−50
0
50
100
Short 1m, Long 7m
ET
N V
alu
e
0 0.5 1 1.5 2 2.5 30
20
40
60
80
VIX
Figure : A simulated long-short in VXX and VXY.
56 / 85
2nd Summary
I Complacency and the risk of being too sure are of concernright now (April 2017)
I ETNs are a bi-product of successful bets on low vol (and ingeneral the ETN craze)
I ETN’s losses are due to negative roll yield, which is similar tothe insurance premium for low-strike SPX put options.
57 / 85
The Distribution of Fear
Figure : By buying and selling VIX options, traders contribute towards acollective distribution on the Fear Index.
58 / 85
VIX Tail Hedge (VXTH) Enter/Exit Strategy
A portfolio to hedge against tail events:
I 1% of portfolio weight in 30-∆ VIX call options if VIX inrange of 15%-30%; rest in S&P500 stocks
I 1/2% of portfolio weight in 30-∆ VIX call options if VIX inrange of 30%-50%; rest in S&P500 stocks
I 0% of portfolio weight in VIX options if VIX outside range of15%-50%; all in S&P500 stocks
Big losses in the S&P500 coincide with spikes in VIX. VXTHprovides positive cash flow on these days.
S&P 500 gained about 15% between 2006 and 2011; VXTHgained about 40%
59 / 85
VIX Tail Hedge (VXTH)
60 / 85
Non-Orthogonality of Markets
61 / 85
Time-Spread Portfolio [Papanicolaou, 2016]
A future on VIXsquared: Vt,T = EtVIX2T
Vt,T =2er(T+τ−t)
τ
∫ ∞0
(P(t,K ,T + τ)− P(t,Ke−rτ ,T )
) dKK 2
i.e. Vt,T is not only an index, but an asset that can be held,
Vt,T = 2
∫ ∞0
Cvix(t,K ,T )dK ,
where Cvix(t,K ,T ) denotes a VIX call option.
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Prediction
Figure : The VIX premium swells when there is unknown.....
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Past VIX Events
I the Russia crisis 8/98
I the dotcom collape 3/00
I market euphoria begins 1/06
I the credit crunch 8/07
I Lehman collapse 9/08
I Greece debt 5/10
I Eurozone debt/US downgrade 8/11
Brexit and Trump election didn’t have much effect.
Ambiguity and Overconfidence [Brenner et al., 2011].
What if France exits???
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French Politics
It’s going to be how it was,..... which was pretty OK.
Or look out!! Will French exithave consequences???
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I don’t get it??
I We have convinced ourselves that VIX is stationary
I ..... that the VX curve is stationary
I ..... and we even have a reasonable model
Then how is it that Marine Le Pen means all bets are off?
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What is a Model for Risk?
I All this time researchingand hedging,
I but then a jockey walksby at the bettingwindow....
I Hey Sherlock!! Look upfor just a second and seethat this jockey is ajuicer!!
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How to Store?
Figure : Where can I purchase the Smell of Fear?
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Fear in a Bottle
How can we buy tomorrow’s volatility?
I Think about cash-and-carry strategies for contango instorables (e.g oil or gas).
I Already talked about complacency and unhedged shorting oflong-dated VX.
I There is no carriable form of volatility.
I How can we implement a cash and carry?
It’s not clear how to buy tomorrow’s volatility.... today.
Suitable instruments for constructing future contracts:
I time-spread portfolio
I forward-start options
I compound options.
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In Terms of Greeks
Let Θ denote option sensitivity to changes in time to maturity,
Θ(t,K ,T ) = − ∂
∂TP(t,K ,T ) .
Time-spread portfolio can be written as
Vt,T = −2
τ
∫ ∞0
∫ τ
0Θ(t,K ,T + u)du
dK
K 2
P(t,T + τ,K )− P(t,T ,K ) = −∫ τ
0Θ(t,K ,T + u)du .
VOLATILITY RISK QUANTIFIED AS EXPOSURE TOTIME
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K40 42 44 46 48 50 52 54 56
price
0
1
2
3
4
5
6
7
8
P(t, T + τ,K)−P(t, T,K) = −
∫τ
0 Θ(t, T + u,K)du
Height of Shaded Area = −
∫τ
0 Θ(t, T + u,K)du
Intrinsic Value
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Can Zero-Θ Instruments be Used to Construct Storage?
I We see the usefulness of instruments that do not decay due topassage of time,
I Time Spread portfolio is a contract on future volatility..... nota carry of today’s.
It seems really hard to carry fear.
An equivalent trade to cash-and-carry for VX has yet to bemade.
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The Thrill of Making Money
Figure : “The Snap”, the most iconic moment in surfing, Tom Carrollputting it all on the line @ the Banzai Pipeline, North Shore of Oahu ’91.The complacent line was to go straight, but Tom saw differently.
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Thank You!
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Generalized OU
We use a factor model where the factors are given by an Rd -valuedprocess Xt that is mean reverting and in its stationary state,
Xt =
∫ t
−∞e−κ(t−s)dLs ,
where κ > 0 is a positive definite matrix with
‖κ−1‖ = decorrelation time,
and Lt is an Rd -valued stable Levy process having triple (a, σ2, ν)and Levy-Khintchine representation (see Chapter 1.2 in[Applebaum, 2004])
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Generalized OU
logEe i〈u,L1〉 = i 〈u, a〉 − ‖σu‖2
2
+
∫Rd\0
(e i〈x ,a〉 − 1− i 〈x , a〉 1|x |<1
)ν(dx) ,
where σ is an invertible volatility matrix for a diffusion component,and ν(x) is an intensity measure with
∫Rd\0(1 ∧ |x |2)ν(dx) <∞.
Using the model from[Bergomi, 2005, Bergomi, 2008, Ould Aly, 2014], the termstructure is
V τt = V∞ee
−κτXt− 12E[[e−κτXt ]] , (4)
where [[ · ]] denotes quadratic (cross) variation of a vector-valuedprocess.
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Generalized OUAssuming the moment generating function (MGF) exists,
Λ1(u) = logEeuL1 <∞ ,
for 0 ≤ u < K <∞. If E log(1 + |L1|
)<∞ then
ΛX (u) = logEeuXt =1
κ
∫ u
0
dz
zΛ1(z) ,
for all u ∈ [0,K ). Now we can construct the VIX futures curve,
V τt = V∞ exp
(e−κτXt − ΛX (e−κτ )
),
so thatVIXt = V∞eXt−ΛX (1) ,
and the relation
V τt = V∞
(VIXt
V∞eΛX (1)
)e−κτ
× ϕ(τ) (5)
ϕ(τ) = e−ΛX (e−κτ ) . (6)
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Generalized OU
The yield is
log
(V τt
VIXt
)= (1− e−κτ ) log
(V∞
VIXt
)+ e−κτΛX (1)− ΛX (e−κτ ) .
(7)
Now let m(x) denote the density of Xt ’s distribution. The Fouriertransform of m is
m(q) =
∫e−ixqm(x)dx = eΛX (−iq) ,
and so the density is given by
m(x) =1
2π
∫e ixu+ΛX (−iu)du .
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Generalized OU
The mode (for a unimodal distribution) is x∗ such that
m′(x∗) =i
2π
∫ue ix
∗u+ΛX (−iu)du = 0 ,
and the most-likely value of VIX in the dull-state is
mode(VIXt) = V∞ex∗−ΛX (1) .
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The Double Nelson Model [Bayer et al., 2013]
Take VIXt = X 1t + X 2
t where
dX 1t = κ1(µ1 − X 1
t )dt + σ1X1t dW
1t
dX 2t = κ1(µ2 − X 2
t )dt + σ2X1t dW
2t .
Has heavy tailedness.
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Kalman Filtering
X θi = Eθ[Xti |Y0:i ] ,
Ωθ = Eθ(Xti − X θi )(Xti − X θ
i )tr ,
and which are given by the Kalman filter,
X θi+1 = AθX θ
i + µθ + K θ(Yi+1 − HAθX θ
ti− G θ
), (8)
where
Ωθ = (I − K θHθ)(HθΩθ(Hθ)tr + R
)−1(9)
K θ = Ωθ(Hθ)tr
(HθΩθ(Hθ)tr + R
)−1(10)
Ωθ = Aθ(Aθ)tr − AθΩθ(Hθ)tr
(HθΩθ(Hθ)tr + R
)−1HθΩθ(Aθ)tr + Qθ .
(11)
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Innovations Process
We denote innovation process as
νθi = Yi − HθAθX θi−1 − G θ ,
which is an iid normal random variable under the null hypothesisthat θ is the true parameter value,
νθi ∼ iidN(
0,HθAθΩθ(HθAθ)tr + HθQθ(Hθ)tr + R).
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Maximum Likelihood Estimation (MLE)Hence there is the log-likelihood function,
L(Y1:N |θ,R)
= −1
2
N∑i=1
∥∥∥∥(HθAθΩθ(HθAθ)tr + HθQθ(Hθ)tr + R)−1/2
νθi
∥∥∥∥2
− 1
2ln∣∣∣HθAθΩθ(HθAθ)tr + HθQθ(Hθ)tr + R
∣∣∣ ,where
∣∣∣ · ∣∣∣ denotes matrix determinant. The maximum likelihood
estimate (MLE) is
(θ, R)mle = arg maxθ,R
L(Y1:N |θ,R) .
In practice filtering makes it difficult to implement code for findingan MLE, so instead there are iterated algorithms such asexpectation maximization (EM) that, while suboptimal, willconverge to reasonable parameter estimate.
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Iterative Scheme
Break the parameter space into the risk neutral and real-worldparameters,
θ = (θ1, θ2)
where
θ1 = (V∞, κ1, κ2, σ1, σ2, ρ)
θ2 = (κp1 , κp2 , µ1, µ2) .
The following is an iteration method that works:
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Algorithm (Parameter Estimation)Initialize with parameter estimates θ(0) = (θ
(0)1 , θ
(0)2 ) and R(0).
1. Compute Kalman Filter using θ(0) and R(0), and re-estimate θ1,
θ1 = arg minθ1
N∑i=1
‖Yi − H θ(0)
X θ(0)
i − G θ‖2 ,
and replace θ(0)1 with θ1;
2. Re-estimate θ2 and matrix R with least-squares estimators,
X θ,lsqi =
((Hθ)trHθ
)−1
(Hθ)tr(Yi − G θ) ,
residuals ηθi = Yi − HθX θ,lsqi with ηθ = 1
N
∑i ηθi and covariance
θ2 = arg minθ2
N∑i=1
∥∥∥(Qθ)−1/2(X lsq
i+1 − AθX lsqi − µ
θ)∥∥∥2
,
R =1
N
N∑i=1
(ηθi − ηθ
)(ηθi − ηθ
)tr.
Replace θ(0)2 with θ2, replace R(0) with R, and repeat from step #1.
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Applebaum, D. (2004).Levy Processes and Stochastic Calculus.CambridgeUniversity Press, Cambridge UK.
Bayer, C., Gatheral, J., and Karlsmark, M. (2013).Fast Ninomiya-Victoir calibration of the double-mean-revertingmodel.Quantitative Finance, 13(11):1813–1829.
Bergomi, L. (2005).Smile dynamics II.Risk, pages 67–73.
Bergomi, L. (2008).Smile dynamics III.Risk, pages 90–96.
Brenner, M., Izhakian, Y., and Sade, O. (2011).Ambiguity and overconfidence.SSRN 2284652.
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Gatheral, J., Jaisson, T., and Rosenbaum, M. (2014).Volatility is rough.Available at SSRN 2509457.
Ould Aly, S. M. (2014).Forward variance dynamics: Bergomi’s model revisited.Applied Mathematical Finance, 21(1):84–107.
Papanicolaou, A. (2016).Analysis of VIX markets with a time-spread portfolio.Applied Mathematical Finance, 23(5):374–408.
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