Consistently Modeling Joint Dynamics of Volatility and Underlying To Enable Effective Hedging

Consistently Modeling Joint Dynamics of Volatility and
Underlying To Enable Eﬀective Hedging
Artur Sepp
Bank of America Merrill Lynch, London
artur.sepp@baml.com
Global Derivatives Trading & Risk Management 2013
Amsterdam
April 16-18, 2013
1

Plan of the presentation
1) Analyze the dependence between returns and volatility in conven-
tional stochastic volatility (SV) models
2) Introduce the beta SV model by Karasinski-Sepp, ”Beta Stochastic
Volatility Model”, Risk, October 2012
3) Illustrate intuitive and robust calibration of the beta SV model to
historical and implied data
4) Mix local and stochastic volatility in the beta SV model to produce
diﬀerent volatility regimes and equity delta
2

References
Some theoretical and practical details for my presentation can be
found in:
Karasinski, P., Sepp, A. (2012) Beta Stochastic Volatility Model, Risk
Magazine October, 66-71
http://ssrn.com/abstract=2150614
Sepp, A. (2013) Empirical Calibration and Minimum-Variance Delta
Under Log-Normal Stochastic Volatility Dynamics, Working paper
Sepp, A. (2013) Log-Normal Stochastic Volatility Model: Pricing of
Vanilla Options and Econometric Estimation, Working paper
3

Empirical analysis I
First, start with some empirical analysis to motivate the choice of
beta SV model
Data:
1) realized one month volatility of daily returns on the S&P500 index
from January 1990 to March 2013 (276 observations) computed from
daily returns within single month
2) the VIX index at the end of each month as a measure of implied
one-month volatility of options on the S&P500 index
4

Empirical analysis II
One month realized volatility is strongly correlated to implied
volatility
Left: time series of realized and implied volatilities
Right: scatter plot of implied volatility versus realized volatility
30%
40%
50%
60%
70%
80%
Realized 1m vol
VIX
0%
10%
20%
30%
1-Feb-90
1-Dec-90
1-Oct-91
1-Aug-92
1-Jun-93
1-Apr-94
1-Feb-95
1-Dec-95
1-Oct-96
1-Aug-97
1-Jun-98
1-Apr-99
1-Feb-00
1-Dec-00
1-Oct-01
1-Aug-02
1-Jun-03
1-Apr-04
1-Feb-05
1-Dec-05
1-Oct-06
1-Aug-07
1-Jun-08
1-Apr-09
1-Feb-10
1-Dec-10
1-Oct-11
1-Aug-12
y = 0.73x + 0.09
R² = 0.78
30%
40%
50%
60%
70%
80%
VIX
0%
10%
20%
30%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90%
Realized 1m volatility
Observation: the level of realized volatility explains 78% of the level
of implied volatility
5

Empirical analysis III
Price returns are negatively correlated with changes in volatility
Left: scatter plot of monthly changes in the VIX versus monthly
returns on the S&P 500 index
Right: the same for the realized volatility
y = -0.67x
R² = 0.49
0%
5%
10%
15%
20%
25%
-25% -20% -15% -10% -5% 0% 5% 10% 15%
MonthlychangeinVIX
-20%
-15%
-10%
-5%
-25% -20% -15% -10% -5% 0% 5% 10% 15%
MonthlychangeinVIX
SP500 monthly return
y = -0.50x
R² = 0.12
0%
10%
20%
30%
40%
Monthlychangein1mrealizedvol
-30%
-20%
-10%
0%
-25% -20% -15% -10% -5% 0% 5% 10% 15%
Monthlychangein1mrealizedvol
SP500 monthly return
Observation: Changes in the S&P 500 index explain over 50% of
variability in the implied volatility (the explanatory power is stronger
for daily and weekly changes, about 80%)
The impact is muted for realized volatility even though the beta co-
eﬃcient is about the same in magnitude
6

Empirical analysis IV
Conclusions for a robust SV model:
1) SV model should be consistent with the dynamics of the realized
volatilities (for short-term vols)
2) SV should describe a robust dependence between the spot and
implied volatility (incorporating of a local vol component, jumps)
Impact on the delta-hedging:
1) leads to estimation of ”correct” gamma P&L (expected implied
vol should be above expected realized vol)
2) leads to estimation of correct vega and vanna, DvegaDspot
7

Conventional SV models I. The dynamics
Start with analysis of dependence between returns and volatility in a
typical SV model:
dS(t)
S(t)
= V (t)dW(0)(t), S(0) = S
dV (t) = a(V )dt + b(V )dW(v)(t), V (0) = V
(1)
with E[dW(0)(t)dW(v)(t)] = ρdt
Here:
V (t) - the instantaneous volatility of price returns
b(V ) - volatility-of volatility that measures the overall uncertainty
about the dynamic of volatility
ρ - the correlation coeﬃcient that measures linear dependence be-
tween returns in spot and changes in volatility
8

Conventional SV models II. Correlation
Correlation is a linear measure of degree of strength between two
variables
Correlation can be suﬃcient for description of static data, but it can
be useless for dynamic data
Question: Given that the correlation between between volatility and
returns is −0.80 and the realized spot return is −1% what is the
expected change in volatility?
Very practical question for risk computation
The concept of correlation alone is of little help for an SV model
9

Volatility beta I
Lets make simple transformation for SV model (1) using decomposi-
tion of Brownian W(v)(t) for volatility process:
dW(v)(t) = ρdW(0)(t) + 1 − ρ2dW(1)(t)
with E[dW(0)(t)dW(1)(t)] = 0
Thus:
dS(t)
S(t)
= V (t)dW(0)(t)
dV (t) = a(V )dt + b(V )ρdW(0)(t) + b(V ) 1 − ρ2dW(1)(t)
(2)
Now
dV (t) = β(V )
dS(t)
S(t)
+ (V )dW(1)(t) + a(V )dt
where
β(V ) =
b(V )ρ
V
, (V ) = b(V ) 1 − ρ2
10

Volatility beta II
The volatility beta β(V ) is interpreted as a rate of change in the
volatility given change in the spot with functional dependence on V :
β(V ) =
b(V )ρ
V
For log-normal volatility (SABR): b(V ) = V so β(V ) is constant:
β(V ) = ρ
For normal volatility (Heston): b(V ) = so β(V ) is inversely propor-
tional to V:
β(V ) =
ρ
V
For quadratic volatility (3/2 SV model): b(V ) = V 2 so β(V ) propor-
tional to V:
β(V ) = (ρ )V
We can measure the elasticity of volatility, α, with b(V ) = V α,
through its impact on the volatility beta
11

Volatility beta III. Empirically, the dependence between the volatility
beta and the level of volatility is weak
Left: scatter plot of VIX beta (regression of daily changes in the VIX
versus daily returns on the S&P500 index within given month) versus
average VIX within given month
Right: the same scatter plot for logarithms of these variables and
corresponding regression model (Heston model would imply the slope
of −1, log-normal - of 0, 3/2 model - of 1)
-1.00
-0.50
0.00
0.50
0% 10% 20% 30% 40% 50% 60% 70%
MonthlyVIXbeta
-2.50
-2.00
-1.50
MonthlyVIXbeta
Average monthly VIX
y = 0.31x + 0.34
R² = 0.06
-1.00
-0.50
0.00
0.50
-250% -200% -150% -100% -50% 0%
log(abs(MonthlyVIXbeta))
-2.00
-1.50
-1.00
log(abs(MonthlyVIXbeta))
log (Average monthly VIX)
This rather implies a log-normal model for the volatility process,
or that the elasticity of volatility α is slightly above one
12

Beta SV model with the elasticity of volatility:
Let us consider the beta SV model with the elasticity of volatility:
dS(t)
S(t)
= V (t)dW(0)(t)
dV (t) = κ(θ − V (t))dt + β[V (t)]α−1dS(t)
S(t)
+ [V (t)]αdW(1)(t)
(3)
Here
β - the constant rate of change in volatility given change in the spot
- the idiosyncratic volatility of volatility
α - the elasticity of volatility
κ - the mean-reversion speed
θ - the long-term level of the volatility
13

Beta SV model. Maximum likelihood estimation I
Let xn denote log-return, xn = ln(S(tn)/S(tn−1)) and
vn volatility, vn = V (tn)
Apply discretization conditional on xn and yn−1 = V (tn−1):
vn − vn−1 = κ(θ − vn−1)dtn + β[vn−1]α−1xn + [vn−1]α
√
dtnζn (4)
where ζn are iid normals
Apply the maximum likelihood estimator for the above model with
the following speciﬁcations:
α = 0 - the normal SV beta model
α = 1 - the log-normal SV beta model
α unrestricted - the SV beta model with the elasticity of volatility
Estimation sample:
Realized vols: compute monthly realized volatility over non-overlapping
time intervals from April 1990 to March 2013 (sample size N = 277)
Implied vols: one-month implied volatilities over weekly non-overlapping
time intervals from October 2007 to March 2013 (N = 272)
Four indices: S&P 500, FTSE 100, NIKKEI 225, STOXX 50
14

S&P500 (RV - estimation using realized vol; IV - estimation using
implied vol)
RV RV RV IV IV IV
α 0 1 1.01 0 1 1.20
β -0.08 -0.55 -0.56 -0.21 -0.91 -1.24
0.19 1.10 1.13 0.17 0.50 0.68
κ 3.00 2.87 2.87 3.27 2.83 2.86
θ 0.18 0.17 0.17 0.22 0.22 0.22
ML(Ω) 1.44 1.64 1.64 2.34 2.82 2.83
Observations:
1) α is close to 1 for both realized and implied vols
2) −β is larger for implied volatility
3) is larger for realized vols
4) κ is about the same for both realized and implied vols
5) θ is higher for implied vols
6) likelihood is larger for the log-normal SV than for normal SV
15

FTSE 100 (RV - estimation using realized vol; IV - estimation using
implied vol)
RV RV RV IV IV IV
α 0 1 0.92 0 1 1.11
β -0.08 -0.48 -0.41 -0.21 -0.92 -1.09
0.19 1.09 0.94 0.19 0.67 0.79
κ 3.40 3.44 3.40 3.84 3.34 3.33
θ 0.17 0.17 0.17 0.22 0.22 0.22
ML(Ω) 1.47 1.63 1.63 2.19 2.53 2.53
Observations:
4) κ is about the same for both realized and implied vols
16

NIKKEI 225 (RV - estimation using realized vol; IV - estimation
using implied vol)
RV RV RV IV IV IV
α 0 1 0.63 0 1 0.74
β -0.08 -0.47 -0.25 -0.18 -0.47 -0.51
0.23 1.27 0.70 0.32 1.27 0.81
κ 3.14 4.23 4.67 4.59 4.23 7.22
θ 0.18 0.23 0.22 0.25 0.23 0.25
ML(Ω) 1.11 1.12 1.15 1.70 1.75 1.81
Observations:
1) α is less than 1 for both realized and implied vols
3) is about the same for both vols
4) κ is higher for implied vols
5) θ is about the same for both vols
17

STOXX 50 (RV - estimation using realized vol; IV - estimation
using implied vol)
RV RV RV IV IV IV
α 0 1 0.72 0 1 1.21
β -0.10 -0.61 -0.35 -0.23 -0.94 -1.26
0.23 1.19 0.70 0.20 0.68 0.90
κ 3.14 2.59 2.76 3.11 3.10 3.14
θ 0.20 0.21 0.20 0.24 0.24 0.24
ML(Ω) 1.28 1.39 1.42 2.05 2.32 2.33
Observations:
4) κ is higher for implied vols
18

Conclusions
The volatility process is closer to being log-normal (α ≈ 1) so that
the log-normal volatility is a robust assumption
The volatility is negatively proportional to changes in spot (to lesser
degree for realized volatility)
The volatility beta has and idiosyncratic volatility have similar mag-
nitude among 4 indices (apart from NIKKEI 225 which typically has
more convexity in implied vol), summarized below
β β
RV IV RV IV
S&P500 -0.55 -0.91 1.10 0.50
FTSE 100 -0.48 -0.92 1.09 0.67
Nikkei 225 -0.47 -0.47 1.27 1.27
STOXX 50 -0.61 -0.94 1.19 0.68
The beta SV model shares universal features across diﬀerent under-
lyings and is robust for both realized and implied volatilities
19

Beta SV model. Option Pricing
Introduce log-normal beta SV model under pricing measure:
dS(t)
S(t)
= µ(t)dt + V (t)dW(0)(t), S(0) = S
dV (t) = κ(θ − V (t))dt + βV (t)dW(0)(t) + εV (t)dW(1)(t), V (0) = V
(5)
where E[dW(0)(t)dW(1)(t)] = 0, µ(t) is the risk-neutral drift
Pricing equation for value function U(t, T, X, V ) with X = ln S(t):
Ut +
1
2
V 2
[UXX − UX] + µ(t)UX
+
1
2
ε2 + β2 V 2UV V + κ(θ − V )UV + βV 2UXV − r(t)U = 0
(6)
The beta SV model is not aﬃne in volatility variable
Nevertheless, I develop an accurate ”aﬃne-like” approximation
20

Beta SV model. Approximation I
Fix expiry time T and introduce mean-adjusted volatility process Y (t):
Y (t) = V (t) − θ , θ ≡ E[V (T)] = θ + (V (0) − θ)e−κT
Consider aﬃne approximation for the moment generating function of
X with second order in Y :
G(t, T, X, Y ; Φ) = exp −ΦX + A(0)(t; T) + A(1)(t; T)Y + A(2)(t; T)Y 2
where Y = V (0) − θ and A(n)(T; T) = 0, n = 0, 1, 2
Substituting into (6) and keeping only quadratic terms in Y , yields:
A
(0)
t + (1/2)ϑθ
2
2A(2) + (A(1))2 + κ A(1) − ΦβA(1)θ
2
+ (1/2)θ
2
q = 0
A
(1)
t + ϑθ 2A(2) + (A(1))2 + 2ϑθ
2
A(1)A(2) − κA(1) + 2κ A(2)
− 2ΦβA(1)θ − 2ΦβA(2)θ
2
+ θq = 0
A
(2)
t + (1/2)ϑ 2A(2) + (A(1))2 + 4ϑθA(1)A(2) + 2ϑθ
2
(A(2))2 − 2κA(2)
− ΦβA(1) − 4ΦβA(2)θ + (1/2)q = 0
where q = Φ2 + Φ, ϑ = ε2 + β2, = θ − θ
21

Beta SV model. Approximation II
This is system of ODE-s is solved by means of Runge-Kutta fourth
order method in a fast way
Thus, for pricing vanilla options under the beta SV model, we can
apply the standard methods based on Fourier inversion (like in Heston
and Stein-Stein SV models)
In particular, the value of the call option with strike K is computed
by applying Lipton’s formula (Lipton (2001)):
C(t, T, S, Y ) = e− T
t r(t )dt
e
T
t µ(t )dt
S −
K
π
∞
0
G(t, T, x, Y ; ik − 1/2)
k2 + 1/4
dk
where x = ln(S/K) + T
t µ(t )dt
The approximation formula is very accurate with diﬀerences between
it and a PDE solver are less than 0.20% in terms of implied volatility
It is straightforward to incorporate time-dependent model parameters
(but not space-dependent local volatility)
22

Beta SV model. Implied volatility asymptotic I
Now study the short-term implied volatility under the beta SV model
Consider beta SV model with no mean-reversion:
dS(t)
S(t)
= V (t)dW(0)(t)
dV (t) = βV (t)dW(0)(t) + V (t)dW(1)(t) , V (0) = σ0
where dW(0)dW(1) = 0
Follow idea from Andreasen-Huge (2013) and obtain the approxima-
tion for the implied log-normal volatility in the beta SV model:
σIMP (S, K) =
ln(S/K)
f(y)
, y =
ln(S/K)
V (0)
(7)
f(y) =
y
0
J−1(u)du =
1
β2 + 2
ln



J(y) β2 + 2 + (β2 + 2)y − β
β2 + 2 − β



J(y) = 1 + β2 + 2 y2 − 2βy
23

Beta SV model. Implied volatility asymptotic II
Illustration of implied log-normal volatility computed by means PDE
solver (PDE), ODE approximation (ODE) and implied vol asymptotic
(IV Asymptotic) for maturity of one-month, T = 1/12
Left: V0 = θ = 0.12, β = −1.0, κ = 2.8, = 0.5;
Right: V0 = θ = 0.12, β = −1.0, κ = 2.8, = 1.0
13%
15%
17%
19%
21%
23%
Impliedvol
PDE
ODE
IV Asymptotic
5%
7%
9%
11%
80% 85% 90% 95% 100% 105% 110%
Strike %
15%
17%
19%
21%
23%
25%
Impliedvol
PDE
ODE
IV Asymptotic
7%
9%
11%
13%
80% 85% 90% 95% 100% 105% 110%
Strike %
Conclusion: approximation for short-term implied vol is very good for
strikes in range [90%, 110%] even in the presence of mean-revertion
24

Beta SV model. Implied Volatility Asymptotic III
Expand approximation for implied volatility (7) around k = 0, where
k = ln(K/S) is log-strike:
σBSM(k) = σ0 +
1
2
βk +
1
12
−
β2
σ0
+ 2
ε2
σ0
k2
Deﬁne skew (typically s = 5%):
Skews =
1
2s
(σIMP (+s) − σIMP (−s))
Thus:
β = 2Skews
Deﬁne convexity (typically c = 5%):
Convexityc =
1
c2
(σIMP (+c) + σIMP (−c) − 2σIMP (0))
Thus:
ε = 3σIMP (0)Convexityc + 2(Skews)2
25

Beta SV model. Implied volatility asymptotic IV
As a result, volatility beta, β, can be interpreted as twice the implied
volatility skew
Idiosyncratic volatility of volatility, , can be interpreted as propor-
tional to the square root of the implied convexity
Time series of these implied parameters are illustrated below
-1.00
-0.50
11-Oct-07
11-Jan-08
11-Apr-08
11-Jul-08
11-Oct-08
11-Jan-09
11-Apr-09
11-Jul-09
11-Oct-09
11-Jan-10
11-Apr-10
11-Jul-10
11-Oct-10
11-Jan-11
11-Apr-11
11-Jul-11
11-Oct-11
11-Jan-12
11-Apr-12
11-Jul-12
11-Oct-12
11-Jan-13
-2.00
-1.50
implied volatility beta
1.50
2.00
2.50
0.50
1.00
11-Oct-07
11-Jan-08
11-Apr-08
11-Jul-08
11-Oct-08
11-Jan-09
11-Apr-09
11-Jul-09
11-Oct-09
11-Jan-10
11-Apr-10
11-Jul-10
11-Oct-10
11-Jan-11
11-Apr-11
11-Jul-11
11-Oct-11
11-Jan-12
11-Apr-12
11-Jul-12
11-Oct-12
11-Jan-13
implied idiosyncratic volatility of volatility
Observation: volatility beta, β, and idiosyncratic volatility of volatil-
ity, , exhibit range-bounded behavior (ranges are narrower when using
longer dated implied vols to infer β and )
26

Summary so far
I introduced the beta SV model and provided the intuition behind its
key parameter - the volatility beta, β
I showed that the beta SV model can adequately describe the his-
toric dynamics of implied and realized volatilities with stable model
parameters
In terms of quality in ﬁtting the implied volatility surface, the beta SV
model is similar to other SV models - the model can explain the term
structure of ATM volatility and longer-term skews (above 6m-1y) but
it cannot reproduce steep short-term skews
For short-term skews, we can introduce local volatility, jumps, a com-
bination of both
The important consideration is the impact on option delta
In the second part of presentation, I concentrate on diﬀerent volatility
regimes and how to model them using the beta SV model
27

Volatility regimes and sticky rules (Derman) I
1) Sticky-strike:
σ(K; S) = σ0 + Skew ×
K
S0
− 1 , σATM(S) = σ0 + Skew ×
S
S0
− 1
ATM vol increase as the spot declines - typical of range-bounded
markets
2) Sticky-delta:
σ(K; S) = σ0 + Skew ×
K − S
S0
, σATM(S) ≡ σ(S; S) = σ0
The level of the ATM volatility does not depend on spot price -typical
of stable trending markets
3) Sticky local volatility:
σ(K; S) = σ0 + Skew ×
K + S
S0
− 2 , σATM(S) = σ0 + 2Skew ×
S
S0
− 1
ATM vol increase as the spot declines twice as much as in the sticky
strike case - typical of stressed markets
28

Sticky rules II
25%
30%
35%
40%
ImpliedVolatility
S(0) = 1.00 goes down to S(1)=0.95 sticky strike
sticky delta
sticky local vol
sigma(S0,K)
Sticky delta
Old ATM Vol
10%
15%
20%
0.90 0.95 1.00 1.05
ImpliedVolatility
Spot Price
Old S(0)New S(1)
Given: Skew = −1.0 and σATM(0) = 25.00%
Spot change: down by −5% from S(0) = 1.00 to S(1) = 0.95
Sticky-strike regime: the ATM volatility moves along the original
skew increasing by −5% × Skew = 5%
Sticky-local regime: the ATM volatility increases by −5%×2Skew =
10% and the volatility skew moves upwards
Sticky-delta regime: the ATM volatility remains unchanged with
the volatility skew moving downwards
29

Impact on option delta
The key implication of the volatility rules is the impact on option
delta ∆
We can show the following rule for call options:
∆Sticky−Local ≤ ∆Sticky−Strike ≤ ∆Sticky−Delta
As a result, for hedging call options, one should be over-hedged (as
compared to the BSM delta) in a trending market and under-hedged
in a stressed market
Thus, the identiﬁcation of market regimes plays an important role to
compute option hedges
While computation of hedges is relatively easy for vanilla options and
can be implemented using the BSM model, for path-dependent exotic
options, we need a dynamic model consistent with diﬀerent volatility
regimes
30

Stickiness ratio I
Given price return from time tn−1 to tn, X(tn) = (S(tn)−S(tn−1))/S(tn−1),
We make prediction for change in the ATM volatility:
σATM(tn) = σATM(tn−1) + Skew × R(tn) × X(tn)
where the stickiness ratio R(tn) indicates the rate of change in the
ATM volatility predicted by skew and price return
Informal deﬁnition of the stickiness ratio:
R(tn) =
σATM(tn) − σATM(tn−1)
X(tn)Skew5%(tn−1)
To estimate stickiness ratio, R,empirically, we apply regression model:
σATM(tn) − σATM(tn−1) = R × Skew5%(tn−1) × X(tn) + n
where X(tn) is realized return for day n; n are iid normal residuals
We expect that the average value of R, R, as follows:
R = 1 under the sticky-strike regime
R = 0 under the sticky-delta regime
R = 2 under the sticky-local regime
31

Stickiness ratio II
Empirical test is based on using market data for S&P500 (SPX)
options from 9-Oct-07 to 1-Jul-12 divided into three zones
crisis recovery range-bound
start date 9-Oct-07 5-Mar-09 18-Feb-11
end date 5-Mar-09 18-Feb-11 31-Jul-12
number days 354 501 365
start SPX 1565.15 682.55 1343.01
end SPX 682.55 1343.01 1384.06
return -56.39% 96.76% 3.06%
start ATM 1m 14.65% 45.28% 12.81%
end ATM 1m 45.28% 12.81% 15.90%
vol change 30.63% -32.47% 3.09%
start Skew 1m -72.20% -61.30% -69.50%
end Skew 1m -57.80% -69.50% -55.50%
skew change 14.40% -8.20% 14.00%
32

Stickiness ratio III
30%
40%
50%
60%
70%
80%
1000
1200
1400
1600
1mATMvol
S&P500
S&P500
1m ATM vol
CRISIS
RECOVERY
RANGE-BOUND
10%
20%
30%
600
800
Oct-07
Dec-07
Mar-08
Jun-08
Sep-08
Dec-08
Mar-09
Jun-09
Aug-09
Nov-09
Feb-10
May-10
Aug-10
Nov-10
Feb-11
Apr-11
Jul-11
Oct-11
Jan-12
Apr-12
Jul-12
33

Stickiness ratio (crisis) for 1m and 1y ATM vols
y = 1.6327x
R² = 0.7738
0%
5%
10%
15%
-12% -10% -8% -6% -4% -2% 0% 2% 4% 6% 8%
Dailychangein1mATMvol
Stickeness for 1m ATM vol, crisis period Oct 07 - Mar 09
-15%
-10%
-5%
-12% -10% -8% -6% -4% -2% 0% 2% 4% 6% 8%
1m Skew * price return
y = 1.5978x
R² = 0.8158
0%
5%
10%
15%
-4% -3% -2% -1% 0% 1% 2% 3%
Dailychangein1yATMvol
Stickeness for 1y ATM vol, crisis period Oct 07 - Mar 09
-15%
-10%
-5%
-4% -3% -2% -1% 0% 1% 2% 3%
1y Skew * price return
34

Stickiness ratio (recovery) for 1m and 1y ATM vols
y = 1.4561x
R² = 0.6472
0%
5%
10%
15%
-6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5%
Stickeness for 1m ATM vol, recovery period Mar 09-Feb 11
-15%
-10%
-5%
-6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5%
y = 1.5622x
R² = 0.6783
0%
5%
10%
15%
-2% -2% -1% -1% 0% 1% 1% 2%
Stickeness for 1y ATM vol, recovery period Mar 09-Feb 11
-15%
-10%
-5%
-2% -2% -1% -1% 0% 1% 1% 2%
35

Stickiness ratio (range) for 1m and 1y ATM vols
y = 1.3036x
R² = 0.6769
0%
5%
10%
15%
-6% -4% -2% 0% 2% 4% 6% 8%
Stickeness for 1m ATM vol, range-bnd period Feb11-Aug12
-15%
-10%
-5%
-6% -4% -2% 0% 2% 4% 6% 8%
y = 1.4139x
R² = 0.7228
0%
5%
10%
15%
-2% -2% -1% -1% 0% 1% 1% 2% 2% 3%
Stickeness for 1m ATM vol, range-bnd period Feb11-Aug12
-15%
-10%
-5%
-2% -2% -1% -1% 0% 1% 1% 2% 2% 3%
36

Stickiness ratio V. Conclusions
Summary of the regression model:
Stickiness, 1m 1.63 1.46 1.30
Stickiness, 1y 1.60 1.56 1.41
R2, 1m 77% 65% 68%
R2, 1y 82% 68% 72%
1) The concept of the stickiness is statistically signiﬁcant explaining
about 80% of the variation in ATM volatility during crisis period and
about 70% of the variation during recovery and range-bound periods
2) Stickiness ratio is
stronger during crisis period, R ≈ 1.6 (closer to sticky local vol)
less strong during recovery period, R ≈ 1.5
weaker during range-bound period, R ≈ 1.35 (closer to sticky-strike)
3) The volatility regime is typically neither sticky-local nor sticky-
strike but rather a combination of both
37

Stickiness ratio VII. Dynamic models A
Now we consider how to model the stickiness ratio within the dynamic
SV models
The primary driver is change in the spot price, ∆S/S
The key in this analysis is what happens to the level of model volatility
given change in the spot price
The model-consistent hedge:
The level of volatility changes proportional to (approximately):
SkewSV Model × ∆S/S
The model-inconsistent hedge:
The level of volatility remains unchanged
Implication for the stickiness under pure SV models:
R = 2 under the model-consistent hedge
R = 0 under the model-inconsistent hedge
38

Stickiness ratio VII. Dynamic models B
How to make R = 1.5 using an SV model
Under the model-consistent hedge: impossible
Under the model-inconsistent hedge: mix SV with local volatility
Remedy: add jumps in returns and volatility
Under any spot-homogeneous jump model, R = 0
The only way to have a model-consistent hedging that ﬁts the desired
stickiness ratio is to mix stochastic volatility with jumps:
the higher is the stickiness ratio, the lower is the jump premium
the lower is the stickiness ratio, the higher is the jump premium
Jump premium is lower during crisis periods (after a big crash or ex-
cessive market panic, the probability of a second one is lower because
of realized de-leveraging and de-risking of investment portfolios, cen-
tral banks interventions)
Jump premium is higher during recovery and range-bound periods (re-
newed fear of tail events, increased leverage and risk-taking given
small levels of realized volatility and related hedging)
39

Stickiness ratio VII. Dynamic models C
The above consideration explain that the stickiness ratio is
stronger during crisis period, R ≈ 1.6 (closer to sticky local vol)
weaker during range-bound and recovery periods, R ≈ 1.35 (closer to
sticky-strike)
To model this feature within an SV model, we need to specify a
proportion of the skew attributed to jumps (see my 2012 presentation
on Global Derivatives)
During crisis periods, the weight of jumps is about 20%
During range-bound and recovery periods, the weight of jumps is
about 40%
For the rest of my presentation, we assume a model-inconsistent
hedge and apply the beta SV model to model diﬀerent volatility
regimes
Goal: create a dynamic model where R can be model as an input
parameter
40

Beta SV model with CEV vol. Incorporate CEV volatility in the
beta SV model (short-term analysis with no mean-reversion):
dS(t)
S(t)
= V (t)[S(t)]βSdW(0)(t), S(0) = S0
dV (t) = βV
dS(t)
S(t)
+ V (t)dW(1)(t)
= βV V (t)[S(t)]βSdW(0)(t) + V (t)dW(1)(t), V (0) = σ0
(8)
To produce volatility skew and price-vol dependence:
βS is the backbone beta, βS ≤ 0
βV is the volatility beta, βV ≤ 0
Connection to SABR model (Hagan et al (2002)):
V (0) = ˆα , βS = β − 1 , βV [S(t)]βS = νρ , = ν 1 − ρ2
Note that volatility beta, βV , measures the sensitivity of instantaneous
volatility to changes in the spot independent on assumption about the
local volatility of the spot
Term βV [S(t)]βS, if βS < 0, increases skew in put wing
41

Beta SV model with CEV vol. Implied volatility
To obtain approximation for the implied volatility under the beta CEV
SV model, we apply formula (7) for implied vol asymptotic with β =
βV and
y =
1
σ0
S−βS − K−βS
−βS
Limit in k = 0, k = ln(K/S):
σBSM(k) = SβS σ0 +
1
2
(σ0βS + βV ) k +
1
12
σ0β2
S −
β2
V
σ0
+ 2
ε2
σ0
k2
(9)
42

Beta SV model with CEV vol. Implied volatility I
Illustration of implied log-normal volatility computed by means PDE
solver (PDE), implied vol asymptotic (IV Asymptotic) and the second
order expansion (2-nd order) for maturity of one-month, T = 1/12
Left: V0 = θ = 0.12, βV = −1.0, βS = −2.0, κ = 2.8, = 0.5;
Right: V0 = θ = 0.12, βV = −1.0, βS = −2.0, κ = 2.8, = 1.0
15%
20%
25%
30%
Impliedvol
PDE
IV Asymptotic
2-nd order
5%
10%
80% 85% 90% 95% 100% 105% 110%
Strike %
20%
25%
30%
35%
Impliedvol
PDE
IV Asymptotic
2-nd order
5%
10%
15%
80% 85% 90% 95% 100% 105% 110%
Strike %
Conclusion: approximation for short-term implied vol is very good for
strikes in range [90%, 110%] even in the presence of mean-revertion
43

Interpretation of model parameters
Volatility beta βV is a measure of linear dependence between daily
returns and changes in the ATM volatility:
σATM(S(tn)) − σATM(S(tn−1)) = βV
S(tn) − S(tn−1)
S(tn−1)
The backbone beta βS is a measure of daily changes in the logarithm
of the ATM volatility to daily returns on the stock
ln [σATM(S(tn))] − ln σATM(S(tn−1)) = βS
S(tn) − S(tn−1)
S(tn−1)
Next I examine these regression models empirically, assuming:
1) all skew is generated by βV with βS = 0
2) all skew is generated by βS with βV = 0
44

Volatility beta (crisis) for 1m and 1y ATM vols
y = -1.1131x
R² = 0.7603
0%
5%
10%
15%
-15% -10% -5% 0% 5% 10% 15%
Daily change in 1m ATM vol, crisis period Oct 07 - Mar 09
-15%
-10%
-5%
Daily price return
y = -0.3948x
R² = 0.8018
0%
5%
10%
15%
-15% -10% -5% 0% 5% 10% 15%
Daily change in 1y ATM vol, crisis period Oct 07 - Mar 09
-15%
-10%
-5%
-15% -10% -5% 0% 5% 10% 15%
Daily price return
45

Volatility beta (recovery) for 1m and 1y ATM vols
y = -0.9109x
R² = 0.603
0%
5%
10%
15%
-6% -4% -2% 0% 2% 4% 6% 8%
Daily change in 1m ATM vol, recovery period Mar 09-Feb 11
-15%
-10%
-5%
-6% -4% -2% 0% 2% 4% 6% 8%
Daily price return
y = -0.3926x
R² = 0.6475
0%
5%
10%
15%
-6% -4% -2% 0% 2% 4% 6% 8%
Daily change in 1y ATM vol, recovery period Mar 09-Feb 11
-15%
-10%
-5%
-6% -4% -2% 0% 2% 4% 6% 8%
Daily price return
46

Volatility beta (range) for 1m and 1y ATM vols
y = -1.0739x
R² = 0.6786
0%
5%
10%
15%
-8% -6% -4% -2% 0% 2% 4% 6%
Daily change in 1m ATM vol, range-bnd period Feb11-Aug12
-15%
-10%
-5%
-8% -6% -4% -2% 0% 2% 4% 6%
Daily price return
y = -0.4374x
R² = 0.7147
0%
5%
10%
15%
-8% -6% -4% -2% 0% 2% 4% 6%
Daily change in 1y ATM vol, range-bnd period Feb11-Aug12
-15%
-10%
-5%
-8% -6% -4% -2% 0% 2% 4% 6%
Daily price return
47

Volatility beta. Summary
Volatility beta 1m -1.11 -0.91 -1.07
Volatility beta 1y -0.39 -0.39 -0.44
R2 1m 76% 60% 68%
R2 1y 80% 65% 71%
The volatility beta is pretty stable across diﬀerent market regimes
The longer term ATM volatility is less sensitive to changes in the spot
Changes in the spot price explain about:
80% in changes in the ATM volatility during crisis period
60% in changes in the ATM volatility during recovery period (ATM
volatility reacts slower to increases in the spot price)
70% in changes in the ATM volatility during range-bound period
(jump premium start to play bigger role n recovery and range-bound
periods)
48

Backbone beta (crisis) for 1m and 1y ATM vols
y = -2.8134x
R² = 0.6722
0%
10%
20%
30%
-15% -10% -5% 0% 5% 10% 15%
Dailychangeinlog1mATMvol
Daily change in log 1m ATM vol, crisis period Oct07 - Mar09
-30%
-20%
-10%
-15% -10% -5% 0% 5% 10% 15%
Daily price return
y = -1.2237x
R² = 0.7715
0%
10%
20%
30%
-15% -10% -5% 0% 5% 10% 15%
Dailychangein1ylogATMvol
Daily change in log 1y ATM vol, crisis period Oct 07 - Mar 09
-30%
-20%
-10%
-15% -10% -5% 0% 5% 10% 15%
Daily price return
49

Backbone beta (recovery) for 1m and 1y ATM vols
y = -3.6353x
R² = 0.5438
0%
10%
20%
30%
-6% -4% -2% 0% 2% 4% 6% 8%
Daily change in log 1m ATM vol, recov period Mar09-Feb11
-30%
-20%
-10%
-6% -4% -2% 0% 2% 4% 6% 8%
Daily price return
y = -1.4274x
R² = 0.6137
0%
10%
20%
30%
-6% -4% -2% 0% 2% 4% 6% 8%
Dailychangeinlog1yATMvol
Daily change in log 1y ATM vol, recov period Mar09-Feb11
-30%
-20%
-10%
-6% -4% -2% 0% 2% 4% 6% 8%
Daily price return
50

Backbone beta (range) for 1m and 1y ATM vols
y = -4.5411x
R² = 0.6212
0%
10%
20%
30%
-8% -6% -4% -2% 0% 2% 4% 6%
Daily change in log 1m ATM vol, range period Feb11-Aug12
-30%
-20%
-10%
-8% -6% -4% -2% 0% 2% 4% 6%
Daily price return
y = -1.8097x
R² = 0.6978
0%
10%
20%
30%
-8% -6% -4% -2% 0% 2% 4% 6%
Dailychangeinlog1yATMvol
Daily change in log 1y ATM vol, range period Feb11-Aug12
-30%
-20%
-10%
-8% -6% -4% -2% 0% 2% 4% 6%
Daily price return
51

The backbone beta. Summary A
Backbone beta 1m -2.81 -3.64 -4.54
Backbone beta 1y -1.22 -1.43 -1.81
R2 1m 67% 54% 62%
R2 1y 77% 61% 70%
The value of the backbone beta appears to be less stable across
diﬀerent market regimes (compared to volatility beta)
Explanatory power is somewhat less (by 5-7%) for 1m ATM vols
(compared to volatility beta)
Similar explanatory power for 1y volatilities
52

The backbone beta. Summary B
Change in the level of the ATM volatility implied by backbone beta
βS is proportional to initial value of the ATM volatility
High negative value of βS implies a big spike in volatility given a
modest drop in the price - a feature of sticky local volatility model
In the ﬁgure, using estimated parameters βV = −1.07, βS − 4.54 in
range-bound period, σ(0) = 20%
8%
10%
12%
14%
16%
Changeinvollevel
Predicted change in volatility
volatility beta_v=-1.07
backbone beta_s=-4.54
0%
2%
4%
6%
-10% -9% -7% -6% -4% -2% -1%
Changeinvollevel
Spot return %
53

Model implied skew
Using approximation (9) for short-term implied volatility, obtain the
following approximate but accurate relationship between the model
parameters and short-term implied ATM volatility, σATM(S), and skew
Skews:
σ0SβS = σATM(S)
σ0βS + βV = 2Skews
The ﬁrst equation is known as the backbone that deﬁnes the trajec-
tory of the ATM volatility given a change in the spot price:
σATM(S) − σATM(S0)
σATM(S0)
≈ βS
S − S0
S0
(10)
54

Model implied stickiness and volatility regimes
If we insist on model-inconsistent delta (change in spot with volatility
level unchanged):
fit backbone beta βS to reproduce specified stickiness ratio
adjust βV so that the model fits the market skew
Using stickiness ratio R(tn) along with (10), we obtain that empiri-
cally:
βS(tn) =
Skews(tn−1)
σATM(tn−1)
R(tn)
Thus, given an estimated value of the stickiness rate we imply βS
Finally, by mixing parameters βS and βV we can produce different
volatility regimes:
sticky-delta with βS = 0 and βV ≈ 2Skews
sticky-local volatility with βV = 0 and βS ≈ 2Skews/σ0
From the empirical data we infer that, approximately,
βS ≈ 60% × 2Skews/σ0 and βV = 40% × 2Skews
55

Illustration of model implied stickiness
T = 1/12, V0 = 0.12, = 0.5;
wSV = {1.00, 0.75, 0.50, 0.25, 0.00}
βV = {−1.00, −0.75, −0.50, −0.25, 0.00},
βS = {0.00, −2.08, −4.17, −6.25, −8.33}
The initial skew and ATM vol is the same for all values of wSV
Left: skew change given spot return of −5%
Right: corresponding ATM vol (lhs) and stickeness (rhs)
20%
30%
40%
50%
60%
Impliedvolatility
Initial Skew
SV=1
SV=0.75
SV=0.5
SV=0.25
0%
10%
20%
75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125%
K/S(0)
SV=0.25
SV=0
1.50
2.00
2.50
3.00
15%
16%
17%
18%
19%
Stickines
ATMimpliedvol
ATM Vol (left)
Stickiness (right)
0.00
0.50
1.00
11%
12%
13%
14%
Initial Skew SV=1 SV=0.75 SV=0.5 SV=0.25 SV=0
ATMimpliedvol
Conclusion: the stickiness ratio is approximately equal to twice the
weight of the SV implied skew
56

Full beta SV model. Dynamics
Pricing version of the beta SV model is speciﬁed under the pricing
measure in terms of a normalized volatility process Y (t):
dS(t)
S(t)
= µ(t)dt + (1 + Y (t))σdW(0)(t), S(0) = S
dY (t) = −˜κY (t)dt + ˜βV (1 + Y (t))σdW(0)(t) + ˜ε(1 + Y (t))dW(1)(t), Y (0) = 0
(11)
where E[dW(0)(t)dW(1)(t)] = 0
σ is the overall level of volatility: it can be set constant, deterministic,
or local stochastic volatility, σLSV (t, S) (parametric, like CEV, or non-
parametric)
˜βV is the rate of change in the normalized SV process Y (t) to changes
in the spot
For constant volatility σ and θ = σ, parameters of the normalized SV
beta are related as follows:
Y (t) =
V (t)
σ
− 1 , ˜βV =
βV
σ
, ˜κ = κ , ˜ε = ε
57

Beta stochastic volatility model. Calibration
Parameters of SV process, ˜βV , ˜ε and ˜κ are specified before calibration
We calibrate the local volatility σ ≡ σLSV (t, S), using either a para-
metric local volatility (CEV) or non-parametric local volatility, so that
the vanilla surface is matched by construction
For calibration of σLSV (t, S) we apply the conditional expectation
(Lipton 2002):
σ2
LSV (T, K)E (1 + Y (T))2
| S(T) = K = σ2
LV (T, K)
where σ2
LV (T, K) is Dupire local volatility
The above expectation is computed by solving the forward PDE cor-
responding to dynamics (11) using finite-difference methods and com-
puting σ2
LSV (T, K) stepping forward in time (for details, see my GB
presentation in 2011)
Once σLSV (t, S) is calibrated we use either backward PDE-s or MC
simulation for valuation of non-vanilla options
58

Full beta SV model. Monte-Carlo simulation
Simulate process Z(t):
Z(t) = ln (1 + Y (t)) , Z(0) = 0
with dynamics:
dZ(t) = −˜κ 1 +
1
2
(˜βV )2 + (˜ε)2 − e−Z(t) dt + ˜βV σdW(0)(t) + ˜εdW(1)(t)
The domain of deﬁnition:
Y (t) ∈ (−1, ∞) , Z(t) ∈ (−∞, ∞)
Obtain the instanteneous volatility by inversion:
Y (t) = eZ(t) − 1
No problem with boundary at zero that exist in some SV models
59

Full beta SV model. Multi-asset Dynamics I. N-asset dynamics:
dSi(t)
Si(t)
= µi(t)dt + (1 + Yi(t))σidW
(0)
i (t)
dYi(t) = −˜κiYi(t)dt + ˜βV,i(1 + Yi(t))σidW
(0)
i (t) + ˜εi(1 + Yi(t))dW
(1)
i (t)
where E[dW
(0)
i (t)dW
(0)
j (t)] = ρ
(0)
ij dt, {i, j} = 1, ..., N, where ρ
(0)
ij is ith
asset - jth asset correlation
E[dW
(0)
i (t)dW
(1)
i (t)] = 0, i = 1, ..., N
E[dW
(0)
i (t)dW
(1)
j (t)] = 0, {i, j} = 1, ..., N
E[dW
(1)
i (t)dW
(1)
j (t)] = ρ(1)dt, {i, j} = 1, ..., N, idiosyncratic volatilities
are correlated (we take ρ(1) = 1 to avoid de-correlation)
MC simulation: 1) Simulate N Brownian increments dW
(0)
i (t) with
correlation matrix {ρ
(0)
ij };
2) Simulate N Brownian increments dW
(1)
i (t) with correlation matrix
{ρ(1)} (only one Brownian is needed if ρ(1) = 1)
60

Full beta SV model. Multi-asset Dynamics II
Implied instantaneous cross asset-volatility correlation:
ρ
dSi(t)
Si(t)
, dYj(t) =
˜βV,jσjρ
(0)
ij
(˜βV,jσj)2 + (˜εj)2
≈ −ρ
(0)
ij +
1
2
˜εj
˜βV,jσj
2
so the correlation is bounded from below by −ρ
(0)
ij and declines less
for large ˜βV,j and small ˜εj
Instantaneous volatility-volatility correlation (taking ρ(1) = 1):
ρ dYi(t), dYj(t) =
˜βV,i ˜βV,jσiσjρ
(0)
ij + ˜εi˜εjρ(1)
(˜βV,i)2σ2
i + (˜εi)2 (˜βV,j)2σ2
j + (˜εj)2
≈ ρ
(0)
ij

1 −
1
2
˜εi
˜βV,iσi
+
˜εj
˜βV,jσj
2

 + ρ(1) ˜εi
˜βV,iσi
˜εj
˜βV,jσj
≤ ρ
(0)
ij +
1
2
˜εi
˜βV,iσi
+
˜εj
˜βV,jσj
2
so volatility-volatility correlation is increased if ρ(1) = 1
61

Summary
1) Presented the beta SV model and illustrated that the model can
describe very well the dynamics of both implied and realized volatilities
2) Obtained an accurate short-term asymptotic for the implied volatil-
ity in the beta SV model and showed how to express the key model
parameters, the volatility beta and idiosyncratic volatility, in terms of
the market implied skew and convexity
3) Derived an accurate closed form solution for pricing vanilla option
in the SV beta model using Fourier inversion method
4) Presented the beta SV model with CEV local volatility to model
diﬀerent volatility regimes and compute appropriate option delta
5) Extended the beta SV model for multi-asset dynamics
62

Final words
I am thankful to the members of BAML Global Quantitative Analytics
The opinions and views expressed in this presentation are those of
the author alone and do not necessarily reﬂect the views and policies
of Bank of America Merrill Lynch
Thank you for your attention!
63

References
Andreasen, J. and B Huge (2013). “Expanded forward volatility”,
Risk, January, 101-107
Hagan, P., Kumar, D., Lesniewski, A., and Woodward, D. (2002) ”,
Managing smile risk, Wilmott Magazine, September, 84-108
Karasinski, P and A. Sepp (2012). “Beta Stochastic Volatility Model”,
Risk, October, 66-71 (http://ssrn.com/abstract=2150614)
Lipton, A. (2001). “Mathematical Methods for Foreign Exchange: a
Financial Engineer’s Approach,” World Scientiﬁc, Singapore
Lipton, A. (2002). “The vol smile problem”, Risk, February, 81-85
Sepp, A. (2012), “Achieving Consistent Modeling Of VIX and Equities
Derivatives”, Global Derivatives conference in Barcelona
64

Sepp, A. (2011), “Eﬃcient Numerical PDE Methods to Solve Cali-
bration and Pricing Problems in Local Stochastic Volatility Models”,
Global Derivatives conference in Paris
Sepp, A., (2013), “When You Hedge Discretely: Optimization of
Sharpe Ratio for Delta-Hedging Strategy under Discrete Hedging and
Transaction Costs,” The Journal of Investment Strategies 3(1), 19-
59
Sepp, A., (2014), “Empirical Calibration and Minimum-Variance Delta
Under Log-Normal Stochastic Volatility Dynamics,” Working paper,
Sepp, A., (2014), “Log-Normal Stochastic Volatility Model: Pric-
ing of Vanilla Options and Econometric Estimation,” Working paper,

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