Consistently Modeling Joint Dynamics of Volatility and Underlying To Enable Effective Hedging
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Consistently Modeling Joint Dynamics of Volatility and Underlying To Enable Effective Hedging Artur Sepp Bank of America Merrill Lynch, London [email protected]Global Derivatives Trading & Risk Management 2013 Amsterdam April 16-18, 2013 1
Consistently Modeling Joint Dynamics of Volatility and Underlying To Enable Effective Hedging
1) Analyze the dependence between returns and volatility in conventional 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
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1. Consistently Modeling Joint Dynamics of Volatility and
Underlying To Enable Eective Hedging Artur Sepp Bank of America
Merrill Lynch, London [email protected] Global Derivatives
Trading & Risk Management 2013 Amsterdam April 16-18, 2013
1
2. 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 dierent volatility regimes and equity delta
2
3. 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 http://ssrn.com/abstract=2387845
Sepp, A. (2013) Ane Approximation for Moment Generating Func- tion
of Log-Normal Stochastic Volatility Model, Working paper
http://ssrn.com/abstract=2522425 3
4. 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
5. 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 R = 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
6. 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 R = 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 R = 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- ecient is about the same in magnitude 6
7. 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
8. 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)] = dt Here: V (t)
- the instantaneous volatility of price returns b(V ) -
volatility-of volatility that measures the overall uncertainty
about the dynamic of volatility - the correlation coecient that
measures linear dependence be- tween returns in spot and changes in
volatility 8
9. Conventional SV models II. Correlation Correlation is a
linear measure of degree of strength between two variables
Correlation can be sucient 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
10. 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) = dW(0)(t) + 1 2dW(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
)dW(0)(t) + b(V ) 1 2dW(1)(t) (2) Now dV (t) = (V ) dS(t) S(t) + (V
)dW(1)(t) + a(V )dt where (V ) = b(V ) V , (V ) = b(V ) 1 2 10
11. Volatility beta II The volatility beta (V ) is interpreted
as a rate of change in the volatility given change in the spot with
functional dependence on V : (V ) = b(V ) V For log-normal
volatility (SABR): b(V ) = V so (V ) is constant: (V ) = For normal
volatility (Heston): b(V ) = so (V ) is inversely propor- tional to
V: (V ) = V For quadratic volatility (3/2 SV model): b(V ) = V 2 so
(V ) propor- tional to V: (V ) = ( )V We can measure the elasticity
of volatility, , with b(V ) = V , through its impact on the
volatility beta 11
12. 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 R = 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 is slightly above one 12
13. 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) = ( V (t))dt + [V (t)]1dS(t) S(t) + [V
(t)]dW(1)(t) (3) Here - the constant rate of change in volatility
given change in the spot - the idiosyncratic volatility of
volatility - the elasticity of volatility - the mean-reversion
speed - the long-term level of the volatility 13
14. Beta SV model. Maximum likelihood estimation I Let xn
denote log-return, xn = ln(S(tn)/S(tn1)) and vn volatility, vn = V
(tn) Apply discretization conditional on xn and yn1 = V (tn1): vn
vn1 = ( vn1)dtn + [vn1]1xn + [vn1] dtnn (4) where n are iid normals
Apply the maximum likelihood estimator for the above model with the
following specications: = 0 - the normal SV beta model = 1 - the
log-normal SV beta model 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
15. S&P500 (RV - estimation using realized vol; IV -
estimation using implied vol) RV RV RV IV IV IV 0 1 1.01 0 1 1.20
-0.08 -0.55 -0.56 -0.21 -0.91 -1.24 0.19 1.10 1.13 0.17 0.50 0.68
3.00 2.87 2.87 3.27 2.83 2.86 0.18 0.17 0.17 0.22 0.22 0.22 ML()
1.44 1.64 1.64 2.34 2.82 2.83 Observations: 1) is close to 1 for
both realized and implied vols 2) is larger for implied volatility
3) is larger for realized vols 4) is about the same for both
realized and implied vols 5) is higher for implied vols 6)
likelihood is larger for the log-normal SV than for normal SV
15
16. FTSE 100 (RV - estimation using realized vol; IV -
estimation using implied vol) RV RV RV IV IV IV 0 1 0.92 0 1 1.11
-0.08 -0.48 -0.41 -0.21 -0.92 -1.09 0.19 1.09 0.94 0.19 0.67 0.79
3.40 3.44 3.40 3.84 3.34 3.33 0.17 0.17 0.17 0.22 0.22 0.22 ML()
1.47 1.63 1.63 2.19 2.53 2.53 Observations: 1) is close to 1 for
both realized and implied vols 2) is larger for implied volatility
3) is larger for realized vols 4) is about the same for both
realized and implied vols 5) is higher for implied vols 6)
likelihood is larger for the log-normal SV than for normal SV
16
17. NIKKEI 225 (RV - estimation using realized vol; IV -
estimation using implied vol) RV RV RV IV IV IV 0 1 0.63 0 1 0.74
-0.08 -0.47 -0.25 -0.18 -0.47 -0.51 0.23 1.27 0.70 0.32 1.27 0.81
3.14 4.23 4.67 4.59 4.23 7.22 0.18 0.23 0.22 0.25 0.23 0.25 ML()
1.11 1.12 1.15 1.70 1.75 1.81 Observations: 1) is less than 1 for
both realized and implied vols 2) is larger for implied volatility
3) is about the same for both vols 4) is higher for implied vols 5)
is about the same for both vols 6) likelihood is larger for the
log-normal SV than for normal SV 17
18. STOXX 50 (RV - estimation using realized vol; IV -
estimation using implied vol) RV RV RV IV IV IV 0 1 0.72 0 1 1.21
-0.10 -0.61 -0.35 -0.23 -0.94 -1.26 0.23 1.19 0.70 0.20 0.68 0.90
3.14 2.59 2.76 3.11 3.10 3.14 0.20 0.21 0.20 0.24 0.24 0.24 ML()
1.28 1.39 1.42 2.05 2.32 2.33 Observations: 1) is close to 1 for
both realized and implied vols 2) is larger for implied volatility
3) is larger for realized vols 4) is higher for implied vols 5) is
higher for implied vols 6) likelihood is larger for the log-normal
SV than for normal SV 18
19. Conclusions The volatility process is closer to being
log-normal ( 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 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 dierent under- lyings and is
robust for both realized and implied volatilities 19
20. Beta SV model. Option Pricing Introduce log-normal beta SV
model under pricing measure: dS(t) S(t) = (t)dt + V (t)dW(0)(t),
S(0) = S dV (t) = ( V (t))dt + V (t)dW(0)(t) + V (t)dW(1)(t), V (0)
= V (5) where E[dW(0)(t)dW(1)(t)] = 0, (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] + (t)UX + 1 2 2 + 2 V 2UV V + ( V
)UV + V 2UXV r(t)U = 0 (6) The beta SV model is not ane in
volatility variable Nevertheless, I develop an accurate ane-like
approximation 20
21. Beta SV model. Approximation I Fix expiry time T and
introduce mean-adjusted volatility process Y (t): Y (t) = V (t) ,
E[V (T)] = + (V (0) )eT Consider ane approximation for the moment
generating function of X with second order in Y : G(t, T, X, Y ; )
= exp X + A(0)(t; T) + A(1)(t; T)Y + A(2)(t; T)Y 2 where Y = V (0)
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) 2 2A(2) +
(A(1))2 + A(1) A(1) 2 + (1/2) 2 q = 0 A (1) t + 2A(2) + (A(1))2 + 2
2 A(1)A(2) A(1) + 2 A(2) 2A(1) 2A(2) 2 + q = 0 A (2) t + (1/2)
2A(2) + (A(1))2 + 4A(1)A(2) + 2 2 (A(2))2 2A(2) A(1) 4A(2) + (1/2)q
= 0 where q = 2 + , = 2 + 2, = 21
22. 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 Liptons formula
(Lipton (2001)): C(t, T, S, Y ) = e T t r(t )dt e T t (t )dt S K 0
G(t, T, x, Y ; ik 1/2) k2 + 1/4 dk where x = ln(S/K) + T t (t )dt
The approximation formula is very accurate with dierences 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
23. 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) = V (t)dW(0)(t) + V (t)dW(1)(t) , V (0) = 0 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:
IMP (S, K) = ln(S/K) f(y) , y = ln(S/K) V (0) (7) f(y) = y 0
J1(u)du = 1 2 + 2 ln J(y) 2 + 2 + (2 + 2)y 2 + 2 J(y) = 1 + 2 + 2
y2 2y 23
24. 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 = =
0.12, = 1.0, = 2.8, = 0.5; Right: V0 = = 0.12, = 1.0, = 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
25. Beta SV model. Implied Volatility Asymptotic III Expand
approximation for implied volatility (7) around k = 0, where k =
ln(K/S) is log-strike: BSM(k) = 0 + 1 2 k + 1 12 2 0 + 2 2 0 k2
Dene skew (typically s = 5%): Skews = 1 2s (IMP (+s) IMP (s)) Thus:
= 2Skews Dene convexity (typically c = 5%): Convexityc = 1 c2 (IMP
(+c) + IMP (c) 2IMP (0)) Thus: = 3IMP (0)Convexityc + 2(Skews)2
25
26. Beta SV model. Implied volatility asymptotic IV As a
result, volatility beta, , 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, , and idiosyncratic
volatility of volatil- ity, , exhibit range-bounded behavior
(ranges are narrower when using longer dated implied vols to infer
and ) 26
27. Summary so far I introduced the beta SV model and provided
the intuition behind its key parameter - the volatility beta, 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 tting 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 dierent volatility regimes and how
to model them using the beta SV model 27
28. Volatility regimes and sticky rules (Derman) I 1)
Sticky-strike: (K; S) = 0 + Skew K S0 1 , ATM(S) = 0 + Skew S S0 1
ATM vol increase as the spot declines - typical of range-bounded
markets 2) Sticky-delta: (K; S) = 0 + Skew K S S0 , ATM(S) (S; S) =
0 The level of the ATM volatility does not depend on spot price
-typical of stable trending markets 3) Sticky local volatility: (K;
S) = 0 + Skew K + S S0 2 , ATM(S) = 0 + 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
29. 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 ATM(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
30. 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: StickyLocal StickyStrike
StickyDelta 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 identication 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 dierent volatility
regimes 30
31. Stickiness ratio I Given price return from time tn1 to tn,
X(tn) = (S(tn)S(tn1))/S(tn1), We make prediction for change in the
ATM volatility: ATM(tn) = ATM(tn1) + 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 denition of
the stickiness ratio: R(tn) = ATM(tn) ATM(tn1) X(tn)Skew5%(tn1) To
estimate stickiness ratio, R,empirically, we apply regression
model: ATM(tn) ATM(tn1) = R Skew5%(tn1) 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
32. 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
34. Stickiness ratio (crisis) for 1m and 1y ATM vols y =
1.6327x R = 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 R = 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
35. Stickiness ratio (recovery) for 1m and 1y ATM vols y =
1.4561x R = 0.6472 0% 5% 10% 15% -6% -5% -4% -3% -2% -1% 0% 1% 2%
3% 4% 5% Dailychangein1mATMvol 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% 1m Skew * price return y = 1.5622x R = 0.6783 0% 5% 10%
15% -2% -2% -1% -1% 0% 1% 1% 2% Dailychangein1yATMvol Stickeness
for 1y ATM vol, recovery period Mar 09-Feb 11 -15% -10% -5% -2% -2%
-1% -1% 0% 1% 1% 2% 1y Skew * price return 35
36. Stickiness ratio (range) for 1m and 1y ATM vols y = 1.3036x
R = 0.6769 0% 5% 10% 15% -6% -4% -2% 0% 2% 4% 6% 8%
Dailychangein1mATMvol Stickeness for 1m ATM vol, range-bnd period
Feb11-Aug12 -15% -10% -5% -6% -4% -2% 0% 2% 4% 6% 8% 1m Skew *
price return y = 1.4139x R = 0.7228 0% 5% 10% 15% -2% -2% -1% -1%
0% 1% 1% 2% 2% 3% Dailychangein1yATMvol Stickeness for 1m ATM vol,
range-bnd period Feb11-Aug12 -15% -10% -5% -2% -2% -1% -1% 0% 1% 1%
2% 2% 3% 1y Skew * price return 36
37. Stickiness ratio V. Conclusions Summary of the regression
model: crisis recovery range-bound 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 signicant
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
38. 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
39. 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 ts 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
40. 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 dierent volatility regimes Goal: create a
dynamic model where R can be model as an input parameter 40
41. 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)]SdW(0)(t), S(0) = S0 dV (t) = V dS(t) S(t)
+ V (t)dW(1)(t) = V V (t)[S(t)]SdW(0)(t) + V (t)dW(1)(t), V (0) = 0
(8) To produce volatility skew and price-vol dependence: S is the
backbone beta, S 0 V is the volatility beta, V 0 Connection to SABR
model (Hagan et al (2002)): V (0) = , S = 1 , V [S(t)]S = , = 1 2
Note that volatility beta, V , measures the sensitivity of
instantaneous volatility to changes in the spot independent on
assumption about the local volatility of the spot Term V [S(t)]S,
if S < 0, increases skew in put wing 41
42. 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 = V and
y = 1 0 SS KS S Limit in k = 0, k = ln(K/S): BSM(k) = SS 0 + 1 2
(0S + V ) k + 1 12 02 S 2 V 0 + 2 2 0 k2 (9) 42
43. 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 = = 0.12, V = 1.0, S = 2.0, = 2.8, = 0.5; Right: V0 = =
0.12, V = 1.0, S = 2.0, = 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
44. Interpretation of model parameters Volatility beta V is a
measure of linear dependence between daily returns and changes in
the ATM volatility: ATM(S(tn)) ATM(S(tn1)) = V S(tn) S(tn1) S(tn1)
The backbone beta S is a measure of daily changes in the logarithm
of the ATM volatility to daily returns on the stock ln [ATM(S(tn))]
ln ATM(S(tn1)) = S S(tn) S(tn1) S(tn1) Next I examine these
regression models empirically, assuming: 1) all skew is generated
by V with S = 0 2) all skew is generated by S with V = 0 44
45. Volatility beta (crisis) for 1m and 1y ATM vols y =
-1.1131x R = 0.7603 0% 5% 10% 15% -15% -10% -5% 0% 5% 10% 15%
Dailychangein1mATMvol Daily change in 1m ATM vol, crisis period Oct
07 - Mar 09 -15% -10% -5% Daily price return y = -0.3948x R =
0.8018 0% 5% 10% 15% -15% -10% -5% 0% 5% 10% 15%
Dailychangein1yATMvol 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
46. Volatility beta (recovery) for 1m and 1y ATM vols y =
-0.9109x R = 0.603 0% 5% 10% 15% -6% -4% -2% 0% 2% 4% 6% 8%
Dailychangein1mATMvol 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 R = 0.6475 0% 5% 10% 15% -6% -4% -2% 0% 2% 4%
6% 8% Dailychangein1yATMvol 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
47. Volatility beta (range) for 1m and 1y ATM vols y = -1.0739x
R = 0.6786 0% 5% 10% 15% -8% -6% -4% -2% 0% 2% 4% 6%
Dailychangein1mATMvol 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 R = 0.7147 0% 5% 10% 15% -8% -6% -4% -2% 0% 2%
4% 6% Dailychangein1yATMvol 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
48. Volatility beta. Summary crisis recovery range-bound
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 dierent 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
49. Backbone beta (crisis) for 1m and 1y ATM vols y = -2.8134x
R = 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 R = 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
50. Backbone beta (recovery) for 1m and 1y ATM vols y =
-3.6353x R = 0.5438 0% 10% 20% 30% -6% -4% -2% 0% 2% 4% 6% 8%
Dailychangeinlog1mATMvol Daily change in log 1m ATM vol, recov
period Mar09-Feb11 -30% -20% -10% -6% -4% -2% 0% 2% 4% 6% 8%
Dailychangeinlog1mATMvol Daily price return y = -1.4274x R = 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
51. Backbone beta (range) for 1m and 1y ATM vols y = -4.5411x R
= 0.6212 0% 10% 20% 30% -8% -6% -4% -2% 0% 2% 4% 6%
Dailychangeinlog1mATMvol 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 R = 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
52. The backbone beta. Summary A crisis recovery range-bound
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 dierent 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
53. The backbone beta. Summary B Change in the level of the ATM
volatility implied by backbone beta S is proportional to initial
value of the ATM volatility High negative value of S implies a big
spike in volatility given a modest drop in the price - a feature of
sticky local volatility model In the gure, using estimated
parameters V = 1.07, S 4.54 in range-bound period, (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
54. 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, ATM(S), and skew Skews: 0SS = ATM(S) 0S + V =
2Skews The rst equation is known as the backbone that denes the
trajec- tory of the ATM volatility given a change in the spot
price: ATM(S) ATM(S0) ATM(S0) S S S0 S0 (10) 54
55. Model implied stickiness and volatility regimes If we
insist on model-inconsistent delta (change in spot with volatility
level unchanged): t backbone beta S to reproduce specied stickiness
ratio adjust V so that the model ts the market skew Using
stickiness ratio R(tn) along with (10), we obtain that empiri-
cally: S(tn) = Skews(tn1) ATM(tn1) R(tn) Thus, given an estimated
value of the stickiness rate we imply S Finally, by mixing
parameters S and V we can produce dierent volatility regimes:
sticky-delta with S = 0 and V 2Skews sticky-local volatility with V
= 0 and S 2Skews/0 From the empirical data we infer that,
approximately, S 60% 2Skews/0 and V = 40% 2Skews 55
56. Illustration of model implied stickiness T = 1/12, V0 =
0.12, = 0.5; wSV = {1.00, 0.75, 0.50, 0.25, 0.00} V = {1.00, 0.75,
0.50, 0.25, 0.00}, S = {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
57. Full beta SV model. Dynamics Pricing version of the beta SV
model is specied under the pricing measure in terms of a normalized
volatility process Y (t): dS(t) S(t) = (t)dt + (1 + Y (t))dW(0)(t),
S(0) = S dY (t) = Y (t)dt + V (1 + Y (t))dW(0)(t) + (1 + Y
(t))dW(1)(t), Y (0) = 0 (11) where E[dW(0)(t)dW(1)(t)] = 0 is the
overall level of volatility: it can be set constant, deterministic,
or local stochastic volatility, LSV (t, S) (parametric, like CEV,
or non- parametric) V is the rate of change in the normalized SV
process Y (t) to changes in the spot For constant volatility and =
, parameters of the normalized SV beta are related as follows: Y
(t) = V (t) 1 , V = V , = , = 57
58. Beta stochastic volatility model. Calibration Parameters of
SV process, V , and are specied before calibration We calibrate the
local volatility LSV (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 LSV
(t, S) we apply the conditional expectation (Lipton 2002): 2 LSV
(T, K)E (1 + Y (T))2 | S(T) = K = 2 LV (T, K) where 2 LV (T, K) is
Dupire local volatility The above expectation is computed by
solving the forward PDE cor- responding to dynamics (11) using
nite-dierence methods and com- puting 2 LSV (T, K) stepping forward
in time (for details, see my GB presentation in 2011) Once LSV (t,
S) is calibrated we use either backward PDE-s or MC simulation for
valuation of non-vanilla options 58
59. Full beta SV model. Monte-Carlo simulation Simulate process
Z(t): Z(t) = ln (1 + Y (t)) , Z(0) = 0 with dynamics: dZ(t) = 1 + 1
2 (V )2 + ()2 eZ(t) dt + V dW(0)(t) + dW(1)(t) The domain of
denition: 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
60. Full beta SV model. Multi-asset Dynamics I. N-asset
dynamics: dSi(t) Si(t) = i(t)dt + (1 + Yi(t))idW (0) i (t) dYi(t) =
iYi(t)dt + V,i(1 + Yi(t))idW (0) i (t) + i(1 + Yi(t))dW (1) i (t)
where E[dW (0) i (t)dW (0) j (t)] = (0) ij dt, {i, j} = 1, ..., N,
where (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)] = (1)dt, {i, j} = 1,
..., N, idiosyncratic volatilities are correlated (we take (1) = 1
to avoid de-correlation) MC simulation: 1) Simulate N Brownian
increments dW (0) i (t) with correlation matrix { (0) ij }; 2)
Simulate N Brownian increments dW (1) i (t) with correlation matrix
{(1)} (only one Brownian is needed if (1) = 1) 60
61. Full beta SV model. Multi-asset Dynamics II Implied
instantaneous cross asset-volatility correlation: dSi(t) Si(t) ,
dYj(t) = V,jj (0) ij (V,jj)2 + (j)2 (0) ij + 1 2 j V,jj 2 so the
correlation is bounded from below by (0) ij and declines less for
large V,j and small j Instantaneous volatility-volatility
correlation (taking (1) = 1): dYi(t), dYj(t) = V,i V,jij (0) ij +
ij(1) (V,i)22 i + (i)2 (V,j)22 j + (j)2 (0) ij 1 1 2 i V,ii + j
V,jj 2 + (1) i V,ii j V,jj (0) ij + 1 2 i V,ii + j V,jj 2 so
volatility-volatility correlation is increased if (1) = 1 61
62. 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
dierent volatility regimes and compute appropriate option delta 5)
Extended the beta SV model for multi-asset dynamics 62
63. 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
reect the views and policies of Bank of America Merrill Lynch Thank
you for your attention! 63
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