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Beta stochastic volatility model
Artur SeppBank of America Merrill Lynch, London
Financial Engineering WorkshopCass Business School, London
October 24, 2012
1
Introduction. Quoting Jesper Andreasen from ”Risk 25: No moreheroes in quantitative finance?”, Risk Magazine, August 2012:”Quants have hundreds of models and, even in one given asset class,a quant will have 10 models that can fit the smile. The questionis, which is the right delta? That’s still an open question, evenrestricting it to vanilla business. It’s one reason why there’s solittle activity in the interest rate options markets.”
Quoting Jim Gatheral from the same article: ”With less trading inexotics and vanillas moving to exchanges, we need to focus ongenerating realistic price dynamics for underlyings.”
I present a stochastic volatility model that:1) can be made consistent with different volatility regimes (thus,potentially computing a correct delta)2) is consistent with observed dynamics of the spot and its volatility3) has very intuitive model parameters
The model is based on joint article with Piotr Karasinski:Karasinski P and Sepp A, Beta Stochastic Volatility Model, RiskMagazine, pp. 66-71, October 2012
2
Plan of the presentation
1) Discuss existing volatility models and their limitations
2) Discuss volatility regimes observed in the market
3) Introduce beta stochastic volatility (SV) model
4) Emphasize intuitive and robust calibration of the beta SV model
5) Case study I: application of the beta SV model to model thecorrelation skew
6) Case study II: application of the beta SV model to model theconditional forward skew
3
Motivation I. Applications of volatility models1) Interpolators for implied volatility surface⊗ Represent functional forms for implied volatility at different strikesand maturities⊗ Assume no dynamics for the underlying (”apart from the SABRmodel”)⊗ Applied for marking vanilla options and serve as inputs for calibrationof dynamic volatility models (local vol, stochastic vol)
2) Hedge computation for vanilla options⊗ Apply deterministic rules for changes in model parameters given changein the spot⊗ The most important is the volatility backbone - the change in theATM volatility (and its term structure) given change in the spot price⊗ Applied for computation of hedges for vanilla and exotic books
3) Dynamics models (local vol, stochastic vol, local stochastic vol)⊗ Compute the present value and hedges of exotic options given in-puts from 1) and 2)
4
Motivation II. Missing points1) Interpolators⊗ Do not assume any specific dynamics⊗ Provide a tool to compute the market observables from given snap-shot of market data: at-the-money (ATM) volatility, skew, convexity,and term structures of these quantities
2) Vanilla hedge computation⊗ Assume specific functional rules for changes in market observablesgiven changes in market data
3) Dynamic models (local vol, Heston) for pricing exotic options⊗ No explicit connection to market observables and their dynamics⊗ Apply ”blind” non-linear and non-intuitive fitting methods for cal-ibration of model parameters (correlation, vol-of-vol, etc)
We need a dynamic volatility model that couldconnect all three tools in a robust and intuitive way!
5
Motivation III. Beta stochastic volatility modelPropose the beta stochastic volatility model that:
1) In its simplified form, the model can be used as an interpolator⊗ Takes market observables (ATM volatility and skew) for model cal-ibration with model parameters easily interpreted in terms of marketobservable - volatility skew (with a good approximation)
2) The model is consistent with vanilla hedge computations -it has a model parameter to replicate the volatility backbone (with agood approximation)⊗ The model assumes the dynamics of the ATM volatility specifiedby the volatility backbone
3) The model provides robust dynamics for exotic options:⊗ It produces steep forward skews, mean-reversion⊗ The model has a mean-reversion and volatility of volatility
6
Implied volatility skew IFirst I describe implied volatility skew and volatility regimes
For time to maturity T , the implied volatility, which is applied to valuevanilla options using the Black-Scholes-Merton (BSM) formula, canbe parameterized by a linear function σ(K;S0) of strike price K:
σ(K;S0) = σ0 + β
(K
S0− 1
)β, β < 0, is the slope of the volatility skew near the ATM strike
Lets take strikes at K± = (1± α)S0, where typically α = 5%:
β =1
2α(σ((1 + α)S0;S0)− σ((1− α)S0;S0)) ≡ Skewα
where Skewα is the implied skew normalized by strike width α
The equity volatility skew is negative, as consequence of the fact that,relatively, it is more expensive to buy an OTM put option than anOTM call option.
7
Implied volatility skew II for 1m options on the S&P 500 index fromOctober 2007 to July 2012
40%
50%
60%
70%
80%
-60%
-40%
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1m
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Sk
ew
skew 1m
1m ATM vol
10%
20%
30%
-120%
-100%
-80%
Left (red): 1m 105%− 95% skew, Skew5%(tn)Right (green): 1m ATM implied volatility
β = −1.0 means that the implied volatility of the put struck at 95% ofthe spot price is −5%× β = 5% higher than that of the ATM option
8
Sticky rules (Derman) I1) Sticky-strike:
σ(K;S) = σ0 + β
(K
S0− 1
), σATM(S) ≡ σ(S;S) = σ0 + β
(S
S0− 1
)ATM vol increase as the spot declines - typical of range-boundedmarkets2) Sticky-delta:
σ(K;S) = σ0 + β
(K − SS0
), σATM(S) = σ0
The level of the ATM volatility does not depend on spot price -typicalof stable trending markets3) Sticky local volatility:
σ(K;S) = σ0 + β
(K + S
S0− 2
), σATM(S) = σ0 + 2β
(S
S0− 1
)ATM vol increase as the spot declines twice as much as in the stickystrike case - typical of stressed markets
9
Sticky rules II
25%
30%
35%
40%
Imp
lie
d V
ola
tili
ty
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
Imp
lie
d V
ola
tili
ty
Spot Price
Old S(0)New S(1)
Given: β = −1.0 and σATM(0) = 25.00%Spot change: down by −5% from S(0) = 1.00 to S(1) = 0.95Sticky-strike regime: the ATM volatility moves along the originalskew increasing by −5%× β = 5%Sticky-local regime: the ATM volatility increases by −5% × 2β =10% and the volatility skew moves upwardsSticky-delta regime: the ATM volatility remains unchanged withthe volatility skew moving downwards
10
Impact on option delta
The key implication of the volatility rules is the impact on optiondelta ∆
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 (ascompared to the BSM delta) in a trending market and under-hedgedin a stressed market
Thus, the identification of market regimes plays an important role tocompute option hedges
While computation of hedges is relatively easy for vanilla options andcan be implemented using the BSM model, for path-dependent exoticoptions, we need a dynamic model consistent with different volatilityregimes
11
Stickiness ratio ITo identify volatility regimes we introduce the stickiness ratio
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) + βR(tn)X(tn)
where the stickiness ratio R(tn) indicates the rate of change in theATM volatility predicted by the skew and price return
Stickiness ratio R is a model-dependent quantity, informally:
R ≈1
β
∂
∂SσATM(S)
We obtain that:R = 1 under sticky-strikeR = 0 under sticky-deltaR = 2 under sticky-local vol
12
Stickiness ratio IIEmpirical 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-boundstart date 9-Oct-07 5-Mar-09 18-Feb-11end date 5-Mar-09 18-Feb-11 31-Jul-12
number days 354 501 365start SPX 1565.15 682.55 1343.01end 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%
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Stickiness ratio III
30%
40%
50%
60%
70%
80%
1000
1200
1400
1600
1m
AT
M v
ol
S&
P5
00
S&P500
1m ATM vol
CRISIS
RECOVERY
RANGE-BOUND
10%
20%
30%
600
800
Oc
t-0
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c-0
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r-0
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Stickiness ratio IVTo test Stickiness empirically, we apply the regression model for pa-rameter R within each zone using daily changes:
σATM(tn)− σATM(tn−1) = R× Skew5%(tn−1)X(tn) + εn
where X(tn) is realized return for day nσATM(tn) and Skewα(tn) are the ATM volatility and skew observed atthe end of the n-th dayεn is iid normal residuals
Informal definition of the stickiness ratio:
R(tn) =σATM(tn)− σATM(tn−1)
X(tn)Skew5%(tn−1)
We expect that the average value of R, R, as follows:R = 1 under the sticky-strike regimeR = 0 under the sticky-delta regimeR = 2 under the sticky-local regime
15
Stickiness ratio (crisis) for 1m and 1y ATM vols
y = 1.6327xR² = 0.7738
0%
5%
10%
15%
-12% -10% -8% -6% -4% -2% 0% 2% 4% 6% 8%
Da
ily c
ha
ng
e in
1m
AT
M v
ol
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 = 1.5978xR² = 0.8158
0%
5%
10%
15%
-4% -3% -2% -1% 0% 1% 2% 3%
Da
ily c
ha
ng
e in
1y A
TM
vo
l
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
16
Stickiness ratio (recovery) for 1m and 1y ATM vols
y = 1.4561xR² = 0.6472
0%
5%
10%
15%
-6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5%
Da
ily c
ha
ng
e in
1m
AT
M v
ol
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 = 1.5622xR² = 0.6783
0%
5%
10%
15%
-2% -2% -1% -1% 0% 1% 1% 2%
Da
ily c
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TM
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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
17
Stickiness ratio (range) for 1m and 1y ATM vols
y = 1.3036xR² = 0.6769
0%
5%
10%
15%
-6% -4% -2% 0% 2% 4% 6% 8%
Da
ily c
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1m
AT
M v
ol
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 = 1.4139xR² = 0.7228
0%
5%
10%
15%
-2% -2% -1% -1% 0% 1% 1% 2% 2% 3%
Da
ily c
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e in
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TM
vo
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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
18
Stickiness ratio V. ConclusionsSummary of the regression model:
crisis recovery range-boundStickiness, 1m 1.63 1.46 1.30Stickiness, 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 explainingabout 80% of the variation in ATM volatility during crisis period andabout 70% of the variation during recovery and range-bound periods
2) Stickiness ratio isstronger during crisis period, R ≈ 1.6 (closer to sticky local vol)less strong during recovery period, R ≈ 1.5weaker 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
19
Stickiness ratio VI. Time series
1.75
2.00
2.25
Sti
ckin
ess
1m Stickeness, 60d average
1.25
1.50
Oct-
07
Dec-0
7
Mar-
08
Ju
n-0
8
Sep
-08
Dec-0
8
Mar-
09
Ju
n-0
9
Au
g-0
9
No
v-0
9
Feb
-10
May-1
0
Au
g-1
0
No
v-1
0
Feb
-11
Ap
r-11
Ju
l-11
Oct-
11
Jan
-12
Ap
r-12
Ju
l-12
20
Stickiness ratio VII. Dynamic models ANow we consider how to model the stickiness ratio within the dynamicSV 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 volatilitygiven change in the spot price (for a very nice discussion see ”A Noteon Hedging with Local and Stochastic Volatility Models” by Mercurio-Morini, on ssrn.com)
The model-consistent hedge:The level of volatility changes by (approximately): Skew×∆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 hedgeR = 0 under the model-inconsistent hedge
21
Stickiness ratio VII. Dynamic models BHow to make R = 1.5 using SV models?Under the model-consistent hedge: impossible?Under the model-inconsistent hedge: mix SV with local volatility
Remedy: add jump processUnder any spot-homogeneous jump model, R = 0
The only way to have a model-consistent hedging that fits the desiredstickiness ratio is to mix stochastic volatility with jumps:the higher is the stickiness ratio, the lower is the jump premiumthe 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 becauseof 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 givensmall levels of realized volatility and related hedging)
22
Stickiness ratio VII. Dynamic models CThe above consideration explain that the stickiness ratio isstronger during crisis period, R ≈ 1.6 (closer to sticky local vol)weaker during range-bound and recovery periods, R ≈ 1.35 (closer tosticky-strike)
To model this feature within an SV model, we need to specify aproportion of the skew attributed to jumps (see my 2011 presentationfor Risk Quant congress and 2012 presentation for Global derivatives)
During crisis periods, the weight of jumps is about 20%During range-bound and recovery periods, the weight of jumps isabout 40%
23
Beta stochastic volatility IFirst, I present a simplified version of the beta stochastic volatil-ity model introduced in Karasinski and Sepp (2012) with no mean-reversion and volatility-of-volatility:
dS(t)
S(t)= σ(t)(S(t))βSdW (t), S(0) = S0
dσ(t) = βVdS(t)
S(t), σ(0) = σ0
(1)
where S(t) is the spot priceσ(t) is instantaneous volatility and σ0 is initial level of ATM volatilityW (t) is a Brownian motion - the only source of randomness
To produce the volatility skew and the dependence between the priceand implied volatility, the model relies on the two parameters:βS is the backbone betaβV is the volatility beta
Estimates of βS and βV are easily inferred from implied/historical data
24
Volatility beta
We replicate σ(t) by short-term ATM volatility, σ(t) = σATM(S(t)) toestimate model parameters by the regression model
Volatility beta βV is a measure of linear dependence between dailyreturns and changes in the ATM volatility:
σATM(S(tn))− σATM(S(tn−1)) = βVS(tn)− S(tn−1)
S(tn−1)
Next we examine this regression model empirically
25
Volatility beta (crisis) for 1m and 1y ATM vols
y = -1.1131xR² = 0.7603
0%
5%
10%
15%
-15% -10% -5% 0% 5% 10% 15%
Da
ily c
ha
ng
e in
1m
AT
M v
ol
Daily change in 1m ATM vol, crisis period Oct 07 - Mar 09
-15%
-10%
-5%
Daily price return
y = -0.3948xR² = 0.8018
0%
5%
10%
15%
-15% -10% -5% 0% 5% 10% 15%
Da
ily c
ha
ng
e in
1y A
TM
vo
l
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
26
Volatility beta (recovery) for 1m and 1y ATM vols
y = -0.9109xR² = 0.603
0%
5%
10%
15%
-6% -4% -2% 0% 2% 4% 6% 8%
Da
ily c
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e in
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AT
M v
ol
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 = -0.3926xR² = 0.6475
0%
5%
10%
15%
-6% -4% -2% 0% 2% 4% 6% 8%
Da
ily c
ha
ng
e in
1y A
TM
vo
l
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
27
Volatility beta (range) for 1m and 1y ATM vols
y = -1.0739xR² = 0.6786
0%
5%
10%
15%
-8% -6% -4% -2% 0% 2% 4% 6%
Da
ily c
ha
ng
e in
1m
AT
M v
ol
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 = -0.4374xR² = 0.7147
0%
5%
10%
15%
-8% -6% -4% -2% 0% 2% 4% 6%
Da
ily c
ha
ng
e in
1y A
TM
vo
l
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
28
Volatility beta. Summary
crisis recovery range-boundVolatility beta 1m -1.11 -0.91 -1.07Volatility 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 period60% in changes in the ATM volatility during recovery period (ATMvolatility 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-boundperiods)
29
The backbone beta
The backbone beta βS is a measure of daily changes in the logarithmof 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 we examine this regression model empirically
30
Backbone beta (crisis) for 1m and 1y ATM vols
y = -2.8134xR² = 0.6722
0%
10%
20%
30%
-15% -10% -5% 0% 5% 10% 15%Da
ily c
ha
ng
e in
lo
g 1
m A
TM
vo
l 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 = -1.2237xR² = 0.7715
0%
10%
20%
30%
-15% -10% -5% 0% 5% 10% 15%Da
ily c
ha
ng
e in
1y lo
g A
TM
vo
l 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
31
Backbone beta (recovery) for 1m and 1y ATM vols
y = -3.6353xR² = 0.5438
0%
10%
20%
30%
-6% -4% -2% 0% 2% 4% 6% 8%
Da
ily c
ha
ng
e in
lo
g 1
m A
TM
vo
l
Daily change in log 1m ATM vol, recov period Mar09-Feb11
-30%
-20%
-10%
-6% -4% -2% 0% 2% 4% 6% 8%
Da
ily c
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e in
lo
g 1
m A
TM
vo
l
Daily price return
y = -1.4274xR² = 0.6137
0%
10%
20%
30%
-6% -4% -2% 0% 2% 4% 6% 8%Da
ily c
ha
ng
e in
lo
g 1
y A
TM
vo
l 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
32
Backbone beta (range) for 1m and 1y ATM vols
y = -4.5411xR² = 0.6212
0%
10%
20%
30%
-8% -6% -4% -2% 0% 2% 4% 6%Da
ily c
ha
ng
e in
lo
g 1
m A
TM
vo
l 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 = -1.8097xR² = 0.6978
0%
10%
20%
30%
-8% -6% -4% -2% 0% 2% 4% 6%Da
ily c
ha
ng
e in
lo
g 1
y A
TM
vo
l 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
33
The backbone beta. Summary A
crisis recovery range-boundBackbone beta 1m -2.81 -3.64 -4.54Backbone 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 acrossdifferent 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
34
The backbone beta. Summary BChange in the level of the ATM volatility implied by backbone betaβS is proportional to initial value of the ATM volatilityHigh negative value of βS implies a big spike in volatility given amodest drop in the price - a feature of sticky local volatility model
In the figure, using estimated parameters βV = −1.07, βS − 4.54 inrange-bound period, σ(0) = 20%
8%
10%
12%
14%
16%
Ch
an
ge
in
vo
l le
ve
l
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%
Ch
an
ge
in
vo
l le
ve
l
Spot return %
35
Connection to the SABR model (Hagan et al (2002) )
Model parameters are related to the SABR model as follows:
a = σ0 , ρ = −1 , ν = −βV , β = βS + 1
Using formula (3.1a) in Hagan et al (2002) for a short maturity andsmall log-moneynes k, k = ln(K/S0), we obtain the following relation-ship for the BSM implied volatility σIMP (k):
σIMP (k) =σ0
SβS
1 +1
2
(βS +
βVσ0
)k +
1
12
β2S −
(βVσ0
)2 k2
Thus, in a simple case, the model can be directly linked to the impliedvolatility interpolator represented by the SABR model
36
Model implied skewWe obtain the following approximate but accurate relationship be-tween the model parameters and short-term implied ATM volatility,σATM(S), and skew Skewα:
σ0SβS = σATM(S)
βS +βVσ0
=2Skewα
σATM(S)≡ Λ
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(2)
37
Model implied stickiness and volatility regimesIf we insist on model-inconsistent delta (change in spot with volatilitylevel unchanged):fit backbone beta βS to reproduce specified stickiness ratioadjust βV so that the model fits the market skew
Using stickiness ratio R(tn) along with (2), we obtain that empirically:
βS(tn) =Skewα(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 differentvolatility regimes:sticky-delta with βS = 0 and βV ≈ 2Skewαsticky-local volatility with βV = 0 and βS ≈ Λ
From the empirical data we infer that, approximately,βS ≈ 70%Λ and βV = 30%× 2Skewα
38
Beta stochastic volatility modelLet me consider pure SV beta expressed in terms of normalized volatil-ity factor Y (t) (this version is applied in practice for beta SV with localvolatility):
dS(t)
S(t)= (1 + Y (t))σdW (t), S(0) = S0
dY (t) = βVdS(t)
S(t), Y (0) = 0
(3)
where σ is the overall level of the volatility (can be deterministic orlocal σ(t, S))Y (t) is the normalized volatility factor fluctuation around zero
Volatility parameter βV can be implied from short term ATM volatilityσATM and skew Skewα:
βV =2Skewα
σATM(4)
The goal now is to investigate the dynamics of the skew
39
Beta stochastic volatility model. SkewInverting the above equation:
Skew(t) =1
2βV σATM(t) (5)
Dynamically, using (4):
dSkew(t) =1
2βV dσATM(t) ∝
1
2βV σATM(t)dY (t)
∝1
2σATM(t)
(βV)2 dS(t)
S(t)
To test the above equation empirically, we apply the regression modelfor coefficient q:
Skew(tn)− Skew(tn−1) = q
[2(Skew(tn−1)
)2 1
σATM(tn−1)
S(tn)− S(tn−1)
S(tn−1)
](6)
First, we test (5)
40
Skew vs ATM volatility (crisis) for 1m and 1y ATM vols
y = -0.2452x - 0.584R² = 0.1023
-40%
-20%
0%
0% 10% 20% 30% 40% 50% 60% 70% 80%
1m ATM vol
1m Skew
1m Skew vs ATM vol, crisis period Oct 07 - Mar 09
-100%
-80%
-60%
y = -0.0147x - 0.2473R² = 0.0052
-20%
0%
10% 15% 20% 25% 30% 35% 40% 45% 50% 55%
1m ATM vol
1y Skew
1y Skew vs ATM vol, crisis period Oct 07 - Mar 09
-40%
41
Skew vs ATM volatility (recovery) for 1m and 1y ATM vols
y = -0.005x - 0.6058R² = 6E-06
-60%
-40%
-20%
0%
0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%
1m ATM vol
1m Skew
1m Skew vs ATM vol, recovery period Mar 09 - Feb 11
-120%
-100%
-80%
y = 0.2671x - 0.3333R² = 0.1335
-20%
0%
10% 15% 20% 25% 30% 35% 40% 45%
1m ATM vol
1y Skew
1y Skew vs ATM vol, recovery period Mar 09 - Feb 11
-40%
42
Skew vs ATM volatility (range) for 1m and 1y ATM vols
y = -1.2322x - 0.4895R² = 0.5236-60%
-40%
-20%
0%
0% 5% 10% 15% 20% 25% 30% 35% 40% 45%
1m ATM vol
1m Skew
1m Skew vs ATM vol, range-bound period Feb 11 - Aug 12
-120%
-100%
-80%
y = -0.0968x - 0.2942R² = 0.0347-20%
0%
10% 15% 20% 25% 30% 35%
1m ATM vol
1y Skew
1y Skew vs ATM vol, range-bound period Feb 11 - Aug 12
-40%
43
Skew vs ATM volatility. Summary
crisis recovery range-boundSkew-vol beta 1m -0.25 -0.01 -1.23Skew-vol beta 1y -0.01 -0.27 -0.10
R2 1m 10% 0% 52%R2 1y 1% 13% 3%
Empirically, in general, a high level of ATM volatility implies a higherlevel of the skew but the relationship is not strong and is mixed
For short-term skew, the relationship is stronger in crisis and range-bound periods
For longer-term skew, the relationship is stronger in recovery periods
Next, we test (6) for relationship between changes in the skew andspot returns
44
Skew vs price return (crisis) for 1m and 1y skews
y = -0.1373xR² = 0.0575
0%
5%
10%
15%
-30% -20% -10% 0% 10% 20% 30%
Da
ily c
ha
ng
e in
1m
Sk
ew
Daily change in 1m Skew, crisis period Oct 07 - Mar 09
-15%
-10%
-5%
-30% -20% -10% 0% 10% 20% 30%
Daily price return
y = -0.1582xR² = 0.0582
0%
5%
10%
15%
-4% -3% -2% -1% 0% 1% 2% 3% 4% 5%
Da
ily c
ha
ng
e in
1y S
ke
w
Daily change in 1y Skew, crisis period Oct 07 - Mar 09
-15%
-10%
-5%
-4% -3% -2% -1% 0% 1% 2% 3% 4% 5%
Daily price return
45
Skew vs price return (recovery) for 1m and 1y skews
y = 0.3685xR² = 0.1837
0%
5%
10%
15%
-40% -30% -20% -10% 0% 10% 20% 30% 40%
Da
ily c
ha
ng
e in
1m
Sk
ew
Daily change in 1m Skew, recovery period Mar 09-Feb 11
-15%
-10%
-5%
-40% -30% -20% -10% 0% 10% 20% 30% 40%
Daily price return
y = -0.0309xR² = 0.0023
0%
5%
10%
15%
-3% -2% -1% 0% 1% 2% 3%
Da
ily c
ha
ng
e in
1y S
ke
w
Daily change in 1y Skew, recovery period Mar 09-Feb 11
-15%
-10%
-5%
-3% -2% -1% 0% 1% 2% 3%
Daily price return
46
Skew vs price return (range) for 1m and 1y skews
y = 0.2089xR² = 0.1266
0%
5%
10%
15%
-50% -40% -30% -20% -10% 0% 10% 20% 30% 40%
Da
ily c
ha
ng
e in
1m
Sk
ew
Daily change in 1m Skew, range-bnd period Feb11-Aug12
-15%
-10%
-5%
-50% -40% -30% -20% -10% 0% 10% 20% 30% 40%
Daily price return
y = -0.0023xR² = -4E-04
0%
5%
10%
15%
-5% -4% -3% -2% -1% 0% 1% 2% 3% 4%
Da
ily c
ha
ng
e in
1y S
ke
w
Daily change in 1y Skew, range-bnd period Feb11-Aug12
-15%
-10%
-5%
-5% -4% -3% -2% -1% 0% 1% 2% 3% 4%
Daily price return
47
Skew vs price return. Summary
crisis recovery range-boundSkew-return beta 1m -0.14 0.37 0.21Skew-return beta 1y -0.16 -0.03 0.00
R2 1m 6% 18% 13%R2 1y 6% 0% 0%
The short-term skew appears to be somewhat dependent on spotchanges:during crisis periods, negative returns decrease the skew (de-leveragingreduces need for downside protection)during recovery and range-bound periods, negative returns increasethe skew (risk-aversion is high especially during recovery period)
The long-term skew does not appear to depend on spot returns
48
Skew vs price return. Conclusions
The skew does not seem to depend on either volatility or spot dy-namics (especially for longer maturities)
About 20% of variations in the short-term skew can be attributed tochanges in the spot
Only jumps appear to have a reasonable explanation for the skew(the fear of a crash does not (or little) depend on current values ofvariables)
49
Full beta stochastic volatility model IThe pricing version of the beta model is specified as follows:
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) + εdW (1)(t), Y (0) = 0(7)
where:βV (βV < 0) is the rate of change in the volatility corresponding tochange in the spot priceε is idiosyncratic volatility of volatilityκ is the mean-reversion rateW (0)(t) and W (1)(t) are two Brownians with dW (0)(t)dW (1)(t) = 0µ(t) is the risk-neutral drift
σ is the overall level of volatilityσ is set to either constant volatility σCV or deterministic volatilityσDV (t), or local stochastic volatility σLSV (t, S)σ = {σCV , σDV (t), σLSV (t, S)}
50
Beta stochastic volatility model. II
Using dynamics (7), for the log-spot, X(t) = ln(S(t)S(0)
)we obtain:
dX(t) = µ(t)dt−1
2σ2(1 + Y (t))2dt+ σ(1 + Y (t))dW (0)(t), X(0) = 0
dY (t) = βσ(1 + Y (t))dW (0)(t)− κY (t)dt+ εdW (1)(t), Y (0) = 0(8)
with
dY (t)dY (t) =(ε2 + β2σ2(1 + Y (t))2
)dt
dX(t)dY (t) = βσ2(1 + Y (t))2dt
The pricing equation for value function U(t, T,X, Y ) has the form:
Ut +1
2σ2(1 + 2Y + Y 2) [UXX − UX] + µ(t)UX
+1
2
(ε2 + β2σ2
(1 + 2Y + Y 2
))UY Y − κY UY
+ βσ2(1 + 2Y + Y 2
)UXY − r(t)U = 0
(9)
where r(t) is the discount rate and subscripts denote partial derivatives
51
Beta stochastic volatility model. IIIThe parameters of the stochastic volatility, β, ε and κ are specifiedbefore the calibration
We calibrate the local volatility σ ≡ σLSV (t, S), using either a para-metric local volatility (CEV) or non-parametric local volatility, so thatthe vanilla surface is matched by construction
For calibration of σLSV (t, S) we apply the conditional expectation(Lipton A, The vol smile problem, Risk, February 2002):
σ2LSV (T,K)E
[(1 + Y (T ))2 |S(T ) = K
]= σ2
LV (T,K)
where σ2LV (T,K) is the local Dupire volatility
The above expectation is computed by solving the forward PDE cor-responding to pricing PDE (9) using finite-difference methods andcomputing σ2
LSV (T,K) stepping forward in time
Once σLSV (t, S) is calibrated we use either backward PDE-s or MCsimulation for valuation of exotic options
52
Beta SV model. Approximation for call price II propose an affine approximation for pricing equation (9) with con-stant or deterministic volatility σ:
G(t, T,X, Y ; Φ) = exp{−ΦX +A(0) +A(1)Y +A(2)Y 2
}with A(n)(T ;T ) = 0, n = 0,1,2
By substitution this into PDE (9) and collecting terms proportionalto Y and Y 2 only, we obtain a system of ODE-s for A(n)(t):
A(0)t + v0A
(2) +1
2v0(A(1))2 −ΦA(1)c0 +
1
2q = 0
A(1)t +
1
2v1(A(1))2 + 2v0A
(1)A(2) + v1A(2) − κA(1) −Φ
(2c0A
(2) + c1A(1))
+ q = 0
A(2)t +
1
2v2(A(1))2 + 2v0(A(2))2 + 2v1A
(1)A(2) + v2A(2) − 2κA(2) −Φ
(2c1A
(2) + c2A(1))
+1
2q = 0
where
q = σ2(Φ2 + Φ
), v0 = ε2 + β2σ2, v1 = 2β2σ2, v2 = β2σ2, c0 = βσ2, c1 = 2βσ2, c2 = βσ2
This is system is solved by means of Runge-Kutta methodsIt is straightforward to incorporate time-dependent model parameters(but not space-dependent local volatility σLSV (t,X))
53
Beta SV model. Approximation for call price IIAs a result, for pricing vanilla options, we can apply the standardmethods based on the Fourier inversion
The value of the call option with strike K is computed by applyingLipton-Lewis formula:
C(t, T, S, Y ) = e−∫ Tt r(t′)dt′
(e∫ Tt µ(t′)dt′S −
K
π
∫ ∞0<[G(t, T, x, Y ; ik − 1/2)
k2 + 1/4
]dk
)
where x = ln(S/K) +∫ Tt µ(t′)dt′
54
Beta SV model. Approximation for call price III
10%
15%
20%
25%
30%
35%
40%
70% 80% 90% 100% 110% 120%
Strike %
Imp
lied
Vo
lApproximation, T=1m
PDE, T=1m
Approximation, T=1y
PDE, T=1y
Approximation, T=2y
PDE, T=2y
Implied model volatilities computed by approximation formula vs nu-merical PDE using β = −7.63, ε = 0.35, κ = 4.32 and constantvolatilities: σCV (1m) = 19.14%, σCV (1y) = 23.96%, σCV (2y) =24.21%
55
Properties of the volatility process, assuming constant vol σCVInstantaneous variance of Y (t) is given by:
dY (t)dY (t) =(β2σ2
CV (1 + Y (t))2 + ε2)dt
which has systemic part proportional to Y (t) and idiosyncratic part ε
In a stress regime, for large values of Y (t), the variance is dominatedby β2σ2
CV Y2(t) (close to a log-normal model for volatility process)
The volatility process has steady-state variance (so that the volatil-ity approaches stationary distribution in the long run):
E[Y 2(t) |Y (0) = 0
]=ε2 + β2σ2
CV
2κ− β2σ2CV
(1− e−(2κ−β2σ2
CV )t)
Effective mean-reversion for the volatility of variance is:
2κ− β2σ2CV
Steady state variance of volatility is
ε2 + β2σ2CV
2κ− β2σ2CV
56
Instantaneous correlation between dY (t) and dX(t):
ρ(dX(t)dY (t)) =βσ2
CV (1 + Y (t))2√(ε2 + β2σ2
CV (1 + Y (t))2)√σ2CV (1 + Y (t))2
With high volatility Y (t) is large so letting Y (t)→∞ we obtain that
ρ(dX(t)dY (t))|Y (t)≈∞ = −1
In a normal regime, Y (t) ≈ 0, so that obtain:
ρ(dX(t)dY (t))|Y (t)≈0 = −1√(
ε2
β2σ2CV
+ 1)
The beta SV model introduces state-dependent spot-volatility cor-relation, with high volatility leading to absolute negative correlation
In contrast, Heston and Ornstein-Uhlenbeck based SV models alwaysassume constant instantaneous correlation
57
The steady state densitySteady state density function G(Y ) of volatility factor Y (t) in dynam-ics (7) solves the following equation:
1
2
[(ε2 + β2σ2
CV
(1 + 2Y + Y 2
))G]Y Y
+ [κY G]Y = 0
We can show that G(Y ) exhibits the power-like behavior for largevalues of Y :
limY→+∞
G(Y ) = Y −α , α = 2
(1 +
κ
(βσCV )2
)This power-like behavior contrasts with Heston and exponential volatil-ity models which imply exponential tails for the steady-state densityof the volatility
Thus, the beta SV model predicts higher probabilities of large valuesof instantaneous volatility
58
The steady state density. Tails
0%
1%
2%
3%
4%
5%
6%
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Y_inf
Pro
bab
ility
(Y
>Y_i
nf)
Beta SV model
OU SV model
Tails in the beta SV model (green) vs Ornstein-Uhlenbeck (OU) ex-ponential SV model (red) with α = 5 in beta SV model and equivalentvol-of-vol in OU SV model εOU = 0.61
59
Case Study I: Correlation skewApply experiment:1) Compute implied volatilities from options on the index (say, S&P500)2) Using implied volatilities (probability density function) of stocks inthis index, and stock-stock Gaussian correlations, compute optionprices on the index, compute the implied volatility from these prices
Empirical observation (correlation skew):The index skew computed in 1) is steeper than that computed in 2)Explanation:Stocks become strongly correlated during big sell-offsIndex skew reflects premium for buying puts on a basket of stocks
Modelling approach:Correlation skew cannot be replicated using Gaussian correlationStochastic and/or local correlations can be appliedBut only SV model with jumps can produce realistic dynamicsand reproduce the correlation skew
Next we augment the beta SV model with jumps and apply it toreproduce the correlation skew
60
SPY and select sector ETF-sConsider:SPY - the ETF tracking the S&P500 indexSelect sector ETF-s - ETF-s tracking 9 sectors of the S&P500 index
ETF SPY weight Sector1 XLK 20.46% INFORMATION TECHNOLOGY2 XLF 14.88% FINANCIALS3 XLV 12.38% HEALTH CARE4 XLP 11.66% CONSUMER STAPLES5 XLY 11.31% CONSUMER DISCRETIONARY6 XLE 11.15% ENERGY7 XLI 10.81% INDUSTRIALS8 XLU 3.84% UTILITIES9 XLB 3.51% MATERIALS
61
Index and sector vols and skewsTerm structure of ATM volatilities (left) and 105%-95% skews (right)SPY ATM vol (black line) can be viewed as a weighted averageof sector ATM volsSPY skew (black line) is steeper than weighted average skews ofsectors
10%
15%
20%
25%
3m 6m 9m 12m 15m 18m 21m 24m 27m 30m 33m 36m
T
AT
M v
ol
SPY, ATM vol XLK, ATM volXLF, ATM vol XLV, ATM volXLP, ATM vol XLY, ATM volXLE, ATM vol XLI, ATM volXLU, ATM vol
-0.50
-0.40
-0.30
-0.20
-0.103m 6m 9m 12m 15m 18m 21m 24m 27m 30m 33m 36m
T
Ske
w
SPY, Market Skew XLK, Market Skew
XLF, Market Skew XLV, Market Skew
XLP, Market Skew XLY, Market Skew
XLE, Market Skew XLI, Market Skew
XLU, Model Skew
62
Calibration of the beta SV model IFirst, calibrate the beta SV model with constant SV parametersCalibration is based on intuition and experience with the model
Volatility beta, β, is set by
β =σIMP (6m,5%)− σIMP (6m,−5%)
0.05σIMP (6m,0%)=
2Skew5%(6m)
σATM(6m)
where σIMP (6m, k%) is 6m implied vol for forward-based log-strike kIdiosyncratic volatility ε is set according to:
ε2 = σ2IMP (6m,0%)β21− (ρ∗)2
(ρ∗)2
ρ∗ is spot-vol correlation for 6m vol implied by SV model with Orstein-Uhlenbeck process for SV driver
Reversion speed κ is adjusted to fit term structure of 1y-3y 105%−95% skew
Term structure of model level vols σDV (t) are calibrated by construc-tion (by root search) so that the ATM implied vol is fitted exactly
63
Calibration of the beta SV model IICalibrated parameters of the beta SV model to SPY and sector ETF-s
SPY XLK XLF XLV XLP XLY XLE XLI XLU XLBβ -4.77 -3.72 -2.78 -5.34 -5.31 -3.93 -2.75 -3.06 -5.55 -2.89ε 0.40 0.39 0.67 0.35 0.33 0.42 0.55 0.41 0.38 0.39κ 1.45 1.60 1.30 1.25 1.15 1.40 1.40 1.30 1.25 1.45ρ∗ -0.81 -0.78 -0.60 -0.80 -0.80 -0.76 -0.66 -0.72 -0.79 -0.73
Next we illustrate plots of the term structure of market and modelimplied 105%−95% skew and 1y implied vols accross range of strikes
Typically, if the beta SV model is fits 105%− 95% skew, then it willfit the skew accross different strikes
Beta SV model is similar to one-factor SV models - the model fits welllonger-term skews (above one-year) while it is unable to fit short-termskews (up to one year) unless beta parameter β is large
By actual pricing, small discrepancies in implied vols are eliminatedby local vol part
64
Calibrated parameters
Volatility Beta-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00SPY XLK XLF XLV XLP XLY XLE XLI XLU XLB Idiosyncratic vol-of-vol
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
SPY XLK XLF XLV XLP XLY XLE XLI XLU XLB
Reversion Speed
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
SPY XLK XLF XLV XLP XLY XLE XLI XLU XLBSpot-Vol Corr
-0.90
-0.80
-0.70
-0.60
-0.50
-0.40
-0.30
-0.20
-0.10
0.00SPY XLK XLF XLV XLP XLY XLE XLI XLU XLB
65
SPY
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
skew
T
SPY, Market Skew
-0.50
-0.40
-0.30SPY, Market Skew
SPY, Model Skew
20%
30%
40%
Imp
l V
ol
1y Implied vols
SPY, Market impl vol
SPY, Model impl vol
10%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
66
XLK - information technology
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
ske
w
T
XLK, Market Skew
-0.50
-0.40
-0.30XLK, Market Skew
XLK, Model Skew
20%
30%
40%
Imp
l V
ol
1y Implied vols
XLK, Market impl vol
XLK, Model impl vol
10%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
67
XLF - financials
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
ske
w
T
XLF, Market Skew
-0.30
-0.20XLF, Market Skew
XLF, Model Skew
30%
40%
50%
Imp
l V
ol
1y Implied vols
XLF, Market impl vol
XLF, Model impl vol
10%
20%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
68
XLV - health care
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
ske
w
T
XLV, Market Skew
-0.50
-0.40
-0.30XLV, Market Skew
XLV, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLV, Market impl vol
XLV, Model impl vol
5%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
69
XLP - consumer staples
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
skew
T
XLP, Market Skew
-0.40
-0.30
XLP, Market Skew
XLP, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLP, Market impl vol
XLP, Model impl vol
5%
50%
60%
70%
80%
90%
100%
110
%
120%
130%
140%
150%
K
70
XLY - consumer discretionary
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLY, Market Skew
-0.50
-0.40
-0.30XLY, Market Skew
XLY, Model Skew
20%
30%
40%
Imp
l V
ol
1y Implied vols
XLY, Market impl vol
XLY, Model impl vol
10%
50%
60%
70%
80%
90%
100
%
110%
120
%
130
%
140
%
150
%
K
71
XLE - energy
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLE, Market Skew
-0.40
-0.30
XLE, Market Skew
XLE, Model Skew
30%
40%
50%
Imp
l V
ol
1y Implied vols
XLE, Market impl vol
XLE, Model impl vol
10%
20%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
72
XLI - industrials
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLI, Market Skew
-0.40
-0.30
XLI, Market Skew
XLI, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLI, Market impl vol
XLI, Model impl vol
5%
15%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
73
XLU - utilities
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
skew
T
XLU, Market Skew
-0.50
-0.40
-0.30XLU, Market Skew
XLU, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLU, Market impl vol
XLU, Model impl vol
5%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
74
XLB - materials
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLB, Market Skew
-0.40
-0.30
XLB, Market Skew
XLB, Model Skew
30%
40%
50%
Imp
l V
ol
1y Implied vols
XLB, Market impl vol
XLB, Model impl vol
10%
20%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
75
Multi-asset beta SV modelThe SV model without jumps cannot reproduce the correlation skew
We present multi-asset beta SV model with simultaneous jumps inassets and their volatilities:
dSn(t)
Sn(t)= µn(t)dt+ (1 + Yn(t))σndWn(t) + (eνn − 1) (dN(t)− λdt)
dYn(t) = −κnYn(t)dt+ βn(1 + Yn(t))σndWn(t) + εndW(1)(t) + ηndN(t)
where n = 1, ..., NdWn(t) are Brownians for asset prices with specified correlation matrixW (1)(t) is the joint driver for idiosyncratic volatilitiesN(t) is the joint Poisson process with intensity λ for simultaneousshocks in prices and volatilitiesνn, νn < 0, are constant jump amplitudes in log-priceηn, ηn > 0, are constant jump amplitudes in volatilities
76
Jump calibrationBased on my presentation for Global Derivatives in Paris, 2011The idea is based on linear impact of jumps on the short-term impliedskew:
σimp(K) ≈ σ −λν
σln (S/K)
Specify wjd - the percentage of the skew attributed to jumpsSet jump intensity as follows:
λ =
(Skew5%(1y)
)2
wjd
The jump size is implied as follows:
ν = −√wjdσATM(1y)
√λ
=wjdσATM(1y)
Skew5%(1y)
Jump size in volatility, η, can be calibrated to options on the VIXskew or options on the realized varianceEmpirically, η ≈ 2
77
Jump calibration for ETF. ISet wjd = 50% and imply jump intensity λSPY from 1y 5% skew forSPY ETFIndividual jump sizes are set using λSPY and sector specific ATMvolatility σATM,n(1y):
νn = −√wjdσATM,n(1y)√λSPY
Jump size in volatility, η, is set uniformly η = 2 (realized jump inATM volatility will be proportional to ATM volatility of sector ETF)
Previously specified β, ε and κ are reduced by 25%
Next we illustrate plots of term structure of market and model implied105%− 95% skew and 1y implied volatilities accross range of strikes
Beta SV model with jumps produces steep forward skews for shortmaturities and is consistent with term structure of skew
Again, by actual pricing, small discrepancies in implied vols are elimi-nated by local vol part
78
Jump calibration for ETF. IIFor recent data, λ = 0.17 and jump sizes are shown in table and figure
SPY XLK XLF XLV XLP XLY XLE XLI XLU XLB-0.32 -0.37 -0.35 -0.28 -0.26 -0.35 -0.41 -0.37 -0.26 -0.41
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
SPY XLK XLF XLV XLP XLY XLE XLI XLU XLB
-0.45
-0.40
-0.35
-0.30
jump size
79
SPY
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
skew
T
SPY, Market Skew
-0.50
-0.40
-0.30SPY, Market Skew
SPY, Model Skew
20%
30%
40%
Imp
l V
ol
1y Implied vols
SPY, Market impl vol
SPY, Model impl vol
10%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
80
XLK - information technology
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
ske
w
T
XLK, Market Skew
-0.50
-0.40
-0.30XLK, Market Skew
XLK, Model Skew
20%
30%
40%
Imp
l V
ol
1y Implied vols
XLK, Market impl vol
XLK, Model impl vol
10%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
81
XLF - financials
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
ske
w
T
XLF, Market Skew
-0.40
-0.30
XLF, Market Skew
XLF, Model Skew
30%
40%
50%
Imp
l V
ol
1y Implied vols
XLF, Market impl vol
XLF, Model impl vol
10%
20%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
82
XLV - health care
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
ske
w
T
XLV, Market Skew
-0.50
-0.40
-0.30XLV, Market Skew
XLV, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLV, Market impl vol
XLV, Model impl vol
5%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
83
XLP - consumer staples
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
skew
T
XLP, Market Skew
-0.60
-0.50
-0.40XLP, Market Skew
XLP, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLP, Market impl vol
XLP, Model impl vol
5%
50%
60%
70%
80%
90%
100%
110
%
120%
130%
140%
150%
K
84
XLY - consumer discretionary
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLY, Market Skew
-0.50
-0.40
-0.30XLY, Market Skew
XLY, Model Skew
20%
30%
40%
Imp
l V
ol
1y Implied vols
XLY, Market impl vol
XLY, Model impl vol
10%
50%
60%
70%
80%
90%
100
%
110%
120
%
130
%
140
%
150
%
K
85
XLE - energy
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLE, Market Skew
-0.50
-0.40
-0.30XLE, Market Skew
XLE, Model Skew
30%
40%
50%
Imp
l V
ol
1y Implied vols
XLE, Market impl vol
XLE, Model impl vol
10%
20%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
86
XLI - industrials
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLI, Market Skew
-0.50
-0.40
-0.30XLI, Market Skew
XLI, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLI, Market impl vol
XLI, Model impl vol
5%
15%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
87
XLU - utilities
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12
m
15
m
18
m
21
m
24
m
27
m
30
m
33
m
36
m
skew
T
XLU, Market Skew
-0.60
-0.50
-0.40XLU, Market Skew
XLU, Model Skew
15%
25%
35%
Imp
l V
ol
1y Implied vols
XLU, Market impl vol
XLU, Model impl vol
5%
50
%
60
%
70
%
80
%
90
%
100%
110%
120%
130%
140%
150%
K
88
XLB - materials
-0.30
-0.20
-0.10
0.00
3m
6m
9m
12m
15m
18m
21m
24m
27m
30m
33m
36m
sk
ew
T
XLB, Market Skew
-0.50
-0.40
-0.30XLB, Market Skew
XLB, Model Skew
30%
40%
50%
Imp
l V
ol
1y Implied vols
XLB, Market impl vol
XLB, Model impl vol
10%
20%
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
K
89
Correlation matrix for sector ETF-sSector-wise correlation are estimated using time series
XLK XLF XLV XLP XLY XLE XLI XLU XLBXLK 100%XLF 85% 100%XLV 64% 80% 100%XLP 69% 77% 77% 100%XLY 92% 87% 84% 83% 100%XLE 85% 84% 75% 48% 83% 100%XLI 90% 89% 86% 81% 94% 88% 100%
XLU 67% 66% 74% 82% 72% 66% 72% 100%XLB 89% 87% 80% 73% 88% 88% 91% 63% 100%
90
Correlation skew. IllustrationNext we compare the index skew implied from:1) SPY options (SPY)2) basket of ETF priced using local volatility with Gaussian correla-tions (Local Vol)3) basket of ETF priced using beta SV model (Beta LSV)4) basket of ETF priced using beta SV model with jumps (BetaLSV+Jumps)
Correlation skew 1y
10%
15%
20%
25%
30%
35%
40%
50% 70% 90% 110% 130% 150%
K %
Imp
l vo
l
SPY
Basket ETF, Local Vol
Basket ETF, Beta LSV
Basket ETF, Beta LSV+Jumps
Correlation skew 2y
10%
15%
20%
25%
30%
35%
50% 70% 90% 110% 130% 150%
K %
Imp
l vo
l
SPY
Basket ETF, Local Vol
Basket ETF, Beta LSV
Basket ETF, Beta LSV+Jumps
91
Correlation skew. Conclusion
Local volatility and stochastic volatility models without jumps cannotreproduce the correlation skew
Only jumps can introduce the correlation skew in a robust way
In my example, I calibrated jumps to 1y skew so the model fits 1ycorrelation skew, but model correlation skew flattens for 2y (problemwith data for long-dated ETF options?)
Perhaps more elaborate jump process is necessary (probably thoughspot- and volatility-dependent intensity process)
92
Case Study II: Conditional forward skewWe imply forward volatility by computing forward-start option condi-tional that S(τ) starts in range (D −∆, D + ∆), typically ∆ = 5%,with the following pay-off:
1{D−∆<S(τ)<D+∆}
(S(T )
S(τ)−K
)+
In the BSM model, the variables S(τ) and S(T )S(τ) are independent so
that the BSM value of this pay-off is the probability of S(τ) hittingthe range times the value of the forward start call
Under alternative models, we compute the above expectation, PV,by means of MC simulations and in addition compute the hittingprobability, P , P = E
[1{D−∆<S(τ)<D+∆}
]Then we imply the conditional volatility using the BSM inversion forcall with strike K, time to maturity T − τ , and value PV/P
We compare two models: local volatility (LV) and beta SV with localvol (LSV)
93
Conditional forward skew II
6m6m, 105%-115%
10%
15%
20%
25%
30%
35%
40%
75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125%
Strike %
Imp
lied
Vo
l
Implied Vol,6m
Local Vol Model
LSV
6m6m, 85%-95%
10%
15%
20%
25%
30%
35%
40%
75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125%
Strike %
Imp
lied
Vo
l
Implied Vol, 6m
Local Vol Model
LSV
6m6m conditional skew for D = 110% (left) and D = 90% (right)
94
Conditional forward skew III
6m6m, 120%-130%
10%
15%
20%
25%
30%
35%
40%
75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125%
Strike %
Imp
lied
Vo
l
Implied Vol, 1y
Local Vol Model
LSV
6m6m, 70%-80%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125%
Strike %
Imp
lied
Vo
l
Implied Vol, 1y
Local Vol Model
LSV
6m6m conditional skew for D = 125% (left) and D = 75% (right)
95
Conditional forward skew IV
6m6m, 135%-145%
10%
15%
20%
25%
30%
35%
40%
75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125%
Strike %
Imp
lied
Vo
l
Implied Vol, 1y
Local Vol Model
LSV
6m6m, 55%-65%
10%
20%
30%
40%
50%
60%
70%
75% 80% 85% 90% 95% 100% 105% 110% 115% 120% 125%
Strike %
Imp
lied
Vo
l
Implied Vol, 1y
Local Vol Model
LSV
6m6m conditional skew for D = 140% (left) and D = 60% (right)
96
ConclusionsI presented the beta stochastic volatility model that
1) Has intuitive parameters (volatility beta) that can be explainedusing empirical data
2) Calibration of parameters for SV process and jumps is straightfor-ward and intuitive (no non-linear optimization methods are necessary)
3) Allows to mix parameters to reproduce different regimes of volatil-ity and the equity skew
4) Equipped with jumps, allows to reproduce correlation skew formulti-underlyings
5) Produces very steep forward skews
6) The driver for the instantaneous volatility has nice properties: fattails and level dependent spot-volatility correlations
97
Open questions1) Better understanding of relationship between model parameters ofmarket observables (ATM vol, skew, their term structures)
2) Model implied risk incorporating the stickiness ratio
3) Numerical methods (numerical PDE, analytic approximations)
4) Calibration of jumps and correlation skew
5) Illustrate/proof that only SV model with jumps is consistent withobserved empirical features:A) Stickiness ratio is between 1 and 2B) Steep correlation skew
Models with local volatility and correlation may be consistent withA) and B) but they are not consistent with observed dynamics thusproducing wrong hedges
Thank you for your attention!
98
Disclaimer
The opinions and views expressed in this presentation are those ofthe author alone and do not necessarily reflect the views and policiesof Bank of America Merrill Lynch
99