1. No category

Pricing and Hedging under Log-normal
Stochastic Volatility Dynamics
Artur Sepp
Bank of America Merrill Lynch, London
[email protected]
2015
1
Headlines
1. Empirical evidence for log-normality of realized and implied volatilities
2. Hedging in the log-normal stochastic volatility (SV) model consistently with empirical dynamics of realized and implied vols
Introduce the volatility beta and volatility skew-beta
3. Closed-form solution for the moment generating function in the lognormal SV model
4. Model applications for estimating statistical dynamics of implied and
realized vols and generating alpha from volatility trading strategies
2
Risk-neutral and econometric volatility models
1) Risk-neutral models
• Local volatility (Dupire (1994)) and its extensions to stochastic volatility (LSV) and jumps
• Stochastic volatility (SABR (2002), Heston (1993))
2) Econometric models
• Garch (Engle and Bollerslev)
• Affine state stochastic vol jump-diffusions (Duffie-Pan-Singleton (2000))
Applications of these models are primarily driven by:
• Analytical tractability, not empirical consistency
• Fitting of risk-neutral models to market implied distributions, not statistical distributions
Instead, I present the factor model for volatility dynamics based on
my work with Piotr Karasinski (Sepp-Karasinski (2012)) - the beta
stochastic volatility model
Primary goal is to fit the empirical dynamics of implied and realized vols
and optimize delta-hedging P&L
3
Volatility models for alpha generation
Alpha generation arises from selling protection at high risk-premiums by
• volatility trading strategies (proprietary views)
• structured products (traditional sell-side approach)
For volatility trading strategies:
1) For delta-hedge, we need to project changes in primary risk drivers
(the S&P 500 index) into secondary drivers (the S&P 500 implied vols)
2) To generate trading signals, we need statistical measures for dynamics
of implied and realized volatilities
For sell-side trading and hedging models:
1) How the implied and realized risk factors (volatility and skew) affect
realized P&L
2) Develop valuation and hedging model consistent with the empirical
dynamics to optimize realized P&L
4
The basis for volatility of volatility, like normal or lognormal, is important for stationary model with timeindependent parameters
Compute the empirical frequency of one-month implied at-the-money
(ATM) volatility proxied by the VIX index for last 20 years
Daily observations normalized to have zero mean and unit variance
Left figure: empirical frequency of the VIX - it is definitely not normal
Right figure: the frequency of the logarithm of the VIX - it does look
like the normal density, especially for the right tail!
Empirical frequency of
normalized VIX
5%
4%
3%
Frequency
6%
Empirical
Standard Normal
2%
1%
0%
VIX
-4
-3
-2
-1
0
1
2
3
4
7%
6%
5%
4%
3%
2%
1%
0%
Empirical frequency of
normalized logarithm of the VIX
Empirical
Standard Normal
Frequency
7%
Log-VIX
-4
-3
-2
-1
0
1
2
3
5
4
Empirical frequency of realized vol is also log-normal
Compute one-month realized volatility of daily returns on the S&P 500
index for each month over non-overlapping periods for last 60 years from
1954
Below is the empirical frequency of normalized historical volatility
Left figure: frequency of realized vol - it is definitely not normal
Right figure: frequency of the logarithm of realized vol - again it does
look like the normal density, specially for the right tail!
4%
Frequency
8%
6%
10%
Frequency of Historic 1m
Volatility of S&P500 returns
Frequency of Logarithm of
Historic 1m Volatility of S&P500
8%
Empirical
Standard Normal
6%
4%
2%
Frequency
10%
Empirical
Standard Normal
2%
Log-Vol
Vol
0%
0%
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
6
4
Beta SV model is factor model for changes in vol V (tn)
predicted by returns in price
S(tn): #
"
S(tn) − S(tn−1)
V (tn) − V (tn−1) = β
+ V (tn−1)n
S(tn−1)
(1)
iid normal residuals n are scaled by vol V (tn−1) due to log-normality
Left figure: scatter plot of daily changes in the VIX vs returns on S&P
500 for past 14 years and estimated regression model
Volatility beta β explains about 80% of variations in volatility
Right: time series of empirical residuals n of regression model (1)
Regression model is stable across different estimation periods
-10%
20%
Change in
VIX vs Return on S&P500
15%
10%
5%
0%
-5% -5% 0%
-10%
-15%
-20%
30%
Time Series of Residual Volatility
20%
y = -1.08x
R² = 67%
10%
0%
-10%
5%
10%
-20%
-30%
Return % on S&P 500
Dec-99
Dec-00
Jan-02
Jan-03
Jan-04
Jan-05
Jan-06
Jan-07
Jan-08
Jan-09
Jan-10
Jan-11
Jan-12
Jan-13
Change in VIX
Volatility beta β: expected change in ATM vol predicted by price return
For return of −1%: expected change in vol = −1.08 × (−1%) = 1.08%
7
Beta stochastic volatility model (Karasinski-Sepp 2012):
is obtained by summarizing our empirical findings for
dynamics of index price St and volatility Vt:
(0)
dSt = VtStdWt
dSt
(1)
dVt = β
+ εVtdWt + κ(θ − Vt)dt
St
(2)
• Vt is either returns (realized) vol or short-term ATM implied vol
(0)
• Wt
(1)
and Wt
are independent Brownian motions
• β is volatility beta - sensitivity of volatility to changes in price
• ε is residual vol-of-vol - standard deviation of residual changes in vol
• Mean-reversion rate κ and mean θ are added for stationarity of volatility
8
From model dynamics to hedging in practice: the minimum variance delta is applied to hedge against changes
in spot and ATM vol in the least-squares sense by minimizing the variance of delta-hedging P&L
For delta computation, choice of any particular model for stochastic
or local vol is irrelevant if models are calibrated to same implied
vols (Sepp (2014))
In presence of the risk-aversion, implied distributions will significantly
differ from statistical distributions especially for fat-tailed distributions
(Bakshi-Kapadia-Madan (2003))
Risk-neutral volatility models take market prices as inputs and extrapolate
the implied dynamics - they produce wrong hedges if the implied dynamics
are different from the realized (statistical) dynamics
9
Modeling challenge is to have a model is consistent with
both the statistical dynamics of implied and realized vols
and with the implied market skew
Step I: log-normal model is a proper basis for volatility dynamics
Step II: The concept of volatility skew-beta (for log-normal model) and
model calibration consistently with empirical dynamics
10
Volatility skew-beta for computing correct option delta
Figure 1) Apply regression model
(1) for time series of ATM vols for
maturities T = {1m, 3m, 6m, 12m, 24m}
(m=month) to estimate regression
volatility beta βREGRES (T ) using
S&P500 returns:
δσAT M (T ) = βREGRES (T ) × δS
0.0
-0.2
-0.4
Regression Volatility Beta(T)
y = 0.19*ln(x) - 0.37
R² = 99%
-0.6
Regression Vol Beta(T)
Decay of Vol Beta in ln(T)
-0.8
-1.0
0.08
Volatility beta β for SV dynamics is
instantaneous beta for very small T
0.0
-0.2
Regression vol beta decays in log-T
due to mean-reversion: long-dated
ATM vols are less sensitive in absolute values to price-returns
-0.4
Maturity T
1.00
2
y = 0.16*ln(x) - 0.29
R² = 99%
-0.6
-0.8
2.0
0.50
Implied Volatility Skew (T)
1m
Figure 2) Implied vol skew for maturity T has similar decay in log-T
0.25
3m
Vol Skew (T)
Decay of Skew in ln(T)
Maturity, T
6m
1y
2y
Volatility Skew-Beta(T)
1.5
Figure 3) Volatility skew-beta is
1.0
regression beta divided by skew
0.5
Skew-Beta(T ) ∝ βREGRES (T )/SKEW(T )
0.0
It is nearly maturity-homogeneous
y = 0.06*ln(x) + 1.41
R² = 80%
Vol Skew-Beta (T)
Decay of Vol Skew-Beta in ln(T)
1m
3m
6m
Maturity T
1y
2y 11
BSM volatility and delta-hedging P&L
Vanilla options are marked using Black-Scholes-Merton (BSM) implied
vol σBSM (K) in %-strike K relative to price S(0) (any SV model implies
quadratic form for implied vols near ATM strikes so generic approach):
σBSM (K; S) = σAT M (S) + SKEW × Z(K; S)
• Z(K; S) is log-moneyness relative to current price S:
Z(K; S) = ln (K × S(0)/S )
• SKEW < 0 is inferred from spread between call and put implied vols
In practice, this form is augmented with extras for convexity and tails
Volatility P&L arises from change in volatility induced by the change in
spot price S → S {1 + δS}:
δσBSM (K; S) ≡ σBSM (K; S {1 + δS}) − σBSM (K; S)
= δσAT M (S) + SKEW × δZ(K; S)
• Stochastic contributor to P&L: change in ATM vol δσAT M (S):
δσAT M (S) = σAT M (S {1 + δS}) − σAT M (S)
• Deterministic contributor to P&L: change in log-moneyness relative
to skew:
δZ(K; S) = − ln(1 + δS) ≈ −δS
12
Changes in the skew are not correlated to changes in
price and ATM vols - we can account only to changes
in ATM vol for correct predict of vol P&L
Empirical observations yet again confirm log-normality dynamics!
(Using S&P500 data from January 2007 to December 2013)
Change in 0.3
1m skew vs Price Return
0.2
0.1
0
-0.1
-0.2
Price return
-0.3
-15%
-5%
y = 0.13x - 0.00
R² = 0%
5%
15%
Change in0.04
1y skew vs Price Return
Change in Skew
Change in Skew
Figure 1: weekly changes in 100% − 95% skew vs price returns for
maturity of one month, 1m, (left) and one year, 12m (right)
0.02
0
-0.02
-0.04
Price return -0.06
-15%
-5%
y = 0.03x - 0.00
R² = 1%
5%
15%
Change in0.3
1m skew vs 1m ATM vol
0.2
0.1
0
-0.1
-0.2
Change in ATM vol
-0.3
-15%
-5%
y = -0.15x - 0.00
R² = 0%
5%
15%
Change in0.04
1y skew vs 1y ATM vol
Change in Skew
Change in Skew
Figure 2: weekly changes in 100% − 95% skew vs changes in ATM
vols for maturity of one month, 1m, (left) and one year, 12m (right)
0.02
0
-0.02
-0.04
Change in ATM-0.06
vol
-15%
-5%
y = -0.09x - 0.00
R² = 2%
5%
13
15%
Volatility skew-beta combines the skew and volatility
P&L together - positive change in ATM vol from negative return is reduced by skew
Given price return δS: S → S {1 + δS}
1) For ATM vol change
• Volatility P&L follows change in ATM vol predicted by regression beta
and vol skew-beta:
δσAT M (S) = βREGRESS × δS
= SKEWBETA × SKEW × δS
2) For BSM vol at fixed strikes
• Log-moneyness changes by δZ(K; S) ≈ −δS
• Volatility P&L is change in ATM vol adjusted for skew P&L:
δσBSM (K) ≡ δσAT M (S) − SKEW × δS
= [SKEWBETA − 1] × SKEW × δS
BSM vol at S0
BSM vol at S1, ATM vol shift
BSM at S1, fixed strikes
Change in ATM vol
Change in vols at fixed strikes
0.8%
18%
0.5%
15%
0.3%
Strike K%
0.0%
90%
95%
100%
BSM vol(K)
21%
1.0%
12%
105%
90%
Strike K%
95%
100%
14
105%
Volatility skew-beta under minimum-variance approach
is applied to compute min-var delta ∆ for hedging against
changes in price and price-induced changes in implied vol
A) We adjust option delta for change in BSM implied vol at fixed strikes
B) The adjustment is proportional to option vega at this strike:
∆(K, T ) = ∆BSM (K, T ) + [SKEWBETA(T ) − 1] × SKEW(T ) × VBSM (K, T )/
∆BSM (K, T ) is BSM delta for strike K and maturity T
VBSM (K, T ) is BSM vega, both evaluated at volatility skew
I classify volatility regimes using vol skew-beta for delta-adjustments:


∆BSM (K, T ) + SKEW(T ) × VBSM (K, T )/S,




∆
BSM (K, T ),
∆(K, T ) =

∆BSM (K, T ) − SKEW(T ) × VBSM (K, T )/S,




1
∆
BSM (K, T ) + 2 SKEW(T ) × VBSM (K, T )/S,
Sticky local
Sticky strike
Sticky delta
Empirical S&P500
”Shadow” delta is obtained using ratio O (may be different from 1/2):
∆(K, T ) = ∆BSM (K, T ) + O × SKEW(T ) × VBSM (K, T )/S
which is traders’ ad-hoc adjustment of option delta
15
Volatility empirical skew-beta and vol regimes are applied for
 model calibration
SkewBeta =


2,



1,
0,
Sticky local regime: minimum-variance delta in SV and LV
Sticky strike regime: BSM delta evaluated at implied skew
Sticky delta regime: model delta in space-homogeneous SV
Using beta SV model with empirical estimate of vol beta and adding
risk-premiums to match the skew premium, we can fit empirical vol
skew-beta and compute correct model delta
1) S&P 500 and STOXX 50: empirical skew-beta of about 1.5
2) NIKKEI: weak skew-beta is about 0.5
Empirical estimates for skew-beta and its lower and upper bounds are
found by empirical regression model
Vol Skew-Beta for S&P500
2.00
0.00
1.50
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
1.00
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
0.50
T in months 0.00
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
0.50
Vol Skew-Beta for NIKKEI
2.00
1.50
1.00
2.50
T in months
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
2.50
16
Estimation and stationarity of skew-beta: skew-beta β (T ) is
estimated empirically (for each vol maturity) by regression of changes in
(T )
(T ) :
ATM vols σATM(tn) predicted by price-return
times
skew
Skew
#
"
S(tn) − S(tn−1)
(T )
(T )
(T
)
× Skew(T )(tn−1)
σATM(tn) − σATM(tn−1) = β
×
S(tn−1)
The skew-beta is range-bound between 1 and 2 with average of
about 1.5, with weak dependence on maturity time
The regression has explain of about 70% for Euro Stoxx 50 vols
Figure: skew-beta of 6m ATM vol for Euro Stoxx 50 using rolling window
of 1 and 12 months
4
Volatility Skew Beta
1m
12m
3
2
1
11-Jan-15
11-Jan-14
11-Jan-13
11-Jan-12
11-Jan-11
11-Jan-10
11-Jan-09
-1
11-Jan-08
0
17
Empirical skew beta applied to vol trading strategies
can significantly increase performance - P&L for deltahedging short 6m straddle on EuroStoxx50
Back-test of monthly rolls into new straddle with maturity of 6m from
2007 to 2015
Delta-hedge daily using specific rules for option delta
Month-end trade close: account for delta-hedge P&L and vol P&L (significant for straddles due to large vega!), 96 monthly P&L-s
Hedging rules:
• StickyStrike - hedging at sticky strike vol (BSM delta),
• StickyLocalVol - using minimum variance hedge,
• Empirical - using skew-beta with estimation window of past xx days
Transaction costs = 5bp
Only empirical hedge produces positive Sharpe ratio
SkewBeta
Sharpe Ex-Costs
Sharpe Post-Costs
StickyStrike
1
0.03
-0.12
StickyLocalVol
2
-0.06
-0.31
Empirical
Dynamic
0.25
0.12
18
P&L for delta-hedging short 6m straddle on EuroStoxx50
with trading signal!
Back-test of monthly rolls into new vols with maturity of 6m from 2007
to 2015 subject to a proprietary trading signal
Delta-hedging using empirical skew-beta outperforms significantly
SkewBeta
Sharpe Ex-Costs
Sharpe Post-Costs
%
1.0
StickyStrike
1
0.57
0.54
62%
StickyLocalVol
2
0.51
0.47
84%
Empirical
Dynamic
0.74
0.87
Sharpe Ratio Post-Costs
0.87
0.8
0.6
0.54
0.47
0.4
0.2
0.0
StickyStrike
StickyLocalVol
Empirical
19
Risk-Premiums in the dynamic SV model
• Empirical skew-beta can be applied for trading strategies in vanilla
options within BSM model with implied vol marking
• Positive expected Sharpe ratio from short vol strategies is created by
risk-premiums in out-of-the-money puts
• Risk premiums arise from heavy tailed distributions of index returns
along with risk-aversion of investors
• Minimum variance hedging using implied measure overestimates the
skew so over-hedges option deltas
• To value and hedge structured products using a dynamic SV model,
we need:
1) Incorporate risk-premiums into model dynamics
2) Calibrate model to both the implied skew and empirical skewbeta
20
Beta SV model with jumps is fitted to empirical&implied
dynamics for computing correct delta (Sepp 2014):
dSt
(0)
= (µ − λ(eη − 1)) dt + VtdWt + (eη − 1)dNt
St
(0)
dVt = κ(θ − Vt)dt + βVtdWt
(1)
+ εVtdWt
+ βηdNt
1) Consistent with empirical dynamics of implied ATM volatility by
specifying empirical volatility beta β
2) Has jumps from Poisson process Nt with intensity λ, as proxy for the
risk-premium, to make model fit to both the empirical dynamics and
the skew-premium - only one parameter fitted to implied skew
For valuation kernel, the distribution of jumps does not matter jumps are only needed to fit the skew risk-premium
For modeling implied vols, we can think of jumps as risk-premiums
for pricing kernel (they have no connection to jumps under statistical
measure)
21
Closed-form solution for log-normal Beta SV (Sepp 2014):
Mean-reverting log-normal SV models are not analytically tractable
I derive a very accurate exp-affine approximation for moment generating function (details in my paper)
Idea comes from information theory: apply Kullback-Leibler relative entropy for unknown PDF p(x) and test PDF q(x) with moment constraints:
R k
R k
x p(x)dx = x q(x)dx, k = 1, 2, ...
Thinking in terms of moment function:
[i] MGF for Beta SV model with normal driver for SV (as in Stein-Stein
SV model) has exact solution, which has exp-affine form
[ii] Correction for log-normal SV has an exp-affine form
35%
30%
Implied vol for 1y S&P500
options, beta SV, NO JUMPS
35%
30%
25%
25%
20%
20%
15%
10%
5%
Analytic for Normal SV
Closed-form for Log-normal SV
Monte-Carlo for Log-normal SV
Strike
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
15%
10%
5%
Implied vol for 1y S&P500
options, beta SV, WITH JUMPS
Analytic for Normal SV
Closed-form for Log-normal SV
Monte-Carlo for Log-normal SV
Strike
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
22
Proof that closed-form MFG for log-normal model produces theoretically consistent probability density
1) Derive solutions for excepted values, variances, and covariances of the
log-price and quadratic variance (QV) by solving PDE directly
2) Prove that moments derived using approximate MGF equal to theoretical moments derived in 1)
Using closed-form MFG for log-normal model, we apply standard valuation
methods for affine SV models
Difference: system of ODE-s is solved numerically by Runge-Kutta
Matlab code for experiments is available (link in my paper)
Implementation of closed-form moment function (MGF), MC, and
PDE pricers produce values of vanilla options on equity and quadratic
variance that are equal within numerical accuracy of these methods
35%
30%
Implied vol for 1y S&P500
options, beta SV, NO JUMPS
35%
30%
25%
25%
20%
20%
15%
Closed-form MGF
15%
10%
Monte-Carlo
10%
5%
PDE, numerical solver
Strike
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
5%
Implied vol for 1y S&P500
options, beta SV, WITH JUMPS
Closed-form MGF
Monte-Carlo
PDE, numerical solver
Strike
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
23
Robust features of Beta model with risk-premiums
1) the model has ability to fit empirical vol skew-beta and produce correct
option delta without re-calibration
Figure: delta from SVJ model fits empirical backbone
1.00
Delta for 1y call option on
S&P500
0.80
0.60
0.40
SV model
SV with Min Var hedge
0.20
SV re-calibrated to Market backbone
0.00
SVJ (without re-calibration)
0.70
0.80
0.90
1.00
1.10
1.20
Strike
2) Consistent with the empirical distributions of implied and realized
volatilities, which are very close to log-normal
3) It has clear intuition behind the key model parameters:
Volatility beta is sensitivity to changes in short-term ATM vol
Residual vol-of-vol is volatility of idiosyncratic changes in ATM vol
4) P&L explain is possible in terms of implied and realized quantities
of key model parameter - volatility beta
24
Calibration of beta SV model is based on econometric
and implied approaches without large-scale non-linear
and non-intuitive calibrations
1) Parameters of SV part are estimated from time series
2) Jump/risk-aversion params are fitted to empirical vol skew-beta
Params in 1) & 2) are updated only following changes in volatility regime
3) Small mis-calibrations of the SV part and jumps are corrected
using local vol (LV) part
Contribution to skew from LV part is kept small (10-20%)
Local vol part is re-calibrated on the fly to reproduce small variations in
some parts of implied vol surface, which are caused by temporary supplydemand factors specific to that part
It is also robust to compute bucketed vega risk in this way
In practical terms:
1) Local volatility part accounts for the noise from idiosyncratic
changes in implied volatility surface
2) Stochastic volatility and jumps serve as time- and space-homogeneou
factors for the dynamics and shape of the implied vol surface
25
Conclusions
• Illustrate the relationship between empirical volatility skew-beta
and delta-hedging P&L and their applications for trading strategies
Accounting for empirical skew-beta can considerably improve realized
P&L of delta-hedging strategies
• Present empirical evidence for log-normality and implied and
realized volatilities
Closed-form solution for moment-generating function in the log-normal
SV model
Apply for maximum likelihood estimation of dynamics of realized and
implied volatilities and generate trade signals
• Introduce the beta stochastic volatility model with risk-premiums
Make the model consistent with both implied skew and empirical skewbeta
26
References:
Beta stochastic volatility model:
Karasinski, P., Sepp, A., (2012), “Beta stochastic volatility model,” Risk,
October, 67-73
http://ssrn.com/abstract=2150614
Computing option delta consistently with empirical dynamics:
Sepp, A., (2014), “Empirical Calibration and Minimum-Variance Delta
Under Log-Normal Stochastic Volatility Dynamics”
http://ssrn.com/abstract=2387845
Sepp, A. (2014), “Realized and implied index skews and minimum-variance
hedging”, Global Derivatives conference in Amsterdam 2014
kodu.ut.ee/~spartak/papers/ArturSeppGlobalDerivatives2014.pdf
Closed-form solution for log-normal model:
Sepp, A., (2014), “Affine Approximation for Moment Generating Function of Log-Normal Stochastic Volatility Model”
http://ssrn.com/abstract=2522425
27
Disclaimer
All statements in this presentation are the authors personal views and not
necessarily those of Bank of America Merrill Lynch.
28