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

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

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

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

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

65. Sepp, A. (2011), Ecient Numerical PDE Methods to Solve Cali- bration and Pricing Problems in Local Stochastic Volatility Models, Global Derivatives conference in Paris Sepp, A., (2013), When You Hedge Discretely: Optimization of Sharpe Ratio for Delta-Hedging Strategy under Discrete Hedging and Transaction Costs, The Journal of Investment Strategies 3(1), 19- 59 Sepp, A., (2014), Empirical Calibration and Minimum-Variance Delta Under Log-Normal Stochastic Volatility Dynamics, Working paper, http://ssrn.com/abstract=2387845 Sepp, A., (2014), Ane Approximation for Moment Generating Function of Log-Normal Stochastic Volatility Model, Working pa- per, http://ssrn.com/abstract=2522425