1. Modelling Volatility Skews Bruno Dupire Bloomberg bdupire@bloomberg.net London, November 17, 2006

2. OUTLINE 1.Generalities 2.Leverage and jumps 3.Break-even volatilities 4.Volatility models 5.Forward Skew 6.Smile arbitrage

3. I.Generalities

4. Bruno Dupire 4 Market Skews Dominating fact since 1987 crash: strong negative skew on Equity Markets Not a general phenomenon Gold: FX: We focus on Equity Markets K KK

5. Bruno Dupire 5 Skews Volatility Skew: slope of implied volatility as a function of Strike Link with Skewness (asymmetry) of the Risk Neutral density function ? MomentsStatisticsFinance 1ExpectationFWD price 2VarianceLevel of implied vol 3SkewnessSlope of implied vol 4KurtosisConvexity of implied vol

6. Bruno Dupire 6 Why Volatility Skews? Market prices governed by –a) Anticipated dynamics (future behavior of volatility or jumps) –b) Supply and Demand To “ arbitrage” European options, estimate a) to capture risk premium b) To “arbitrage” (or correctly price) exotics, find Risk Neutral dynamics calibrated to the market K Market Skew Th. Skew Supply and Demand

7. Bruno Dupire 7 Modeling Uncertainty Main ingredients for spot modeling Many small shocks: Brownian Motion (continuous prices) A few big shocks: Poisson process (jumps) t S t S

8. Bruno Dupire 8 To obtain downward sloping implied volatilities –a) Negative link between prices and volatility Deterministic dependency (Local Volatility Model) Or negative correlation (Stochastic volatility Model) –b) Downward jumps K 2 mechanisms to produce Skews (1)

9. Bruno Dupire 9 2 mechanisms to produce Skews (2) –a) Negative link between prices and volatility –b) Downward jumps

10. Leverage and Jumps

11. Bruno Dupire 11 Dissociating Jump & Leverage effects Variance : Skewness : t0t0 t1t1 t2t2 x = S t1 -S t0 y = S t2 -S t1 Option prices Δ Hedge FWD variance Option prices Δ Hedge FWD skewness Leverage

12. Bruno Dupire 12 Dissociating Jump & Leverage effects Define a time window to calculate effects from jumps and Leverage. For example, take close prices for 3 months Jump: Leverage:

13. Bruno Dupire 13 Dissociating Jump & Leverage effects

14. Bruno Dupire 14 Dissociating Jump & Leverage effects

15. Break Even Volatilities

16. Bruno Dupire 16 Theoretical Skew from Prices Problem : How to compute option prices on an underlying without options? For instance : compute 3 month 5% OTM Call from price history only. 1)Discounted average of the historical Intrinsic Values. Bad : depends on bull/bear, no call/put parity. 2)Generate paths by sampling 1 day return recentered histogram. Problem : CLT => converges quickly to same volatility for all strike/maturity; breaks autocorrelation and vol/spot dependency. ? =>

17. Bruno Dupire 17 Theoretical Skew from Prices (2) 3)Discounted average of the Intrinsic Value from recentered 3 month histogram. 4) Δ-Hedging : compute the implied volatility which makes the Δ- hedging a fair game.

18. Bruno Dupire 18 Theoretical Skew from historical prices (3) How to get a theoretical Skew just from spot price history? Example: 3 month daily data 1 strike –a) price and delta hedge for a given within Black-Scholes model –b) compute the associated final Profit & Loss: –c) solve for –d) repeat a) b) c) for general time period and average –e) repeat a) b) c) and d) to get the “theoretical Skew” t S K

19. Bruno Dupire 19 Strike dependency BE volatility is an average of returns, weighted by the Gammas, which depend on the strike

20. Bruno Dupire 20 Theoretical Skew from historical prices (4)

21. Bruno Dupire 21 Theoretical Skew from historical prices (4)

22. Bruno Dupire 22 Theoretical Skew from historical prices (4)

23. Bruno Dupire 23 Theoretical Skew from historical prices (4)

24. Local Volatility Model

25. Bruno Dupire 25 One Single Model We know that a model with dS =  (S,t)dW would generate smiles. –Can we find  (S,t) which fits market smiles? –Are there several solutions? ANSWER: One and only one way to do it.

26. Bruno Dupire 26 Diffusions Risk Neutral Processes Compatible with Smile The Risk-Neutral Solution But if drift imposed (by risk-neutrality), uniqueness of the solution

27. Bruno Dupire 27 Forward Equation BWD Equation: price of one option for different FWD Equation: price of all options for current Advantage of FWD equation: –If local volatilities known, fast computation of implied volatility surface, –If current implied volatility surface known, extraction of local volatilities:

28. Bruno Dupire 28 Forward Equations (2) Several ways to obtain them: –Fokker-Planck equation: Integrate twice Kolmogorov Forward Equation –Tanaka formula: Expectation of local time –Replication Replication portfolio gives a much more financial insight

29. Bruno Dupire 29 K,T fixed. C 0 price with LVM Real dynamics: Ito Taking expectation: Equality for all (K,T)  Volatility Expansion

30. Bruno Dupire 30 Summary of LVM Properties is the initial volatility surface compatible with local vol compatible with (calibrated SVM are noisy versions of LVM) deterministic function of (S,t) (if no jumps) future smile = FWD smile from local vol Extracts the notion of FWD vol (Conditional Instantaneous Forward Variance)

31. Stochastic Volatility Models

32. Bruno Dupire 32 Heston Model Solved by Fourier transform:

33. Bruno Dupire 33 Role of parameters Correlation gives the short term skew Mean reversion level determines the long term value of volatility Mean reversion strength –Determine the term structure of volatility –Dampens the skew for longer maturities Volvol gives convexity to implied vol Functional dependency on S has a similar effect to correlation

39. Bruno Dupire 39 2 ways to generate skew in a stochastic vol model -Mostly equivalent: similar (S t,  t  ) patterns, similar future evolutions -1) more flexible (and arbitrary!) than 2) -For short horizons: stoch vol model  local vol model + independent noise on vol. Spot dependency STST  STST 

40. Bruno Dupire 40 SABR model F: Forward price with correlation 

46. Bruno Dupire 46 The SABR Claim

47. Bruno Dupire 47 Market behavior

48. Bruno Dupire 48 SABR fallacy SABR claims to dissociate vol dynamics from Skew fitting Many banks manage their vol risk with SABR BUT if 2 SABR models fit the same skew, they generate essentially the same vol dynamics, which coincide in average with LVM dynamics!

49. Bruno Dupire 49 Calibration according to SABR Backbone: as a function of F with frozen depends only on Many banks calibrate by: 1) Estimating from historical backbone 2) Then adjusting to fit the skew

50. Bruno Dupire 50 Problems Freezing ignores which actually impacts the average backbone SABR can be rewritten with the lognormal volatility In this parameterization instead of, freezing gives : the backbone is then constant! The backbone depends on the parameterization, it is not intrinsic to the model: it is a flawed concept

51. Bruno Dupire 51 Message from LVM In particular for short maturities so average backbone ~ local vols! In the absence of jumps, the skew is due to average levels of vols higher on the downside If you were in that region increases

52. Bruno Dupire 52 2 fitting models SABR A SABR B calibrated to A Same skew Different backbones A0.10 0 B 10.175-0.58

53. Bruno Dupire 53 Comparison Scattered plot Average backbone Same skew similar vol dynamics = LVM vol dynamics in average

54. Bruno Dupire 54 SABR Conclusion To dissociate vol dynamics from skew fitting, jumps are needed In a smile (as opposed to skew) dominated market, it is less clear as and cancel their first order impact Banks may be unaware of the inconsistency of their risk management

55. Forward Skew

56. Bruno Dupire 56 Forward Skews In the absence of jump : model fits market This constrains a) the sensitivity of the ATM short term volatility wrt S; b) the average level of the volatility conditioned to S T =K. a) tells that the sensitivity and the hedge ratio of vanillas depend on the calibration to the vanilla, not on local volatility/ stochastic volatility. To change them, jumps are needed. But b) does not say anything on the conditional forward skews.

57. Bruno Dupire 57 Many products depend on short term skew in the future Example: Napoleon, globally/locally capped/floored cliquet What does current vol surface tell us on future short term skew? Controlling Fwd Skew Local cap/floor: Global cap/floor:

58. Bruno Dupire 58 flat 1 month skew but if decreases: generate initial skew Toy model but flat future skew A) : freeze vol at beginning of month

59. Bruno Dupire 59 Toy model but not flat future skew B) (or slightly increasing in )

60. Bruno Dupire 60 Limitations Behavior of future 1 month skew very different if start date is mid month instead of beginning of month The amplitude of jumps is a measure of the wrongness of the model LVM corresponds to the jumpless case

61. Bruno Dupire 61 Sensitivity of ATM volatility / S At t, short term ATM implied volatility ~ σ t. As σ t is random, the sensitivityis defined only in average: In average, follows. Optimal hedge of vanilla under calibrated stochastic volatility corresponds to perfect hedge ratio under LVM.

62. Bruno Dupire 62 Market Model of Implied Volatility Implied volatilities are directly observable Can we model directly their dynamics? where is the implied volatility of a given Condition on dynamics?

63. Bruno Dupire 63 Drift Condition Apply Ito’s lemma to Cancel the drift term Rewrite derivatives of gives the condition that the drift of must satisfy. For short T, we get the Short Skew Condition (SSC): close to the money:  Skew determines u 1

64. Bruno Dupire 64 Optimal hedge ratio Δ H : BS Price at t of Call option with strike K, maturity T, implied vol Ito: Optimal hedge minimizes P&L variance: BS Delta BS Vega Implied Vol sensitivity

65. Bruno Dupire 65 Optimal hedge ratio Δ H II With  Skew determines u 1, which determines Δ H

66. Smile Arbitrage

67. Bruno Dupire 67 S0S0 K T1T1 T2T2 t0 Deterministic future smiles It is not possible to prescribe just any future smile If deterministic, one must have Not satisfied in general

68. Bruno Dupire 68 Det. Fut. smiles & no jumps => = FWD smile If stripped from Smile S.t Then, there exists a 2 step arbitrage: Define At t0 : Sell At t: gives a premium = PLt at t, no loss at T Conclusion: independent of from initial smile S t0t0 tT S0S0 K

69. Bruno Dupire 69 Consequence of det. future smiles Sticky Strike assumption: Each (K,T) has a fixed independent of (S,t) Sticky Delta assumption: depends only on moneyness and residual maturity In the absence of jumps, – Sticky Strike is arbitrageable – Sticky Δ is (even more) arbitrageable

70. Bruno Dupire 70 Example of arbitrage with Sticky Strike Each CK,T lives in its Black-Scholes ( )world P&L of Delta hedge position over dt: If no jump !

71. Bruno Dupire 71 Arbitrage with Sticky Delta In the absence of jumps, Sticky-K is arbitrageable and Sticky-Δ even more so. However, it seems that quiet trending market (no jumps!) are Sticky-Δ. In trending markets, buy Calls, sell Puts and Δ-hedge. Example: S Vega > Vega PF Vega < Vega SPF Δ-hedged PF gains from S induced volatility moves.

72. Bruno Dupire 72 Conclusion Both leverage and asymmetric jumps may generate skew but they generate different dynamics The Break Even Vols are a good guideline to identify risk premia The market skew contains a wealth of information and in the absence of jumps, –The spot correlated component of volatility –The average behavior of the ATM implied when the spot moves –The optimal hedge ratio of short dated vanilla –The price of options on RV If market vol dynamics differ from what current skew implies, statistical arbitrage

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