WIXLIB

Skew ModelingBruno DupireBloomberg LPbdupire@bloomberg.netColumbia University, New YorkSeptember 12, 2005

I.Generalities

Market SkewsDominating fact since 1987 crash: strong negative skew onEquity Marketsσ implKNot a general phenomenonGold: σ implFX: σ implKKWe focus on Equity MarketsBruno Dupire3

Skews Volatility Skew: slope of implied volatility as afunction of Strike Link with Skewness (asymmetry) of the RiskNeutral density function ϕ ?Moments1234Bruno sFinanceFWD priceLevel of implied volSlope of implied volConvexity of implied vol4

Why Volatility Skews? Market prices governed by– a) Anticipated dynamics (future behavior of volatility or jumps)– b) Supply and Demandσ implSupMarket SkewplyandDemandTh. SkewK To “ arbitrage” European options, estimate a) to capturerisk premium b) To “arbitrage” (or correctly price) exotics, find RiskNeutral dynamics calibrated to the marketBruno Dupire5

Modeling UncertaintyMain ingredients for spot modeling Many small shocks: Brownian Motion(continuous prices) St A few big shocks: Poisson process (jumps)StBruno Dupire6

2 mechanisms to produce Skews (1) To obtain downward sloping implied volatilitiesσimpK– a) Negative link between prices and volatility Deterministic dependency (Local Volatility Model) Or negative correlation (Stochastic volatility Model)– b) Downward jumpsBruno Dupire7

2 mechanisms to produce Skews (2)– a) Negative link between prices and volatilityS1S2– b) Downward jumpsS1Bruno DupireS28

Leverage and Jumps

Dissociating Jump & Leverage effectst0t1t2x St1-St0 Variance :y St2-St1(x y) x 2xy y222Option pricesFWD variance Hedge Skewness :(x y)3 x3 3x2 y 3xy2 y3LeverageOption prices HedgeBruno DupireFWD skewness10

Dissociating Jump & Leverage effectsDefine a time window to calculate effects from jumps andLeverage. For example, take close prices for 3 months Jump: (δS )3tii Leverage: (Sti)( ) S t1 δS ti2iBruno Dupire11

Dissociating Jump & Leverage effectsBruno Dupire12

Dissociating Jump & Leverage effectsBruno Dupire13

Break Even Volatilities

Theoretical Skew from Prices? Problem : How to compute option prices on an underlying withoutoptions?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 allstrike/maturity; breaks autocorrelation and vol/spot dependency.Bruno Dupire15

Theoretical Skew from Prices (2)3) Discounted average of the Intrinsic Value from recentered 3 monthhistogram.4) -Hedging : compute the implied volatility which makes the hedging a fair game.Bruno Dupire16

Theoretical Skewfrom historical prices (3)How to get a theoretical Skew just from spot pricehistory?SKExample:ST3 month daily datatT1T21 strike K k ST1– a) price and delta hedge for a given σ within Black-Scholes1––––modelb) compute the associated final Profit & Loss: PL(σ )c) solve for σ k / PL σ k 0d) repeat a) b) c) for general time period and averagee) repeat a) b) c) and d) to get the “theoretical Skew”Bruno Dupire( )( ( ))17

Theoretical Skewfrom historical prices (4)Bruno Dupire18

Theoretical Skewfrom historical prices (4)Bruno Dupire19

Theoretical Skewfrom historical prices (4)Bruno Dupire20

Theoretical Skewfrom historical prices (4)Bruno Dupire21

Barriers as FWD Skew trades

Beyond initial vol surface fitting Need to have proper dynamics of implied volatility– Future skews determine the price of Barriers andOTM Cliquets– Moves of the ATM implied vol determine the ofEuropean options Calibrating to the current vol surface do not imposethese dynamicsBruno Dupire23

Barrier Static HedgingDown & Out Call Strike K, Barrier L, r 0 : With BS: DOCK ,L CK K L PL2KIf S t L ,unwind hedge, at 0costIf not touched, IV’s are equalK2LKLKL2KLLKLK With normal modelDOCK , L C K P2 L KdS σdWBruno Dupire2L-K124

Static Hedging: Model Dominance Back to DOCK , LLK An assumption as the skew at L corresponds toan affine modeldS (aS b )dW (displaced LN) DOCK ,L priced as in BS with shifted K and L givesnew hedging PF which is 0 when L is touched ifSkew assumption is conservativeBruno Dupire25

Skew Adjusted Barrier HedgesdS (aS b )dWDOC K , LaK b CK PaL2 b (2 L K )aL baK bUOC K , La aK bC aL2 b (2 L K ) C K (L K ) 2 Dig L CL aL b aL b aK bBruno Dupire26

Local Volatility Model

One Single Model We know that a model with dS σ(S,t)dWwould generate smiles.– Can we find σ(S,t) which fits market smiles?– Are there several solutions?ANSWER: One and only one way to do it.Bruno Dupire28

The Risk-Neutral SolutionBut if drift imposed (by risk-neutrality), uniqueness of the th SmileBruno Dupiresought diffusion(obtained by integrating twiceFokker-Planck equation)29

Forward Equations (1) BWD Equation:price of one option C (K 0 ,T0 ) for different (S, t ) FWD Equation:price of all options C (K , T ) for current (S 0 ,t0 ) Advantage of FWD equation:– If local volatilities known, fast computation of impliedvolatility surface,– If current implied volatility surface known, extraction oflocal volatilities,– Understanding of forward volatilities and how to lockthem.Bruno Dupire30

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 financialinsightBruno Dupire31

Fokker-Planck If dx b( x, t )dW ϕ 1 2 (b 2ϕ ) Fokker-Planck Equation: t 2 x 2 2C Where ϕ is the Risk Neutral density. As ϕ K 22 C t x 22 2C 2 2 C 2 b2 K 1 K t x 22 Integrating twice w.r.t. x: C b2 2C t 2 K 2Bruno Dupire32

Volatility Expansion K,T fixed. C0 price with LVM σ 0 ( S t , t ) : dS t σ 0 ( S t , t ) dWt Real dynamics: dSt σt dWt2 C C0 21 20dS (σ σ Ito ( ST K ) C0 ( S 0 ,0) 0 ( S t , t )) dtt2 S2 0 S0TT Taking expectation:1C ( S 0 ,0) C0 ( S 0 ,0) Γ0 ( S , t ) Ε σ t2 St S σ 02 ( S , t ) ϕ ( S , t )dSdt222 Equality for all (K,T) Ù Ε σ t S t S σ 0 ( S , t )[Bruno Dupire([]])33

Summary of LVM PropertiesΣ0 is the initial volatility surface σ (S,t ) compatible with Σ0 σ σ (ω ) compatible with Σ E[σ0σ̂ k,T Bruno Dupire2local volST K ] (local vol)²deterministic function of (S,t) (if no jumps)future smile FWD smile from local vol34

Stochastic Volatility Models

Heston Model dS S µ dt v dW dv κ v 2 v dt η v dZ()dW , dZ ρ dtSolved by Fourier transform:FWDx lnτ T tKC K ,T ( x, v,τ ) e x P1 ( x, v,τ ) P0 ( x, v,τ )Bruno Dupire36

Role of parameters Correlation gives the short term skew Mean reversion level determines the long termvalue 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 effectto correlationBruno Dupire37

Spot dependency2 ways to generate skew in a stochastic vol model1) σ t xt f (S , t ), ρ (W , Z ) 02)σρ(W , Z ) 0S0σSTSTS0-Mostly equivalent: similar (St,σt ) patterns, similarfutureevolutions-1) more flexible (and arbitrary!) than 2)-For short horizons: stoch vol model Ù local vol model independent noise on vol.Bruno Dupire43

SABR model F: Forward priceβdF F σ t dWdσσ α dZ With correlation ρBruno Dupire44

Smile Dynamics

Smile dynamics: Local Vol Model (1) Consider, for one maturity, the smiles associatedto 3 initial spot valuesSkew caseLocal volsSmile S Smile S 0Smile S S S0 S K– ATM short term implied follows the local vols– Similar skewsBruno Dupire51

Smile dynamics: Local Vol Model (2) Pure Smile caseLocal volsSmile S Smile S Smile S 0S S0S K– ATM short term implied follows the local vols– Skew can change signBruno Dupire52

Smile dynamics: Stoch Vol Model (1)Skew case (r 0)Local volsσSmile S Smile S 0Smile S S S S0K- ATM short term implied still follows the local vols(E [σ2T])S T K σ 2 (K , T )- Similar skews as local vol model for short horizons- Common mistake when computing the smile for anotherspot: just change S0 forgetting the conditioning on σ :if S : S0 S where is the new σ ?Bruno Dupire53

Smile dynamics: Stoch Vol Model (2) Pure smile case (r 0)σLocal volsSmile S Smile S Smile S 0S S0S K ATM short term implied follows the local vols Future skews quite flat, different from local volmodel Again, do not forget conditioning of vol by SBruno Dupire54

Smile dynamics: Jump ModelSkew caseLocal volsSmile S Smile S 0S Smile S S0 S K ATM short term implied constant (does not follows thelocal vols) Constant skew Sticky Delta modelBruno Dupire55

Smile dynamics: Jump ModelPure smile caseLocal volsSmile S 0Smile S Smile S S S0S K ATM short term implied constant (does not follows thelocal vols) Constant skew Sticky Delta modelBruno Dupire56

Smile dynamicsWeighting scheme imposessome dynamics of the smile fora move of the spot:For a given strike K,S1S0KS σ K (we average lower volatilities)Smile today (Spot St)&Smile tomorrow (Spot St dt)in sticky strike modelt2625.525Smile tomorrow (Spot St dt)if σATM constant24.5Smile tomorrow (Spot St dt)in the smile model23.5Bruno Dupire24St dtSt57

Volatility Dynamics of different models Local Volatility Model gives future short termskews that are very flat and Call lesser thanBlack-Scholes. More realistic future Skews with:– Jumps– Stochastic volatility with correlation and meanreversion To change the ATM vol sensitivity to Spot:– Stochastic volatility does not help much– Jumps are requiredBruno Dupire58

ATM volatility behavior

Forward SkewsIn the absence of jump :model fits market K , T2E[σ T2 ST K ] σ loc(K ,T )This constrainsa) the sensitivity of the ATM short term volatility wrt S;b) the average level of the volatility conditioned to ST K.a) tells that the sensitivity and the hedge ratio of vanillas depend on thecalibration 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.Bruno Dupire60

Sensitivity of ATM volatility / SAt t, short term ATM implied volatility σt.As σt is random, the sensitivityEt [σ2t δt σ t S2is defined only in average:2 σ loc(S , t ) σ Sδt St δS ] σ ( St δS , t δt ) σ ( St , t ) δS S2t2loc2loc22In average, σ ATMfollows σ loc.Optimal hedge of vanilla under calibrated stochastic volatility corresponds toperfect hedge ratio under LVM.Bruno Dupire61

Market Model of Implied Volatility Implied volatilities are directly observable Can we model directly their dynamics? (r 0) dS S σ dW1 dσˆ α dt u dW u dW1122 σˆwhere σ̂ is the implied volatility of a given C K ,T Condition on σˆ dynamics?Bruno Dupire62

Drift ConditionC(S,σˆ , t ) Apply Ito’s lemma to Cancel the drift term Rewrite derivatives ofC(S,σˆ , t )gives the condition that the driftα of dσˆσˆmust satisfy.For short T, we get the Short Skew Condition (SSC):22K K 2σˆ σ u1 ln( ) u2 ln( ) S S K2close to the money: σˆ σ u1 ln( )SÎ Skew determines u1Bruno DupireC ( S ,σ ,̂ t)t T63

Optimal hedge ratio H C ( S , σˆ , t ) : BS Price at t of Call option withstrike K, maturity T, implied vol Ito: dC ( S , σˆ , t ) 0dt CS dS Cσˆ dσˆ Optimal hedge minimizes P&L variance:dC.dSdσˆ .dSH CS Cσˆ22(dS )(dS )Implied Volσ̂BS DeltaBruno DupireBS Vegasensitivity64

Optimal hedge ratio H IIdσˆ .dS CS Cσˆ(dS ) 2H dS S σdW1With dσˆ αdt u dW u dW1122 σˆdσˆ .dS u1σS (dW1 ) 2 u1σˆ 222σS(dS )(σS ) (dW1 )Î Skew determines u1, which determines HBruno Dupire65

Smile Arbitrage

Deterministic future smilesIt is not possible to prescribe just any futuresmileIf deterministic, one must haveC K ,T (S 0 , t 0 ) ϕ (S 0 , t 0 , S , T1 ) C K ,T (S , T1 )dS22Not satisfied in generalKS0ϕt0Bruno DupireT1T267