Unifying Volatility Models

The University of Reading THE BUSINESS SCHOOL FOR FINANCIAL MARKETS Unifying Volatility Models Carol Alexander (Co-authored works with E. Lazar and L. Nogueira) ISMA Centre, University of Reading Email: c.alexander@ismacentre.rdg.ac.uk Pdfs: www.ismacentre.rdg.ac.uk/alexander Newton Institute, Cambridge, May 2005

I Stochastic Local Volatility 2

Model Acronyms BS: Black-Scholes SV: Stochastic Volatility LV: Local Volatility SD: Sticky Delta ST: Sticky Tree SIV: Scale Invariant Volatility SLV: Stochastic Local Volatility SABR: Stochastic Alpha-Beta-Rho CEV: Constant Elasticity of Variance MM: Market Model of implied volatility 3

Claim Characteristics : k: claim strike T: maturity of claim Notation Model Price of Claim : f = f s, σ, λ s k,t k,t ( ) 0 0 0 : underlying price at time 0 0 σ : instantaneous volatility at time 0 0 λ0: vector of volatility parameters at time t0 t 0 : calibration time t t 4

Scale Invariant Volatility (SIV) s 1 f, σ, λ = f s, σ, λ k k ( ) 0 1,T 0 0 k,t 0 0 0 s 0 k s 0 /k 1 t 0 T time t 0 T time 5

Which Models are Scale Invariant? Most stochastic volatility models: even if price and/or volatility driven by Levy process Local volatility models: but only sticky delta models, not sticky tree models All BS mixture models, e.g.: Merton s (1976) jump diffusion Brigo-Mercurio s (2001) normal mixture diffusion 6

Example: Stochastic Volatility (SV) Heston (1993) is SIV: ds S ( ) σ( ) = r q dt + t db ( ) ( 2 2 2 2 ( ) ( ) )( ) d σ = 1 ς ρξ + ψ ς 1 ς ξ ω σ dt + ξσdz < db,dz > = ρdt SABR Hagan et al. (2002) is not SIV (unless β = 0): ds β = ( r q) dt + αs db S dα = ζ dz α < db,dz > = ρdt 7

Example: Local Volatility (LV) Dupire (1993), Derman and Kani (1994), Rubinstein (1994): ds ( r q) dt σ( t,s ) db S = + LV is SIV σ t,s is a function of time and S/S 0 only. However many parametric deterministic functions σ t,s have the sticky delta property that implied volatility is a function of delta and residual time to maturity only: σ 2 ( t,s ) t = T,s = k ( ) f 2 T = f + ( r q) k + q f k 2 2 f k 2 k S θk,t ( t,s ) = θ,t t k ( ) Sticky Delta (SD) models are SIV, Sticky Tree (ST) models are not SIV. 8

Relationship Between SV & LV σ 0 σ( t,s ) Equivalent Local Volatility Surface t T S(t)=s t 0 σ s 0 2 2 ( ) =E σ ( ) ( ) t,s t S t = s 9

Scale Invariance of Local Volatility How does σ 0 change as we change s 0? σ( t,s ) t T σ 0 S(t )=s Consider the local volatility curve at time t that is equivalent to a scale invariant SV model: t 0 s 0 10

Independence of σ 0 and S 0 ( t ) σ =E σ is the conditional expectation at 2 2 0 un t 0 σ( t,s ) S S/k σ 0 The local volatility depends only on the ratio S/S 0 (and time) Hence when we change the calibration value of S, the local volatility surface moves as shown Hence σ 0 is independent of S 0 S 11

Stochastic Local Volatility (SLV) Dupire (1996), Kani, Derman & Kamal (1997) Local variance calibrated at time t 0 < t expectation of a stochastic instantaneous variance: Q ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 L = = I 0 t = I0 Ω σ t, s E [σ t,s, x S t s, ]= σ t,s, x h xs t s, dx where x = x(t, S ) is a vector of all sources of uncertainty that influence the instantaneous volatility process. t 12

Parametric Stochastic Local Volatility Alexander & Nogueira (2004) studies the general parametric form: ds S ( ) σ ( λ ) = r q dt + t,s, db ( λ ) ( λ ) d λ = µ t, dt + ζ t, dz i i i i < dz,db > = ρ dt < dz,dz > = ρ dt i is i j ij Usual regularity conditions for drift and volatility in the parameter diffusions Strict no-arbitrage conditions 13

Example: CEV-SLV & SABR Model An example of a SLV model is the CEV-SLV: ds β = ( r q) dt + αs db S dα = ζ dz α < db,dz > = ρdt Same parameterization as SABR model! Different no-arbitrage conditions 14

Equivalence with Market Model (MM) Schönbucher (1999): market model of implied volatilities This has the same implied volatility dynamics as the general SLV model n d θ = ξ dt + ψ db + ζ dz Hence the SLV and MM models have the same claim prices and hedge ratios k,t k,t k,t j,k,t j j = 1 But MM has many option specific parameters 15

II Hedging: Theoretical and Empirical Results 16

Hedging with SIV Models Proposition : When a model has been calibrated to the market implied volatilty surface: S0 fk,t ( S 0, σ0, λ0 ) = k f 1,T, σ0, λ0 k S0 θk,t ( S 0, σ0, λ0 ) = θ1,t, σ 0, λ0 k θ S k,t 0 : model implied volatility, calibrated at time : underlying price at the calibration time t 0 17

Proof S0 fk,t ( S 0, σ0, λ0 ) = k f 1,T, σ0, λ0 k f k,t fk,t ( S 0, σ0, λ0 ) = S0 + k S θk,t k θ = S S k 0 0 k,t S 0 θk,t = θ1,t, σ 0, λ0 k 0 f k,t k Euler s Theorem Differentiate f k,t w.r.t S 0 f BS k,t and k : Trivial BS S0 S0 : f 1,T, θk,t f 1,T, σ0, λ0 k k see Fouque et al. (2000). 18

SIV Partial Price Sensitivities are Model Free fk,t fk,t ( S 0, σ0, λ0 ) = S0 + k S 0 2 2 2 k,t 1 k,t k,t k k,t = S0 fk,t k and = 2 2 S0 k S0 S0 k f Partial price sensitivities obtained directly from market data! Hence the only differences in partial price sensitivities, between all SIV models, is in their fit to the smile!!!! k,t k f f f f 19

Relationship with BS Delta We can relate the first (and second) order partial price sensitivities to the BS delta (and gamma): Calibration to smile means setting f k,t f BS k,t Differentiating this: BS BS k,t k,t k,t k,t f f f θ = + S S θ S 0 0 k,t 0 But in any SIV model Hence: f k θ S S k k,t BS BS = δk,t νk,t 0 0 θk,t k θ = S S k 0 0 k,t k,t Slope of smile 20

Models with Model Free Delta In deterministic volatility models, delta partial price sensitivity This holds for SD and deterministic mixture models But these models are also SIV Hence their delta (and gamma) are model free. Conclusions: 1. All of the SD and deterministic mixture models have the same delta (and gamma) 2. This delta is inappropriate in most equity markets 21

Sticky Tree (ST) Delta Sticky tree models are not scale invariant, indeed f f S,, S k ST k,t k,t k,t k,t 0 ( σ λ ) = + S 0 0 0 0 0 S0 k ST BS BS k,t k,t k,t δ = δ ν k S θ ST ST ST k,t k f ST f k,t σ + σ S 0 0 0 0 f σ σ S 0 0 Model Free < 0 in Equity 22

SV Delta At time t > t 0 : ( ) SV SV SV k, T k, T k, T df t, S,σ f f σ = + ds S σ S We define delta at the calibration time t 0 as: δ SV k, T ( t, S,σ) Example: Heston model SV SV k, T k, T SV k, T df f f = lim = + t t ds S σ 0 0 0 t = t 0 lim t t σ S δ SV k, T SV SV 1 SV f k, T f, 0 k, T ξρ k T + 2 k σ0 = S f k ξ: vol of vol ρ: price-vol correlation Model Free < 0 in Equity 23

SLV Delta Delta is the standard local volatility delta, but adjusted for stochastic movements in local volatility surface: δ SLV k,t LV βiρi = δk,t + i σs LV,S f k,t 0 λ i Similar adjustments for gamma and theta Bias and efficiency? Both δ LV and δ SLV delta hedged portfolios have zero expectation but δ SLV hedged portfolio has smaller hedging error variance. Hence SLV model may be useful for hedging 24

Summary of Deltas Model Model Delta BS Delta Model Delta Sticky Delta & Mixtures 1 f S0 fk,t k k k,t Model Free ν BS k,t k S 0 θ k,t k Sticky Tree f S f k f σ 1 k,t k,t 0 BS 0 k, T + ν k,t k σ0 s0 S0 k θ k,t k f σ k,t σ s 0 0 0 Stochastic Volatility S 1 0 f k, T f k, T f k + k σ k, T σ lim S k θk, T f lim S k S k, T σ t t k, T 0 t = t 0 0 0 σ t t t = t0 ν BS Stochastic Local Volatility: Sticky Delta Stochastic Local Volatility: Sticky Tree 1 fk,t ζ ρ S0 fk,t k + k i σs i i,s f k,t BS k θk,t ζiρi,s f k, T νk, T 0 λ s i 0 k i σs0 λi 1 fk,t fk,t σ ζ ρ 0 i i,s fk,t S 0 fk,t k BS k θk,t f σ ζ ρ f 0 + + νk,t k σ0 S0 i σs0 λ S0 k σ0 s0 i σs0 λ i k,t i i,s k,t i 25

Empirical Results Data on June 2004 European options on SP500: Daily close prices from 02 Jan 2004 to 15 June 2004 (111 business days) Strikes from 1005 to 1200 (i.e. 34 different strikes) All strikes within ±10% of index level used for daily calibration 1180 SP500 Future 1160 1140 1120 1100 1080 16-Jan-04 30-Jan-04 13-Feb-04 27-Feb-04 12-Mar-04 26-Mar-04 09-Apr-04 23-Apr-04 07-May-04 21-May-04 04-Jun-04 26

Hedging Race 7 Models 1. Black & Scholes (BS) 2. Sticky Delta (CEV) 3. Normal Mixture (NM) 4. Sticky Tree (CEV_LV) 5. Heston (SQRT) 6. SABR (CEV_SLV) 7. NM with stochastic parameters: (NM_SLV) 2 Strategies 1. Delta Neutral: Short one call on each strike and delta hedge with underlying 2. Delta-Gamma Neutral: Buy 1125 strike to zero gamma then delta hedge Rebalance daily from 16 Jan 2004 onwards (100 business days) assuming zero transactions costs 27

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Comparison of Delta 28 K/S 0.905 0.915 0.925 0.935 0.945 0.955 0.965 0.975 0.985 0.995 1.005 1.015 1.025 1.035 1.045 1.055 1.065 1.075 1.085 1.095 CEV delta NM delta CEV_LV delta SQRT delta BS delta

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 29 K/S 0.905 0.915 0.925 0.935 0.945 0.955 0.965 0.975 0.985 0.995 1.005 1.015 1.025 1.035 1.045 1.055 1.065 1.075 1.085 1.095 Comparison of Delta CEV_SLV delta NM_SLV delta CEV_LV delta SQRT delta BS delta

Hedged Portfolio P&L Stats Average Std.Dev. Skewness XS Kurtosis Delta Hedge CEV_ST 0.1462 0.5847-0.3424 0.7820 SQRT 0.1370 0.6103-0.5705 1.6737 CEV_SLV 0.1393 0.6280-0.5701 1.6041 NM_SLV 0.1399 0.6329-0.5224 1.2351 BS 0.1401 0.7451-0.7029 2.0370 CEV 0.1367 1.1035-0.6525 1.6691 SI 0.1446 1.1448-0.5395 1.3708 NM 0.1373 1.1788-0.5928 1.4834 Delta-Gamma Hedge BS -0.0014 0.2612-0.4353 2.5297 CEV_ST 0.0098 0.2691-0.0291 3.0850 NM_SLV 0.0182 0.2773 0.1673 3.2069 SQRT 0.0111 0.2789 0.1929 3.6019 CEV_SLV 0.0154 0.2832 0.1304 4.1414 CEV 0.0382 0.4102 0.2501 4.0340 SI 0.0370 0.4205 0.1438 3.7003 NM 0.0428 0.4548 0.0208 4.0123 30

III GARCH from Discrete to Continuous Time 31

GARCH(1,1) Bollerslev (1986) Generalized Engle s (1982) ARCH model to GARCH(1,1): ε ( ) ln S /S = µ + ε t t 1 t 2 t t 1 0 σt I ~ N(, ) 2 2 2 t = + t 1 + t 1 σ ω αε βσ ω > 0, α, β 0, α + β 1 σ 0 2 2 2 t = + t 1 + t 1 σ ω αε βσ ε t 2 t 32

GARCH Option Pricing Duan (1995), Siu et al. (2005) Discrete time, numerical methods, Esscher transform Heston & Nandi (2000) Closed form, continuous limit is Heston SV model with perfect price-volatility correlation Continuous time limit of GARCH(1,1) conflicting results Nelson (1990) Diffusion limit model Corradi (2000) Deterministic limit model Which? 33

GARCH Diffusion Nelson (1990) Continuous limit of GARCH(1,1) Nelson assumed the following finite limits exist and are positive: ω 1 β α ω h θ h lim lim h = = h 0 h h 0 h α 2 2 h α = lim h 0 h σ 0 ( ) 2 2 2 dσ = ω θσ dt + 2ασ dw where h ω, h α, h β are the parameters using returns with step-length h t 34

Why No Diffusion? 1. GARCH has only one source of uncertainty in discrete time (Heston-Nandi, 2000) 2. The discretization of Nelson s limit model does not return the original GARCH model: it is ARV (Taylor, 1986) 3. GARCH diffusion and GARCH have non-equivalent likelihood functions (Wang, 2002) 4. Nelson s assumptions cannot be extended to other GARCH models 5. Nelson s assumptions imply that the GARCH model is not timeaggregating (Drost and Nijman, 1993). Only the assumptions made by Corradi (2000) are consistent with time-aggregation 6. Related papers: Brown et al. (2002) Duan et al. (2005) 35

Deterministic Limit? Corradi (2000) Continuous limit of GARCH(1,1) (revisited) Different assumptions: h ω 1 hβ ω = lim ψ = lim h 0 h h 0 h +X dσ 2 = ω θσ 2 dt 2ασ 2 dw σ 0 ( ) α h α = lim h 0 h where θ = ψ α t 36

Multi-State GARCH Alexander & Lazar (2004), Haas et al. (2004) y t S S ( S / S ) t t 1 = ln t t 1 St 1 2 2 ( ) ε I ~ NM π,..., π ; µ,..., µ ; σ,..., σ t t 1 1 K 1 K 1t Kt yt = µ + ε t K π = 1 i = 1 i K π µ = 0 i = 1 i i GARCH(1,1): A-GARCH: GJR: σ σ σ 2 it 2 2 = ω i + αiεt 1 + βi σ it 1 2 2 2 it = ωi + αi ( εt 1 λ i ) + βi σ it 1 2 it 2 2 2 = ωi + α iεt 1 + λ id t 1εt 1 + βi σ it 1 + parameter restrictions K K 2 2 2 t = i it + i i i = 1 i = 1 σ π σ π µ for volatility states i = 1,, K 37

Volatilities and Mixing Law SP500 200% 1 150% 0.5 100% 0 volatility 1 volatility 2 p 1 50% -0.5 0% Jan-2001 Jul-2001 Jan-2002 Jul-2002 Jan-2003-1 38

Conditional Skewness and Kurtosis SP500 Jan-2001 Jul-2001 Jan-2002 Jul-2002 Jan-2003 0 11 9 7-0.5 5 3 skewness kurtosis 1-1 -1 39

Features of Multi-State GARCH Endogenous time varying higher moments endogenous term structure in volatility skew surfaces Different component means long-run persistence in volatility skew Different GARCH volatility components regime specific meanreversion of volatility Asymmetric GARCH components regime specific leverage effect (slope of short term volatility skew) NM GARCH is better fit to historical data than single state GARCH models (Alexander & Lazar, 2004) 40

Volatility State Dependence σ p 1 if system is in state i ( st = i ) = ( p 1, p 2,..., p ) : p = otherwise t,t,t n,t i,t For continuous limit, introduce step-length h : p 1 if system is in state i ( h skh = i ) = ( p 1, p 2,..., p ) : p = otherwise h kh h,hk h,hk h n,hk h i,kh Define K continuous volatility states: K ( 1 2 ) ( ) ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 2 K t = σ t, σ t,, σ with overall volatility σ π σ π µ 2 K t ' t = i i t + i i i = 1 i = 1 Define the state indicator process: ( ) p t = lim p where p = p for t [ kh, ( k + 1) h ) h t h t h kh h 0 41

Transition Matrix State probabilities p are given by the (conditional) expectation of the state indicator vector: ( ) ( ) ( ) ( ) P h skh = i I k 1 h = E h pkh I k 1 h = h p kh, say Introduce state dependence via transition matrix for step-length h: ( ) ( ) ( k 1) h Q q = Ρ s = i s = j ; = q with q = 1, q 0 h ij h kh h h h ij h ij h ij i = 1 K Then and p = Q p ( 1) h kh h h k h h Q ϖ = ϖ Where the unconditional state probabilities are the mixing law, π 42

Transition Rate Matrix Define limit of state probabilities: And the transition rate matrix: Λ ( t ) p = lim p where p = p lim h 0 h = = h 0 h t h t h kh Q h I ( λ ij ) Then the continuous time dynamics of the state probability vector in the physical measure are given by: ( ) = Λ p ( ) d p t t dt And the number of volatility jumps follow a Poisson process: k ( λ jt ) P(k jumps from state j in an interval of length t ) = exp ( λ jt ) k! where λ = λ is the instantaneous probability of a jump from state j. j i j ij 43

GARCH Jump Assume: ω α 1 β ω = lim α = lim ψ = lim µ = lim µ h h h Continuous limit of Markov Switching GARCH(1,1) is: Risk neutral measure: ds S = rdt + σdb * d p = Λ p dt * ( ) h i h i h i i i i i h i h 0 h 0 h 0 h 0 * (( ) ' ) Λ = Λ 1 + η 1 2 2 2 d σ = p' d σ + σ ' d p ( σ ) 2 2 2 dσ = ω + α ψ σ η: vector of conditional state risk premia dt Physical measure: ds = p ' µ dt + σdb S d p = Λ p dt 44

Example: Two Volatility States ( ) ( ) ( ) ( ) 1 1 1 1 2 2 2 2 1 2 2 2 2 2 2 2 2 dσ = p ω + α σ ψ σ dt + 1 p ω + α σ ψ σ dt + σ σ dp p σ ( ) ( ) 1 ( ) 2 ( ) dp t ( t ) ( ) ( ) 1 if s t = 1 t = ( p t, p t ) p ( t ) = 0 if s t = 2 1 if state changes from 2 to 1 = 0 if state does not change 1 if state changes from 1 to 2 2 σ 1 ( t ) in state 1 = 2 σ2 ( t ) in state 2 45

Realised Volatility 30% 20% 10% 0% vol 1 vol 2 realized vol 0 100 200 300 400 500 46

Related Work Option pricing and hedging with Markov switching volatility processes: Naik (1993): Theoretical results in continuous time, stochastic jump arrival rates, constant volatilities and perfectly correlated jumps in price Elliot et al. (2005): MS version of GARCH model of Heston & Nandi (2000) based on Esscher transform 47

Acknowledgements Co-authors: Emese Lazar, ISMA Centre Leonardo Nogueira, ISMA Centre Helpful discussions: Jacques Pézier, ISMA Centre Hyungsok Ahn, Commerzbank London Bruno Dupire, Bloomberg Peter Carr, Bloomberg and NYU Participants at Risk 2005 Europe and at Newton Institute, 2005. 48