Realized Volatility with Leverage and Jumps: A New Model for Option Pricing

Transcription

1 Realized Volailiy wih Leverage and Jumps: A New Model for Opion Pricing Dario Aliab 1, Giacomo Bormei 1,2, Fulvio Corsi 3,4, and Adam A. Majewski 1 1 Scuola Normale Superiore, Pisa 2 QUANTLab, Pisa 3 Ca Foscari Universiy of Venice 4 Ciy Universiy of London XVI Workshop on Quaniaive Finance Parma, 30 January 2015

2 Moivaion Sochasic volailiy models: inroduced o reproduce well-esablished sylized facs abou asse reurns and volailiy ime series (fa ails, leverage effec, high persisence); Enhancing volailiy predicion is a key poin for opion pricing; No considering hese models generaes: misfiing skew/smile profile of implied volailiy; significan mispricing of ou-of-he-money opions in paricular for shor mauriy;

3 Moivaion Sochasic volailiy models: inroduced o reproduce well-esablished sylized facs abou asse reurns and volailiy ime series (fa ails, leverage effec, high persisence); Enhancing volailiy predicion is a key poin for opion pricing; No considering hese models generaes: misfiing skew/smile profile of implied volailiy; significan mispricing of ou-of-he-money opions in paricular for shor mauriy; We wan o include he jump componen in an analyical model, esimae i and evaluae he impac on implied volailiies.

4 Sylized facs: Volailiy high persisence SampleIAuocorrelaionIS&P500IFu.IAbs.IReurns S&PI500 ConfIInerval Lag Figure: Sample auocorrelaion funcion of absolue daily log-reurns of S&P 500 Fuures from 28h April 1982 o 4h February The confidence limis represen he 95% confidence inerval of a Gaussian random walk.

5 Sylized facs: Leverage effec Leverage Effec a Differen Lags Negaive Reurns Posiive Reurns Correlaion Lag Figure: Lagged correlaion funcion beween pas negaive daily log-reurns and curren volailiy (red line) and beween pas posiive daily log-reurns and curren volailiy (blue line) for he S&P 500 Fuures from 28h April 1982 o 4h February 2009.

6 Sylized facs: Volailiy cascade Asymmeric propagaion of informaion: volailiy over longer ime inervals have sronger influence on ha a shorer ime inervals han he converse; Possible explanaion: heerogeneiy of expecaions and risk profiles of marke agens. Figure: Lagged correlaion beween he mean absolue daily reurn on USD-DEM in a week and he absolue reurn over a full week (Muller e al. (1997)).

7 Jumps presence in ime series and smiles profile Real Daa Smiles Y 0.22 Y 0.45 Y 0.97 Y Implied Volailiy Moneyness Figure: (Lef) Percen conribuion of daily jump o he oal quadraic variaion measured over a moving window of 3-monh (doed line) and 1-year (solid line) for he S&P 500 fuures from 28 April 1982 o 4 February 2009 (Corsi, F. and Renò, R., (2012)). (Righ) Implied volailiy profile for opions wrien on S&P 500 index wih differen mauriies. The pronounced smile for shor-mauriy opions is regarded as an evidence for he jump fear of he marke agens.

8 Shor review of available volailiy models Coninuous ime Sochasic Volailiy (like Heson (1993)): GARCH-ype: GARCH(p,q) (Bollerslev, T., (1986)); CGARCH (muli-componen by Chrisoffersen (2008)) Realized Volailiy (RV) (Barndorff-Nielsen, O.E., Shephard, N. (2001)) HAR (Corsi (2009)): heerogeneous auoregressive srucure; HARGL (Corsi e al. (2012)): binary leverage; no closed-form pricing formula (Mone Carlo pricing); LHARG (Majewski, Bormei, Corsi (2013)): analyically racable heerogeneous leverage; closed-form pricing formula.

9 Shor review of available volailiy models Coninuous ime Sochasic Volailiy (like Heson (1993)): GARCH-ype: GARCH(p,q) (Bollerslev, T., (1986)); CGARCH (muli-componen by Chrisoffersen (2008)) Realized Volailiy (RV) (Barndorff-Nielsen, O.E., Shephard, N. (2001)) HAR (Corsi (2009)): heerogeneous auoregressive srucure; HARGL (Corsi e al. (2012)): binary leverage; no closed-form pricing formula (Mone Carlo pricing); LHARG (Majewski, Bormei, Corsi (2013)): analyically racable heerogeneous leverage; closed-form pricing formula. Jump dynamics has been negleced by above RV models

10 Our proposal: JLHARG-RV model General daily asse log-reurn dynamics: y +1 = r + λl(f +1 ) + L(f +1 )ɛ +1 r: risk-free rae; λ: marke price of risk; L : R k R + : linear funcion of he variance; f +1 : k-dimensional vecor of facors; ɛ : i.i.d. N (0, 1).

11 Our proposal: JLHARG-RV model Measure of variance: Realized Variance (RV); Choice of 2-dimensional facor: f = (RV c, RV j ) RV c : coninuous componen of realized variance; RV j : disconinuous jump componen of realized variance. The linear funcion of variance L given by L(f ) = f (1) + f (2) = RV c + RV j = RV Log-reurns evoluion according o his choice: y +1 = r + λrv +1 + RV +1 ɛ +1,

12 Our proposal: JLHARG-RV model Dynamics of coninuous componen of RV modelled wih noncenral gamma disribuion: RV c +1 F γ(δ, Θ(RV c, L ), θ) (a) γ (x δ =5, θ =1) Θ Θ Θ =0 =5 =25 = x

13 Our proposal: JLHARG-RV model Θ(, ) =

14 Our proposal: JLHARG-RV model Θ(RV c, ) = β d RV c (d)

15 Our proposal: JLHARG-RV model Θ(RV c, ) = β d RV c (d) AuocorrelaiongSimulaedgAbs.gReurnsgOne Scale Daily Scale ConfgInerval Lag

16 Our proposal: JLHARG-RV model Θ(RV c, ) = β d RV c (d) + β w RV c (w) + β m RV c (m) AuocorrelaionMSimulaedMAbs.MReurnsMMuli Scale Muli Scale Daily Scale ConfMInerval Lag

17 Our proposal: JLHARG-RV model Θ(RV c, ) = β d RV c (d) + β w RV c (w) + β m RV c (m)

18 Our proposal: JLHARG-RV model Θ(RV c, ) = β d RV c (d) + β w RV c (w) + β m RV c (m) 0.05 Skewness No Leverage Effec Time (Day)

19 Our proposal: JLHARG-RV model Θ(RV c, L ) = β d RV c (d) + α d l (d) + β w RV c (w) + β m RV c (m) ( ) 2 l (d) = ɛ γ RV c + RVj Skewness 1.8 x,10 4 Model,Leverage,Effec,as,a,funcion,ofε No Leverage Effec Time (Day) Cov [ε,,rv +1 ] ε

20 Our proposal: JLHARG-RV model Θ(RV c, L ) = β d RV c (d) + α d l (d) + β w RV c (w) + β m RV c (m) ( ) 2 l (d) = ɛ γ RV c + RVj Skewness 1.8 x,10 4 Model,Leverage,Effec,as,a,funcion,ofε No Leverage Effec Daily Scale Leverage Time (Day) Cov [ε,,rv +1 ] ε

21 Our proposal: JLHARG-RV model Θ(RV c, L ) = β d RV c (d) + α d l (d) + β w RV c (w) + α w l (w) + α m l (m) + β m RV c (m) ( ) 2 l (d) = ɛ γ RV c + RVj Skewness 1.8 x,10 4 Model,Leverage,Effec,as,a,funcion,ofε NoMLeverageMEffec Daily ScaleMLeverage Muli ScaleMLeverage TimeM(Day) Cov [ε,,rv +1 ] ε

22 Our proposal: JLHARG-RV model Dynamics of disconinuous componen of RV modelled wih a compound Poisson process: n +1 Y i RV j +1 F i=1 n +1 P( Θ): Poisson variable wih inensiy Θ; Y i Γ( δ, θ): i.i.d. size variables sampled from a gamma disribuion wih shape δ and scale θ. We label his model Heerogeneous AuoRegressive Gamma wih Jumps and Leverage (JLHARG); Analyical compuaion of Momen Generaing Funcion of log-reurns (MGF) under physical measure for his model (exponenial-affine form).

23 Sochasic Discoun Facor for JLHARG The perspecive of pricing imposes he change o a risk neural measure via a Sochasic Discoun Facor (SDF): E Q s [Y ] = E P s [M s,s+1 Y ] wih Y F s+1 ) exp ( ν 1 crvc s+1 ν j 1 RVj s+1 ν 2y s+1 M s,s+1 = E [exp P ( ν c 1 RVc s+1 ν j 1 RVj s+1 ν 2y s+1 ) F s ] Exponenial affine form; Compensaing for hree risks: RV c s+1, RV j s+1, y s+1; Clear idenificaion of risk premia: ν1 c : variance risk premium (coninuous componen); ν j 1 : variance risk premium (disconinuous componen); ν 2 : equiy risk premium.

24 Resuls under risk neural measure No-arbirage condiion: JLHARG model wih he previous SDF is compaible wih no-arbirage condiion provided ha ν 2 = λ + 1/2, while ν1 c and νj 1 remain free parameer; MGF under Q: Under risk-neural change of measure wih he previous SDF, he MGF of log-reurns preserves he exponenial affine form. I is deermined by ime recursive relaions of is affine coefficiens; Mapping P Q: A one-o-one mapping exiss from physical parameers o risk-neural ones such ha he RV is sill a JLHARG process under Q.

25 Opion Pricing wih JLHARG Pricing procedure: Model esimaion under ( P via Maximum ) Likelihood Esimaor using hisorical series y, RV c, RV j ; Calibraion of ν c 1 and ν j 1 (variance risk-premia) on real daa; Mapping of he parameers P Q; Pricing formula in closed-form via risk-neural MGF and numerically implemened using Fang-Ooserlee (2008) efficien scheme. Pricing performance evaluaion via Roo Mean Squared Error (RMSE) (Renaul (1997), Corsi e al. (2012), Majewski e al. (2013)): RMSE IV = N i=1 ( IV mod i ) IVi mk 2, N LHARG model by Majewski e al. as naural compeior.

26 Pricing performance IV RMSE Raios - OTM S&P 500 Opions Global Performance Moneyness Model 0.9 < m < 1.1 JLHARG/LHARG JLHARG/LHARG (*) Only pricing daes preceded by a Jump even. Deailed Performance Mauriy Moneyness (K/S) τ < τ 160 IV RMSE Raios JLHARG/LHARG 0.9 < m < m < m

27 Variance Risk Premium 4 VarianceRiskPremium Marke 2 0 AnnualizedVariance(%) Marke Variance Risk Premium (VRP) following Bollerslev e al. (2014): VRP Mk (, 22) = k=1 E P [RV +k ] VIX 2. where E P [RV +k ] is based on HAR model by Corsi (2009).

28 Variance Risk Premium 4 2 VarianceRiskPremium Marke JLHARG 0 AnnualizedVariance(%) Modelled Variance Risk Premium wih JLHARG: ( [ VRP Mod (, 22) = 1 22 ] E P RV +k E Q 22 k=1 [ 22 k=1 RV +k ]).

29 Variance Risk Premium ExpecedmvariancemovermnexmmonhmundermmeasuremQ VIX JLHARG 16 AnnualizedmVarianceHXR JLHARG expeceed variance under Q measure: [ T ] 1 22 EQ RV +k. k=1

30 Variance Risk Premium ExpecedmvariancemovermnexmmonhmundermmeasuremQ VIX Coninuous Jump AnnualizedmVariance(X) JLHARG expeceed variance under Q measure: [ 1 T ( +k)] 22 EQ RV c +k + RVj. k=1

31 Conclusions JLHARG-RV model: improvemen of LHARG models inroducing jump dynamics; exponenial-affine form for MGF under P and Q measure; RV-dependen exponenial-affine SDF including risk premia for coninuous and disconinuous RV componens; esimaion of model parameers via Maximum Likelihood Esimaors; analyical pricing formula in closed-form via MGF. Pricing performance: beer performance for shor-mauriy opions; abiliy of he model o cach higher skewness in implied reurns disribuion by including jumps and relaive risk premia.

32 Perspecive for fuure sudies Sudy of he dynamics of variance risk premia (coninuous and disconinuous componens): Perform a condiional calibraion; Sudy he premia when a jump has occurred on a previous day; Verifying if our variance-dependen pricing kernel is U-shaped: Pricing kernel as he raio beween risk-neural and physical probabiliy densiies of reurns; Empirical evidences of he U-shape paern (Chrisoffersen e al. (2012)); Explanaion of he faer ails of he implied reurn disribuion relaive o he physical one; Enrich JLHARG models o accoun for overnigh effec: so far RV ime-series adjused by a mean overnigh effec; need of a model for overnigh dynamics and a new esimaion procedure;

33 Proposiion Under P, he MGF for JLHARG model has he following form φ P (, T, z) = E P [ e zy ] p q +1 F = exp a + b c,i RVc +1 i + c,i l +1 i i=1 i=1 where a s = a s+1 + zr 1 2 log(1 2c s+1,1) + dv(x c s+1, θ) δw(xc s+1, θ) + ΘJ (x j s+1, θ) { b c b c s,i = s+1,i + V(x c s+1, θ)β i for 1 i p 1 V(x c s+1, θ)β i for i = p { c c s,i = s+1,i + V(x c s+1, θ)α i for 1 i q 1 V(x c s+1, θ)α i for i = q where x l s+1 = zλ + bl s+1, z 2 + γ 2 c s+1,1 2c s+1,1 γz 1 2c s+1,1 l = c, j. The funcions V, W and J are defined as V(x, θ) = θx 1 θx, W(x, θ) = ln(1 xθ), J (x, θ, δ) = 1 (1 δ θx) (1 θx) δ and he erminal condiion are a T = b l T,i = c T,j = 0 for i = 1, 2,..., p and j = 1, 2,..., q.

34 Corollary Under Q, he MGF for JLHARG model has he following form p q φ Q ν 1 ν (, T, z) = exp a 2 + b c,i RVc +1 i + c,i l +1 i i=1 i=1 where a s = a s+1 + zr 1 2 log(1 2c s+1,1 ) + dv(xc s+1, θ) dv(yc s+1, θ) δw(x c s+1, θ) + δw(yc s+1, θ) + ΘJ (x j s+1, θ) ΘJ (y j s+1, θ) ( ) b c b c s+1,i + V(x c s+1, θ) V(yc s+1 s,i =, θ) β i for 1 i p 1 ( ) V(xs+1 c, θ) V(yc s+1, θ) β i for i = p ( ) c s+1,j + V(x c s+1, θ) V(yc s+1, θ) α j for 1 j q 1 c s,j = ( V(x c s+1, θ) V(yc s+1, θ) ) α j for j = q where x l s+1 = (z ν 2)λ + b l s+1,1 νl (z ν 2 ) 2 + γ 2 c s+1,1 2c s+1,1 γ(z ν 2) 1 2c s+1,1 y l s+1 = ν 2λ ν l ν2 2, wih l = c, j and he erminal condiion are a T = bl T,i = c T,j = 0 for i = 1, 2,..., p and j = 1, 2,..., q.

35 Corollary Under Q he log-reurns and he realized variance sill follow a JLHARG process wih wih equiy risk premium λ = 0.5 and parameers d, δ, θ, Θ, δ, θ γ, αl, βl for l = d, w, m given by he following one-o-one mapping from he physical ones: β d βd = 1 θy, c β w = 1 θy, c β m = 1 θy, c αd α d = 1 θy, α w c α w = 1 θy, α m c α m = 1 θy, c θ θ = 1 θy, c δ = δ, γ = γ + λ + 1 2, d d = 1 θy, δ = δ, Θ c Θ =, ) δ (1 θy j θ = θ 1 θy j, where y c = λ 2 /2 ν c and yj = λ 2 /2 ν j β w β m

36 Calibraed variance risk premia Variance Risk Premia Model ν1 c ν j 1 HARG LHARG ZM-LHARG JHARG JLHARG ZM-JLHARG Table: Coninuous and disconinuous variance risk premia for differen models calibraed according o new calibraion procedure.

37 Pricing performance Implied Volailiy RMSE Moneyness Model 0.9 < m < < m < 1.2 ZM-LHARG JLHARG/JHARG ZM-JLHARG/JHARG ZM-JLHARG/JLHARG JHARG/HARG JLHARG/LHARG ZM-JLHARG/ZM-LHARG Table: Global opion pricing performance on S&P500 OTM opions from January 1, 1996 o December 31, 2004.

38 Appendix on pricing performance Mauriy Moneyness τ < τ < τ < τ Panel A ZM-LHARG Implied Volailiy RMSE 0.7 m < m < m < m < m Panel B JHARG/HARG Implied Volailiy RMSE 0.7 m < m < m < m < m

39 Appendix on pricing performance Mauriy Moneyness τ < τ < τ < τ Panel C JLHARG/LHARG Implied Volailiy RMSE 0.7 m < m < m < m < m Panel D ZM-JLHARG/ZM-LHARG Implied Volailiy RMSE 0.7 m < m < m < m < m