Smiling Twice: The Heston++ Model

Smiling twice: The Heston++ model

C. Pacati1 G. Pompa2 R. Renò3

1Dipartimento di Economia Politica e StatisticaUniversità di Siena, Italy

2,IMT School for Advanced Studies Lucca, Italy

3Dipartimento di Scienze EconomicheUniversità degli Studi di Verona, Italy

XVII Workshop on Quantitative Finance, Pisa 2016

C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model

The problem

There is growing demand, and correspondingly a liquid market, for tradingvolatility derivatives and managing volatility risk.

SPX and VIX derivatives both provide informations on the same volatilityprocess, a model which is able to price one market, but not the other, isinherently misspecified.

There is need of a pricing framework for consistent pricing both equityderivatives and volatility derivatives;

Affine models are unable to reproduce VIX Futures and Options features;

Non-affine models are often analitically intractable and computationally heavy.

We tackle the problem of jointly fit the IV surface of SPX indexoptions, together with the term structure of VIX futures and thesurface of VIX options, leveraging on an affinity-preservingdeterministic shift extension of the volatility process.


The problem

There is growing demand, and correspondingly a liquid market, for tradingvolatility derivatives and managing volatility risk.

SPX and VIX derivatives both provide informations on the same volatilityprocess, a model which is able to price one market, but not the other, isinherently misspecified.

There is need of a pricing framework for consistent pricing both equityderivatives and volatility derivatives;

Affine models are unable to reproduce VIX Futures and Options features;

Non-affine models are often analitically intractable and computationally heavy.

We tackle the problem of jointly fit the IV surface of SPX indexoptions, together with the term structure of VIX futures and thesurface of VIX options, leveraging on an affinity-preservingdeterministic shift extension of the volatility process.


VIX: the Fear Index

Since 1993, VIX reflects the 30-day expected risk-neutralS&P500 index volatility.

Leverage effect: inverse relationship SPX-VIX (2004-2016)

Jan04 Jan06 Jan08 Jan10 Jan12 Jan14 Jan160

250

500

750

1000

1250

1500

1750

2000

S&P

500

0

10

20

30

40

50

60

70

80

90

VIX

DailyClose


VIX: the Fear Index

Since 1993, VIX reflects the 30-day expected risk-neutralS&P500 index volatility.

Positively skewed and leptokurtic (2004-2016)

0 10 20 30 40 50 60 70 80 900

0.02

0.04

0.06

0.08

0.1

0.12

VIX Dai ly Close

EmpiricalPDF


VIX Futures: the term structure of Fear

Traded since 2004, convey market visions on volatility ofS&P500 (2004-2014).

0

1

2

3

4

5

6

7

Nov−14Feb−13

May−11Jul−09

Oct−07Jan−06

Mar−04

0

10

20

30

40

50

60

70

DateTenor (months)


VIX Futures: the term structure of Fear

Traded since 2004, convey market visions on volatility ofS&P500 (2004-2014).

Humped term structure (June 29, 2009)


VIX Options: the cross-section of Fear

Traded since 2006, provide insurance from equity marketdownturns: S&P500 vanilla below (June 29, 2009)


VIX Options: the cross-section of Fear

Traded since 2006, provide insurance from equity marketdownturns: VIX options below (June 29, 2009).


Modeling VIX and VIX derivatives: literature review

Standalone approach: volatility is directly modeled,separated from the underlying stock price process. Whaley1993 (GBM), Grünbichler and Longstaff 1996 (SQR),Detemple and Osakwe 2000 (LOU), Mencia and Sentana2013 (CTLOUSV ), Goard and Mazur 2013 (3/2).


Modeling VIX and VIX derivatives: literature review

Standalone approach: volatility is directly modeled,separated from the underlying stock price process. Whaley1993 (GBM), Grünbichler and Longstaff 1996 (SQR),Detemple and Osakwe 2000 (LOU), Mencia and Sentana2013 (CTLOUSV ), Goard and Mazur 2013 (3/2).Consistent approach: VIX is derived from the specificationof SPX dynamics. Sepp 2008 (SVVJ), Lian Zhu 2013(SVCJ), Lo et al. 2013 (2-SVCJ), Bardgett et al. 2013(2-SMRSVCJ), Branger et al. 2014 (2-SVSVJ), Baldeauxand Badran 2014 (3/2J), Pacati et al. 2015 (2-SVCVJ++).


The 2-SVCVJ++ model: motivation

++ extension of an affine processes:Brigo and Mercurio (2001): CIR++ fits observed termstructure of forward rates.Pacati, Renò and Santilli (2014): Heston++ reproducesATM term structure of FX options.

Two sources of jumps:CO- market downturns correlated with volatility spikes(Todorov and Tauchen, 2011, Bandi and Renò, 2015).Idiosyncratic- direct channel for right skewness of volatility.


The 2-SVCVJ++ model

dSt

St−= (r − q − λµ̄) dt +

√σ2

1,t + φtdW S1,t + σ2,tdW S

2,t + (ecx − 1)dNt

dσ21,t = α1(β1 − σ2

1,t )dt + Λ1σ1,tdWσ1,t + cσdNt + c′σdN ′t

dσ22,t = α2(β2 − σ2

2,t )dt + Λ2σ2,tdWσ2,t

under Q, where φ0 = 0, φt ≥ 0, cx ∼ N(µx + ρJcσ, δ2

x)|cσ,

cσ ∼ E(µco,σ) and c′σ ∼ E(µid ,σ). The model is affine provided

corr(dW S1,t ,dW σ

1,t ) = ρ1

√√√√ σ21,t

σ21,t + φt

corr(dW S2,t ,dW σ

2,t ) = ρ2


The 2-SVCVJ++ model: nested models

Taxonomy of the H++ models used in the empirical analysis.All models have two factors.

jumps in price volatility volatility displacement(idiosyncratic) (co-jumps) φt

2-SVJ X2-SVJ++ X X2-SVCJ X X2-SVCJ++ X X X2-SVVJ X X2-SVVJ++ X X X2-SVCVJ X X X2-SVCVJ++ X X X X


The 2-SVCVJ++ model: affinity

Lemma (Conditional Characteristic Functions)Under the H++ models, the conditional characteristic function ofreturns fH++

x (z) = EQ [eizxT∣∣Ft]

and of the two stochastic volatility

factors fH++σ (z1, z2) = EQ

[eiz1σ

21,T +iz2σ

22,T

∣∣∣Ft

]are given by:

fH++x (z; xt , σ

21,t , σ

22,t , t ,T , φ) = fHx (z; xt , σ

21,t , σ

22,t , τ)e−

12 z(i+z)Iφ(t ,T )

fH++σ (z1, z2;σ2

1,t , σ22,t , τ) = fHσ (z1, z2;σ2

1,t , σ22,t , τ)

where τ = T − t , z, z1, z2 ∈ C and Iφ(t ,T ) =∫ T

t φsds.


The 2-SVCVJ++ model: S&P500 Options

Proposition (Price of SPX Options)Under the H++ models, the arbitrage-free price at time t of aEuropean call option on the underlying St , with strike price Kand time to maturity τ = T − t , is given by (Lewis 2000, 2001)

CH++SPX (K , t ,T )

= St e−qτ −1π

√St K e−

12 (r+q)τ

∫ ∞0

Re[

eiuk fHx

(u −

i2

)]e−(

u2+ 14

)Iφ(t,T )

u2 + 14

du

where k = log(

StK

)+ (r − q)τ .


The 2-SVCVJ++ model: VIX Index

Proposition (VIX Index)Under the H++ models,

(VIXH++

t,τ̄

100

)2

=

(VIXHt,τ̄100

)2

+1τ̄

Iφ(t , t + τ̄)

where τ̄ = 30 days, VIXHt ,τ̄ is the corresponding quotation underH models, which is an affine function of the volatility factors σ2

1,tand σ2

2,t (VIXHt,τ̄100

)2

=1τ̄

∑k=1,2

ak (τ̄)σ2k,t + bk (τ̄)

where Iφ(t , t + τ̄) =

∫ t+τ̄t φsds.


The 2-SVCVJ++ model: VIX Futures and Options

Proposition (Price of VIX derivatives)Under H++ models, the time t value of a futures on VIXt,τ̄ settled at time T and thearbitrage-free price at time t of a call option on VIXt,τ̄ , with strike price K and time tomaturity τ = T − t are given respectively by

FH++VIX (t, T )

100=

1

2√π

∫ ∞0

Re

fHσ

(−z

a1(τ̄)

τ̄,−z

a2(τ̄)

τ̄

) e−iz

(∑k=1,2 bk (τ̄)+Iφ(T ,T +τ̄)

)/τ̄

(−iz)3/2

d Re(z)

and

CH++VIX (K , t, T )

100=

e−rτ

2√π

∫ ∞0

Re[

fHσ

(−z

a1(τ̄)

τ̄,−z

a2(τ̄)

τ̄

)

×e−iz

(∑k=1,2 bk (τ̄)+Iφ(T ,T +τ̄)

)/τ̄ (

1− erf(K/100√−iz)

)(−iz)3/2

d Re(z)

where z = Re(z) + i Im(z) ∈ C, 0 < Im(z) < ζc(τ) and erf(z) = 2√π

∫ z0 e−s2

ds.


SPX Vanilla (September 2, 2009)

800 1000

20

25

30

35

40

45

50

55

Strike

Vol (%

)

17 days

Calls

Puts

900 1000 1100

20

25

30

35

Strike

Vol (%

)

28 days

2−SVCVJ

2−SVCVJ++

600 800 1000 1200

20

30

40

50

60

Strike

Vol (%

)

45 days

600 800 1000 1200

20

25

30

35

40

45

50

55

Strike

Vol (%

)

80 days

600 800 1000 120020

25

30

35

40

45

50

55

Strike

Vol (%

)

108 days

500 1000

20

25

30

35

40

45

50

Strike

Vol (%

)

199 days

500 1000

20

25

30

35

40

45

50

Strike

Vol (%

)

290 days


VIX Futures (September 2, 2009)

0 20 40 60 80 100 120 140 160 180 200

29

29.5

30

30.5

31

31.5

32

32.5

33

Tenor (days)

Settle

Price (

US

$)

Data

2−SVCVJ2−SVCVJ++


VIX Options (September 2, 2009)

34 36 38 40 42 44

90

100

110

120

130

Strike

Vol (%

)

14 days

Data

30 40 50

60

65

70

75

80

85

90

95

Strike

Vol (%

)

49 days

2−SVCVJ2−SVCVJ++

30 40 50 60 70 80

60

70

80

90

100

Strike

Vol (%

)

77 days

20 30 40 5050

55

60

65

70

75

80

85

Strike

Vol (%

)

105 days


Calibration Errors (in %)

2-SVJ 2-SVJ++ 2-SVCJ 2-SVCJ++ 2-SVVJ 2-SVVJ++ 2-SVCVJ 2-SVCVJ++Panel A: RMSE

RMSESPX 1.17 0.99 1.04 0.86 0.99 0.77 0.90 0.65(6.01) (3.75) (4.11) (2.42) (4.28) (3.15) (4.28) (1.64)

RMSEFut 0.70 0.49 0.59 0.34 0.59 0.31 0.53 0.22(3.49) (1.85) (1.62) (1.32) (1.66) (1.19) (1.50) (1.07)

RMSEVIX 5.73 3.82 4.12 2.45 4.06 2.32 3.39 1.64(27.91) (17.58) (17.66) (9.03) (15.55) (8.76) (14.70) (4.03)

RMSEAll 2.20 1.56 1.70 1.16 1.64 1.07 1.42 0.82(8.80) (4.84) (5.44) (3.14) (7.12) (3.97) (4.57) (2.11)

Panel B: RMSRE

RMSRESPX 4.06 3.30 3.55 2.73 3.42 2.51 3.07 2.02(16.79) (9.29) (10.93) (6.04) (11.31) (8.25) (11.31) (3.95)

RMSREFut 2.32 1.61 2.01 1.13 1.98 1.02 1.81 0.74(9.11) (5.01) (6.48) (3.73) (6.14) (2.92) (6.13) (2.60)

RMSREVIX 7.38 4.66 5.69 3.12 5.59 2.88 4.78 2.04(28.32) (16.50) (25.14) (13.11) (23.66) (12.98) (23.56) (4.34)

RMSREAll 4.63 3.51 3.91 2.80 3.77 2.56 3.34 2.01(15.75) (9.90) (10.54) (6.15) (10.70) (7.94) (10.70) (3.94)


Agenda

Infer the true dynamics of S&P500: change of measurebetween P and Q measure, non-standard Kalman filteringof latent variables, pricing kernel and variance risk-premiaestimation.Understanding the meaning of φt :

1 Is it an affine approximation of some non-affine (true)model?

2 Is it (and to what extent) an additionally volatility statevector?


Thanks for your attention!


References I

Chicago Board Options Exchange. The CBOE volatility index-VIX. White Paper(2009).

Grünbichler, A., and Longstaff, F. A. Valuing futures and options on volatility.Journal of Banking & Finance 20 (6), 985-1001.

Mencía, J. and Sentana, E. Valuation of VIX derivatives. Journal of FinancialEconomics 108 (2), 367-391.

Bardgett, C., Gourier, E., and Leipold, M. Inferring volatility dynamics and riskpremia from the S&P 500 and VIX markets. Working paper.

Cont, R., and Kokholm, T. A consistent pricing model for index options andvolatility derivatives. Mathematical Finance 23.2 (2013): 248-274.

Sepp, A. Pricing options on realized variance in the Heston model with jumps inreturns and volatility. Journal of Computational Finance 11 (4), 33Ð70.

Sepp, A. VIX option pricing in a jump-diffusion model. Risk (April), 84-89.

Pacati, C., Renò, R. and Santilli, M. (2014). Heston Model: shifting on thevolatility surface. Risk (November), 54-59.

Brigo, D-, and Mercurio, F. A deterministic-shift extension of analytically-tractableand time-homogeneous short-rate models. Finance & Stochastics 5, 369-388.


References II

Bakshi, G., and Madan, D. Spanning and derivative-security valuation. Journal ofFinancial Economics 55.2 (2000): 205-238.

Schoutens, W. Levy processes in Finance: Pricing Financial Derivatives. Wiley,2003.

Duffie, D., Pan, J., and Singleton, K. Transform analysis and asset pricing foraffine jump-diffusions. Econometrica 68.6 (2000): 1343-1376.

Zhu, S.-P., and Lian, G.-H. An analytical formula for VIX futures and itsapplications. Journal of Futures Markets 32.2 (2012): 166-190.

Lian, G.-H., and Zhu S.-P. Pricing VIX options with stochastic volatility andrandom jumps. Decisions in Economics and Finance 36.1 (2013): 71-88.

Heston, S. L. A closed-form solution for options with stochastic volatility withapplications to bond and currency options. Review of financial studies 6.2 (1993):327-343.

Christoffersen, P., Heston, S., and Jacobs, K.. The shape and term structure ofthe index option smirk: Why multifactor stochastic volatility models work so well.

Bates, D. S. Jumps and stochastic volatility: Exchange rate processes implicit indeutsche mark options. Review of financial studies 9.1 (1996): 69-107.


References III

Branger, N., and Völkert, C. The fine structure of variance: Consistent pricing ofVIX derivatives. Working paper.


Science

Smiling Twice: The Heston++ Model