smiling twice: the heston++ model
TRANSCRIPT
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.
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.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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)
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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)
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
VIX Options: the cross-section of Fear
Traded since 2006, provide insurance from equity marketdownturns: S&P500 vanilla below (June 29, 2009)
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
VIX Options: the cross-section of Fear
Traded since 2006, provide insurance from equity marketdownturns: VIX options below (June 29, 2009).
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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).
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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++).
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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 .
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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++
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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)
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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?
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
Thanks for your attention!
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
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.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model
References III
Branger, N., and Völkert, C. The fine structure of variance: Consistent pricing ofVIX derivatives. Working paper.
C. Pacati, G. Pompa, R. Renò Smiling twice: The Heston++ model