consistent pricing of vix derivatives and spx options with the heston++ model

IntroductionOur approach

Empirical investigationConclusions

Consistent Pricing of VIX Derivativesand SPX Options

with the Heston++ model

G. Pompa1 C. Pacati2 R. Renò2

1IMT Institute for Advanced Studies Lucca, Italy

2Dipartimento di Economia Politica e StatisticaUniversità di Siena, Italy

XVI Workshop on Quantitative Finance, Parma 2015

G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

Outline1 Introduction

The problemVIX & Co.Standard approaches

2 Our approachThe basic Heston++ modelMultifactor ++ extensionsSPX Vanilla pricingVIX index modelingVIX Futures and Options pricing

3 Empirical investigationDataSPX Vanilla + VIX Futures calibrationVIX options pricing out-of-sample

4 Conclusions

The problem

The growing demand for trading volatility and managingvolatility risk has lead today to a liquid market for derivatives onrealized variance, such as VIX options and VIX futures. Theseare derivatives written on S&P500 volatility index (VIX):

there is need of a pricing framework for consistent pricingboth equity derivatives and volatility derivatives;since SPX and VIX derivatives both provide informationson the same volatility process, a model which is able toprice one market, but not the other, is inherentlymisspecified;if a model is misspecified, inferred dynamics andrisk-premia are unreliable.

we tackle the problem of jointly fit the IV surface of SPX indexoptions, together with the term structure of VIX futures.

The problem

VIX: stylized facts

Introduced in 1993, the VIX quotation is computed by CBOE asa model-free replication of the S&P500 realized volatility overthe following 30 days (CBOE VIX white paper, 2003):

Leverage effect: inverse relationship SPX-VIX

VIX: stylized facts

Introduced in 1993, the VIX quotation is computed by CBOE asa model-free replication of the S&P500 realized volatility overthe following 30 days (CBOE VIX white paper, 2003):

Positively skewed and leptokurtic distribution (years1990-2013 plotted)

Modeling VIX and VIX derivatives: literature review

Standalone approach: volatility is directly modeled,separated from the underlying stock price process (Whaley1993, Grünbichler and Longstaff 1996, Detemple andOsakwe 2000, Mencia and Sentana 2013);Consistent approach: VIX is derived from the specificationof SPX dynamics (Bardgett, Gourier and Leippold 2013,Cont Kokholm 2013).

Gatheral (2008): inadequacy of Heston model in reproducingpositive skew of VIX options IV

Figure : Call options on VIX, 29/06/2009.

Gatheral (2008): inadequacy of Heston model in reproducingpositive skew of VIX options IV⇒ Sepp (2008 a,b) addsvolatility jumps

Figure : Call options on VIX, 29/06/2009.

The basic Heston++ modelMultifactor ++ extensionsSPX Vanilla pricingVIX index modelingVIX Futures and Options pricing

The basic Heston++ model

Pacati, Renò and Santilli (2014) consider a deterministic shiftextension (Brigo and Mercurio in short rates modeling, 2001) ofthe SV of the Heston class of models: the Heston++ model(H1f++) is the basic example

St= (r − q)dt +

√σ2

t + φtdW St

dσ2t = α(β − σ2

t )dt + ΛσtdW σt

under Q, where φ0 = 0, φt ≥ 0 is called the displacement andthe model is affine provided that

corr(dW St ,dW σ

t ) = ρ

√σ2

σ2t + φt

dt (2)

The displacement φ increases the flexibility in fitting the ATMterm structure

H1f Vs H1f++ e/US$ FX options, July 3, 2009. Source: Pacati, C., Renò, R. and

Santilli, M. (2014). Heston Model: shifting on the volatility surface. Risk (2014)

The displacement φ increases the flexibility in fitting the ATMterm structure of IV surface⇒ eases the fit of the whole surface

H1f Vs H1f++ e/US$ FX options, July 3, 2009. Source: Pacati, C., Renò, R. and

Santilli, M. (2014). Heston Model: shifting on the volatility surface. Risk (2014)G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

Multifactor extensions

We consider several multifactor affine specifications for theS&P500 dynamics (φt ≡ 0), along with their displacedcounterparts (φt ≥ 0):

classical H1f Heston (1993);two factor H2f (Christoffelsen, Heston and Jacob, 2009);models with jump in price only: Bates (1996) like modelH1fj and corresponding two factor version H2fj;H1fcoj model with synchronous correlated jumps in priceand in volatility (Duffie, Pan and Singleton, 2000) and twofactor version H2fcoj;

The Heston 2-factor and co-jumps ++ model

The most general specification for the S&P500 dynamics thatwe consider is the H2fcoj++ model, under Q:

St−= (r − q − λµ̄) dt +

√σ2

1,t + φtdW S1,t + σ2,tdW S

2,t + (ezx − 1)dNt

dσ21,t = α1(β1 − σ2

1,t )dt + Λ1σ1,tdW σ1,t + z1dNt

dσ22,t = α2(β2 − σ2

2,t )dt + Λ2σ2,tdW σ2,t

where jumps are (zx , z1) ∼ N(µx + ρJz1, δ

2x)× E(µ1) and

corr(dW S1,t ,dW σ

1,t ) = ρ1

√√√√ σ21,t

σ21,t + φt

corr(dW S2,t ,dW σ

2,t ) = ρ2dt

SPX vanilla option pricing in the H2fcoj++ model

Under H2fcoj++, the arbitrage free price of a Call on S&P500(Bakshi Madan 2000 and Schoutens 2003 if φt ≡ 0)

C(t ,T ,K ) = Ste−qτQ1 − Ke−rτQ2 (1)

Q1 =12

∫ ∞0

e−iz log K f (z − i)izf (−i)

Q2 =12

∫ ∞0

e−iz log K f (z)

where f (z) = EQ[eiz log ST |Ft ] is the risk-neutral conditional CFof log-index at maturity (τ = T − t , Iφ(t1, t2) =

∫ t2t1φtdt)

f (z; log St , σ21,t , σ

22,t , t ,T ) = f H(z; log St , σ

21,t , σ

22,t , τ)︸︷︷︸

H2fcoj CF

++ correction︷︸︸︷e−

12 z(i+z)Iφ(t ,T )

SPX vanilla option pricing in the H2fcoj++ model

Under H2fcoj++, the arbitrage free price of a Call on S&P500(Bakshi Madan 2000 and Schoutens 2003 if φt ≡ 0)

C(t ,T ,K ) = Ste−qτQ1 − Ke−rτQ2 (1)

Q1 =12

∫ ∞0

e−iz log K f (z − i)izf (−i)

Q2 =12

∫ ∞0

e−iz log K f (z)

where f (z) = EQ[eiz log ST |Ft ] is the risk-neutral conditional CFof log-index at maturity (τ = T − t , Iφ(t1, t2) =

∫ t2t1φtdt)

f (z; log St , σ21,t , σ

22,t , t ,T ) = f H(z; log St , σ

21,t , σ

22,t , τ)︸︷︷︸

H2fcoj CF

++ correction︷︸︸︷e−

12 z(i+z)Iφ(t ,T )

VIX index in the H2fcoj++ model

The squared VIXt is an affine function of the volatility statevector Σt = (σ1,t , σ2,t )

>:(VIXt

= Aφ(t , τ̄) + B(τ̄) · Σt (3)

where τ̄ = 30/365 and

Aφ(t , τ̄) = A(τ̄)︸︷︷︸affinity

++ correction︷︸︸︷1τ̄

Iφ(t , t + τ̄) (4)

Coefficients A(τ̄) and B(τ̄) depend on the VIX time scale τ̄ only.

VIX Futures & Options in the H2fcoj++ model

Price of a tenor T VIX futures is (Zhu and Lian 2012, if φt ≡ 0)

12√π

∫ ∞0

1− e−sAφ(T ,τ̄)F (isB(τ̄))

s3/2 ds

and for a Call option on VIX (Lian and Zhu 2013, if φt ≡ 0)

C(t ,T ,K )

e−rτ

2√π

∫ ∞0

e−izAφ(T ,τ̄)F (−zB(τ̄))1− erf (K/100

√−iz)

(−iz)3/2

)dRe(z)

Volatility factor CF does not depend on displacement φ

F (Z1,Z2;σ21,t , σ

22,t , t ,T ) = EQ[eiZ1σ

21,T +iZ2σ

22,T |Ft ] =

∏k=1,2

Fk (Zk , σ2k ,t , τ)

factorizes in 1-factor CFs (Duffie, Pan and Singleton, 2000).G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

VIX Futures & Options in the H2fcoj++ model

Price of a tenor T VIX futures is (Zhu and Lian 2012, if φt ≡ 0)

12√π

∫ ∞0

1− e−sAφ(T ,τ̄)F (isB(τ̄))

s3/2 ds

and for a Call option on VIX (Lian and Zhu 2013, if φt ≡ 0)

C(t ,T ,K )

e−rτ

2√π

∫ ∞0

e−izAφ(T ,τ̄)F (−zB(τ̄))1− erf (K/100

√−iz)

(−iz)3/2

)dRe(z)

Volatility factor CF does not depend on displacement φ

F (Z1,Z2;σ21,t , σ

22,t , t ,T ) = EQ[eiZ1σ

21,T +iZ2σ

22,T |Ft ] =

∏k=1,2

Fk (Zk , σ2k ,t , τ)

factorizes in 1-factor CFs (Duffie, Pan and Singleton, 2000).G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

DataSPX Vanilla + VIX Futures calibrationVIX options pricing out-of-sample

Data: SPX Vanilla & VIX Futures

Figure : Implied volatility surface, European calls and puts on S&P500, 29/06/2009.G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

Figure : VIX index and VIX Futures term structure, 29/06/2009.

We optimize parameters of each model on the SPX volatility surface and VIX futuresterm structure minimizing the normalized SSE:

loss(π) =∑{Vanilla}

MKT − IV %model (π)

NVanilla

NVIX-Futures

∑{VIX-Futures}

(FMKT−Fmodel (π)

SPX + VIX Futures

Figure : Fit error on SPX Vanilla surface and VIX Futures term structure separately,calibration on {SPX Vanilla, VIX Futures}, 29/06/2009.

SPX + VIX Futures: H1f Vs H1f++

SPX + VIX Futures: H1fcoj Vs H1fcoj++

SPX + VIX Futures

Figure : Fit error on SPX Vanilla surface and VIX Futures term structure separately,calibration on {SPX Vanilla, VIX Futures}, 29/06/2009.

SPX + VIX Futures: H1f Vs H1f++

VIX options pricing out-of-sample

Figure : Call options on VIX, 29/06/2009.G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

VIX options pricing out-of-sample

Figure : Filters for Call options on VIX, 29/06/2009.

VIX options pricing out-of-sample: H2fcoj Vs H2fcoj++

Figure : Call options on VIX, 29/06/2009. Calibration on {Vanilla, VIX Futures} onlyG. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

Summary and Conclusions

we have characterized a class of Heston-like displaced modelsfinding pricing formulas for SPX Vanilla (Pacati, Reno’, Santilli2014), VIX Futures (new) and VIX Options (new); theintroduction of displacement does not alter the affinity of themodel and comes almost at no additional computational costw.r.t. non-displaced model;

we have calibrated several Heston-like affine models on theS&P500 Vanilla surface together with the VIX Futures termstructure; the displacement looks promising as:

1 improves the fit of SPX surface, especially long-term options;2 provides an - almost exact - fit of the VIX futures term structure;3 in out-of-sample exercise it seems to better capture the positive skew of

VIX options surface. In-sample exercises (not shown today) suggest thatdisplaced models keep doing better over non-displaced models. Work inprogress...

Agenda

confirm the effectiveness of the displacement with VIXoptions calibrated consistently with SPX vanilla and VIXfutures;confirm the effectiveness of the displacement on a widertime domain;evaluate the effectiveness of the displacement on differentaffine specifications, e.g. stochastic mean reverting level(Bardgett, Gourier and Leippold, 2013), stochasticvol-of-vol (Branger and Volkert, 2012);test the time consistency of the displacement (one functionφt per dataset)⇒ estimate risk premia.

Agenda

Thanks for your attention!

Appendix References

References I

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

Grünbichler, Andreas, and Francis A. Longstaff. Valuing futures and options onvolatility. Journal of Banking & Finance 20.6 (1996): 985-1001.

Wang, Zhiguang, and Robert T. Daigler. The performance of VIX option pricingmodels: empirical evidence beyond simulation. Journal of Futures Markets 31.3(2011): 251-281.

Mencia, Javier, and Enrique Sentana. Valuation of VIX derivatives. Journal ofFinancial Economics 108.2 (2013): 367-391.

Bardgett, Chris, Elise Gourier, and Markus Leipold. Inferring volatility dynamicsand risk premia from the S&P 500 and VIX markets. Swiss Finance InstituteResearch Paper 13-40 (2013).

Cont, Rama, and Thomas Kokholm. A consistent pricing model for index optionsand volatility derivatives. Mathematical Finance 23.2 (2013): 248-274.

Appendix References

References II

Gatheral, Jim. Consistent modeling of SPX and VIX options. Bachelier Congress.2008.

Sepp, Artur. Pricing options on realized variance in the Heston model with jumpsin returns and volatility. Journal of Computational Finance 11.4 (2008): 33.

Sepp, Artur. VIX option pricing in a jump-diffusion model. Risk magazine (2008):84-89.

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

Brigo, Damiano, and Fabio Mercurio. A deterministicÐshift extension ofanalyticallyÐtractable and timeÐhomogeneous shortÐrate models. Finance andStochastics 5.3 (2001): 369-387.

Bakshi, Gurdip, and Dilip Madan. Spanning and derivative-security valuation.Journal of Financial Economics 55.2 (2000): 205-238.

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

Appendix References

References III

Duffie, Darrell, Jun Pan, and Kenneth Singleton. Transform analysis and assetpricing for affine jump-diffusions. Econometrica 68.6 (2000): 1343-1376.

Zhu, Song-Ping, and Guang-Hua Lian. An analytical formula for VIX futures andits applications. Journal of Futures Markets 32.2 (2012): 166-190.

Lian, Guang-Hua, and Song-Ping Zhu. Pricing VIX options with stochasticvolatility and random jumps. Decisions in Economics and Finance 36.1 (2013):71-88.

Heston, Steven 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, Peter, Steven Heston, and Kris Jacobs. The shape and termstructure of the index option smirk: Why multifactor stochastic volatility modelswork so well.

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

Appendix References

References IV

Branger, Nicole, and Clemens Völkert. The fine structure of variance: Consistentpricing of VIX derivatives. Paris December 2012 Finance MeetingEUROFIDAI-AFFI Paper. 2013.

consistent pricing of vix derivatives and spx options with the heston++ model

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