Consistent Pricing of VIX Derivatives and SPX Options with the Heston++ model

1. Introduction Our approach Empirical investigation Conclusions Consistent Pricing of VIX Derivatives and SPX Options with the Heston++ model G. Pompa1 C. Pacati2 R. Renò2 1IMT Institute for Advanced Studies Lucca, Italy 2Dipartimento di Economia Politica e Statistica Università di Siena, Italy XVI Workshop on Quantitative Finance, Parma 2015 G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

2. Introduction Our approach Empirical investigation Conclusions Outline 1 Introduction The problem VIX & Co. Standard approaches 2 Our approach The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing 3 Empirical investigation Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample 4 Conclusions G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

3. Introduction Our approach Empirical investigation Conclusions The problem VIX & Co. Standard approaches The problem The growing demand for trading volatility and managing volatility risk has lead today to a liquid market for derivatives on realized variance, such as VIX options and VIX futures. These are derivatives written on S&P500 volatility index (VIX): there is need of a pricing framework for consistent pricing both equity derivatives and volatility derivatives; since SPX and VIX derivatives both provide informations on the same volatility process, a model which is able to price one market, but not the other, is inherently misspecified; if a model is misspecified, inferred dynamics and risk-premia are unreliable. we tackle the problem of jointly fit the IV surface of SPX index options, together with the term structure of VIX futures. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

8. Introduction Our approach Empirical investigation Conclusions The problem VIX & Co. Standard approaches VIX: stylized facts Introduced in 1993, the VIX quotation is computed by CBOE as a model-free replication of the S&P500 realized volatility over the following 30 days (CBOE VIX white paper, 2003): Leverage effect: inverse relationship SPX-VIX G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

9. Introduction Our approach Empirical investigation Conclusions The problem VIX & Co. Standard approaches VIX: stylized facts Introduced in 1993, the VIX quotation is computed by CBOE as a model-free replication of the S&P500 realized volatility over the following 30 days (CBOE VIX white paper, 2003): Positively skewed and leptokurtic distribution (years 1990-2013 plotted) G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

10. Introduction Our approach Empirical investigation Conclusions The problem VIX & Co. Standard approaches Modeling VIX and VIX derivatives: literature review Standalone approach: volatility is directly modeled, separated from the underlying stock price process (Whaley 1993, Grünbichler and Longstaff 1996, Detemple and Osakwe 2000, Mencia and Sentana 2013); Consistent approach: VIX is derived from the speciﬁcation of SPX dynamics (Bardgett, Gourier and Leippold 2013, Cont Kokholm 2013). G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

11. Introduction Our approach Empirical investigation Conclusions The problem VIX & Co. Standard approaches Modeling VIX and VIX derivatives: literature review Gatheral (2008): inadequacy of Heston model in reproducing positive skew of VIX options IV Figure : Call options on VIX, 29/06/2009. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

12. Introduction Our approach Empirical investigation Conclusions The problem VIX & Co. Standard approaches Modeling VIX and VIX derivatives: literature review Gatheral (2008): inadequacy of Heston model in reproducing positive skew of VIX options IV ⇒ Sepp (2008 a,b) adds volatility jumps Figure : Call options on VIX, 29/06/2009. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

13. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing The basic Heston++ model Pacati, Renò and Santilli (2014) consider a deterministic shift extension (Brigo and Mercurio in short rates modeling, 2001) of the SV of the Heston class of models: the Heston++ model (H1f++) is the basic example dSt St = (r − q)dt + σ2 t + φt dWS t dσ2 t = α(β − σ2 t )dt + Λσt dWσ t (1) under Q, where φ0 = 0, φt ≥ 0 is called the displacement and the model is afﬁne provided that corr(dWS t , dWσ t ) = ρ σ2 t σ2 t + φt dt (2) G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

14. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing The basic Heston++ model The displacement φ increases the ﬂexibility in ﬁtting the ATM term 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) G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

15. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing The basic Heston++ model The displacement φ increases the flexibility in fitting the ATM term 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

16. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing Multifactor extensions We consider several multifactor affine specifications for the S&P500 dynamics (φt ≡ 0), along with their displaced counterparts (φt ≥ 0): classical H1f Heston (1993); two factor H2f (Christoffelsen, Heston and Jacob, 2009); models with jump in price only: Bates (1996) like model H1fj and corresponding two factor version H2fj; H1fcoj model with synchronous correlated jumps in price and in volatility (Duffie, Pan and Singleton, 2000) and two factor version H2fcoj; G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

17. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing The Heston 2-factor and co-jumps ++ model The most general speciﬁcation for the S&P500 dynamics that we consider is the H2fcoj++ model, under Q: dSt St− = (r − q − λ¯µ) dt + σ2 1,t + φt dWS 1,t + σ2,t dWS 2,t + (ezx − 1)dNt dσ2 1,t = α1(β1 − σ2 1,t )dt + Λ1σ1,t dWσ 1,t + z1dNt dσ2 2,t = α2(β2 − σ2 2,t )dt + Λ2σ2,t dWσ 2,t where jumps are (zx , z1) ∼ N µx + ρJz1, δ2 x × E(µ1) and corr(dWS 1,t , dWσ 1,t ) = ρ1 σ2 1,t σ2 1,t + φt dt corr(dWS 2,t , dWσ 2,t ) = ρ2dt G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

18. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing 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) = St e−qτ Q1 − Ke−rτ Q2 (1) Q1 = 1 2 + 1 π ∞ 0 Re e−iz log K f(z − i) izf(−i) dz Q2 = 1 2 + 1 π ∞ 0 Re e−iz log K f(z) iz dz (2) where f(z) = EQ[eiz log ST |Ft ] is the risk-neutral conditional CF of log-index at maturity (τ = T − t, Iφ(t1, t2) = t2 t1 φt dt) f(z; log St , σ2 1,t , σ2 2,t , t, T) = fH (z; log St , σ2 1,t , σ2 2,t , τ) H2fcoj CF × ++ correction e−1 2 z(i+z)Iφ(t,T) G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

19. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing 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) = St e−qτ Q1 − Ke−rτ Q2 (1) Q1 = 1 2 + 1 π ∞ 0 Re e−iz log K f(z − i) izf(−i) dz Q2 = 1 2 + 1 π ∞ 0 Re e−iz log K f(z) iz dz (2) where f(z) = EQ[eiz log ST |Ft ] is the risk-neutral conditional CF of log-index at maturity (τ = T − t, Iφ(t1, t2) = t2 t1 φt dt) f(z; log St , σ2 1,t , σ2 2,t , t, T) = fH (z; log St , σ2 1,t , σ2 2,t , τ) H2fcoj CF × ++ correction e−1 2 z(i+z)Iφ(t,T) G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

20. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing VIX index in the H2fcoj++ model The squared VIXt is an affine function of the volatility state vector Σt = (σ1,t , σ2,t ) : VIXt 100 2 = Aφ(t, ¯τ) + B(¯τ) · Σt (3) where ¯τ = 30/365 and Aφ(t, ¯τ) = A(¯τ) affinity + ++ correction 1 ¯τ Iφ(t, t + ¯τ) (4) Coefficients A(¯τ) and B(¯τ) depend on the VIX time scale ¯τ only. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

21. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing VIX Futures & Options in the H2fcoj++ model Price of a tenor T VIX futures is (Zhu and Lian 2012, if φt ≡ 0) FT t 100 = 1 2 √ π ∞ 0 1 − e−sAφ(T,¯τ)F(isB(¯τ)) s3/2 ds and for a Call option on VIX (Lian and Zhu 2013, if φt ≡ 0) C(t, T, K) 100 = e−rτ 2 √ π ∞ 0 Re e−izAφ(T,¯τ) F(−zB(¯τ)) 1 − erf(K/100 √ −iz) (−iz)3/2 dRe(z) Volatility factor CF does not depend on displacement φ F(Z1, Z2; σ2 1,t , σ2 2,t , t, T) = EQ [eiZ1σ2 1,T +iZ2σ2 2,T |Ft ] = k=1,2 Fk (Zk , σ2 k,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

22. Introduction Our approach Empirical investigation Conclusions The basic Heston++ model Multifactor ++ extensions SPX Vanilla pricing VIX index modeling VIX Futures and Options pricing VIX Futures & Options in the H2fcoj++ model Price of a tenor T VIX futures is (Zhu and Lian 2012, if φt ≡ 0) FT t 100 = 1 2 √ π ∞ 0 1 − e−sAφ(T,¯τ)F(isB(¯τ)) s3/2 ds and for a Call option on VIX (Lian and Zhu 2013, if φt ≡ 0) C(t, T, K) 100 = e−rτ 2 √ π ∞ 0 Re e−izAφ(T,¯τ) F(−zB(¯τ)) 1 − erf(K/100 √ −iz) (−iz)3/2 dRe(z) Volatility factor CF does not depend on displacement φ F(Z1, Z2; σ2 1,t , σ2 2,t , t, T) = EQ [eiZ1σ2 1,T +iZ2σ2 2,T |Ft ] = k=1,2 Fk (Zk , σ2 k,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

23. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX 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

24. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample Data: SPX Vanilla & VIX Futures Figure : VIX index and VIX Futures term structure, 29/06/2009. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

25. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample Data: SPX Vanilla & VIX Futures We optimize parameters of each model on the SPX volatility surface and VIX futures term structure minimizing the normalized SSE: loss(π) = {Vanilla} IV% MKT − IV% model (π) 2 + NVanilla NVIX-Futures {VIX-Futures} FMKT −Fmodel (π) 2 G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

26. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample 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. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

27. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample SPX + VIX Futures: H1f Vs H1f++ G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

28. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample SPX + VIX Futures: H1fcoj Vs H1fcoj++ G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

30. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample 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. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

31. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample SPX + VIX Futures: H1f Vs H1f++ G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

34. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample 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

35. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample VIX options pricing out-of-sample Figure : Filters for Call options on VIX, 29/06/2009. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

36. Introduction Our approach Empirical investigation Conclusions Data SPX Vanilla + VIX Futures calibration VIX options pricing out-of-sample VIX options pricing out-of-sample: H2fcoj Vs H2fcoj++ Figure : Call options on VIX, 29/06/2009. Calibration on {Vanilla, VIX Futures} only G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

37. Introduction Our approach Empirical investigation Conclusions Summary and Conclusions we have characterized a class of Heston-like displaced models finding pricing formulas for SPX Vanilla (Pacati, Reno’, Santilli 2014), VIX Futures (new) and VIX Options (new); the introduction of displacement does not alter the affinity of the model and comes almost at no additional computational cost w.r.t. non-displaced model; we have calibrated several Heston-like affine models on the S&P500 Vanilla surface together with the VIX Futures term structure; 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 that displaced models keep doing better over non-displaced models. Work in progress... G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

43. Introduction Our approach Empirical investigation Conclusions Agenda confirm the effectiveness of the displacement with VIX options calibrated consistently with SPX vanilla and VIX futures; confirm the effectiveness of the displacement on a wider time domain; evaluate the effectiveness of the displacement on different affine specifications, e.g. stochastic mean reverting level (Bardgett, Gourier and Leippold, 2013), stochastic vol-of-vol (Branger and Volkert, 2012); test the time consistency of the displacement (one function φt per dataset) ⇒ estimate risk premia. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

48. Introduction Our approach Empirical investigation Conclusions Thanks for your attention! G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

49. 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 on volatility. Journal of Banking & Finance 20.6 (1996): 985-1001. Wang, Zhiguang, and Robert T. Daigler. The performance of VIX option pricing models: empirical evidence beyond simulation. Journal of Futures Markets 31.3 (2011): 251-281. Mencia, Javier, and Enrique Sentana. Valuation of VIX derivatives. Journal of Financial Economics 108.2 (2013): 367-391. Bardgett, Chris, Elise Gourier, and Markus Leipold. Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets. Swiss Finance Institute Research Paper 13-40 (2013). Cont, Rama, and Thomas Kokholm. A consistent pricing model for index options and volatility derivatives. Mathematical Finance 23.2 (2013): 248-274. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

50. 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 jumps in 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 the volatility surface. Risk (2014), November, pp 54-59 Brigo, Damiano, and Fabio Mercurio. A deterministicÐshift extension of analyticallyÐtractable and timeÐhomogeneous shortÐrate models. Finance and Stochastics 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. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

51. Appendix References References III Duffie, Darrell, Jun Pan, and Kenneth Singleton. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68.6 (2000): 1343-1376. Zhu, Song-Ping, and Guang-Hua Lian. An analytical formula for VIX futures and its applications. Journal of Futures Markets 32.2 (2012): 166-190. Lian, Guang-Hua, and Song-Ping Zhu. Pricing VIX options with stochastic volatility 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 with applications to bond and currency options. Review of financial studies 6.2 (1993): 327-343. Christoffersen, Peter, Steven Heston, and Kris Jacobs. The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Bates, David S. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of financial studies 9.1 (1996): 69-107. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

52. Appendix References References IV Branger, Nicole, and Clemens Völkert. The ﬁne structure of variance: Consistent pricing of VIX derivatives. Paris December 2012 Finance Meeting EUROFIDAI-AFFI Paper. 2013. G. Pompa, C. Pacati, R. Renò Consistent Pricing of VIX Derivatives and SPX Options

Consistent Pricing of VIX Derivatives and SPX Options with the Heston++ model

Recommended

Recommended

More Related Content

Similar to Consistent Pricing of VIX Derivatives and SPX Options with the Heston++ model

Similar to Consistent Pricing of VIX Derivatives and SPX Options with the Heston++ model (20)

Recently uploaded

Recently uploaded (20)

Consistent Pricing of VIX Derivatives and SPX Options with the Heston++ model