Advertisement

Review of Quantitative Finance and Accounting

, Volume 35, Issue 3, pp 245–269 | Cite as

A jump diffusion model for VIX volatility options and futures

  • Dimitris Psychoyios
  • George Dotsis
  • Raphael N. Markellos
Original Research

Abstract

Volatility indices are becoming increasingly popular as a measure of market uncertainty and as a new asset class for developing derivative instruments. Although jumps are widely considered as a salient feature of volatility, their implications for pricing volatility options and futures are not yet fully understood. This paper provides evidence indicating that the time series behaviour of the VIX index is well approximated by a mean reverting logarithmic diffusion with jumps. This process is capable of capturing stylized facts of VIX dynamics such as fast mean-reversion at higher levels, level effects of volatility and large upward movements during times of market stress. Based on the empirical results, we provide closed-form valuation models for European options written on the spot and forward VIX, respectively.

Keywords

Implied volatility Jump diffusion Option pricing Volatility risk 

JEL Classification

G13 C51 C52 

References

  1. Bakshi G, Cao C (2006) Risk-neutral kurtosis, jumps, and option pricing: evidence from most actively traded firms on the CBOE. Working Paper, University of MarylandGoogle Scholar
  2. Bakshi G, Madan D (2000) Spanning and derivative-security valuation. J Financ Econ 55:205–238CrossRefGoogle Scholar
  3. Bakshi G, Ju N, Ou-Yang H (2006) Estimation of continuous-time models with an application to equity volatility dynamics. J Financ Econ 82:227–249CrossRefGoogle Scholar
  4. Black F (1976) The pricing of commodity contracts. J Financ Econ 3:167–179CrossRefGoogle Scholar
  5. Blair BJ, Poon S, Taylor SJ (2001) Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns. J Econom 105:5–26CrossRefGoogle Scholar
  6. Brenner M, Galai D (1989) New financial instruments for hedging changes in volatility. Financ Anal J 4:61–65CrossRefGoogle Scholar
  7. Brenner M, Galai D (1993) Hedging volatility in foreign currencies. J Deriv 1:53–59CrossRefGoogle Scholar
  8. Brenner M, Ou E, Zhang J (2006) Hedging volatility risk. J Bank Financ 30:811–821CrossRefGoogle Scholar
  9. Carr P, Lee R (2005) Robust replication of volatility derivatives working paper, Courant Institute, NYU and University of ChicagoGoogle Scholar
  10. Carr P, Wu L (2006) A tale of two indices. J Deriv 13:13–29CrossRefGoogle Scholar
  11. Chan KC, Karolyi GA, Longstaff FA, Sanders AB (1992) An empirical comparison of alternative models of the short-term interest rate. J Financ 47:1209–1227CrossRefGoogle Scholar
  12. Chriss N, Morokoff W (1999) Market risk of variance swaps. RISK 12:55–59Google Scholar
  13. Christoffersen P, Jacobs K, Mimouni K (2006) Models for S&P500 dynamics: evidence from realized volatility, daily returns, and option prices. Working Paper, McGill UniversityGoogle Scholar
  14. Corrado CJ, Miller T (2005) The forecast quality of CBOE implied volatility indexes. J Futur Mark 25:339–373CrossRefGoogle Scholar
  15. Cox JC, Ingersoll JE, Ross SA (1981) The relationship between forward prices and futures prices. J Financ Econ 9:321–346CrossRefGoogle Scholar
  16. Cox JC, Ingersoll JE, Ross SA (1985) A theory of the term structure of interest rates. Econometrica 53:385–408CrossRefGoogle Scholar
  17. Daouk H, Guo JQ (2004) Switching asymmetric GARCH and options on a volatility index. J Futur Mark 24:251–282CrossRefGoogle Scholar
  18. Das RS, Sundaram R (1999) Of smiles and smirks: a term structure perspective. J Financ Quant Anal 34:211–240CrossRefGoogle Scholar
  19. Demeterfi K, Derman E, Kamal M, Zou J (1999) More than you ever wanted to know about volatility swaps. J Deriv 6:9–32CrossRefGoogle Scholar
  20. Detemple J, Osakwe C (2000) The valuation of volatility options. Eur Financ Rev 4:21–50CrossRefGoogle Scholar
  21. Dotsis G, Psychoyios D, Skiadopoulos G (2007) An empirical comparison of continuous-time models of implied volatility indices. J Bank Financ 31:3584–3603CrossRefGoogle Scholar
  22. Duffie D, Pan J, Singleton K (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68:1343–1376CrossRefGoogle Scholar
  23. Engle RF, Ng VK (1993) Measuring and testing the impact of news and volatility. J Financ 48:1749–1778CrossRefGoogle Scholar
  24. Fleming J, Ostdiek B, Whaley RE (1995) Predicting stock market volatility: a new measure. J Futur Mark 15:265–302CrossRefGoogle Scholar
  25. Giot P (2005) Relationships between implied volatility indices and stock index returns. J Portf Manage 31:92–100CrossRefGoogle Scholar
  26. Grünbichler A, Longstaff F (1996) Valuing futures and options on volatility. J Bank Financ 20:985–1001CrossRefGoogle Scholar
  27. Heston S (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6:327–343CrossRefGoogle Scholar
  28. Heston S, Nandi S (2000a) Derivatives on volatility: some simple solutions based on observables. Working Paper, Federal Reserve Bank of AtlantaGoogle Scholar
  29. Heston S, Nandi S (2000b) A closed-form GARCH option pricing model. Rev Financ Stud 13:585–626CrossRefGoogle Scholar
  30. Jones C (2003) The dynamics of stochastic volatility: evidence from underlying and options markets. J Econom 116:181–224CrossRefGoogle Scholar
  31. Kou SG (2002) A jump-diffusion model for option pricing. Manage Sci 48:1086–1101CrossRefGoogle Scholar
  32. Lin Y (2007) Pricing VIX futures: evidence from integrated physical and risk-neutral probability measures. J Futur Mark 27:1175–1217CrossRefGoogle Scholar
  33. Nelson DB (1990) ARCH models as diffusion approximations. J Econom 45:7–39CrossRefGoogle Scholar
  34. Pan J (2002) The jump-risk premia implicit in options: evidence from an integrated time-series study. J Financ Econ 63:3–50CrossRefGoogle Scholar
  35. Phillips PCB, Yu J (2005) Jackknifing bond option prices. Rev Financ Stud 18:707–742CrossRefGoogle Scholar
  36. Scott LO (1987) Option pricing when the variance changes randomly: theory, estimation, and an application. J Financ Quant Anal 22:419–438CrossRefGoogle Scholar
  37. Sepp A (2008) VIX option pricing in a jump-diffusion model. RISK 21:84–89Google Scholar
  38. Simon DP (2003) The Nasdaq volatility index during and after the Bubble. J Deriv 11:9–24CrossRefGoogle Scholar
  39. Singleton KJ (2001) Estimation of affine asset pricing models using the empirical characteristic function. J Econom 102:111–141CrossRefGoogle Scholar
  40. Stein E, Stein J (1991) Stock price distributions with stochastic volatility: an analytic approach. Rev Financ Stud 4:727–752CrossRefGoogle Scholar
  41. Vuong Q (1989) Likelihood ratio tests for model selection and nonnested hypotheses. Econometrica 57:307–333CrossRefGoogle Scholar
  42. Wagner N, Szimayer A (2004) Local and Spillover shocks in implied market volatility: evidence for the US and Germany. Res Int Bus Financ 18:237–251CrossRefGoogle Scholar
  43. Whaley RE (1993) Derivatives on market volatility: hedging tools long overdue. J Deriv 1:71–84CrossRefGoogle Scholar
  44. Whaley RE (2000) The investor fear gauge. J Portf Manage 26:12–17CrossRefGoogle Scholar
  45. Zhu Y, Zhang JE (2007) Variance term structure and VIX futures pricing. Int J Theor Appl Financ 10:111–127CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Dimitris Psychoyios
    • 1
    • 2
  • George Dotsis
    • 3
  • Raphael N. Markellos
    • 4
    • 5
  1. 1.Department of Industrial ManagementUniversity of PiraeusPiraeusGreece
  2. 2.CAIR, Manchester Business SchoolUniversity of ManchesterManchesterUK
  3. 3.Essex Business School and Essex Finance CentreUniversity of EssexColchesterUK
  4. 4.Department of Management Science and TechnologyAthens University of Economics and BusinessAthensGreece
  5. 5.Centre for Research in International Economics and Finance (CIFER)Loughborough UniversityLoughboroughUK

Personalised recommendations