Pricing Central Tendency in Volatility Stanislav Khrapov NES Anniversary, Moscow December 14...
Motivation and Contribution The Model ResultsMarket Retur...
Motivation and Contribution The Model ResultsVolatili...
Motivation and Contribution The Model ResultsPersistence ...
Motivation and Contribution The Model ResultsThick Tails ...
Motivation and Contribution The Model ResultsContribution Two-component vo...
Motivation and Contribution The Model ResultsContribution Two-component vo...
Motivation and Contribution The Model ResultsContribution Two-component...
Motivation and Contribution The Model ResultsContribution Two-component...
Motivation and Contribution The Model ResultsThe Model Historical: ...
Motivation and Contribution The Model ResultsThe Model Risk-neutral: ...
Motivation and Contribution The Model ResultsThe Model Risk-neutral: ...
Motivation and Contribution The Model ResultsData Objecti...
Motivation and Contribution The Model ResultsData Objecti...
Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt+h - ...
Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt...
Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt...
Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt...
Motivation and Contribution The Model ResultsMoment Conditions First moment: ...
Motivation and Contribution The Model ResultsMoment Conditions First moment: ...
Motivation and Contribution The Model ResultsMoment Conditions Second mom...
Motivation and Contribution The Model ResultsPremia Volatility pre...
Motivation and Contribution The Model ResultsPremia Volatility pre...
Motivation and Contribution The Model ResultsPremia Volatility pre...
Motivation and Contribution The Model ResultsParameter estimates ...
Motivation and Contribution The Model ResultsParameter estimates ...
Motivation and Contribution The Model ResultsParameter estimates ...
Motivation and Contribution The Model ResultsParameter estimates ...
Motivation and Contribution The Model ResultsParameter estimates ...
Motivation and Contribution The Model ResultsVolatility Premia ...
Motivation and Contribution The Model ResultsConclusion Joint estimation of volatility m...
Thank you!
Adrian, Tobias, and Joshua Rosenberg, 2008, Stock Returns and Volatility: Pricing the Short-Run and Long-Run Components ...
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Pricing Central Tendency in Volatility

NES 20th Anniversary Conference, Dec 13-16, 2012 Pricing Central Tendency in Volatility (based on the article presented by Stanislav Khrapov at the NES 20th Anniversary Conference). Author: Stanislav Khrapov, NES
Published on: Mar 4, 2016
Source: www.slideshare.net


Transcripts - Pricing Central Tendency in Volatility

  • 1. Pricing Central Tendency in Volatility Stanislav Khrapov NES Anniversary, Moscow December 14, 2012
  • 2. Motivation and Contribution The Model ResultsMarket Returns S&P500 index 1600 1400 SPX 1200 1000 800 600 400 S&P500 log returns 15 10 logR 5 0 −5 −10 97 99 01 03 05 07 09 11 19 19 20 20 20 20 20 20
  • 3. Motivation and Contribution The Model ResultsVolatility Volatility measures 140 120 RV 100 80 VIX 60 40 20 0 Difference 60 40 RV-VIX 20 0 −20 −40 7 9 1 3 5 7 9 1 199 199 200 200 200 200 200 201
  • 4. Motivation and Contribution The Model ResultsPersistence Autocorrelation function 1.0 0.8 0.6 0.4 0.2 VIX 0.0 RV logR −0.2 0 10 20 30 40 50 60 70 80 90 Lags, days
  • 5. Motivation and Contribution The Model ResultsThick Tails Min Max Mean Std Skewness Kurtosis logR -9.47 10.96 0.01 1.32 -0.25 7.98 VIX 9.89 80.86 21.69 8.83 2.09 7.40 RV 2.38 118.75 13.37 8.61 3.41 20.12
  • 6. Motivation and Contribution The Model ResultsContribution Two-component volatility (central tendency) Engle and Lee (1996), Andersen and Lund (1997), Balduzzi, Das, and Foresi (1998), Reschreiter (2010, 2011)
  • 7. Motivation and Contribution The Model ResultsContribution Two-component volatility (central tendency) Engle and Lee (1996), Andersen and Lund (1997), Balduzzi, Das, and Foresi (1998), Reschreiter (2010, 2011) Both volatility risks are priced Adrian and Rosenberg (2008), Todorov (2010)
  • 8. Motivation and Contribution The Model ResultsContribution Two-component volatility (central tendency) Engle and Lee (1996), Andersen and Lund (1997), Balduzzi, Das, and Foresi (1998), Reschreiter (2010, 2011) Both volatility risks are priced Adrian and Rosenberg (2008), Todorov (2010) Explicit expressions for innovations, moments, etc Bollerslev and Zhou (2002), Eraker (2009), Todorov (2010)
  • 9. Motivation and Contribution The Model ResultsContribution Two-component volatility (central tendency) Engle and Lee (1996), Andersen and Lund (1997), Balduzzi, Das, and Foresi (1998), Reschreiter (2010, 2011) Both volatility risks are priced Adrian and Rosenberg (2008), Todorov (2010) Explicit expressions for innovations, moments, etc Bollerslev and Zhou (2002), Eraker (2009), Todorov (2010) Joint estimation under P and Q Chernov and Ghysels (2000), Garcia, Lewis, Pastorello, and Renault (2011), Bollerslev, Gibson, and Zhou (2011)
  • 10. Motivation and Contribution The Model ResultsThe Model Historical: dpt = (r + µπ ) dt + σt dWtr dσt2 = κσ yt − σt2 dt + ησ σt dWtσ √ dyt = κy (µ − yt ) dt + ηy yt dWty pt - log price σt2 - stochastic volatility yt - central tendency
  • 11. Motivation and Contribution The Model ResultsThe Model Risk-neutral: dpt ˜ = r dt + σt d Wtr ˜ dσt2 = κσ yt − σt2 dt + ησ σt d Wtσ ˜ ˜ ˜ d yt ˜ µ ˜ = κy (˜ − yt ) dt + ηy ˜ yt d Wty ˜ ˜ pt - log price σt2 - stochastic volatility yt - central tendency
  • 12. Motivation and Contribution The Model ResultsThe Model Risk-neutral: dpt ˜ = r dt + σt d Wtr ˜ dσt2 = κσ yt − σt2 dt + ησ σt d Wtσ ˜ ˜ ˜ d yt ˜ µ ˜ = κy (˜ − yt ) dt + ηy ˜ yt d Wty ˜ ˜ κσ = κσ − λσ ησ , ˜ κy = κy − λy ηy ˜ pt - log price σt2 - stochastic volatility yt - central tendency
  • 13. Motivation and Contribution The Model ResultsData Objective measure: n 2 a.s. RVt,1 ≡ rt+ j−1 ,t+ j −→ Vt,1 n n j=1
  • 14. Motivation and Contribution The Model ResultsData Objective measure: n 2 a.s. RVt,1 ≡ rt+ j−1 ,t+ j −→ Vt,1 n n j=1 Risk-neutral measure: VIXt,22 = EtQ Vt,22
  • 15. Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt+h - VAR(1)-type
  • 16. Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt+h - VAR(1)-type 1 ´ t+h 2 1 ´ t+h Vt,h ≡ h t σs ds, Yt,h ≡ h t ys ds
  • 17. Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt+h - VAR(1)-type 1 ´ t+h 2 1 ´ t+h Vt,h ≡ h t σs ds, Yt,h ≡ h t ys ds Vt,h , Yt,h - VARMA(1,1)-type
  • 18. Motivation and Contribution The Model ResultsDiscretization 2 σt+h , yt+h - VAR(1)-type 1 ´ t+h 2 1 ´ t+h Vt,h ≡ h t σs ds, Yt,h ≡ h t ys ds Vt,h , Yt,h - VARMA(1,1)-type Vt,h - ARMA(2,2)-type
  • 19. Motivation and Contribution The Model ResultsMoment Conditions First moment: EtP 1 − Ay L × (1 − Aσ L) × Vt+2h,h = Const h h
  • 20. Motivation and Contribution The Model ResultsMoment Conditions First moment: EtP 1 − Ay L × (1 − Aσ L) × Vt+2h,h = Const h h EtP Vt+2h,h − ρ0 − ρ1 Vt+h,h − ρ2 Vt,h = 0
  • 21. Motivation and Contribution The Model ResultsMoment Conditions Second moment: (1 − γ1 L) ×    (1 − γ2 L) ×  EtP     (1 − γ3 L) ×  = Const   (1 − γ4 L) ×  (1 − γ5 L) 2 × Vt+5h,h
  • 22. Motivation and Contribution The Model ResultsPremia Volatility premium VPt,H = EtP Vt,H − EtQ Vt,H
  • 23. Motivation and Contribution The Model ResultsPremia Volatility premium VPt,H = EtP Vt,H − EtQ Vt,H Central tendency premium CPt,H = EtP Yt,H − EtQ Yt,H ˜
  • 24. Motivation and Contribution The Model ResultsPremia Volatility premium VPt,H = EtP Vt,H − EtQ Vt,H Central tendency premium CPt,H = EtP Yt,H − EtQ Yt,H ˜ Transient premium TPt,H = VPt,H − CPt,H
  • 25. Motivation and Contribution The Model ResultsParameter estimates µ 0.0046 (0.0005) κσ 0.8989 (0.0057) κy 0.0178 (0.0038) ησ 0.1041 (0.0225) ηy 0.0073 (0.0033) λσ 0.2013 (0.0786) λy 1.0929 (0.4835)
  • 26. Motivation and Contribution The Model ResultsParameter estimates µ 0.0046 (0.0005) κσ 0.8989 (0.0057) κy 0.0178 (0.0038) ησ 0.1041 (0.0225) ηy 0.0073 (0.0033) λσ 0.2013 (0.0786) λy 1.0929 (0.4835) µ - unconditional mean
  • 27. Motivation and Contribution The Model ResultsParameter estimates µ 0.0046 (0.0005) κσ 0.8989 (0.0057) κy 0.0178 (0.0038) ησ 0.1041 (0.0225) ηy 0.0073 (0.0033) λσ 0.2013 (0.0786) λy 1.0929 (0.4835) κ - speed of mean reversion
  • 28. Motivation and Contribution The Model ResultsParameter estimates µ 0.0046 (0.0005) κσ 0.8989 (0.0057) κy 0.0178 (0.0038) ησ 0.1041 (0.0225) ηy 0.0073 (0.0033) λσ 0.2013 (0.0786) λy 1.0929 (0.4835) η - instantaneous SD
  • 29. Motivation and Contribution The Model ResultsParameter estimates µ 0.0046 (0.0005) κσ 0.8989 (0.0057) κy 0.0178 (0.0038) ησ 0.1041 (0.0225) ηy 0.0073 (0.0033) λσ 0.2013 (0.0786) λy 1.0929 (0.4835) λ - price of a shock
  • 30. Motivation and Contribution The Model ResultsVolatility Premia 1 0 Mean premium, var units −1 −2 −3 VP CP TP −4 5 10 15 20 Forecast horizon, days
  • 31. Motivation and Contribution The Model ResultsConclusion Joint estimation of volatility model Long-term mean is changing Corresponding risk has a price Corresponding premium is large
  • 32. Thank you!
  • 33. Adrian, Tobias, and Joshua Rosenberg, 2008, Stock Returns and Volatility: Pricing the Short-Run and Long-Run Components of Market Risk, Journal of Finance 63, 2997–3030.Andersen, Torben G, and Jesper Lund, 1997, Stochastic Volatility and Mean Drift in the Short Rate Diffusion: Sources of Steepness, Level and Curvature in the Yield Curve, .Balduzzi, Pierluigi, Sanjiv Ranjan Das, and Silverio Foresi, 1998, The Central Tendency: A Second Factor in Bond Yields, Review of Economics and Statistics 80, 62–72.Bollerslev, Tim, Michael Gibson, and Hao Zhou, 2011, Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities, Journal of Econometrics 160, 235–245.Bollerslev, Tim, and Hao Zhou, 2002, Estimating stochastic volatility diffusion using conditional moments of integrated volatility, Journal of Econometrics 109, 33 – 65.Chernov, Mikhail, and Eric Ghysels, 2000, A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation,

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