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Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options

Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options. Peter Christoffersen, Kris Jacobs and Chayawat Ornthanalai McGill University . FDIC Risk Management Conference April 12 th 2008. 1/18. Objectives . To understand how should jump dynamics be specified.

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Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options

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  1. Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options Peter Christoffersen, Kris Jacobs and Chayawat Ornthanalai McGill University FDIC Risk Management Conference April 12th 2008

  2. 1/18 Objectives • To understand how should jump dynamics be specified. • Should jump intensity be time-varying? • How should the jump and normal components be specified jointly in the returns of S&P500 index? • To investigate the implication of different jump specifications on option pricing • The implication of our results on the existing estimates of the continuous-time models that are based on the jump-diffusion framework

  3. 1/18 Objectives • To understand how should jump dynamics be specified. • Should jump intensity be time-varying? • How should the jump and normal components be specified jointly in the returns of S&P500 index? • To investigate the implication of different jump specifications on option pricing • The implication of our results on the existing estimates of the continuous-time models that are based on the jump-diffusion framework

  4. 1/18 Objectives • To understand how should jump dynamics be specified. • Should jump intensity be time-varying? • How should the jump and normal components be specified jointly in the returns of S&P500 index? • To investigate the implication of different jump specifications on option pricing • The implication of our results on the existing estimates of the continuous-time models that are based on the jump-diffusion framework

  5. The Compound Poisson (CP) process is distributed as For the Merton jump: The number of jumps that arrive over a finite interval is: 2/18 The structure of Merton jumps

  6. 3/18 Background/Motivations (1) • Intuitions behind the time-varying jump intensity? • Jumps clustering: Maheu and Mccurdy (2004) • The likelihood of jumps (Fear of crash): Bates (1991) • Evidence for time-varying (TV) jump intensities in the returns dynamics are mixed. • Andersen, Benzoni, and Lund (ABL, 2002) - Find no time series evidence for the TV jump intensity. • Bates (2006) – Uses his AML method and finds evidence for the TV jump intensity. • In the mix of opinions regarding the TV jump intensity • Recent studies have become more supportive of the model with jumps in the volatility: SVCJ, Eraker, Johannes, and Polson (EJP 2003) • Li, Wells and Yu (2006), Broadie, Chernov, and Johannes (BCJ 2007,2008), and Johannes, Polson, and Stroud (2008)

  7. 3/18 Background/Motivations (1) • Intuitions behind the time-varying jump intensity? • Jumps clustering: Maheu and Mccurdy (2004) • The likelihood of jumps (Fear of crash): Bates (1991) • Evidence for time-varying (TV) jump intensities in the returns dynamics are mixed. • Andersen, Benzoni, and Lund (ABL, 2002) - Find no time series evidence for the TV jump intensity. • Bates (2006) – Uses his AML method and finds evidence for the TV jump intensity. • In the mix of opinions regarding the TV jump intensity • Recent studies have become more supportive of the model with jumps in the volatility: SVCJ, Eraker, Johannes, and Polson (EJP 2003) • Li, Wells and Yu (2006), Broadie, Chernov, and Johannes (BCJ 2007,2008), and Johannes, Polson, and Stroud (2008)

  8. 3/18 Background/Motivations (1) • Intuitions behind the time-varying jump intensity? • Jumps clustering: Maheu and Mccurdy (2004) • The likelihood of jumps (Fear of crash): Bates (1991) • Evidence for time-varying (TV) jump intensities in the returns dynamics are mixed. • Andersen, Benzoni, and Lund (ABL, 2002) - Find no time series evidence for the TV jump intensity. • Bates (2006) – Uses his AML method and finds evidence for the TV jump intensity. • In the mix of opinions regarding the TV jump intensity • Recent studies have become more supportive of the model with jumps in the volatility: SVCJ, Eraker, Johannes, and Polson (EJP 2003) • Li, Wells and Yu (2006), Broadie, Chernov, and Johannes (BCJ 2007,2008), and Johannes, Polson, and Stroud (2008)

  9. 4/18 Summary of selected jump-diffusion models in the literature Notes on the abbreviations: Bakshi,Cao and Chen (BCC,1997), Andersen, Bezoni, and Lund (ABL, 2002), Chernov, Gallant, Ghysels, and Tauchen (CGGT, 2003), Eraker, Johannes, and Polson (EJP, 2002), Broadie, Chernov, and Johannes (BCJ,2007), and Duan, Ritchken, and Sun (DRS,2006).

  10. 4/18 Summary of selected jump-diffusion models in the literature Notes on the abbreviations: Bakshi,Cao and Chen (BCC,1997), Andersen, Bezoni, and Lund (ABL, 2002), Chernov, Gallant, Ghysels, and Tauchen (CGGT, 2003), Eraker, Johannes, and Polson (EJP, 2002), Broadie, Chernov, and Johannes (BCJ,2007), and Duan, Ritchken, and Sun (DRS,2006).

  11. 5/18 Why do we use a discrete-time framework? It is far from simple to estimate continuous-time models that are based on the jump-diffusion framework. Challenge: the presence of latent factors. • Estimation of complex jump models are often based on option prices. This is dangerous when the latent factors are treated as part of the structural parameters in the estimation. • Time-series estimates of the SVCJ from returns rely on discrete-time approximation. EJP, Eraker (2004), and Li, Wells, and Yu (2007) approximate CP using the Bernoulli jump process.

  12. 5/18 Why do we use a discrete-time framework? It is far from simple to estimate continuous-time models that are based on the jump-diffusion framework. Challenge: the presence of latent factors. • Estimation of complex jump models are often based on option prices. This is dangerous when the latent factors are treated as part of the structural parameters in the estimation. • Time-series estimates of the SVCJ from returns rely on discrete-time approximation. EJP, Eraker (2004), and Li, Wells, and Yu (2007) approximate CP using the Bernoulli jump process.

  13. 5/18 Why do we use a discrete-time framework? It is far from simple to estimate continuous-time models that are based on the jump-diffusion framework. Challenge: the presence of latent factors. • Estimation of complex jump models are often based on option prices. This is dangerous when the latent factors are treated as part of the structural parameters in the estimation. • Time-series estimates of the SVCJ from returns rely on discrete-time approximation. EJP, Eraker (2004), and Li, Wells, and Yu (2007) approximate CP using the Bernoulli jump process.

  14. 6/18 Main Findings • Yes! It is important to have time-varying jump intensities in modeling the returns dynamic. • Option pricing results favour the model with jump intensity that is linear with the variance of the normally distributed return component. • The estimates from our model support the presence of multiple jumps per day. Hence, parameters that are estimated from the studies that use Bernoulli distribution to approximate the CP process are likely to be biased. • In order to produce significant improvement in option valuation, jump models must allow for sizable magnitude of jump risk premia.

  15. 6/18 Main Findings • Yes! It is important to have time-varying jump intensities in modeling the returns dynamic. • Option pricing results favour the model with jump intensity that is linear with the variance of the normally distributed return component. • The estimates from our model support the presence of multiple jumps per day. Hence, parameters that are estimated from the studies that use Bernoulli distribution to approximate the CP process are likely to be biased. • In order to produce significant improvement in option valuation, jump models must allow for sizable magnitude of jump risk premia.

  16. 6/18 Main Findings • Yes! It is important to have time-varying jump intensities in modeling the returns dynamic. • Option pricing results favour the model with jump intensity that is linear with the variance of the normally distributed return component. • The estimates from our model support the presence of multiple jumps per day. Hence, parameters that are estimated from the studies that use Bernoulli distribution to approximate the CP process are likely to be biased. • In order to produce significant improvement in option valuation, jump models must allow for sizable magnitude of jump risk premia.

  17. 6/18 Main Findings • Yes! It is important to have time-varying jump intensities in modeling the returns dynamic. • Option pricing results favour the model with jump intensity that is linear with the variance of the normally distributed return component. • The estimates from our model support the presence of multiple jumps per day. Hence, parameters that are estimated from the studies that use Bernoulli distribution to approximate the CP process are likely to be biased. • In order to produce significant improvement in option valuation, jump models must allow for sizable magnitude of jump risk premia.

  18. 7/18 The general setup: physical measure The conditional equity premium is

  19. 8/18 Various nested specifications

  20. 8/18 Various nested specifications

  21. 8/18 Various nested specifications

  22. 8/18 Various nested specifications

  23. 8/18 Various nested specifications

  24. 9/18 MLE results: overview J-GARCH models

  25. 9/18 MLE results: overview J-GARCH models

  26. 9/18 MLE results: overview J-GARCH models

  27. 10/18 MLE results: conditional jump intensity

  28. 11/18 Option Pricing Results: At a fixed level of the equity premium • We price call options on each Wednesday from 1996-2005 • We set the long-run equity premium to be 6% across all models.

  29. 11/18 Option Pricing Results: At a fixed level of the equity premium • We price call options on each Wednesday from 1996-2005 • We set the long-run equity premium to be 6% across all models.

  30. 12/18 Option Pricing Results: Changing the level of the equity premium

  31. 12/18 Option Pricing Results: Changing the level of the equity premium

  32. 13/18 Option Pricing Results: The impact of changes in the equity premium level on the IV smirk

  33. 14/18 Further Analysis : Decomposition of daily returns by particle filter (1/2)

  34. 15/18 Further Analysis : Decomposition of daily returns by particle filter (2/2)

  35. 16/18 Further Analysis : The impact of using Bernoulli approximation Result: no time-series evidence for time-varying jump intensities

  36. 17/18 Further Analysis : What is the bias from applying Bernoulli approximation to J-GARCH(3)? • With Bernoulli approximation, jumps are perceived as extremely large and rare events.

  37. 18/18 Conclusions • Models that are based on the finite-activity Merton Jump should allow for time-varying jump intensity. • Jumps arrive in clusters, and their arrival rate is dependent on the level of the market variance. • Option pricing models require the presence of jump risk premium for explaining the smirk patterns in the implied volatilities at the reasonable equity premium levels. • It is not acceptable to approximate a Compound Poisson process using a Bernoulli distribution. Studies that assume this will produce biases in their estimates.

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