際際滷

際際滷Share a Scribd company logo

Dynamic Jump Intensity Dynamic GARCH Volatility

Amit Sinha
Amit Sinha

This document summarizes a study modeling dynamic behavior of daily soybean prices using a GARCH-Jump model. The model was estimated using 20 years of CBOT soybean futures data and found evidence of jumps in both returns and volatility. Parameter estimates were provided and results showed the jump component accounted for around 16% of total variance. Applications of the model for option pricing, simulation, and risk analysis were discussed.

1 of 21
Download to read offline
Jumps in Soybean Prices Evidence and Applications Quant Team Ruchi Agri-Trading Singapore April 24, 2013
Overview Objective Introduction Model Description Data and Model Estimation Estimation Results Applications
Objective To study and model Dynamic behaviour of daily soybean prices by 鍖nding strong evidence for conditional volatility(GARCH) and conditional jump behaviour. To use modeling framework for simulations and Option pricing in a trading environment.
Introduction Q:What are volatility models? A:Models used to forecast and measure volatility.
Introduction Simplest Model : Equally weighted volatility rt is the excess return, 2 t = 1 N + 1 N j=0 r2 tj 1) all observations from t-N to t are given equal weight 2) all observations before t-N are given no weight 3) the choice of N is left to the trader.
Clustering in Financial Time Series
Introduction GARCH GENERALIZED - more general than ARCH model AUTOREGRESSIVE-depends on its own past CONDITIONAL-variance depends upon past information HETEROSKEDASTICITY- fancy word for non-constant variance rt = ht t GARCH(1, 1) where t N(0, ht) ht = + 硫ht1 + 留r2 t1 a constant variance yesterdays forecast yesterdays news
Introduction GARCH-JUMP model Q:Why incorporate Jumps in GARCH? A1:There is empirical evidence of jumps in both returns and volatility. A2:An innovation/news may arrive in a way which cannot be modelled completely within traditional GARCH framework
Introduction GARCH-JUMP model Q:How to incorporate Jump? A:Compound Poisson process Q:What does this mean? Jumps arrive randomly Size of jumps is also random : J(了, 慮, 隆2 ) where : 了 is jump intensity or expected number of jumps on a given day 慮 is the mean jump size 隆 is the variance of jump size
Model Description: DVDJ Model Daily Return Dynamics Rt+1 log St+1 St = r +(了z 1 2 )hz,t+1 +(了y 両)hy,t+1 +zt+1 +yt+1 Where St+1 denotes asset price at close of day t + 1 r denotes risk free rate zt+1 denotes normal component of daily shocks distributed as N(0, hz,t+1) yt+1 denotes jump component of daily shocks distributed by a compound Poisson process J(hy,t+1, 慮, 隆2) (了z 1 2) and (了y 両) are mathematical adjustments required for option pricing
Model Description: DVDJ Model Daily Variance Dynamics hz,t+1 = wz + bzhz,t + az hz,t (zt czhz,t)2 + dz(yt ez)2 Daily Jump Intensity hy,t+1 = wy + by hy,t + ay hz,t (zt cy hz,t)2 + dy (yt ez)2 Total variance of Rt+1 is given by: Variance(Rt+1) = hz,t+1 + (隆2 + 慮2 )hy,t+1
Data and Model Estimation We estimate our model using CBOT Soybean November futures for last 20 years(1993-2012) We cut o鍖 each series 20 trading days before expiry Each future series contributes 1 year daily prices Model requires estimation of 11 parameters: Parameters of the GARCH [了z , 了y , wz , b, a, c, d, e] Parameters of the jump [wy ,慮, 隆] Model is estimated using optimization of standard maximum likelihood
Estimation and Results Table 1 : DVDJ Model- GARCH Parameters 了z 了y wz b a c d e 1.9707 -0.0046 -5.6069e-06 0.9780 8.6808e-06 -11.333 0.0670 -0.0012 Table 2 : DVDJ Model -Jump Parameters wy 慮 隆 0.0909 -0.0022 0.0218 Table 3 : LogLikihood(lower is better) GARCH(1,1) DVDJ Model -14201.66 -15217.42 Table 4 : Vol properties AVG. Annual Vol-GARCH(1,1) AVG. Annual-Vol DVDJ Model Normal Comp of Vol Jump Comp of Vol 20.67 % 21.1 % 84.03% 15.97%
Estimation and Results SX 12: DJI Model Vol
Estimation and Results SX 12: DJI vs GARCH(1,1) Vol Comparison
Estimation and Results SX 12: Expected number of jumps
Estimation and Results Contribution of Jump Component to Returns
Estimation and Results 2008 vs. 2012
Application Option Pricing
Application Option Pricing
Application Simulation Framework To ex-Ante predict impact of information based jump on Volatility A full probability model to incorporate known information to provide more accurate con鍖dence intervals VaR Calculation Stress and Scenario Testing

More Related Content

Dynamic Jump Intensity Dynamic GARCH Volatility

  • 1. Jumps in Soybean Prices Evidence and Applications Quant Team Ruchi Agri-Trading Singapore April 24, 2013
  • 2. Overview Objective Introduction Model Description Data and Model Estimation Estimation Results Applications
  • 3. Objective To study and model Dynamic behaviour of daily soybean prices by 鍖nding strong evidence for conditional volatility(GARCH) and conditional jump behaviour. To use modeling framework for simulations and Option pricing in a trading environment.
  • 4. Introduction Q:What are volatility models? A:Models used to forecast and measure volatility.
  • 5. Introduction Simplest Model : Equally weighted volatility rt is the excess return, 2 t = 1 N + 1 N j=0 r2 tj 1) all observations from t-N to t are given equal weight 2) all observations before t-N are given no weight 3) the choice of N is left to the trader.
  • 7. Introduction GARCH GENERALIZED - more general than ARCH model AUTOREGRESSIVE-depends on its own past CONDITIONAL-variance depends upon past information HETEROSKEDASTICITY- fancy word for non-constant variance rt = ht t GARCH(1, 1) where t N(0, ht) ht = + 硫ht1 + 留r2 t1 a constant variance yesterdays forecast yesterdays news
  • 8. Introduction GARCH-JUMP model Q:Why incorporate Jumps in GARCH? A1:There is empirical evidence of jumps in both returns and volatility. A2:An innovation/news may arrive in a way which cannot be modelled completely within traditional GARCH framework
  • 9. Introduction GARCH-JUMP model Q:How to incorporate Jump? A:Compound Poisson process Q:What does this mean? Jumps arrive randomly Size of jumps is also random : J(了, 慮, 隆2 ) where : 了 is jump intensity or expected number of jumps on a given day 慮 is the mean jump size 隆 is the variance of jump size
  • 10. Model Description: DVDJ Model Daily Return Dynamics Rt+1 log St+1 St = r +(了z 1 2 )hz,t+1 +(了y 両)hy,t+1 +zt+1 +yt+1 Where St+1 denotes asset price at close of day t + 1 r denotes risk free rate zt+1 denotes normal component of daily shocks distributed as N(0, hz,t+1) yt+1 denotes jump component of daily shocks distributed by a compound Poisson process J(hy,t+1, 慮, 隆2) (了z 1 2) and (了y 両) are mathematical adjustments required for option pricing
  • 11. Model Description: DVDJ Model Daily Variance Dynamics hz,t+1 = wz + bzhz,t + az hz,t (zt czhz,t)2 + dz(yt ez)2 Daily Jump Intensity hy,t+1 = wy + by hy,t + ay hz,t (zt cy hz,t)2 + dy (yt ez)2 Total variance of Rt+1 is given by: Variance(Rt+1) = hz,t+1 + (隆2 + 慮2 )hy,t+1
  • 12. Data and Model Estimation We estimate our model using CBOT Soybean November futures for last 20 years(1993-2012) We cut o鍖 each series 20 trading days before expiry Each future series contributes 1 year daily prices Model requires estimation of 11 parameters: Parameters of the GARCH [了z , 了y , wz , b, a, c, d, e] Parameters of the jump [wy ,慮, 隆] Model is estimated using optimization of standard maximum likelihood
  • 13. Estimation and Results Table 1 : DVDJ Model- GARCH Parameters 了z 了y wz b a c d e 1.9707 -0.0046 -5.6069e-06 0.9780 8.6808e-06 -11.333 0.0670 -0.0012 Table 2 : DVDJ Model -Jump Parameters wy 慮 隆 0.0909 -0.0022 0.0218 Table 3 : LogLikihood(lower is better) GARCH(1,1) DVDJ Model -14201.66 -15217.42 Table 4 : Vol properties AVG. Annual Vol-GARCH(1,1) AVG. Annual-Vol DVDJ Model Normal Comp of Vol Jump Comp of Vol 20.67 % 21.1 % 84.03% 15.97%
  • 14. Estimation and Results SX 12: DJI Model Vol
  • 15. Estimation and Results SX 12: DJI vs GARCH(1,1) Vol Comparison
  • 16. Estimation and Results SX 12: Expected number of jumps
  • 17. Estimation and Results Contribution of Jump Component to Returns
  • 21. Application Simulation Framework To ex-Ante predict impact of information based jump on Volatility A full probability model to incorporate known information to provide more accurate con鍖dence intervals VaR Calculation Stress and Scenario Testing