dynamic jump intensity dynamic garch volatility
TRANSCRIPT
Jumps in Soybean PricesEvidence and Applications
Quant Team
Ruchi Agri-TradingSingapore
April 24, 2013
Overview
I Objective
I Introduction
I Model Description
I Data and Model Estimation
I Estimation Results
I Applications
Objective
I To study and model Dynamic behaviour of daily soybeanprices by finding strong evidence for conditionalvolatility(GARCH) and conditional jump behaviour.
I To use modeling framework for simulations and Option pricingin 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,
σ2t =1
N + 1
N∑j=0
r2t−j
1) all observations from t-N to t are given equal weight2) all observations before t-N are given no weight3) the choice of N is left to the trader.
Clustering in Financial Time Series
Introduction
GARCH
I GENERALIZED - more general than ARCH model
I AUTOREGRESSIVE-depends on its own past
I CONDITIONAL-variance depends upon past information
I HETEROSKEDASTICITY- fancy word for non-constantvariance
rt =√
htεt
GARCH(1, 1) where εt N(0, ht)
ht = ω + βht−1 + αr2t−1
I a constant variance
I yesterday’s forecast
I yesterday’s news
Introduction
GARCH-JUMP modelQ:Why incorporate Jumps in GARCH?A1:There is empirical evidence of jumps in both returns andvolatility.A2:An innovation/news may arrive in a way which cannot bemodelled completely within traditional GARCH framework
Introduction
GARCH-JUMP modelQ:How to incorporate Jump?A:Compound Poisson process
Q:What does this mean?
I Jumps arrive randomly
I Size of jumps is also random :
J(λ, θ, δ2)
where :
I λ is jump intensity or expected number of jumps on a givenday
I θ is the mean jump size
I δ is the variance of jump size
Model Description: DVDJ Model
Daily Return Dynamics
Rt+1 ≡ logSt+1
St= r +(λz−
1
2)hz,t+1+(λy −ξ)hy ,t+1+zt+1+yt+1
Where
I St+1 denotes asset price at close of day t + 1
I r denotes risk free rate
I zt+1 denotes normal component of daily shocks distributed asN(0, hz,t+1)
I yt+1 denotes jump component of daily shocks distributed by acompound Poisson process J(hy ,t+1, θ, δ
2)
I (λz − 12) and (λy − ξ) are ”mathematical adjustments”
required for option pricing
Model Description: DVDJ Model
Daily Variance Dynamics
hz,t+1 = wz + bzhz,t +azhz,t
(zt − czhz,t)2 + dz(yt − ez)2
Daily Jump Intensity
hy ,t+1 = wy + byhy ,t +ayhz,t
(zt − cyhz,t)2 + dy (yt − ez)2
Total variance of Rt+1 is given by:
Variance(Rt+1) = hz,t+1 + (δ2 + θ2)hy ,t+1
Data and Model Estimation
I We estimate our model using CBOT Soybean Novemberfutures for last 20 years(1993-2012)
I We cut off each series 20 trading days before expiry
I Each future series contributes 1 year daily pricesI Model requires estimation of 11 parameters:
I Parameters of the GARCH [λz , λy , wz , b, a, c, d, e]I Parameters of the jump [wy ,θ, δ]I Model is estimated using optimization of standard maximum
likelihood
Estimation and Results
Table 1 : DVDJ Model- GARCH Parameters
λz λy 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 θ δ
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
I To ex-Ante predict impact of information based jump onVolatility
I A full probability model to incorporate known information toprovide more accurate confidence intervals
I VaR Calculation
I Stress and Scenario Testing