3. Objective
To study and model Dynamic behaviour of daily soybean
prices by finding strong evidence for conditional
volatility(GARCH) and conditional jump behaviour.
To use modeling framework for simulations and Option pricing
in a trading environment.
5. Introduction
Simplest Model : Equally weighted volatility
rt is the excess return,
σ2
t =
1
N + 1
N
j=0
r2
t−j
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 = ω + βht−1 + αr2
t−1
a constant variance
yesterday’s forecast
yesterday’s 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(λ, θ, δ2
)
where :
λ is jump intensity or expected number of jumps on a given
day
θ is the mean jump size
δ is the variance of jump size
10. Model Description: DVDJ Model
Daily Return Dynamics
Rt+1 ≡ log
St+1
St
= r +(λz −
1
2
)hz,t+1 +(λy −ξ)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, θ, δ2)
(λz − 1
2) and (λy − ξ) 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 + (δ2
+ θ2
)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 off 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 [λz , λy , wz , b, a, c, d, e]
Parameters of the jump [wy ,θ, δ]
Model is estimated using optimization of standard maximum
likelihood
13. 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%
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 confidence intervals
VaR Calculation
Stress and Scenario Testing
