dynamic jump intensity dynamic garch volatility

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