UNIVERSITY OF CALGARY
Risk neutral measures and GARCH model calibration
by
Sheng Li
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER of SCIENCE
DEPARTMENT OF MATHEMATICS and STATISTICS
CALGARY, ALBERTA
September, 2012
c Sheng Li 2012
UNIVERSITY OF CALGARY
FACULTY OF GRADUATE STUDIES
The undersigned certify that they have read, and recommend to the Faculty of Graduate
Studies for acceptance, a thesis entitled “Risk neutral measures and GARCH model calibration” submitted by Sheng Li in partial fulfillment of the requirements for the degree of
MASTER of SCIENCE.
Supervisor, Dr. Alexandru Badescu
Department of Mathematics and Statistics
Dr. Anatoliy Swishchuk
Department of Mathematics and Statistics
Dr. Jean-Francois Wen
Department of Economics
Date
Abstract
Empirical studies have shown that GARCH models can be successfully used to describe
option prices. Pricing such option contracts requires the risk neutral return dynamics of
underlying asset. Since under the GARCH framework the market is incomplete, there is
more than one risk neutral measure. In this thesis, we study the locally risk neutral valuation
relationship, the mean correcting martingale measure, the conditional Esscher transform and
the second order Esscher transform as martingale measure candidates. All these methods lead
to the respective risk neutral return dynamics. We empirically examine in-sample and out-ofsample performance of Gaussian-TGARCH and Normal inverse Gaussian (NIG)-TGARCH
models under these risk neutral measures.
ii
Acknowledgements
I would like to express my gratitude to all those who helped me during the writing of this
thesis.
My deepest gratitude goes first and foremost to Dr. Alexandru Badescu, my supervisor,
for his constant encouragement and guidance. He has walked me through all the stages of
the writing of this thesis. Without his consistent and illuminating instruction, this thesis
could not have reached its present form.
Second, I would like to express my heartfelt gratitude to my friends and my fellow
classmates: Kaijie Cui, Guoqiang Chen, Zheng Yuan, Lifeng Zhang and Shan Zhu. They
gave me their help and time in listening to me and helping me work out my problems during
the difficult course of the thesis. I am also greatly indebted to the professors who taught me
in the past two years at the Department of Mathematics and Statistics.
Last my thanks would go to my beloved family for their loving considerations and great
confidence in me all through these years.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Black-Scholes and GARCH Model . . . . . . . . . . . . . . . . . . . . . . .
2.1 Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 GARCH-in-Mean Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 GARCH Model Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Risk Neutral Measures for GARCH Model . . . . . . . . . . . . . . . . . . .
3.1 Duan’s locally risk neutral valuation relationship . . . . . . . . . . . . . . . .
3.2 Mean Correcting Martingale Measure . . . . . . . . . . . . . . . . . . . . . .
3.3 Conditional Esscher Transform Method . . . . . . . . . . . . . . . . . . . . .
3.4 A discrete version of the Girsanov change of measure . . . . . . . . . . . . .
3.5 Second order Esscher Transform Method . . . . . . . . . . . . . . . . . . . .
3.6 Variance Dependent Pricing Kernel . . . . . . . . . . . . . . . . . . . . . . .
4
Generalized Hyperbolic GARCH Model . . . . . . . . . . . . . . . . . . . . .
4.1 Risk Neutral Dynamic from MCMM under GH-GARCH . . . . . . . . . . .
4.2 Risk Neutral Dynamic from Conditional Esshcer under GH-GARCH . . . . .
4.3 Special Case of Generalized Hyperbolic distribution . . . . . . . . . . . . . .
4.3.1 Risk Neutral Dynamic from MCMM under NIG-GARCH . . . . . . .
4.3.2 Risk Neutral Dynamic from Conditional Esshcer under NIG-GARCH
5
Empirical Analysis for GARCH Model . . . . . . . . . . . . . . . . . . . . .
5.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Simulation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Empirical Martingale Simulation . . . . . . . . . . . . . . . . . . . .
5.2.3 Control Variates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A
First Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
ii
iii
iv
v
vi
1
7
7
10
11
14
15
16
18
22
26
30
35
37
38
41
44
44
46
46
48
49
50
51
52
69
70
List of Tables
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
Number of Call option contracts in 2004 . . . . . . . . . . . . . . . . . . . .
Average Call option prices in 2004 . . . . . . . . . . . . . . . . . . . . . . . .
MLE results for Gaussian-TGARCH and NIG-TGARCH . . . . . . . . . . .
In-sample estimation performance for four competing models based on RMSE
In-sample estimation performance for four competing models based on %RMSE
MLE results for Gaussian innovation . . . . . . . . . . . . . . . . . . . . . .
In-sample estimation of the Guassian-TGARCH . . . . . . . . . . . . . . . .
In-sample estimation of the NIG-TGARCH with MCMM . . . . . . . . . . .
In-sample estimation of the NIG-TGARCH with Esscher transform . . . . .
Out-of-sample RMSE for the three competing models . . . . . . . . . . . . .
Out-of-sample RMSE using MLE for the three competing models . . . . . .
v
48
48
54
55
55
56
58
59
60
64
68
List of Figures and Illustrations
5.1
5.2
5.3
5.4
5.5
5.6
Boxplot of in-sample RMSE for the three competing models . . . . . . . . .
Black-Scholes implied volatilities for the three competing models . . . . . . .
Boxplot of out-of-sample RMSE for the three competing models . . . . . . .
Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for
the Gaussian-TGARCH model . . . . . . . . . . . . . . . . . . . . . . . . . .
Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for
the NIG-TGARCH with MCMM model . . . . . . . . . . . . . . . . . . . . .
Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for
the NIG-TGARCH with Esscher model . . . . . . . . . . . . . . . . . . . . .
vi
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63
65
66
67
67
Chapter 1
Introduction
Option valuation has been one of the major areas of interest in the financial literature.
Generally, the price of an option is determined by the value of the underlying asset, the
strike price, the volatility of this asset, the risk-free interest rate, time to maturity and
the dividends paid. Black and Scholes (1973) and Merton (1973) developed a closed form
solution for European option prices. This formula is known as the Black-Scholes option
pricing formula which assumes the underlying asset price follows a geometric Brownian
motion with a constant volatility, a perfect market and no arbitrage opportunities. It is
widely used by market practitioners for pricing, hedging and risk management of options.
However, many empirical studies on asset price dynamics have shown evidence against
the constant volatility assumption in the Black and Scholes (1973) and Merton (1973) option
pricing models. These studies have shown that the implied volatility displays smile or skew
effects. Various extensions which can accommodate the time variation of volatility have been
proposed. Therefore, many studies proposed models which incorporate a stochastic volatility. There are two types of volatility models: continuous-time stochastic volatility models
and discrete-time Autoregressive Conditional Heteroskedaticity (ARCH) or Generalized Autoregressive Conditional Heteroskedaticity (GARCH) models.
Stochastic volatility models were first introduced by Hull and White (1987). The contribution in the stochastic volatility models includes the works of Wiggings (1987), Scott
(1987), Stein and Stein (1991), Heston (1993). Even if the stochastic volatility models
present advantages in constructing closed form solutions for European option prices, it is
difficult to implement and test them. Although these models assume that volatility is observable, it is impossible to exactly filter a volatility variable from discrete observations of
1
spot asset prices in a continuous-time stochastic volatility model. As a result, it is not possible to compute out-of-sample option valuation errors from history of asset returns. For the
majority of stochastic volatility models, the numerical methods are highly computationally
intensive. Thus, most of the option valuation are based on discrete-time GARCH option
pricing models.
ARCH models were introduced by Engle (1982). Bollerslev (1986) extended the ARCH
model to generalized ARCH (GARCH) models. The advantage of GARCH models is that
they can capture the stylized facts of financial time series. In the GARCH models, option prices are evaluated as discounted expected value of the payoff fucntion under a martingale measure. Many studies compute the option prices using Monte Carlo simulation
technique. However, Heston and Nandi (2000) derived a semi-analytical pricing formula for
European options under a specific form of GARCH model. This model is known as Heston and Nandi GARCH model. In that paper, they examined in-sample and out-of-sample
performance for European options and the conclusion was that their model clearly outperforms the homoskedastic models in explaining option prices data. Christoffersen and Jacobs
(2004) computed option prices using Monte Carlo simulation and compared their in-sample
and out-of-sample performance. They argued that a simple leverage effect in the conditional
variance process outperforms most of the extensions considered in the GARCH option pricing
literature.
Depending on the specific forms of the volatility term, we can have different types of
GARCH models. We can have the general GARCH model proposed by Bollerslev (1986) or
the asymmetric GARCH models. The asymmetric GARCH models includes the Exponential GARCH (EGARCH) of Nelson (1991), the threshold GARCH (TGARCH) of Zakoian
(1994) and GJR-GARCH of Glosten et. al (1993). All these GARCH models can capture
asymmetric volatility. This more flexible volatility can respond to the positive and negative
shocks. Moreover, Awartani and Corradi (2005) provided supportive evidence that GARCH
2
models allowed for asymmetries in volatility produce more accurate volatility predictions.
In this thesis, we will focus on the TGARCH model for the simulation results.
Although GARCH models with Gaussian innovations can depict the typical characteristics of the financial data, these models can not capture the skewness and leptokurtosis of
financial data. The negative skewness and excess leptokurtosis are called the conditional
skewness and conditional leptokuritosis respectively. Many studies extended the Gaussian
innovation framework to a non-Gaussian innovation one. For example, shifted Gamma innovation (Siu et al., 2004), Inverse Gaussian innovation (Christoffersen et al., 2006), Generalized Error innovation (Duan, 1999), α-stable innovation (Menn and Rachev, 2005), Normal
Inverse Gaussian innovation(Stentoft, 2008), Hyperbolic Distribution innovation (Badescu,
et al., 2011), mixture of Normal innovation (Badescu, et al., 2008), and Poisson-normal innovations (Duan, 2006) have been all successfully implemented. The empirical studies show
that these innovation distributions can capture the excess kurtosis of the financial data and
have fatter tails than the Gaussian innovation distribution. Therefore, these innovation distributions may have a significant improvement over the Gaussian innovation. In this thesis,
along with Gaussian noise we will consider the Normal Inverse Gaussian distribution as the
innovation distribution.
Another important issue in the GARCH option pricing model is the equivalent martingale measure considered for pricing purposes. Since the market is incomplete in the GARCH
setup, there is an infinite number of risk neutral measures under which one can price derivatives. Therefore, this leads to more than one possible fair prices, all of which are consistent
with the absence of arbitrage opportunities. Usually, we choose the appropriate price kernel
based on analytical tractability or mathematical convenience. The traditional method for
derivative pricing in the GARCH setup is the Risk Neutral Valuation Relationship (RNVR).
The RNVR was introduced by Rubinstein (1976) and Brennan (1979) for discrete time models with normally distributed asset returns, Duan (1995) introduced the local Risk Neutral
3
Valuation Relationship (LRNVR) and used it to compute European option prices under
the assumption of the Gaussian innovation. Moreover, Duan (1995) considered well-known
constraints on investor preferences, along with non-affine type of GARCH-in-mean models.
The disadvantage of this method is that it does not work when the innovation distribution
is non-Gaussian. Thus other equivalent martingale measures have been developed. For example, Gerber and Shiu (1994) introduced conditional Esscher transform method which can
accommodate for non-Gaussian innovation. This method is widely used for option pricing.
Siu et al (2004) studied the conditional Esscher transform and showed that LRNVR is a special case. Furthermore, Monfort and Pegoraro (2011) made a improvement for conditional
Esscher transform method. They argued that the price kernel function is of quadratic form
instead of linear form. This method is called the second-order Esscher transform method and
it is used to explain the variance risk premium. Christoffersen, Heston and Jacobs (2012)
proposed the variance dependent pricing kernel and found similar results as Monfort and
Pegoraro (2011).
Maximum likelihood estimation (MLE) method is a popular tool for estimating the
GARCH model parameters. Usually, we estimate the GARCH model parameters based on
the historical asset returns by maximizing the specific likelihood function. These GARCH
model parameters are obtained under the historical probability measure. MLE method is typically straightforward and computationally easy. However, an alternative estimation method
can be used to estimate GARCH model parameters directly using the option prices. For instance, one can use this approach by minimizing an objective function to obtain the GARCH
estimates. We shall also use option prices directly rather than using the MLE method for
the GARCH model estimations, since we expect this approach to perform better than the
MLE method for out-of-sample purposes. Christoffersen and Jacobs (2004) proposed the
two approaches for GARCH option valuation with Gaussian innovations. They concluded
that GARCH model parameter estimation using option prices directly works better than the
4
MLE method based on in-sample parameter estimates. In this thesis, we will compare the
out-of-sample performance for the two approaches. We also need to restrict the GARCH
model parameters to satisfy the stationary property.
Monte Carlo simulation technique is widely used for GARCH option pricing. It is very
useful when there is no closed-form solution. Duan (1995) first used this technique to compute option price under normal GARCH models. However, Monte Carlo simulation tends to
be numerically intensive if a high degree of accracy is desired. Duan and Simonato (1998)
developed the Empirical Martingale Simulation (EMS) method to solve this problem. They
showed that the EMS yields substantial variance reduction particularly in- and at-the-money
or longer-maturity options. Since this method can be incorporated easily with a GARCH
framework, it is a popular tool for pricing the GARCH option price. Christoffersen and
Jacobs (2004) also computed the option prices using this method.
In this thesis, we describe the risk neutral measures for our GARCH setup. We investigate the in-sample performance and the out-of-sample performance for Gaussian and
Normal Inverse Gaussian (NIG) innovations for different risk neutral measures. First, we
briefly describe five risk neutral measures in a general GARCH framework: Duan’s local risk
neutral valuation relationship (LRNVR), the mean correcting martingale measure (MCMM),
the conditional Esscher transform (Ess), the second order Esscher transform (2nd-Ess) and
variance dependent pricing kernel. We try to derive the relationship between the risk neutral measures and historical measure. Second, we illustrate the TGARCH model based
on Gaussian innovation and derive the risk neutral dynmaic under LRNVR, MCMM, Ess
and 2nd-Ess. We also derive the risk neutral dynamic for NIG-TGARCH under MCMM and
Ess. We use maximum likelihood technique (MLE) to estimate the parameters for GaussianTGARCH and NIG-TGARCH based on historical returns. Then we examine the in-sample
and out-of-sample performance of Gaussian-TGARCH and NIG-TGARCH using these martingale measures. More specifically, we examine the in-sample performance for the Gaussian-
5
TGARCH model, the NIG-TGARCH model with MCMM, the NIG-TGARCH model with
Ess and the Gaussian-TGARCH model with 2nd-Ess for the first data set (April 18th, 2002).
We also examine the in-sample and out-of-sample performance for the Gaussian-TGARCH
model, the NIG-TGARCH model with MCMM and the NIG-TGARCH model with Ess for
the second data set (Jan 7th, 2004 to Dec 29th, 2004). Moreover, we compare the performance between risk neutral estimators and MLE for the out-of-sample exercise.
The remainder of this thesis is organized as follows. In chapter 2 we introduce the BlackScholes and GARCH model. In Chapter 3 we introduce the five risk neutral measures for
GARCH models and derive the risk neutral return dynamics for Gaussian innovations. In
Chapter 4 we derive the risk neutral return dynamic for NIG innovation using the risk neutral
measures in chapter 3. Chapter 5 describes the parameter estimation results and in- and
out-of-sample performance.
6
Chapter 2
Black-Scholes and GARCH Model
2.1 Black-Scholes model
The most famous option pricing formula is the Black-Scholes formula which assumes the
normal distribution of log returns and a constant volatility. We consider a stochastic process
St that follows a Geometric Brownian Motion process which satisfy the following stochastic
differential equation (SDE):
dSt = µSt dt + σSt dWt .
(2.1)
Here µ is the drift, σ is the volatility and Wt is a standard Brownian motion. µ and σ are
assumed to be constant. The solution of (2.1) is provided below.
1
St = S0 exp((µ − σ 2 )t + σWt ).
2
(2.2)
Proof. From SDE (2.1), we can get
dSt
= µdt + σdWt .
St
Using the following Itô-Doeblin formula for an Itô’s process in differential notation:
1
df (t, X(t)) = ft (t, X(t))dt + fx (t, X(t))dX(t) + fxx (t, X(t))dX(t)dX(t).
2
Setting f (t, X(t)) = log St and X(t) = St ,
d log St =
dSt 1 1
−
dSt dSt ,
St
2 St2
Since dtdt = 0, dtdWt = 0 and dWt dWt = dt, dSt dSt = µ2 St2 dtdt+2µσSt2 dtdWt +σ 2 St2 dWt dWt =
σ 2 St2 dt and substituting (2.1) into the above equation we have:
d log St =
µSt dt + σSt dWt 1 2
− σ dt
St
2
7
1
= (µ − σ 2 )dt + σdWt .
2
Integration both sides it follows that
1
log St − log S0 = (µ − σ 2 )t + σWt .
2
Therefore
1
St = S0 exp((µ − σ 2 )t + σWt ).
2
The Black-Scholes option pricing is calculated in the risk neutral world, So we need to
use Girsanov theorem to change the measure from the real world to risk neutral world. The
Girsanov theorem is the analogue of the change of variable formula for integration in calculus.
Theorem 2.1. (The Girsanov Theorem) If Wt is a P -Brownian motion and Xs is an F previsible process, then there exists a measure Q such that
1. Zt = exp(−
Rt
2. WtQ = Wt +
Assume that
0
Xs dWs −
Rt
0
Rt
1
2
0
Xs ds.
Z
E [
P
0
Xs2 ds),
T
Xs2 Zs2 ds] < ∞.
Then Q defined by
dQ
= ZT .
dP
is an equivalent probability measure w.r.t P under which WtQ is also a Brownian motion.
From equation (2.2) and using the Girsanov Theorem
1
St = S0 exp((µ − σ 2 )t + σWt )
2
8
1
= S0 exp((r − σ 2 + µ − r)t + σWt )
2
(µ − r)
1
t))
= S0 exp((r − σ 2 )t + σ(Wt +
2
σ
Z t
(µ − r)
1 2
ds))
= S0 exp((r − σ )t + σ(Wt +
2
σ
0
1
= S0 exp((r − σ 2 )t + σWtQ .
2
Here Xs in the Girsanov theorem is the constant
µ−r
.
σ
Moreover, WtQ is the Brownian motion
under the risk neutral measure. The call option price is computed by the formula:
C(S, t) = e−r(T −t) E Q [max(ST − K, 0)],
Thus we can obtain the famous Black-Scholes formula for European option:
C(S, t) = SN(d1 ) − Ke−r(T −t) N(d2 ),
Z x
1 2
1
e− 2 y dy,
N (x) = √
2π −∞
log(S/K) + (r + 12 σ 2 )(T − t)
√
,
d1 =
σ T −t
log(S/K) + (r − 12 σ 2 )(T − t)
√
d2 =
,
σ T −t
√
we may also write d2 = d1 − σ T − t. The price of a corresponding put option based on
put-call parity is:
P (S, t) = C(S, t) + Ke−r(T −t) − S(t).
Black-Scholes formula is the most commonly used formula in the option pricing literature.
However, empirical evidence contradicts two key aspects: lognormal distribution of the asset
return and constant volatility. This evidence show that the distribution of the asset returns
does not behave as a lognormal random variable and volatility changes stochastically over
time. Therefore, models with stochastic volatility have been proposed.
9
2.2 GARCH-in-Mean Model
Consider a discrete time economy with a risk-free asset and a risky asset. We define a
complete filtered probability space (Ω, F , {Ft}, P ) to model uncertainty. P is the historical
(physical) measure and F = {Ft } , t = 0, 1, ...T (T < ∞), is a filtration, or a family
of increasing σ-field information sets, representing the resolution of uncertainty based on
information generated by the market prices up to and including time t. We assume F0 =
σ{∅, Ω} and FT = F . We assume the following general GARCH(p,q) model for the return
yt := log(St /St−1 ), where St is the stock price at time t.
yt = mt + σt εt ,
q
p
X
X
2
2
2
bi σt−i
.
αi σt−i ϕ(εt−i ) +
σt = α0 +
(2.3)
(2.4)
i=1
i=1
There are some assumptions we make:
• {εt }0≤t≤T is a sequence of independent and identically distributed (i.i.d) random variables with common distribution D(0, 1); the mean and variance for
the distribution is 0 and 1;
• the conditional mean return mt is assumed to be an Ft -predictable process.
In many studies, mt is assumed to be a function of the conditional variance σt
of the return and a risk premium quantifier at time t;
• the function ϕ(·) describes the impact of random shock of return εt on the
conditional variance σt2 . This is called the news impact curve;
• α0 , α1 and b1 are the coefficients of the GARCH model, where α0 > 0 and
α1 , b1 ≥ 0, and these parameters and function ϕ(·) are such that the conditional
variance dynamics are covariance stationary.
10
Furthermore, throughout the thesis, we assume the conditional cumulant function of εt given
Ft−1 under P exists and is given by:
κPεt (u) = log E P [euεt |Ft−1 ] < ∞ ,
u ∈ R.
(2.5)
Since the GARCH(1,1) model has often performed just as well as the GARCH(p,q) model,
we restrict our attention to the simple GARCH(1,1) models. For equation (2.3), mt is
set to be the constant r or if we add the heteroskedastic term in the conditional mean
equation, then mt = r + λσt , where r is one-period risk-free rate of return (continuously
compounded) and λ is the market price of risk. This extended GARCH model is often
referred to as GARCH-in-Mean or GARCH-M model. Thus the GARCH(1,1)-M model for
the stock return yt := log(St /St−1 ) under P has the following dynamic:
yt = mt + σt εt ,
(2.6)
2
2
σt2 = α0 + α1 σt−1
ϕ(εt−1 ) + b1 σt−1
,
(2.7)
mt = r + λσt .
(2.8)
The assumptions for the GARCH-M model are the same as in the standard GARCH model.
Moreover, the conditional mean and variance of yt are:
mt = E[yt |Ft−1 ],
σt2 = V ar[yt |Ft−1 ].
(2.9)
(2.10)
In equation (2.9), we can see that the conditional mean of yt is dependent on the square root
of the variance.
2.3 GARCH Model Extensions
Different GARCH models are mainly characterized by differences in the function ϕ. If we
consider p=q=1 and ϕ(εt−1 ) = ε2t−1 in equation (2.4), we can get the conditional variance
11
dynamic for the basic GARCH model proposed by Bollerslev (1986).
2
2
σt2 = α0 + α1 σt−1
ε2t−1 + b1 σt−1
.
(2.11)
To ensure the covariance stationarity, α1 + b1 < 1 is generally required. In many cases,
the basic GARCH model (2.11) provides a reasonably good model for analyzing financial
time series and estimating conditional volatility. However, some of the characteristics can
not be captured by the basic GARCH model. This led to extend the basic GARCH model
to exponential GARCH (EGARCH) model, GJR-GARCH model, TGARCH model, Power
GARCH (PGARCH) model or NGARCH model. These models are more flexible than the
basic GARCH model.
EGARCH Model
Nelson (1991) introduced the following EGARCH model specified as follows:
ht = α0 + α1
|εt−1 | + γεt−1
+ b1 ht−1 .
σt−1
(2.12)
Here ht = log σt2 . When there is good news, εt−1 is positive and the total effect of εt−1 is
(1+γ)εt−1. When there is bad news, εt−1 is negative and the total effect of εt−1 is (1−γ)εt−1 .
Bad news has thus a larger impact on volatility.
TGARCH model
The TGARCH (threshold) model was proposed by Zakoian (1994) and the similar GJRGARCH model was studied by Glosten.et.al (1993). In the TGARCH model, the state of
the world is determined by an observable TGARCH variable, while the conditional variance
follows a GARCH process with each state. The conditional variance for TGARCH model is:
2
2
σt2 = α0 + (α1 + γI(εt−1 < 0))ε2t−1 σt−1
+ b1 σt−1
.
(2.13)
Here γ is the positive parameter and I(εt−1 < 0) is the indicator function. The stationary
covariance property requires α1 + b1 +
γ
2
< 1. Depending on whether εt−1 is above or below
12
zero, ε2t−1 have different effects on the conditional variance σt2 . If there is good news, εt−1 ≥ 0
such that I(εt−1 < 0) = 0, the total effect is α1 ε2t−1 on the next period conditional variance.
If there is bad news, εt−1 < 0 such that I(εt−1 < 0) = 1, the total effect is (α1 + γ)ε2t−1 on
the next period conditional variance. So bad news will have larger impact on the conditional
variance.
PGARCH Model
Ding, Granger and Engle (1993) introduced PGARCH (power GARCH) model. The conditional variance has the form:
d
σt2 = α0 + α1 (|εt−1 | + γεt−1 )d + b1 σt−1
.
(2.14)
Here d and γ are the positive parameters. Ding, Garnger and Engle (1993) showed that the
basic GARCH model is the special case of the PGARCH model.
NGARCH model
The NGARCH model was proposed by Engle and Ng (1993). NGARCH model allows for
asymmetric behaviour in the volatility. When we set ϕ(εt−1 ) = (εt−1 − γ)2 and p = q = 1 in
equation (2.4), we can get the conditional variance in the following form:
2
2
σt2 = α0 + α1 σt−1
(εt−1 − γ)2 + b1 σt−1
.
(2.15)
Here parameter γ is positive and the stationary covariance constraint for NGARCH model
is α1 (1 + γ 2 ) + b1 < 1. Notice that the negative value of εt−1 will generate a larger value of
conditional variance in the next period.
13
Chapter 3
Risk Neutral Measures for GARCH Model
The Fundamental theorem for asset pricing proposed by Harrison and Kreps (1979) and
Harrison and Pliska (1981) states that the absence of arbitrage opportunities is equivalent
to the existence of an equivalent martingale measure. Typically, the price of European style
options is calculated in the risk neutral world. Therefore, the main purpose for this chapter
is to find an equivalent martingale measure with respect to the historical measure P . Recall
that r is the continuously compounded one-period risk-free rate of return, and let S̃t be the
discounted stock price at time t, so S̃t = e−rt St .
Definition 3.1. A probability measure Q is an equivalent martingale measure with respect
to P if:
• Q ≈ P (∀B ∈ F , Q(B) = 0 ⇔ P (B) = 0);
• the discounted price process S̃t is a martingale under Q with respect to Ft , that
h
i
Q
is E S̃t |Ft−1 = S̃t−1 .
Remark 3.1. The martingale condition for the discounted stock price can be rewritten as:
h
i
E Q eyt |Ft−1 = er .
Proof. From second condition in Definition 3.1, we have
h
i
E Q S̃t |Ft−1
i
h
⇔ E Q e−rt St |Ft−1
"
#
S
t ⇔ EQ
Ft−1
St−1
h
i
⇔
E Q eyt |Ft−1
14
= S̃t−1
= e−r(t−1) St−1
= er
= er .
(3.1)
If the market is complete, there should be a unique martingale measure. However, the
assumption of market completeness is questionable in the real-world securities market. Since
under discrete time GARCH model the market is incomplete, there is more than one equivalent martingale measure. In the next subsections, we will discuss five martingale measures
in the GARCH framework mentioned in the previous section.
3.1 Duan’s locally risk neutral valuation relationship
Duan (1995) proposed the locally risk neutral valuation relationship (LRNVR) to accommodate heteroskedasticity of the asset return process, and this was the first time the theoretical
foundation for option valuation in the context of GARCH models was provided.
Definition 3.2. A pricing measure Q is said to satisfy the locally risk neutral valuation
relationship (LRNVR) if measure Q is mutually absolutely continuous with respect to measure
P and satisfies the following conditions:
• yt |Ft−1 is normally distributed under Q;
h
i
t
• E Q SSt−1
|Ft−1 = er ;
h
i
h
i
t
t
• V ar Q log( SSt−1
)|Ft−1 = V ar P log( SSt−1
)|Ft−1 almost surely with respect to
measure P .
From the previous proof, the second condition in LRNVR is the same martingale condit
tion as for equation (3.1). In the definition of LRNVR, the conditional variance of log( SSt−1
)
is invariant under change of probability measures from P to Q almost surely.
The first condition in LRNVR ensures that the asset returns are normally distributed
under the risk neutral measure Q. In order to implement Duan’s LRNVR in our GARCH
setup, we assume εt ∼ N(0, 1), and then yt |Ft−1 ∼ N(mt , σt2 ). Therefore, the return dynamic
under P are equations from (2.6) to (2.8). We can specify as follows:
yt = mt + σt εt ,
15
(3.2)
2
2
σt2 = α0 + α1 σt−1
ϕ(εt−1 ) + b1 σt−1
,
(3.3)
mt = r + λσt .
(3.4)
Theorem 3.1. Suppose the asset return process y := {yt }t∈T satisfies (3.2)-(3.4) under P .
Under the risk neutral LRNVR QLRN V R the dynamics of the return are given by:
1
yt = r − σt2 + σt ε0t ,
2
ε0t ∼ N(0, 1),
1
2
2
σt2 = α0 + α1 σt−1
ϕ(ε0t−1 − σt−1 − λ) + b1 σt−1
.
2
Here ε0t are i.i.d distributed and if ϕ(ε0t−1 − 12 σt−1 − λ) is given by:
γI(ε0t−1 − 21 σt−1 − λ < 0)
1
1
),
ϕ(ε0t−1 − σt−1 − λ) = (ε0t−1 − σt−1 − λ)2 (1 +
2
2
α1
the conditional variance of return process follows a TGARCH model under the risk neutral
measure QLRN V R :
1
1
2
2
σt2 = α0 + (α1 + γI(ε0t−1 − σt−1 − λ < 0))(ε0t−1 − σt−1 − λ)2 σt−1
+ b1 σt−1
.
2
2
3.2 Mean Correcting Martingale Measure
The idea of the mean correcting martingale measure (MCMM) is based on a Girsanov-type
transformation which keeps the variance unchanged after the measure changes, but the return
distribution shifts the mean mt under P when the measure changes from P to Q. Therefore,
the discounted asset price becomes a martingale under the new probability measure. The
last two conditions of Definition 3.2 are still valid. However, the MCMM is more efficient
than the LRNVR since it does not require the asset returns to be conditionally Gaussian
t
distributed. The main idea is to identify the quantity mshif
, which represents the shift to
t
the conditional mean return mt such that the discounted stock price is an {Ft }-martingale
under probability measure Qm . Consequently, under Qm , we have:
t
yt = mt + mshif
+ σt ε0t ,
t
16
ε0t ∼ D(0, 1).
(3.5)
t
We implement MCMM in our approach, and mshif
will be determined in the following
t
theorem.
Theorem 3.2. Suppose the asset return process y := {yt }t∈T satisfies (3.2)-(3.4) under P .
Under the risk neutral MCMM Qm the dynamics of the return are given by:
yt = r − κPεt (σt ) + σt ε0t ,
ε0t ∼ D(0, 1),
σt2
= α0 +
2
α1 σt−1
ϕ(ε0t−1
t
mshif
t−1
2
) + b1 σt−1
.
+
σt−1
Proof. Using martingale equation (3.1):
EQ
⇔ EQ
(m)
shif t
[emt +mt
shif t
⇔ emt +mt
⇔
EQ
(m)
[eyt |Ft−1 ] = er
+σt ε0t
|Ft−1 ] = er
0
[eσt εt |Ft−1 ] = er
(m)
t
mt +mshif
+κQ0
t
e
(σt )
= er
+κP
εt (σt )
= er
εt
shif t
emt +mt
⇔
⇔
(m)
t
mt + mshif
+ κPεt (σt ) = r.
t
t
Thus, the shift term mshif
is :
t
t
mshif
= r − mt − κPεt (σt ).
t
(3.6)
We use equation (3.5) and (3.6) to obtain the asset return dynamic in Theorem 3.2. Furthermore, by equalizing the returns dynamic under historical measure P and risk neutral measure
Q, as shown in equation (3.2) and (3.5), we can get the following relationship between εt
and ε0t :
ε0t
t
mshif
t
= εt −
.
σt
17
Substituting this relationship into equation (3.3), we can get the conditional variance
under risk neutral measure in Theorem 3.2. Again, when we set
mshif t
γI(ε0t−1 + σt−1
< 0)
mshif t
mshif t
t−1
ϕ(ε0t−1 + t−1 ) = (ε0t−1 + t−1 )2 (1 +
).
σt−1
σt−1
α1
We can get the TGARCH conditional variance, that is :
σt2
= α0 + (α1 +
γI(ε0t−1
t
t
mshif
mshif
t−1
t−1
0
2
2
+
< 0))(εt−1 +
)2 σt−1
+ b1 σt−1
.
σt−1
σt−1
If we assume εt ∼ N(0, 1) under historical measure P , then κPεt (σt ) =
1 2
σ .
2 t
The return
dynamic under the risk neutral measure Qm is:
1
yt = r − σt2 + σt ε0t , ε0t ∼ N(0, 1).
2
t
mshif
t
0
εt = εt −
σt
r − mt − 12 σt2
= εt −
σt
r − (r + λσt ) − 12 σt2
= εt −
σt
1 2
−λσt − 2 σt
= εt −
σt
1
= εt + λ + σt .
2
Then the conditional variance under risk neutral measure Qm is :
1
2
2
σt2 = α0 + α1 σt−1
ϕ(ε0t−1 − σt−1 − λ) + b1 σt−1
.
2
We notice that the dynamics from MCMM are consistent with the ones given by Duan’s
LRNVR for Gaussian innovation.
3.3 Conditional Esscher Transform Method
The third risk neutral measure is the conditional Esscher transform proposed by Gerber
and Shiu (1994) for option valuation under the GARCH model. Siu, Tong and Yang (2004)
18
used the concept of conditional Esscher transform introduced by Buhlmann et al. (1996)
to identify an equivalent martingale measure and proposed an alternative approach to the
valuation of derivatives under the GARCH models in the context of Gerber-Shiu’s option
pricing model. The advantage of the conditional Esscher transform for picking an equivalent
martingale measure is its capability in incorporating different distributions for the GARCH
innovations. Similar to the mean correcting martingale measure (MCMM) method, the
conditional Esscher transform method does not require the innovation distributions to be
conditionally normal distributed. Kallsen and Shiryaev (2002) mentioned the conditional
Esscher transform enjoyed a desirable mathematical property that it can be computed easily, since one only needs to solve an equation for the Esscher parameter that ensures the
martingale property.
Definition 3.3. Let the process Zt defined by:
Zt =
t
Y
θk yk −κP
y
e
k |Fk−1
(θk )
,
(3.7)
k=1
where θk is Fk -predictable, and we need to solve θk from:
κPyk |Fk−1 (1 + θk ) − κPyk |Fk−1 (θk ) = r,
(3.8)
for all k ∈ {1, 2..., T }. The conditional Esscher transform Qe with respect to P of the process
yt is defined as:
dQe = ZT .
dP FT
(3.9)
The existence of a solution of (3.8) is guaranteed by the existence of the cumulant generating function imposed in Chapter 2. However, Christoffersen (2010) discussed the existence
and uniqueness of solution of (3.8). He proposed that constraints needed to be imposed
on the cumulant generating function when the distribution is non-Normal. The martingale
property of the conditional Esscher transform risk neutral measure is ensured by equation
(3.8). This can be proved by Bayes’ Theorem, which is stated below:
19
Lemma 3.1. Let λ = {λk }k∈T an F -adapted stochastic process and define:
m
Y
dQe λk ,
Fm = Zm =
dP
k=1
For any Fn -measurable random variable h, n ≥ m:
e
E Q [h|Fm ] =
E P [hZn |Fm ]
.
Zm
Consider λk = exp(θk yk − κPyk |Fk−1 (θk )) and h = ezyk . Setting n = k, m = k − 1 and applying
the Bayes’ Theorem. We have:
e
e
Q
[ezyk |Fk−1])
κQ
yk |Fk−1 (z) = log(E
= log E P
k
Q
#!
exp(θt yt − κPyt |Ft−1 (θt )) t=1
ezyk k−1
Fk−1
Q
P
exp(θt yt − κyt |Ft−1 (θt ))
"
t=1
= log(E P [ezyk exp(θk yk − κPyk |Fk−1 (θk ))|Fk−1])
= log(E P [ezyk +θk yk exp(−κPyk |Fk−1 (θk ))|Fk−1])
= log(E P [ezyk +θk yk |Fk−1]) + log(exp[(−κPyk |Fk−1 (θk ))|Fk−1])
= κPyk |Fk−1 (z + θk ) − κPyk |Fk−1 (θk ).
e
When z = 1, we can get r = κQ
yk |Fk−1 (1). Then the martingale equation (3.8) holds. If
εt ∼ N(0, 1), we will use conditional Esscher transform to derive the following theorem.
Theorem 3.3. Suppose the asset return process y := {yt }t∈T satisfies (3.2)-(3.4) under P .
Under the risk neutral conditional Esscher transform measure Qe the dynamics of the return
are given by:
1
yt = r − σt2 + σt ε0t ,
2
ε0t ∼ N(0, 1),
1
2
2
σt2 = α0 + α1 σt−1
ϕ(ε0t−1 − σt−1 − λ) + b1 σt−1
.
2
Proof. We start from the martingale equation in the previous proof. If εt ∼ N(0, 1), then
yt |Ft−1 ∼ N(mt , σt2 ) .
20
e
P
P
κQ
yt |Ft−1 (z) = κyt |Ft−1 (z + θt ) − κyt |Ft−1 (θt )
1
1
= (mt (z + θt )) + (z + θt )2 σt2 ) − (mt θt + θt2 σt2 )
2
2
1 2 2
= mt z + σt z + θt σt2 z
2
1
= (mt + θt σt2 )z + σt2 z 2 .
2
This means
yt |Ft−1 ∼ N(mt + θt σt2 , σt2 ),
(3.10)
under Qe . Solving the above equation for z = 1.
e
r = κQ
yt |Ft−1 (1)
⇒ θt
1
= (mt + θt σt2 ) + σt2 ,
2
r − mt − 12 σt2
.
=
σt2
Substituting θt into (3.10). We have:
1
yt |Ft−1 ∼ N(r − σt2 , σt2 ).
2
The relationship between εt and ε0t :
1
εt = ε0t − σt − λ.
2
Thus the risk neutral dynamic under Qe is:
1
yt = r − σt2 + σt ε0t ,
2
ε0t ∼ N(0, 1),
1
2
2
σt2 = α0 + α1 σt−1
ϕ(ε0t−1 − σt−1 − λ) + b1 σt−1
,
2
where ε0t are i.i.d distributed. These results are also consistent with Duan’s results for
Gaussian GARCH pricing models. Moreover, if ϕ(ε0t−1 − 21 σt−1 − λ) is given by:
γI(ε0t−1 − 21 σt−1 − λ < 0)
1
1
),
ϕ(ε0t−1 − σt−1 − λ) = (ε0t−1 − σt−1 − λ)2 (1 +
2
2
α1
21
the conditional variance of return process follows a TGARCH model under the risk neutral
measure Qe :
1
1
2
2
σt2 = α0 + (α1 + γI(ε0t−1 − σt−1 − λ < 0))(ε0t−1 − σt−1 − λ)2 σt−1
+ b1 σt−1
.
2
2
3.4 A discrete version of the Girsanov change of measure
In this section we show that MCMM and conditional Esscher transform measure can be
obtained by considering a discrete version of the Girsanov change of measure from the
continuous finance.
We recall some results from continuous time. Let Wt , 0 ≤ t ≤ T be a Brownian motion
on a probability space (Ω, F , P ) and let Ft be the filtration for this Brownian motion. As
in the discrete time setting, we consider that T represents the fixed expiration time. Let St
satisfy the following generalized Geometric Brownian motion (generalized GBM) differential
equation:
dSt = µt St dt + σt St dWt .
(3.11)
Here µt and σt are two adapted processes representing the mean rate of return and the
volatility at time t. Assuming that σt is non-negative P almost surely, then the solution of
(3.11) is given by:
St = S0 exp(
Z
t
0
1
(µs − σs2 )ds +
2
Z
t
σs dWs ).
(3.12)
0
Assuming the stock follows the generalized GBM differential equation (3.11), one can obtain
the risk neutral dynamic of the stock price via Girsanov theorem (Theorem 2.1).
dSt = µt St dt + σt St dWt
= (r + µt − r)St dt + σt St dWt
22
= rSt dt + σt St (
µt − r
dt + dWt )
σt
= rSt dt + σt St dWtQ .
In this case, the stochastic process Xt =
µt −r
.
σt
(3.13)
By analogy with this example for the con-
tinuous time, we construct a risk-neutral measure for our GARCH models driven by normal
innovations. We recall that equation (3.2) can be written in the following form:
St = S0 exp(
t
X
mk +
t
X
σk εk ).
(3.14)
k=1
k=1
Comparing (3.14) with its continuous counterpart (3.12) we notice that mt corresponds to
µt − 21 σt2 . We assume that yt follows the Normal-GARCH model given by (3.2)-(3.4), a
discrete version of the Girsanov theorem for the Normal-GARCH model can be defined in
the following:
T
T
X
dQ
1X 2
X )
= ZT = exp(−
Xt ε t −
dP
2 t=1 t
t=1
= exp(−
= exp(−
T
X
µt − r
t=1
T
X
t=1
σt
T
1 X µt − r 2
(
εt −
))
2 t=1 σt
T
mt + 12 σt2 − r
1 X mt + 21 σt2 − r 2
(
εt −
) ).
σt
2 t=1
σt
(3.15)
Based on this observation, we can show that the MCMM and the conditional Esscher have
the same ZT process as a discrete version of Girsanov theorem for the Normal-GARCH
model. In order to do this, we need to introduce the Stochastic Discount Factor (SDF).
A SDF or pricing kernel is denoted by Mt . Let {Mt }0≤t≤T be a positive-valued, {Ft }adapted, stochastic process defined on (Ω, F , P ) such that the following no-arbitrage conditions hold:
E P [Mt |Ft−1 ] = e−r ,
E P [Mt eyt |Ft−1 ] = 1.
(3.16)
(3.17)
We recall that r is the one-period interest rate that is assumed to be constant from time
t − 1 to time t, for any t. Condition (3.16) ensures that the probability measure induced by
23
Mt is well-defined, while condition (3.17) ensures that discounted asset prices are martingale
under this new measure.
A price at time t of a European option with payoff h(ST ) and expiration time T associated
with a SDF {Mt } is given by:
P
ΠM
t (h(ST )) = E [Mt+1 , ..., MT h(ST )|Ft ].
(3.18)
Here E P is expectation under P .
Definition 3.4. Let {Mt }0≤t≤T be a family of SDF satisfying the conditions from (3.16)(3.17). Let Q be a measure defined by:
T
Y
PT
dQ
= ZT = e k=1 r
Mk ,
dP
(3.19)
k=1
For MCMM, we assume the state price process Mt that obeys the no-arbitrage conditions
(3.16)-(3.17) has the following form (Badescu, 2011):
Mt = e−r
ftP (εt + %t )
.
ftP (εt )
(3.20)
Here
• ftP (·) is the conditional probability density of the innovation εt at time t given
Ft−1 under the historical measure P ;
• the market price of risk process, denoted as {%t }, is an {Ft }-predictable process
and is uniquely determined from condition (3.17) for any t ∈ T \{0};
• it is clear by definition that the parametric form of the SDF from (3.20) satisfies
(3.16).
We let %t =
mt +κP
εt (σt )−r
σt
and use Definition 3.4. Under the assumption of εt ∼ N(0, 1), we
can get:
PT
ZT = e
k=1
r
T
Y
Mk
k=1
24
T
Y
ftP (εk + %k )
ftP (εk )
k=1
2
T √1 exp − (εk +%k )
Y
2
2π
=
(εk )2
1
√
exp
−
k=1
2
2π
PT
=e
=
=
r
k=1
T
Y
k=1
T
Y
e−r
exp(−%k εk −
exp(−
k=1
%2k
)
2
mk + κPεk (σk ) − r
1 mk + κPεk (σk ) − r 2
εk − (
))
σk
2
σk
T
T
X
mk + 21 σk2 − r
1 X mk + 12 σk2 − r 2
(
= exp(−
εk −
) ).
σ
2
σ
k
k
k=1
k=1
(3.21)
For conditional Esscher transform, the process ZT has already been defined by equation
(3.7), (3.9) and θt =
r−mt − 21 σt2
.
σt2
Under the assumption of εt ∼ N(0, 1), we can derive ZT as
in the following:
ZT =
=
=
=
T
Y
θk yk −κP
y
e
k |Fk−1 (θk )
k=1
T
Y
1
exp(θk (mk + σk εk ) − (θk mk + θk2 σk2 ))
2
k=1
T
Y
1
exp(θk σk εk − θk2 σk2 )
2
k=1
T
Y
k=1
exp(
r − mk − 21 σk2
1 r − mk − 21 σk2 2 2
(
σ
ε
−
) σk )
k k
σk2
2
σk2
T
T
X
mk + 21 σk2 − r
1 X mk + 21 σk2 − r 2
= exp(−
(
εk −
) ).
σk
2 k=1
σk
k=1
(3.22)
Therefore, the MCMM and the conditional Esscher transform have the same ZT process.
Based on this, the two measures give the same risk neutral dynamic for the Normal-GARCH
model. Moreover, the ZT process of two measures are consistent with the discrete version of
Girsanov theorem for the Normal-GARCH model.
25
3.5 Second order Esscher Transform Method
Monfort and Pegoraro (2011) extended the conditional Esscher transform method to the
second-order Esscher transform method. Unlike the Esscher transform, the second order
Esscher transform implies that the risk neutral conditional variance is different from the
historical one for Gaussian innovation distribution. Therefore, not only the second order
Esscher transform changes the mean, but also changes the conditional variance in the risk
neutral measure. The second order Esscher transform method depends on the second order
log laplace transform, thus we start with the definition of the second order log laplace
transform.
Definition 3.5. The conditional second order log Laplace transform of the asset return
y := {yt }t∈T is given by:
2
κP(yt ,y2 )|Ft−1 (θ1t , θ2t ) = log E P [eθ1t yt +θ2t yt |Ft−1 ],
t
(3.23)
where θ1t and θ2t are Ft -predictable.
2
Here E P [eθ1t yt +θ2t yt |Ft−1 ] is the conditional second order laplace transform under the
historical measure P . In our GARCH setup, If we assume εt ∼ N(0, 1), then yt |Ft−1 ∼
N(mt , σt2 ) under historical measure P . We can derive the following corollary.
Corollary 3.1. The conditional second order log laplace transform for Gaussian distribution
is:
1
mt
m2
σt2
1
κP(yt ,yt2 )|Ft−1 (θ1t , θ2t ) = − log(1 − 2σt2 θ2t ) − t2 + (
)( 2 + θ1t )2 . (3.24)
2
2
2σt
2 1 − 2σt θ2t σt
Proof. Starting from the second order laplace transform. For convenience, we assume y ∼
N(m, σ 2 ). Thus we have:
Z
P θ1 y+θ2 y 2
E [e
] =
f P (y) exp(θ1 y + θ2 y 2 )dy
ZR
(y − m)2
1
√ exp(−
) exp(θ1 y + θ2 y 2 )dy
=
2
2σ
R σ 2π
26
(y − m)2
1
√ exp(−
+ θ1 y + θ2 y 2)dy
2
2σ
σ
2π
ZR
1
y 2 − 2ym + m2
√ exp(−
=
+ θ1 y + θ2 y 2)dy
2
2σ
σ
2π
ZR
1
m
m2
1
√ exp(− 2 ) exp(y 2(θ2 − 2 ) + y(θ1 + 2 ))dy
=
2σ
2σ
σ
R σ 2π
h
i
2
2
1
m
1
σ
m
2
= (1 − 2σ 2 θ2 )− 2 exp − 2 + (
)(
+
θ
)
1
2σ
2 1 − 2σ 2 θ2 σ 2
Z
m
2
σ2
1 − 2σ 2 θ2 1
q
y−
+ θ1
dy
exp
−
×
2
2θ
2
2σ
1
−
2σ
σ
σ2
2
R
2π 1−2σ2 θ2
h m2
i
1
1
σ2
m
2
= (1 − 2σ 2 θ2 )− 2 exp − 2 + (
.
)(
+
θ
)
1
2σ
2 1 − 2σ 2 θ2 σ 2
m
2
1−2σ2 θ2
σ2
exp − 2σ2
y − 1−2σ2 θ2 σ2 + θ1
is the p.d.f of Normal distribu=
Since
1
r
2π
σ2
1−2σ 2 θ2
tion with mean
σ2
1−2σ2 θ2
Z
m
σ2
m
σ2
( 1−2σ
2 θ )( σ 2
2
+ θ1
2
+ θ1 ) and variance
σ2
,
1−2σ2 θ2
R
1
r
R
2π
σ2
1−2σ 2 θ2
exp
−
1−2σ2 θ2
2σ2
y−
dy = 1 and taking the neutral logarithm we have:
1
m2
1
σ2
m
2
log E P [eθ1 y+θ2 y ] = − log(1 − 2σ 2 θ2 ) − 2 + (
)( 2 + θ1 )2 .
2
2
2σ
2 1 − 2σ θ2 σ
Given yt |Ft−1 ∼ N(mt , σt2 ) under historical measure P , we can get equation (3.24). Based
on the second order log laplace transform for Gaussian distribution, we can define the process
for the second order Esscher transform.
Definition 3.6. Let the process Zt defined by:
Zt =
t
Y
θ1k yk +θ2k yk2 −κP
(yk ,y 2 )|Fk−1
k
e
(θ1k ,θ2k )
,
(3.25)
k=1
θ1k and θ2k are the solution of the equation:
κP(yk ,y2 )|Fk−1 (1 + θ1k , θ2k ) − κP(yk ,y2 )|Fk−1 (θ1k , θ2k ) = r,
k
k
(3.26)
for all k ∈ {1, 2..., T }. The second order Esscher transform Q2e with respect to P of the
process yt is defined as:
dQ2e = ZT .
dP FT
27
(3.27)
We can use the martingale condition to derive the equation (3.26). This can also be proved
by Bayes’ Theorem, which is stated below:
Lemma 3.2. Let λ = {λk }k∈T an F -adapted stochastic process and define:
m
Y
dQ2e λk ,
F m = Zm =
dP
k=1
For any Fn -measurable random variable h, n ≥ m:
E
Q2e
E P [hZn |Fm ]
[h|Fm ] =
.
Zm
2
Consider λk = exp(θ1k yk + θ2k yk2 − κP(yk ,y2 )|Fk−1 (θ1k , θ2k )) and h = ez1 yk +z2 yk . Setting n = k,
k
m = k − 1 and applying the Bayes’ Theorem. We have:
2e
2e
2
(z , z ) = log(E Q [ez1 yk +z2 yk |Fk−1])
κQ
(yk ,y 2 )|Fk−1 1 2
k
= log E P
"
k
Q
#!
exp(θ1t yt + θ2t yt2 − κP(yt ,y2 )|Ft−1 (θ1t , θ2t )) t
2 t=1
ez1 yk +z2 yk k−1
Fk−1
Q
2
P
exp(θ1t yt + θ2t yt − κ(yt ,y2 )|Ft−1 (θ1t , θ2t ))
t
t=1
2
= log(E P [ez1 yk +z2 yk exp(θ1k yk + θ2k yk2 − κP(yk ,y2 )|Fk−1 (θ1k , θ2k ))|Fk−1])
k
z1 yk +z2 yk2 +θ1k yk +θ2k yk2
= log(E P [e
exp(−κP(yk ,y2 )|Fk−1 (θ1k , θ2k ))|Fk−1])
k
(z1 +θ1k )yk +(z2 +θ2k )yk2
= log(E P [e
|Fk−1]) + log(exp[(−κP(yk ,y2 )|Fk−1 (θ1k , θ2k ))|Fk−1])
k
= κP(yk ,y2 )|Fk−1 (z1 + θ1k , z2 + θ2k ) − κP(yk ,y2 )|Fk−1 (θ1k , θ2k ).
k
k
2e
When z1 = 1 and z2 = 0, we can get r = κQ
(1, 0). Then the martingale equation
(yk ,y 2 )|Fk−1
k
(3.26) holds. We compare the second order Esscher transform with the conditional Esscher
transform. The conditional Esscher pricing kernel is an exponential linear function. However,
the second order Esscher transform pricing kernel is an exponential non-linear function. The
important feature of using ZT is to derive the risk neutral distribution.
Corollary 3.2. Suppose yt |Ft−1 ∼ N(mt , σt2 ) under the historical measure P . Under the
σ2
m +σ2 θ
t
t
t 1t
risk neutral measure Q, yt |Ft−1 ∼ N( 1−2θ
2 , 1−2θ σ 2 ).
2t σ
2t
t
t
28
Proof. Using distribution relationship between P and Q and without lots of generating.
f Q (yt |Ft−1 ) = f P (yt |Ft−1 ) × Zt
h (y − m )2 i
1
t
t
= p
exp(θ1t yt + θ2t yt2 − κP(yt ,yt2 )|Ft−1 (θ1t , θ2t ))
exp −
2
2
2σ
2πσt
t
h
1
(yt − mt )2 i
exp −
= p
2σt2
2πσt2
mt
m2t
σt2
1
1
2
)( 2 + θ1t )2 ]
× exp[θ1t yt +
+ log(1 − 2σt θ2t ) + 2 − (
2
2
2σt
2 1 − 2σt θ2t σt
i
h
2
1
m2t
1
σt2
mt
(yt − mt )
2
2
= q
+ θ1t yt + θ2t yt + 2 − (
)(
+ θ1t )
exp (−
σt2
2σt2
2σt
2 1 − 2σt2 θ2t σt2
2π
θ2t yt2
1−2θ2t σt2
= q
1
σ2
2π 1−2θt2t σ2
t
h 1 − 2θ σ 2
mt + σt2 θ1t i
2t t
exp (−
)(y
−
) .
t
2σt2
1 − 2θ2t σt2
Therefore under Q,
⇒ yt |Ft−1 ∼ N(
σt2
mt + σt2 θ1t
,
).
1 − 2θ2t σt2 1 − 2θ2t σt2
This property leads to the following theorem.
Theorem 3.4. Suppose the asset return process y := {yt }t∈T satisfies (3.2)-(3.4) under P .
Under the risk neutral second order Esscher transform measure Q2e the dynamics of the
return are given by:
1
yt = r − (σtQ )2 + σtQ ε0t ,
2
σt2
(σtQ )2 =
.
1 − 2θ2t σt2
ε0t ∼ N(0, 1),
Proof. If εt ∼ N(0, 1), then yt |Ft−1 ∼ N(mt , σt2 ) under the historical measure P . Starting
from the equation (3.26).
r = κP(yt ,yt2 )|Ft−1 (1 + θ1t , θ2t ) − κP(yt ,yt2 )|Ft−1 (θ1t , θ2t )
29
mt
m2
σt2
1
1
)( 2 + 1 + θ1t )2
= − log(1 − 2σt2 θ2t ) − t2 + (
2
2
2σt
2 1 − 2σt θ2t σt
h 1
i
2
mt
m
σt2
1
2
− − log(1 − 2σt2 θ2t ) − t2 + (
)(
+
θ
)
1t
2
2σt
2 1 − 2σt2 θ2t σt2
h m
i
σt2
1
mt
t
2
2
(
=
+
1
+
θ
)
−
(
+
θ
)
1t
1t
2 1 − 2σt2 θ2t
σt2
σt2
2m
1
σt2
t
=
+
1
+
2θ
1t
2 1 − 2σt2 θ2t
σt2
1
2m
σt2
σt2
1
t
=
+
+
2θ
1t
2 1 − 2θ2t σt2
2 1 − 2θ2t σt2
σt2
m + σ 2 θ 1
σt2
σt2
t
t 1t
=
+
2
2
2
2 1 − 2θ2t σt
1 − 2θ2t σt
σt
2
2
σt
mt + σt θ1t
1
+
.
=
2
2 1 − 2θ2t σt
1 − 2θ2t σt2
So we can get
mt +σt2 θ1t
1−2θ2t σt2
and
σt2
1−2θ2t σt2
σt2
1
mt + σt2 θ1t
=
r
−
.
1 − 2θ2t σt2
2 1 − 2θ2t σt2
are the mean and variance under Q. We assume (σtQ )2 =
σt2
1−2θ2t σt2
and
use Corollary 3.2. The return dynamics in Theorem 3.4 are obtained.
Moreover, we can get the relationship between θ1t and θ2t . From the previous proof,
m + σ2 θ
1
σt2
t
t 1t
r=
+
2 1 − 2θ2t σt2
1 − 2θ2t σt2
1
⇒ r(1 − 2θ2t σt2 ) = σt2 + (mt + θ1t σt2 )
2
1
⇒
r(1 − 2θ2t σt2 ) − mt − σt2 = θ1t σt2
2
r(1 − 2θ2t σt2 ) − mt − 12 σt2
⇒
θ1t =
.
σt2
For simulation purpose in section 5.3, we set θ2t = C (constant).
3.6 Variance Dependent Pricing Kernel
Christoffersen, Heston and Jacobs (2012) developed a new pricing kernel for Heston-Nandi
(2000) GARCH model. The idea of this new pricing kernel comes from the stochastic volatil30
ity model. They showed that this general pricing kernel is non-monotonic, qualitatively accounting for the stylized facts. They also showed the Heston-Nandi (2000) GARCH model
is the special case of this general pricing kernel. Moreover, they found the mapping between
historical parameters and risk neutral parameters in Heston-Nandi (2000) GARCH model
and estimated the historical and risk-neutral parameters by maximizing the joint likelihood
function consisting of an option based component and a return based component.
Definition 3.7. The variance dependent pricing kernel is defined as:
Nt = N0
S φ
t
S0
exp(δt + η
t
X
s=1
2
σs2 + ξ(σt+1
− σ12 )),
(3.28)
and therefore
S φ
Nt
t
2
=
exp(δ + ησt2 + ξ(σt+1
− σt2 )).
Nt−1
St−1
(3.29)
Where δ and η govern the time-preference and φ and ξ govern the respective aversion to
equity and variance risk.
We implement this pricing kernel in our GARCH setup. From (3.2) to (3.4) we can write
St
= exp(r + λσt + σt εt ),
St−1
2
σt+1
− σt2 = α0 + α1 σt2 ϕ(εt ) + (b1 − 1)σt2 .
Substituting these into (3.29) gives
Nt
= exp(rφ + λσt φ + σt εt φ + δ + ησt2 + ξ(α0 + α1 σt2 ϕ(εt ) + (b1 − 1)σt2 )),
Nt−1
Rearranging gives
Nt
= exp(rφ + λσt φ + δ + ησt2 + ξα0 + ξ(b1 − 1)σt2 + σt εt φ + ξα1σt2 ϕ(εt )).
Nt−1
Depending on different forms of ϕ(εt ), we can have different forms of pricing kernel. If we
assume the NGARCH specification of ϕ(εt ),
ϕ(εt ) = (εt − γ)2 .
31
Then we have:
Nt
= exp(rφ + λσt φ + δ + ησt2 + ξα0 + ξ(b1 − 1)σt2 + σt εt φ + ξα1 σt2 (εt − γ)2 ).
Nt−1
Expanding the square and collecting terms gives
Nt
= exp(rφ + λσt φ + δ + ησt2 + ξα0 + ξ(b1 − 1)σt2 + σt εt φ + ξα1σt2 ε2t − 2ξα1σt2 εt γ + ξα1 σt2 γ 2 )
Nt−1
Nt
= exp(rφ + λσt φ + δ + ησt2 + ξα0 + ξ(b1 − 1)σt2 + ξα1 σt2 γ 2 + (σt φ − 2ξα1σt2 γ)εt + ξα1 σt2 ε2t ).
Nt−1
First, we use the fact that for any initial value σt , the parameters must be consistent with
the Euler equation for the riskless asset.
Et−1
N t
= exp(−r).
Nt−1
Note that
Et−1
N t
= exp(rφ + λσt φ + δ + ησt2 + ξα0 + ξ(b1 − 1)σt2 + ξα1 σt2 γ 2 )
Nt−1
× E(exp((σt φ − 2ξα1 σt2 γ)εt + ξα1 σt2 ε2t ))
Et−1
N t
= exp(rφ + λσt φ + δ + ησt2 + ξα0 + ξ(b1 − 1)σt2 + ξα1 σt2 γ 2 )
Nt−1
× E(exp((φ − 2ξα1 σt γ)σt εt + ξα1 σt2 ε2t )).
We need the following result
1
2a2 b2 E(exp(az 2 + 2abz)) = exp − log(1 − 2a) +
.
2
1 − 2a
For our application we have
a = ξα1,
b=
φ − 2ξα1 σt γ
,
2ξα1
z = σt εt ,
and thus
(φ − 2ξα1 σt γ)2
2a2 b2
=
.
1 − 2a
2(1 − 2ξα1 )
32
(3.30)
Therefore
E(exp((φ − 2ξα1σt γ)σt εt +
ξα1 σt2 ε2t ))
= exp
1
(φ − 2ξα1 σt γ)2 − log(1 − 2ξα1 ) +
,
2
2(1 − 2ξα1 )
and
N t
Et−1
= exp rφ + λσt φ + δ + ησt2 + ξα0 + ξ(b1 − 1)σt2 + ξα1 σt2 γ 2
Nt−1
1
(φ − 2ξα1 σt γ)2 − log(1 − 2ξα1) +
2
2(1 − 2ξα1 )
N 1
φ2
t
= exp rφ + δ + ξα0 − log(1 − 2ξα1 ) +
Et−1
Nt−1
2
2(1 − 2ξα1 )
2φξα1γ
2(ξα1 γ)2 2 + (λφ −
)σt + (η + ξ(b1 − 1) + ξα1 γ 2 +
)σ .
1 − 2ξα1
1 − 2ξα1 t
Using (3.30) we get
φ2
1
exp rφ + δ + ξα0 − log(1 − 2ξα1) +
2
2(1 − 2ξα1)
2φξα1 γ
2(ξα1 γ)2 2 + (λφ −
)σt + (η + ξ(b1 − 1) + ξα1 γ 2 +
)σ = exp(−r)
1 − 2ξα1
1 − 2ξα1 t
φ2
2φξα1γ
1
+ (λφ −
)σt
⇒ rφ + δ + ξα0 − log(1 − 2ξα1) +
2
2(1 − 2ξα1)
1 − 2ξα1
2(ξα1γ)2 2 + (η + ξ(b1 − 1) + ξα1 γ 2 +
)σ = −r
1 − 2ξα1 t
φ2
2φξα1γ
1
+ (λφ −
)σt
⇒ r(φ + 1) + δ + ξα0 − log(1 − 2ξα1 ) +
2
2(1 − 2ξα1 )
1 − 2ξα1
2(ξα1γ)2 2 2
+ (η + ξ(b1 − 1) + ξα1 γ +
)σ = 0.
1 − 2ξα1 t
Thus we must have
φ2
1
= 0,
r(φ + 1) + δ + ξα0 − log(1 − 2ξα1) +
2
2(1 − 2ξα1)
2φξα1 γ
λφ −
= 0,
1 − 2ξα1
2(ξα1 γ)2
η + ξ(b1 − 1) + ξα1 γ 2 +
= 0.
1 − 2ξα1
We can obtain
ξ=
λ
,
2α1 (λ + γ)
33
η = −ξ(b1 − 1) − ξα1 γ 2 −
δ = −r(φ + 1) − ξα0 +
2(ξα1 γ)2
,
1 − 2ξα1
1
φ2
log(1 − 2ξα1 ) −
.
2
2(1 − 2ξα1 )
Although both variance dependent kernel and the second order Esscher transform method
assume that the pricing kernel is quadric form, they are different from each other. The form
of the variance dependent kernel depends on the form of the ϕ(εt ). Different specification
of ϕ(εt ) leads to different forms of variance dependent kernel. However, the second order
Esscher transform method leads to the same pricing kernel for different specifications of
ϕ(εt ). The method used in Christoffersen (2012) to estimate the model parameters is too
complicated for variance dependent kernel. Therefore, we prefer to use the second order
Esscher transform method in simulation purpose.
34
Chapter 4
Generalized Hyperbolic GARCH Model
The generalized hyperbolic distribution (GH) proposed by Barndorff-Nielsin (1977) captures many important features of the financial data. The GH distribution is defined as
the normal variance-mean mixture where the mixing distribution is the generalized inverse
Gaussian(GIG) distribution.
Definition 4.1. If the random variable X follows a Generalized Inverse Gaussian(GIG)
distribution, then the probability density function is given by:
h δ 2 x−1 + γ 2 x i
γ
xλ−1
fGIG (x, λ, δ, γ) = ( )λ
.
exp −
δ 2Kλ (δγ)
2
(4.1)
where x > 0, kλ (·) is the modified Bessel function of the third kind associated with the
parameter λ. The parameters should satisfy the following conditions:
• δ ≥ 0, |γ| > 0 if λ > 0;
• δ > 0, |γ| > 0 if λ = 0;
• δ > 0, |γ| ≥ 0 if λ < 0.
Definition 4.2. Suppose a random variable Y has the GH distribution with parameter
(λ, α, β, δ, µ), y ∼ GH(λ, α, β, δ, µ). Then the probability distribution function is given by:
p
λ
1 (α
K
δ 2 + (y − µ)2 )
γ
λ− 2
√
fGH (y, λ, α, β, δ, µ) =
exp[β(y − µ)], y ∈ R. (4.2)
p
1
2πδ λ Kλ (δγ) ( δ 2 + (y − µ)2 /α) 2 −λ
Where α is the kurtosis, β is the skewness, δ is the scale parameter and µ is the location
parameter. Moreover, γ 2 = α2 − β 2 . The constrain for the parameter is |β| ≤ α. The
cumulant generating function of y provided by Eberlein and Hammerstein (2003) is in the
following:
35
α2 − β 2 λ
log
2
α2 − (β + u)2
p
Kλ (δ α2 − (β + u)2)
p
,
+ log
Kλ (δ α2 − β 2 )
κGH (u) =µu +
|β + u| < α.
(4.3)
The mean and Variance of y area given by:
δβ
Rλ (δγ),
γ
δ
β 2δ2
V ar[y] =
Rλ (δγ) + 2 Sλ (δγ).
γ
γ
E[y] = µ +
(4.4)
(4.5)
Here the functions Rλ and Sλ are defined for all u ∈ R+ by:
Kλ+1 (u)
,
Kλ (u)
2
Kλ+2 (u)Kλ(u) − Kλ+1
(u)
.
Sλ (u) =
2
Kλ (u)
Rλ (u) =
(4.6)
(4.7)
We implement the Generalized Hyperbolic distribution in the GARCH setup described in
the previous section. Consider that under the historical measure P , the stock return process
has the following dynamic:
yt = mt + σt εt ,
(4.8)
εt ∼ GH(λ, α, β, δ, µ),
(4.9)
2
2
σt2 = α0 + α1 σt−1
ϕ(εt−1 ) + b1 σt−1
,
(4.10)
mt = r + λσt .
(4.11)
The innovation process {εt } are i.i.d with common probability density function (4.2). To
standardize the innovation process, we impose the model parameters (λ, α, β, δ, µ) to satisfy
that the mean of εt equals to 0 and the variance of εt equals to 1. That is,
δβ
Rλ (δγ) = 0,
γ
β 2δ2
δ
Rλ (δγ) + 2 Sλ (δγ) = 1.
γ
γ
µ+
36
(4.12)
(4.13)
If εt ∼ GH(λ, α, β, δ, µ), then yt = mt + σt εt ∼ GH(λ, α/σt, β/σt , δσt , δµ + mt ). The
conditional probability density function of yt given Ft−1 under P is given by:
q
t
λ
− µ)2 )
Kλ− 1 (α δ 2 + ( y−m
σt
(α2 − β 2 ) 2
2
P
p
fyt (y) = √
· q
2πσt δ λ Kλ (δ α2 − β 2 ) ( δ 2 + ( y−mt − µ)2 /α) 21 −λ
σt
· exp[β(
y − mt
− µ)], y ∈ R.
σt
(4.14)
4.1 Risk Neutral Dynamic from MCMM under GH-GARCH
We can derive the risk neutral dynamics under the mean correcting martingale measure
(MCMM) for Generalized Hyperbolic GARCH model in the following theorem.
Theorem 4.1. Suppose the asset return process y := {yt }t∈T satisfies equations (4.8)-(4.11)
under P . Under the risk neutral MCMM Qm the dynamics of the return are given by:
α2 − β 2
λ
log 2
2
α − (β + σt )2
p
Kλ (δ α2 − (β + σt )2 )
p
− log
+ σt ε0t ,
2
2
Kλ (δ α − β )
∼ GH(λ, α, β, δ, µ),
mshif t 2
2
,
= α0 + α1 σt−1
ϕ ε0t−1 + t−1 + b1 σt−1
σt−1
yt = r − µσt −
ε0t
σt2
(4.15)
(4.16)
(4.17)
t
Where mshif
is given by:
t
t
mshif
t
α2 − β 2
λ
= r − mt − µσt − log 2
2
α − (β + σt )2
p
Kλ (δ α2 − (β + σt )2 )
p
− log
.
Kλ (δ α2 − β 2 )
Proof. Using the cumulant generating function of εt .
α2 − β 2
λ
log
2
α2 − (β + σt )2
p
Kλ (δ α2 − (β + σt )2 )
p
.
+ log
Kλ (δ α2 − β 2 )
κPεt (σt ) = µσt +
37
(4.18)
By equation (3.6) and Theorem 3.2. We have:
t
mshif
= r − mt − κPεt (σt )
t
yt
λ
α2 − β 2
= r − mt − µσt − log
2
α2 − (β + σt )2
p
Kλ (δ α2 − (β + σt )2 )
p
.
− log
Kλ (δ α2 − β 2 )
= r − κPεt (σt ) + σt ε0t
α2 − β 2
λ
log 2
2
α − (β + σt )2
p
Kλ (δ α2 − (β + σt )2 )
p
− log
+ σt ε0t .
2
2
Kλ (δ α − β )
= r − µσt −
ε0t = εt −
σt2
= α0 +
t
mshif
t
.
σt
2
α1 σt−1
ϕ
t
mshif
t−1
0
2
εt−1 +
+ b1 σt−1
.
σt−1
4.2 Risk Neutral Dynamic from Conditional Esshcer under GH-GARCH
We can also derive the risk neutral dynamics under the conditional Esscher transform measure for Generalized Hyperbolic GARCH model in the following theorem.
Theorem 4.2. Suppose the asset return process y := {yt }t∈T satisfies equations (4.8)-(4.11)
under P . Under the risk neutral conditional Esscher Qe the dynamics of the return are given
by:
yt = mt + σt (µ + ν1t ) + σt ν2t ε0t ,
δ
ν1t ε0t |Ft−1 ∼ GH λ, αν2t , β1t ν2t ,
,−
,
ν2t ν2t
(4.19)
(4.20)
Where {β1t },{ν1t } and {ν2t } are some {Ft }-predictable processes given by:
β1t = β + θt σt ,
(4.21)
38
ν1t = p
δβ1t
q
2
2
Rλ δ α − β1t ,
2
α2 − β1t
q
q
1
2
δ 2 β1t
δ
2 − β2 ) +
2 − β2 ) 2 .
p
S
(δ
R
(δ
α
α
=
λ
1t
1t
2 λ
2
α2 − β1t
α2 − β1t
ν2t
(4.22)
(4.23)
such that for each t ∈ T . ε0t has zero conditional mean and unit conditional variance given
Ft−1 and θt is the unique predictable solution of the following martingale equation:
p
Kλ (δ α2 − (β + θt σt )2 )
λ
α2 − (β + (1 + θt )σt )2
p
= µσt + mt − r.
log
+ log
2
α2 − (β + θt σt )2
Kλ (δ α2 − (β + (1 + θt )σt )2 )
(4.24)
Proof. We know that if εt ∼ GH(λ, α, β, δ, µ), then yt = mt +σt εt ∼ GH(λ, α/σt, β/σt , δσt , δµ+
mt ), and using (4.3) we can get the cumulant function of the returns under the physical measure P in the following:
kyPt (u)
p
Kλ (δ α2 − (β + uσt )2 )
α2 − β 2
λ
p
+ log
,
=(mt + µσt )u + log
2
α2 − (β + uσt )2
Kλ (δ α2 − β 2 )
|β + uσt | < α.
We start by evaluating the conditional cumulant generating function of yt given Ft−1 under
Qe .
e
kyQt (u) = kyPt (u + θt ) − kyPt (θt )
e
λ
α2 − β 2
= (mt + µσt )(u + θt ) + log
2
α2 − (β + (u + θt )σt )2
p
Kλ (δ α2 − (β + (u + θt )σt )2 )
p
− (mt + µσt )θt
+ log
Kλ (δ α2 − β 2 )
p
Kλ (δ α2 − (β + θt σt )2 )
α2 − β 2
λ
p
− log
− log
2
α2 − (β + θt σt )2
Kλ (δ α2 − β 2 )
λ
α2 − (β + θt σt )2
= (mt + µσt )u + log 2
2
α − (β + θt σt + uσt )2
p
Kλ (δ α2 − (β + θt σt + uσt )2 )
p
.
+ log
Kλ (δ α2 − (β + θt σt )2 )
When u = 1, kyQt (1) = r. We can get the martingale equation (4.24). This expression is
well defined provided that |β + θt σt + uσt | < α. From the above expression we can see that
39
under Qe , the conditional distribution of the return yt given Ft−1 is again a GH distribution
as follows:
yt |Ft−1 ∼ GH(λ, α/σt, β1t /σt , δσt , mt + σt µ),
where β1t = β + θt σt . We notice that the conditional return distribution after the measure
change arising from Qe is obtained by shifting only the skewness of the original distribution
with θt . Therefore, the return dynamics under Qe are:
yt = mt + σt ηt ,
ηt |Ft−1 ∼ GH(λ, α, β + θt σt , δ, µ).
In order to represent yt in the form given by (4.19)-(4.20) we denote by ν1t = E P [ηt |Ft−1 ] − µ
and ν2t the conditional standard deviation of ηt and we let ε0t = η/ν2t −(ν1t +µ)/ν2t . Moreover,
we conclude that, under the risk neutral conditional Esscher transform Qe , the conditional
e
e
mean and standard deviation of the returns, mQ and σ Q , are given by:
p
e
σt δ(β + θt σt )
Rλ (δ α2 − (β + θt σt )2 ),
mQ
t = mt + σt µ + p 2
α − (β + θt σt )2
p
e
δ
Rλ (δ α2 − (β + θt σt )2 )
σtQ =σt p
α2 − (β + θt σt )2
! 12
δ 2 (β + θt σt )2
p
.
+
α2 − (β + θt σt )2 Sλ (δ α2 − (β + θt σt )2 )
We remark that the risk neutral return dynamics have no longer a linear GARCH form
since the innovations ε0t are not independent and identically distributed and the conditional
valatility of return can not be updated under the conditional Esscher transform directly.
However, one can still simulate the process {yt } under this new probability measure using
(4.19)-(4.20), where the conditional variance process is filtered according to (4.10).
40
4.3 Special Case of Generalized Hyperbolic distribution
The Hyperbolic (HYP) distribution is a special case of the GH distribution when λ =
1. Furthermore, the Normal Inverse Gaussian (NIG) distribution is also a special case of
GH distribution when λ = −1/2 and the Variance Gamma (VG) distribution is a special
limit case of the GH distribution when δ → 0. For this section, we will focus on the NIG
distribution.
Corollary 4.1. Suppose a random variable Y has the GH distribution with parameter
(λ, α, β, δ, µ), y ∼ GH(λ, α, β, δ, µ). When λ = −1/2, the probability distribution function
of NIG is given by:
p
K−1 (α δ 2 + (y − µ)2 )
γ −1/2
1
p
fN IG y, − , α, β, δ, µ = √
2
2πδ −1/2 K− 1 (δγ) ( δ 2 + (y − µ)2 /α)
2
· exp[β(y − µ)],
y ∈ R.
(4.25)
Where γ 2 = α2 − β 2 .
There are various scale and location invariant parametrizations proposed in the literature.
For numerical purposes we use the following parametrization:
• Parametrization α̃ = αδ, β̃ = βδ.
Corollary 4.2. If we apply parametrization α̃ = αδ, β̃ = βδ and use equation (4.12)-(4.13)
to standardize the NIG distribution(4.25), The probability distribution function of the NIG
is given by:
q
x−µ̃ 2
q
h
i
x − µ̃ K1 α̃ 1 + ( δ̃ )
α̃
q
exp
α̃2 − β̃ 2 + β̃
,
fN IG (x) =
2
π δ̃
δ̃
1 + ( x−µ̃
)
δ̃
(4.26)
δ̃ and µ̃ are given by:
δ̃ =
!3/2
q
α̃2 − β̃ 2
α̃
41
,
(4.27)
β̃
µ̃ = −
α̃
!1/2
q
α̃2 − β̃ 2
.
(4.28)
Proof. Starting from the equation (4.25).
p
K−1 (α δ 2 + (x − µ)2 )
γ −1/2
p
exp[β(x − µ)]
fN IG (x) = √
2πδ −1/2 K− 1 (δγ) ( δ 2 + (x − µ)2 /α)
2
−1/2
p
p
2
2
α −β
K−1 (α δ 2 + (x − µ)2 )
p
p
=√
exp[β(x − µ)]
2πδ −1/2 K− 1 (δ α2 − β 2 ) ( δ 2 + (x − µ)2 /α)
2
q
−1/2
2
p
β̃ 2
α̃
−
2
2
K−1 ( α̃δ δ 2 + (x − µ)2 )
δ
δ
β̃
p
exp[ (x − µ)]
=√
q
2
2
2
2
δ
2πδ −1/2 K− 1 δ α̃δ2 − β̃δ2 ( δ + (x − µ) /(α̃/δ))
2
r
2 q
−1/2
x−µ
2
2
K−1 α̃ 1 + δ
δ
α̃ − β̃
β̃
q
=√
r
2 . exp[ δ (x − µ)]
2πK− 1
α̃2 − β̃ 2 δ 2
α̃
1 + x−µ
2
δ
r
q
−1/2
2
α̃2 − β̃ 2
K−1 α̃ 1 + x−µ
α̃
δ
β̃
r
exp[ (x − µ)]
=
q
q
−1/2
2
√
δ
p
2πδ π2
α̃2 − β̃ 2
1 + x−µ
exp − α̃2 − β̃ 2
δ
2 r
x−µ
q
1
+
K
α̃
h
δ
x−µ i 1
α̃
r
α̃2 − β̃ 2 + β̃(
.
= exp
)
2
πδ
δ
x−µ
1+ δ
By equation (4.12).
µ+
⇒
⇒
⇒
δβ
Rλ (δγ) = 0
γ
δ β̃
µ+ q
2
δ α̃δ2 −
β̃ 2
δ2
1
R− δ
2
s
α̃2 β̃ 2 − 2 =0
δ2
δ
q
δ β̃
µ+ q
R− 1 ( α̃2 − β̃ 2 ) = 0
2
α̃2 − β̃ 2
q
K 1 ( α̃2 − β̃ 2 )
δ β̃
2
q
=0
+
µ+ q
2
2
2
2
α̃ − β̃
K− 1 ( α̃ − β̃ )
2
42
q
K ( α̃2 − β̃ 2 )
δ β̃
q
=0
+
µ+ q
2
2
2
2
α̃ − β̃
K 1 ( α̃ − β̃ )
1
2
⇒
2
δ β̃
µ+ q
= 0.
2
2
α̃ − β̃
⇒
By equation (4.13).
⇒
⇒
⇒
⇒
(4.29)
δ
β 2δ2
Rλ (δγ) + 2 Sλ (δγ) = 1
γ
γ
s
s
β̃ 2 2
2
2
δ
α̃
α̃2 β̃ 2 β̃
δ
δ2
q
1 δ
R− 1 δ
+
−
− 2 =1
S
2
2
β̃ 2 − 2
α̃2
δ2
δ2
δ
δ
β̃ 2
α̃2
−
− δ2
δ2
δ2
δ2
q
q
δ 2 β̃ 2
δ2
2
2
q
S− 1 ( α̃2 − β̃ 2 ) = 1
R− 1 ( α̃ − β̃ ) +
2
2
2
2
α̃ − β̃
α̃2 − β̃ 2
q
q
q
2
2
2
2
2
1
3
α̃
−
β̃
)K
α̃
−
β̃
)
−
K
α̃2 − β̃ 2 )
(
(
(
K
1
−2
−2
δ2
δ 2 β̃ 2
2
q
q
=1
+
2
2
α̃2 − β̃ 2 α̃ − β̃
K−2 1 ( α̃2 − β̃ 2 )
2
q
q
q
2 − β̃ 2 )[1 + √ 1
2 − β̃ 2 ) − K 2 ( α̃2 − β̃ 2 )
1
1
]K
α̃
α̃
K
(
(
1
2
2
δ 2 β̃ 2
δ2
α̃2 −β̃ 2
2
q
q
=1
+
2 − β̃ 2
α̃
2
2
2
2
2
α̃ − β̃
K 1 ( α̃ − β̃ )
2
2
⇒
⇒
⇒
2 2
δ
1
δ β̃
q
q
+
=1
2
2
α̃2 − β̃ 2 α̃ − β̃
α̃2 − β̃ 2
β̃ 2
1
2
q
δ
+
=1
2
2
2
α̃2 − β̃ 2 (α̃ − β̃ ) 3
!3/2
q
δ̃ =
α̃2 − β̃ 2
α̃
.
(4.30)
Substituting equation (4.30) into (4.29), we can get equation (4.28).
In the following subsections, we implement the MCMM and the conditional Esscher transform measure for NIG-GARCH model. We assume that the asset return process y := {yt }t∈T
under the historical measure P are given by:
yt = mt + σt εt ,
(4.31)
43
εt ∼ NIG(α̃, β̃, δ̃, µ̃),
(4.32)
2
2
σt2 = α0 + α1 σt−1
ϕ(εt−1 ) + b1 σt−1
,
(4.33)
mt = r + λσt .
(4.34)
4.3.1 Risk Neutral Dynamic from MCMM under NIG-GARCH
The general case GH-GARCH model can be extended to the special case NIG-GARCH model
under the MCMM in the following.
Corollary 4.3. Suppose the asset return process y := {yt }t∈T satisfies equations (4.31)(4.34) under P . Under the risk neutral MCMM Qm the dynamics of the return are given
by:
yt = r − µ̃σt −
q
α̃2
−
β̃ 2
+
q
α̃2 − (β̃ + σt δ̃)2 + σt ε0t ,
ε0t ∼ NIG(α̃, β̃, δ̃, µ̃),
mshif t 2
2
σt2 = α0 + α1 σt−1
ϕ ε0t−1 + t−1 + b1 σt−1
,
σt−1
(4.35)
(4.36)
(4.37)
t
where mshif
is given by:
t
t
mshif
t
q
q
= r − mt − µ̃σt − α̃2 − β̃ 2 + α̃2 − (β̃ + σt δ̃)2 .
(4.38)
4.3.2 Risk Neutral Dynamic from Conditional Esshcer under NIG-GARCH
Under the conditional Esscher transform measure, the GH-GARCH model can also be extended to the NIG-GARCH model.
Corollary 4.4. Suppose the asset return process y := {yt }t∈T satisfies equations (4.31)(4.34) under P . Under the risk neutral conditional Esscher Qe the dynamics of the return
are given by:
yt = mt + σt (µ̃ + δ̃ν1t ) + σt δ̃ν2t ε0t ,
44
(4.39)
1
ν1t
ε0t ∼ NIG(α̃, β˜1t ,
, − ),
ν2t ν2t
(4.40)
where {β̃1t }, {ν1t }, {ν2t } are some {Ft }-predictable processes given by:
β̃1t = β̃ + θt δ̃σt ,
β̃1t
,
ν1t = q
2
α̃2 − β̃1t
ν2t = q
α̃
α̃2
−
2
β̃1t
(4.41)
(4.42)
32 .
(4.43)
Such that ε0t has zero conditional mean and unit conditional variance given Ft−1 , and θt is
the unique predictable solution of the following martingale equation:
q
α̃2
− (β̃ + δ̃σt + θt δ̃σt
)2
q
− α˜2 − (β̃ + θt δ̃σt )2 = µ̃σt + mt − r,
t ∈ T \{0}.
(4.44)
In this case we can determine an analytical form for the risk neutral Esscher parameter θt :
v
!
u
2
2
(r
−
m
−
σ
µ̃)
1
β̃
4
α̃
1u
t
t
θt = − −
−1 .
(4.45)
− t
2 2
2 σt δ̃ 2
σt δ̃
σt2 δ̃ 2 + (r − mt − σt µ̃)2
45
Chapter 5
Empirical Analysis for GARCH Model
In this chapter we examine the performance of the Gaussian-TGARCH and NIG-TGARCH
models described in the previous sections using the mean correcting martingale measure
(MCMM), the conditional Esscher transform and the second-order Esscher transform. Based
on the recent studies, the class of the GARCH models perform better than the traditional
Black-Scholes model and the TGARCH models outperform the standard GARCH model
in the context option pricing. As a result, we use the TGARCH model in our simulation
studies. In the following sections, we first describe the data set, then introduce the technique
used for option pricing. Finally, we present the simulation results.
5.1 Data description
We use European call options on the S&P 500 (Symbol:SPX) index to test our models. The
S&P 500 index is a value weighted index of the prices of 500 large-cap companies actively
traded in the United States. It is considered a bellwether for American economy. Hundreds
of billions of US dollars have been invested in the index. S&P 500 has been maintained
by the Standard & Poor’s, a division of McGraw-Hill. The components of the S&P 500
index are selected by a committee who intends to pick companies across various industries
to represent the United States economy. The market for S&P 500 index options is one of
the most active index options market in the world.
There are two sets of data used for the simulation. The first one consists of 54 European
call options on the S&P 500 index in market on April 18th, 2002. The strike prices range
from $975 to $1325 and options with maturities T =22,46,109,173 and 234 days. The average
option price is $56.94. The second one is the raw option data retrieved from STRICKNET
46
INC. The data set is sampled every Wednesday from Jan 7th, 2004 to Dec 29th, 2004. If
Wednesday is a holiday, then we take the subsequent trading day. We use the average of the
bid-ask quotes as the option observed prices and the closing pricing as the underlying asset
prices. This data set consists of 1145 European call options on the S&P 500 index in the
market. During the year of 2004, the S&P 500 index ranges from a minimum of 1075.79 to
a maximum of 1213.45, with average of 1131.66.
We divide the second option data into several categories based on maturity and moneyness. The day to maturity (DTM) is defined as the number of trading days to the expiration
time of the option; the moneyness, denoted as M0 , can be expressed as the ratio of the
strike price over the underlying stock price (M0 = K/S). A call option is said to be outof-the-money if the moneyness of the call option is greater than 1 (Mo > 1), and is said
to be in-the-money if its moneyness less than 1 (Mo < 1). According to the data selection
technique proposed by Dumas, Fleming and Whaley (1998), we exclude the option data with
DTM < 7 days, DTM > 200 days, the call option price < $0.5 and |100(M0 − 1)| > 0.1 for
2004 option data set. Many studies have further divided the option data into two additional
categories which are so called deep out-of-the-money and deep in-the-money. However, according to those studies, there is no standard boundary between out-of-the-money and deep
out-of-the-money or between in-the-money and deep in-the-money. In order to examine
closely the accuracy of option pricing results on different level of moneyness, we divide our
option data into seven intervals based on the values of Mo. We discard options with moneyness greater than 1.1 or less than 0.9. The option data has been also classified into four
groups by DTM. According to our classification, an option can be short-term maturity if it
has less than 40 trading days to expire, a medium-term maturity if the number of days to
maturity is between 40 and 80 days, a long-term maturity for the days to maturity between
80 and 180 days, or a very-long-term maturity if the option has 180 to 200 days to expire.
We also set restrictions for the daily volume and daily open interest on our option data
47
in order to eliminate inactive options. Only options with daily trading volume more than
200 in addition of at least 500 open interest will be considered in our data set. The number
of call options and the average option price corresponding to each category considered are
reported in Table 5.1 and Table 5.2.
Table 5.1: Number of Call option contracts in 2004
1
2
3
4
5
6
7
Mo
[0.9, 0.95]
[0.95, 0.975]
[0.975, 0.99]
[0.99, 1.01]
[1.01, 1.025]
[1.025, 1.05]
[1.05, 1.1]
All
0 < DT M ≤40
25
27
62
201
151
171
68
705
40 < DT M ≤80
5
6
19
75
37
57
76
275
80 < DT M ≤180
3
2
8
37
17
32
44
143
180 < DT M ≤ 200
0
1
4
7
2
0
8
22
All
33
36
93
320
207
260
196
1145
180 < DT M ≤ 200
0
93.6
71.98
64.11
56.25
0
27.11
52.71
All
85.67
51.8
34.94
24.05
12.5
8.47
6.88
19.02
Table 5.2: Average Call option prices in 2004
1
2
3
4
5
6
7
Mo
[0.9, 0.95]
[0.95, 0.975]
[0.975, 0.99]
[0.99, 1.01]
[1.01, 1.025]
[1.025, 1.05]
[1.05, 1.1]
All
0 < DT M ≤40
82.69
48.88
27.3
15.73
7.20
3.56
1.34
14.22
40 < DT M ≤80
88.22
52.75
40.99
30.94
19.95
11.82
4.51
20.41
80 < DT M ≤180
106.3
67.4
61.25
47.69
38.21
28.79
15.84
34.8
5.2 Simulation methodology
In this section, we introduce serval techniques used in the simulation. In many existing
studies, option prices have been computed by Monte Carlo simulation. We also implement
the Monte Carlo simulation here. Moreover, several techniques are applied to improve the
standard Monte Carlo simulation.
48
5.2.1 Monte Carlo Simulation
Monte Carlo simulation is a widely used tool for estimating derivative security. It was
proposed by Boyle (1997) to option pricing among many others. Monte Carlo method is
especially useful when one deals with path dependent asset prices and option payoffs. The
price of a derivatives contract in an arbitrage-free economy can be expressed as a discounted
expected payoffs in the risk neutral world. Hence, the Monte Carlo simulation is a natural
tool for approximating this expectation by the sample average. The commonly used Monte
Carlo simulation procedure for option pricing can be described as follows: first, simulate
sample paths for the underlying asset price; second, compute its corresponding option payoff
for each sample path; finally, average the simulated payoffs and discount the average to get
the Monte Carlo price of an option.
We implement the Monte Carlo method in two ways: (i) simulate sample paths for the
underlying asset price under the risk neutral measure Q and then follow the Monte Carlo
simulation procedure mentioned above to get the Monte Carlo option price; (ii) simulate
sample paths for the underlying asset price under the historical measure P and evaluate
option prices as a weighted average of the payoffs for each of the corresponding path, where
the weights are given by the Radon-Nikodym derivative evaluated for this Monte Carlo
simulated path. In addition, method (ii) is very useful when there is an explicit form of the
return dynamic under P but hardly to trace return dynamic under Q, thus making method
(i) difficult to use. Furthermore, method (ii) also reduces the variance of our estimators.
The two methods can be expressed in the following:
n
method(i) :
X
1
(Q)
Ĉt = e−r(T −t)
max(0, ST,i − K),
n
i=1
(5.1)
n
method(ii) :
X
1
dQ
(P )
Ĉt = e−r(T −t)
max(0, ST,i − K) (i),
n
dP
i=1
(5.2)
where T − t is the time to maturity, n is the number of the Monte Carlo simulation and K
is the strike price.
49
5.2.2 Empirical Martingale Simulation
Duan and Simonato (1998) introduced the Empirical Martingale Simulation (EMS) method
for asset prices in the risk neutral framework. This procedure proposed a simple modification
to the standard Monte Carlo simulation procedure for the prices of derivative securities.
The modification imposes the martingale property on the simulated sample paths of the
underlying asset price. In a standard Monte Carlo simulation, the martingale property
almost always fails in the simulated sample. Due to the well known fact that the standard
error of a Monte Carlo simulation is inverse proportional to the square root of the number
of simulated sample paths, if we increase the sample size, we can obtain a high degree
of accuracy. A less known difficulty related to the use of Monte Carlo simulation is the
occurrence of the simulated price violating rational option pricing bounds. This bound
violation could have serious implications. The EMS ensures that the price estimated by
simulation satisfies the rational option pricing bounds. Furthermore, the EMS reduces the
error and can be easily coupled with the standard variance reduction methods. The EMS
procedure can be expressed as:
Let t0 = 0, the current time, and the EMS procedure generates the EMS asset prices at
a sequence of future time points,t1 , t2 , ..., tm , using the following system:
Si∗ (tj , n) = S0
Zi (tj , n)
,
Z0 (tj , n)
(5.3)
where
Zi (tj , n) = Si∗ (tj−1 , n)
Ŝi (tj )
Ŝi (tj−1 )
,
(5.4)
n
X
1
Zi (tj , n).
Z0 (tj , n) = e−rtj
n
i=1
(5.5)
Note that Ŝi (t) is the ith simulated asset price at time t prior ro the EMS adjustment, and
Ŝi (t0 ) and Si∗ (t0 , n) are set equal to S0 . The adjustment steps can be understood as follows.
First, we take the standard simulated return from tj−1 to tj , i.e., Ŝi (tj )/Ŝi (tj−1 ), to create a
50
temporary asset price at time tj , i.e., Zi (tj , n). Second, we compute the discounted sample
average, Z0 (tj , n). Finally, we compute the EMS asset price at time tj by (5.3). After the
EMS correction, the simulation moves on to the next time point, and repeats the whole
process again.
5.2.3 Control Variates
The method of control variates is the most effective and applicable technique for reducing
the variance of Monte Carlo simulation. It can improve the efficiency of the Monte Carlo
simulation and exploit information about the errors in estimates of known quantities to
reduce the error in an estimate of an unknown quantity.
To describe the method, let Y1 , ..., Yn be outputs form n replications of simulation. Suppose that Yi are i.i.d and our objective is to estimate E(Yi ). On each replication we calculate
another output Xi along with Yi . Assume that the pairs (Xi , Yi ), i = 1, ..., n, are i.i.d and
that the expectation E(X) of the Xi is known. Then for any fixed b we can calculate
Yi (b) = Yi − b(Xi − E(X)),
(5.6)
from the ith replication and then compute the sample mean
n
1X
Ȳ (b) = Ȳ − b(X̄ − E(X)) =
(Yi − b(Xi − E(X))).
n i=1
(5.7)
This is a control variate estimator and the observed error X̄ − E(X) serves as a control in
estimating E(Y ). The optimal coefficient b∗ is given by
b∗ =
Cov(X, Y )
σY
ρXY =
,
σX
V ar(X)
(5.8)
In practice, it is hard to know Cov(X, Y ) and V ar(X). So we can estimate these two values
from the data. Replacing b∗ with b̂n , where the latter is given by
b̂n =
Pn
i=1 (Xi − X̄)(Yi −
Pn
2
i=1 (Xi − X̄)
51
Ȳ )
.
(5.9)
Then, we can rewrite equation (5.8) and (5.9) as:
Yi (b̂n ) = Yi − b̂n (Xi − E(X)),
(5.10)
n
1X
Ȳ (b̂n ) = Ȳ − b̂n (X̄ − E(X)) =
(Yi − b̂n (Xi − E(X))).
n i=1
(5.11)
We implement control variates technique in Monte Carlo simulation along with EMS. That
is, for European call options, we use their Black-Scholes counterparts in the control variate
Monte Carlo simulation. Firstly, we simulate the terminal stock price under the risk neutral
√
(i)
measure in Black-Scholes model by ST = S0 exp((r − 12 σ 2 )T + σ T Zi ), i = 1, ..., n, Zi ∼
N(0, 1). Then we can calculate the discounted payoff for each sample path in the BlackScholes model. Let Xi be the discounted payoff for sample path i in Black-Scholes and
E(X) is the Black-Scholes option price. Following the practice in Duan (1995), the BlackScholes option price and terminal stock price are computed using the stationary variance
α0 (1 − α1 − b1 − 0.5γ)−1 of the TGARCH model in the Black-Scholes closed-form formula.
The EMS method can be used to adjust the simulated stock price in the TGARCH model,
and then we use the adjusted simulated stock price to get the discounted payoff function for
each sample path. Let Yi be this discounted payoff function for sample path i in the GARCH
model. By equation (5.9) and (5.10), we can get new payoff function Yi (b̂n ) for each sample
path. Finally, we use equation (5.11) to get the option prices.
5.3 Simulation Results
In this section we present the simulation results for two data sets described in the previous
section. The first one consists of 54 European call options in the market on April 18th, 2002.
This is just one day of options. Four competing models examined for this data set are:
1. TGARCH(1,1) model based on Gaussian innovation;
2. TGARCH(1,1) model based on NIG-distributed innovation using mean correcting martingale measure approach;
52
3. TGARCH(1,1) model based on NIG-distributed innovation using conditional
Esscher transform approach;
4. TGARCH(1,1) model based on Gaussian innovation using Second order Esscher transform approach.
We describe two approaches to estimate the model parameters. The first approach consists
of using the historical S&P 500 return to estimate the parameters under the physical probability measure P . The maximum likelihood estimation (MLE) technique is used in the first
approach. The second approach is to estimate the risk neutral parameters minimizing the
root mean square error (RMSE) and relative root mean square error (%RMSE). The second
approach is referred to as in-sample estimation. We use RMSE and %RMSE to evaluate the
in-sample performance of each model. The RMSE and %RMSE are given by:
v
u
N
u1 X
RMSE θ = t
(C market − Ĉimodel (θ))2 ,
N i=1 i
v
!2
u
N
u1 X
market
model
C
−
Ĉ
(θ)
i
i
%RMSE θ = t
.
market
N i=1
Ci
Where N is the number of option contracts, C market is the option price observed from the
market and Ĉ model is the simulated price for the model considered. θ represents the parameter
set, θ = {α0 , α1 , b1 , λ, γ, α̃, β̃, θ2 }. α0 , α1 , b1 , λ, γ are TGARCH parameters. α̃, β̃ are NIGdistributed parameters. θ2 is the parameter from the second order Esscher transform. In
models 1 and 2, Ĉ model is derived by Monte Carlo method (i) (equation(5.1)), EMS method
and control variates technique. In models 3 and 4, Ĉ model is derived by Monte Carlo method
(ii) (equation(5.2)) and control variates technique. We set the number of the Monte Carlo
simulation M = 50, 000 for all four competing models.
The parameter estimation results for the TGARCH(1,1)-in-Mean models with conditional
distributions Gaussian and NIG are presented in Table 5.3 (Badsecu et al, 2010). The
estimation results are based on the first approach using daily closing prices of S&P 500 from
53
Jan 2th, 1988 to April 17th, 2002, for a total of 3,606 observation.
In Table 5.3, α0 , α1 , b1 , λ and γ are the TGARCH(1,1)-in-Mean model parameters. α̃, β̃
are the NIG-distributed parameters. Standard errors of the parameter estimates are shown
in parentheses. Skw and Kts denote point estimates of the skewness and excess kurtosis of
the standard residuals. l(θ̂) is the log likelihood evaluated at the MLE θ̂. Akaike information
criteria (AIC) and Bayes information criteria (BIC) are standard model selection criteria. A
smaller AIC or BIC, indicates a better model. NIG model is better, since it yields smaller
AIC and BIC.
Table 5.3: MLE results for Gaussian-TGARCH and NIG-TGARCH
Gaussian
1.11 ∗ 10−6
(3.1 ∗ 10−7 )
NIG
8.4 ∗ 10−7
(2.5 ∗ 10−7 )
α1
0.0076
(0.0057)
0.0093
(0.0068)
b1
0.9428
(0.0097)
0.944
(0.0099)
λ
0.0443
(0.0165)
0.042
(0.0167)
γ
0.0721
(0.0141)
0.0738
(0.0160)
α0
α̃
1.6893
(0.2375)
β̃
-0.1916
(0.0682)
Skw
Kts
l(θ̂)
AIC
BIC
0
0
-4697.3
9404.6
9435.6
54
-0.263
1.879
-4551.5
9117.0
9160.3
We use the parameter estimation results in Table 5.3 as the initial value in the second
approach. For Gaussian-2nd Esscher model, we set the initial value of θ2 = 0.25. Table 5.4
and Table 5.5 present the in-sample estimation performance based on RMSE and %RMSE.
Table 5.4: In-sample estimation performance for four competing models based on RMSE
α0
α1
b1
λ
γ
α̃
β̃
θ2
RMSE
Gaussian
1.134 ∗ 10−6
0.0148
0.9169
0.0029
0.1238
1.495
NIG-MCMM
8.458 ∗ 10−7
0.009
0.944
0.0623
0.074
1.5822
NIG-Esscher
1.202 ∗ 10−6
0.0071
0.9416
0.0028
0.0787
1.5318
-0.2404
-0.375
1.455
1.153
Gaussian-2nd Esscher
3.175 ∗ 10−7
0.000067
0.9425
0.1885
0.1026
-507.47
0.569
Table 5.5: In-sample estimation performance for four competing models based on %RMSE
α0
α1
b1
λ
γ
α̃
β̃
θ2
%RMSE
Gaussian
7.139 ∗ 10−7
0.0292
0.895
0.028
0.1437
0.0524
NIG-MCMM
1.442 ∗ 10−6
0.0134
0.942
0.0145
0.055
2.096
NIG-Esscher
1.424 ∗ 10−6
0.0025
0.9435
2.464 ∗ 10−5
0.0736
2.1535
-1.547
-1.1311
0.0434
0.0364
Gaussian-2nd Esscher
3.7 ∗ 10−7
0.0023
0.9427
0.2038
0.098
-634.82
0.0191
In Table 5.4 and Table 5.5, the in-sample parameter estimation results are quite different
from the results in Table 5.3. Of all the four competing models, the value of b1 is very close to
the historical one. The other parameter estimation results are different from the historical
results. In Table 5.4, the Gaussian-2nd Esscher RMSE is the smallest and the Gauissan
RMSE is the largest. Thus the Gaussian-2nd Esscher model performs best based on RMSE.
55
In Table 5.5, the Gaussian-2nd Esscher model has the smallest %RMSE and the Gaussian
model has the largest %RMSE. Therefore, the Gaussian-2nd Esscher model performs best
based on %RMSE.
The second data set we used is sampled every Wednesday from Jan 7th, 2004 to Dec
29th, 2004. This data set consists of 52 weeks. The three competing models examined for
this data set are:
1. TGARCH(1,1) model based on Gaussian innovation;
2. TGARCH(1,1) model based on NIG-distributed innovation using mean correcting martingale measure approach;
3. TGARCH(1,1) model based on NIG-distributed innovation using conditional
Esscher transform approach.
Two approaches are used to estimate the model parameters. The first approach consists of
using the MLE technique to estimate model parameters based on historical return of S&P 500
index. For the Gaussian innovation model, the estimation results use the historical returns
data from Jan 3th, 1988 to Jan 6th, 2004. For the NIG innovation model, we use the MLE
results in Table 5.3 and update the volatility to Jan 6th, 2004 to be the starting volatility.
Table 5.6 presents the estimation results of the Gaussian innovation model. Standard errors
of the parameter estimates are in parentheses.
Table 5.6: MLE results for Gaussian innovation
α0
α1
b1
λ
10.7404 ∗ 10−7
(2 ∗ 10−7 )
0.00631494
(0.005)
0.94328
(0.008)
0.0435936
(0.016)
56
γ
Skw
Kts
0.0761998
(0.013)
0
0
l(θ̂)
AIC
BIC
-5409.55
10829.1
10860.62
The second approach is to use the MLE of Gaussian innovation and NIG innovation
as the initial value to estimate the model parameters by minimizing RMSE. There are 52
weeks in this data set. First, we use the 1st week to the 26th week to estimate the model
parameters by minimizing RMSE. We refer to this as in-sample estimation, and then we use
the in-sample parameter estimation results to compute the RMSE in the 27th week. This is
referred to as out-of-sample RMSE. Following the same procedure, we can get the in-sample
estimation results from the 2nd week to the 27th week and out-of-sample RMSE of the 28th
week. We can also get the in-sample estimation results from the 3rd week to the 28th week
and out-of-sample RMSE of the 29th week. Therefore, we can obtain 26 sets of in-sample
estimation results and 26 out-of-sample RMSE.
Moreover, we use an updating scheme by constructing a series of volatilities using observed returns of 2004. We need to choose the starting volatility of each week. For example,
we use the estimated volatility on Jan 6th, 2004 as initial volatility σ0 for pricing the options
of the first week (Jan 7th, 2004). Then we update this volatility by using the corresponding
TGARCH specification up to Jan 13th, 2004 (one day before the trading day), and then use
this updated volatility as initial volatility σ0 to pricing the options of the second week (Jan
14th, 2004). We keep updating the volatility for every week to ensure the most accurate
initial volatility. We use the corresponding σ0 of each week for in-sample estimation and
out-of-sample RMSE. For example, if we want to get the in-sample-estimation from the 1st
week to the 26th week, the corresponding σ0 of the 1st week to the 26th week is used. If
we want to get the out-of-sample RMSE of the 27th week, the corresponding σ0 of the 27th
week is used.
In models 1 and 2, Ĉ model is derived by Monte Carlo method (i) (equation(5.1)), EMS
method and control variates technique. In model 3, Ĉ model is derived by Monte Carlo method
(ii) (equation(5.2)) and control variates technique. The number of the Monte Carlo simulation is M = 50, 000 for all three competing models.
57
The following tables are in-sample estimation results for the three competing models.
Table 5.7: In-sample estimation of the Guassian-TGARCH
Week
period
1-26
2-27
No. of
weeks
26
26
No. of
contract
532
529
3-28
4-29
26
26
534
540
5-30
6-31
26
26
550
559
7-32
8-33
26
26
567
586
9-34
26
597
10-35
11-36
26
26
597
602
12-37
13-38
26
26
597
599
14-39
15-40
26
26
599
605
16-41
17-42
26
26
608
605
18-43
19-44
26
26
605
607
20-45
21-46
26
26
605
615
22-47
23-48
26
26
614
620
24-49
26
627
25-50
26-51
26
26
616
620
α0
2.685 ∗ 10−6
2.374 ∗ 10−6
α1
0.00077
0.0024
b1
0.8326
0.8477
λ
1.2409
1.2468
γ
0.0576
0.0508
RMSE
1.5931
1.5416
2.141 ∗ 10−6
2.406 ∗ 10−6
0.0507
0.0593
0.8521
0.8493
1.2598
1.0712
0.0003
0.0025
1.4715
1.3746
2.08 ∗ 10−6
1.7506 ∗ 10−6
0.0074
0.0152
0.8655
0.8815
1.1592
1.1193
0.0445
0.0328
1.3349
1.3751
1.833 ∗ 10−6
1.797 ∗ 10−6
0.0193
0.049
0.8889
0.8773
0.9363
1.1164
1.3969
1.3756
1.8834 ∗ 10−6
0.0506
0.8724
1.1211
0.0343
5.605 ∗ 10−12
1.96 ∗ 10
2.072 ∗ 10−6
0.049
0.0037
0.8723
0.8833
1.079
0.7733
0.0036
0.0647
1.3729
1.3098
1.517 ∗ 10−6
1.251 ∗ 10−6
0.0322
0.0371
0.8854
0.8941
1.1312
1.2797
0.0144
0.0002
1.2312
1.2482
1.183 ∗ 10−6
9.233 ∗ 10−7
0.0192
0.0097
0.9008
0.9052
1.1626
1.3037
0.0207
0.0242
1.2813
1.256
8.241 ∗ 10−7
8.97 ∗ 10−7
0.0316
0.013
0.9102
0.9088
1.3067
1.207
0.0004
0.0231
1.2453
1.2144
9.356 ∗ 10−7
8.794 ∗ 10−7
0.0344
0.0275
0.9073
0.9075
1.1921
1.2444
0.0023
0.0077
1.1995
1.1783
7.774 ∗ 10−7
1.185 ∗ 10−6
0.0189
0.0355
0.9179
0.9081
1.1588
0.7873
0.0155
0.0189
1.165
1.253
8.09 ∗ 10−7
7.921 ∗ 10−7
0.0181
0.0151
0.922
0.9235
1.093
1.0673
0.0165
0.02
1.1954
1.178
8.143 ∗ 10−7
0.0311
0.9215
1.102
0.003
1.1664
8.306 ∗ 10−7
8.929 ∗ 10−7
0.0328
0.01
0.922
0.9238
1.1276
1.055
1.305 ∗ 10−11
0.0249
1.1532
1.159
−6
58
4.245 ∗ 10−13
1.3685
Table 5.8: In-sample estimation of the NIG-TGARCH with MCMM
Week
period
1-26
2-27
No. of
weeks
26
26
No. of
contract
532
529
α̃
2.4616
2.6254
β̃
-0.2561
-0.2813
3-28
4-29
26
26
534
540
1.2692
3.5049
-0.0628
-0.6384
5-30
6-31
26
26
550
559
6.9424
4.9366
-2.4878
-2.3388
7-32
8-33
26
26
567
586
4.6376
5.9588
-1.9117
-2.0589
9-34
10-35
26
26
597
597
9.785
9.2537
-6.1269
-6.0745
11-36
26
602
4.8491
-3.206
12-37
13-38
26
26
597
599
4.1035
0.7653
-2.8509
-0.2447
14-39
15-40
26
26
599
605
5.4575
5.4087
-2.5972
-4.0857
16-41
17-42
26
26
608
605
2.3136
3.3775
-1.339
-2.01
18-43
19-44
26
26
605
607
4.4187
1.9672
-2.6642
-0.9244
20-45
21-46
26
26
605
615
2.0477
0.665
-1.0852
-0.207
22-47
23-48
26
26
614
620
3.7469
2.4137
-2.4216
-1.0405
24-49
25-50
26
26
627
616
2.1434
2.319
-0.8117
-0.7941
26-51
26
620
2.3341
-0.7761
α1
0.011
0.0095
b1
0.9431
0.8946
λ
0.0049
0.0058
γ
0.073
0.1501
RMSE
1.8823
1.6294
0.0029
0.0079
0.898
0.8989
0.0096
0.0782
0.1615
0.127
1.5771
1.432
0.0091
0.0009
0.9419
0.9338
0.0069
0.0088
0.0754
0.0918
1.5656
1.4419
0.0061
0.0095
0.939
0.9446
0.0031
0.0014
0.0816
0.0732
1.4894
1.5573
2.375 ∗ 10−6
2.521 ∗ 10−6
0.0095
0.0111
0.9051
0.9013
0.1994
0.1752
0.091
0.0945
1.3703
1.3695
0.0102
0.929
0.0317
0.08
1.2697
2.014 ∗ 10−6
1.542 ∗ 10−6
0.0077
0.0097
0.923
0.9349
0.0446
0.0399
0.0876
0.0781
1.2214
1.2036
1.2602 ∗ 10−6
1.666 ∗ 10−6
0.0097
0.0079
0.9418
0.9338
0.0001
0.0374
0.0736
0.0747
1.3092
1.2368
0.0088
0.0104
0.929
0.9317
0.0025
0.0049
0.0818
0.0793
1.2396
1.2248
1.361 ∗ 10−6
1.519 ∗ 10−6
0.1001
0.0072
0.9387
0.9364
0.0025
0.0044
0.0742
0.0803
1.2315
1.2251
0.0091
0.0095
0.9332
0.9321
0.000011
0.0883
0.079
0.0789
1.1922
1.1921
0.0088
0.0079
0.9281
0.9415
0.0164
0.0013
0.084
0.0778
1.2169
1.2277
0.0036
0.0047
0.9509
0.9431
0.0462
0.06
0.0681
0.0781
1.1992
1.2092
0.0082
0.9437
0.0342
0.0746
1.2301
α0
1.262 ∗ 10−6
2.436 ∗ 10−6
2.369 ∗ 10−6
2.469 ∗ 10−6
1.319 ∗ 10−6
1.828 ∗ 10−6
1.46 ∗ 10−6
1.174 ∗ 10−6
1.776 ∗ 10−6
1.883 ∗ 10−6
1.652 ∗ 10−6
1.661 ∗ 10−6
1.534 ∗ 10−6
1.737 ∗ 10−6
1.194 ∗ 10−6
9.963 ∗ 10−7
1.099 ∗ 10−6
1.058 ∗ 10−6
59
Table 5.9: In-sample estimation of the NIG-TGARCH with Esscher transform
Week
period
1-26
2-27
No. of
weeks
26
26
No. of
contract
532
529
α̃
0.5392
1.6635
β̃
0.0313
0.1385
3-28
4-29
26
26
534
540
1.2752
0.6851
-0.0841
0.1092
5-30
6-31
26
26
550
559
2.3741
0.5641
-0.5858
-0.0201
7-32
8-33
26
26
567
586
0.4407
0.6355
-0.0683
-0.1605
9-34
10-35
26
26
597
597
0.2764
0.4224
0.0398
0.0648
11-36
26
602
0.3561
-0.0101
12-37
13-38
26
26
597
599
1.0718
0.3494
-0.2108
0.0108
14-39
15-40
26
26
599
605
0.6523
1.4011
0.0837
-0.1818
16-41
17-42
26
26
608
605
1.0055
1.4482
-0.1643
-0.252
18-43
19-44
26
26
605
607
1.9829
0.8572
-0.2833
-0.0888
20-45
21-46
26
26
605
615
1.9653
1.7533
-0.5268
-0.1565
22-47
23-48
26
26
614
620
1.7371
2.0898
0.1005
-0.4327
24-49
25-50
26
26
627
616
1.7909
1.5252
-0.5277
-0.3409
26-51
26
620
1.8137
-0.3216
α0
2.125 ∗ 10−6
1.806 ∗ 10−6
2.537 ∗ 10−6
1.771 ∗ 10−6
2.891 ∗ 10−6
2.535 ∗ 10−6
2.276 ∗ 10−6
2.953 ∗ 10−6
1.607 ∗ 10−6
1.601 ∗ 10−6
1.596 ∗ 10−6
2.388 ∗ 10−6
1.55 ∗ 10−6
1.415 ∗ 10−6
1.973 ∗ 10−6
2.114 ∗ 10−6
1.905 ∗ 10−6
1.777 ∗ 10−6
1.618 ∗ 10−6
1.932 ∗ 10−6
1.096 ∗ 10−6
1.151 ∗ 10−6
1.856 ∗ 10−6
2.045 ∗ 10−6
1.84 ∗ 10−6
1.498 ∗ 10−6
60
α1
0.0104
0.0084
b1
0.9118
0.9243
λ
0.234
0.2319
γ
0.0765
0.0752
RMSE
1.7343
1.6917
0.0082
0.0083
0.9003
0.93
0.1771
0.2184
0.0967
0.0736
1.6005
1.5354
0.0035
0.0081
0.901
0.908
0.0704
0.1475
0.1152
0.0899
1.4461
1.4541
0.0094
0.0083
0.9262
0.9073
0.045
0.0534
0.0784
0.0937
1.4588
1.4143
0.0085
0.0095
0.9252
0.9294
0.2107
0.2217
0.0785
0.0727
1.3665
1.4037
0.0079
0.9355
0.0854
0.0759
1.3441
0.0057
0.0074
0.9049
0.9297
0.1051
0.1421
0.1043
0.0791
1.2611
1.2692
0.0065
0.0065
0.9314
0.9096
0.206
0.1432
0.0795
0.0992
1.3184
1.3084
0.0083
0.0069
0.9072
0.9131
0.1358
0.1209
0.0942
0.0967
1.289
1.2752
0.0062
0.0073
0.9169
0.9258
0.1143
0.1048
0.0989
0.0858
1.2877
1.2781
0.0089
0.0093
0.9169
0.9427
0.0546
0.0609
0.098
0.0735
1.254
1.323
0.0082
0.0088
0.9329
0.9108
0.2239
0.1147
0.0741
0.0985
1.277
1.2673
0.0081
0.0085
0.9122
0.9181
0.0586
0.0775
0.1012
0.0925
1.2535
1.2538
0.0091
0.9284
0.0712
0.0849
1.2777
Table 5.7 presents the in-sample estimation results of the Gaussian-TGARCH model.
The estimation results are quite different from results in Table 5.6. Comparing with Table
5.6, some estimates of α0 are not just different, but of a different order of magnitude. All the
estimates of b1 are small and three estimated values of γ are very small. The most important
difference is the estimated value of λ. Almost all the estimated values of λ are greater than
1, whereas the value of λ from the MLE is very small.
Table 5.8 reports the in-sample estimation results of the NIG-TGARCH with MCMM
model. The estimates of α̃ and β̃ are quite different from the MLE in Table 5.3, but the
estimates of β̃ are still have same sign with MLE. Most of the α0 values have smaller order
of magnitude and the other parameters are a little bit different from the MLE.
Table 5.9 shows the in-sample estimation of the NIG-TGARCH with Esscher transform
model. The estimates of α̃ and β̃ are not much different from the MLE, but some estimates
of β̃ have different sign with MLE. Most of the α0 values have smaller order of magnitude.
The estimated values of λ are larger than MLE. The values of b1 and γ are close to the MLE
results.
Figure 5.1 presents the Boxplot of in-sample RMSE for the three competing models.
From this boxplot, we can see that the Gaussian-TGARCH model performs better than the
other two models. The minimum value for the Gaussian-TGARCH is the smallest and for the
NIG-TGARCH with conditional Esscher transform is the largest. The maximum value for
the Gaussian-TGARCH is the smallest and for the NIG-TGARCH with conditional Esscher
transform is the largest. There is one outlier for the NIG-TGARCH with MCMM model and
one outlier for the NIG-TGARCH with conditional Esscher transform model. The outlier
for the NIG-TGARCH with MCMM model is larger than that of the NIG-TGARCH with
conditional Esscher transform model. The distribution of RMSE for the Gaussian-TGARCH
model is more symmetric. Furthermore, the distribution of RMSE for NIG-TGARCH with
MCMM and NIG-TGARCH with conditional Esscher transform models are skewed to the
61
right.
Figure 5.1: Boxplot of in-sample RMSE for the three competing models
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
Gaussian
NIG−MCMM
NIG−Ess
Figure 5.2 shows the Black-Scholes implied volatilities of the market option prices and
the other three models. From this plot, we can see that the implied volatility of the NIGTGARCH with MCMM model performs better than the other two models for moneyness
between 0.9 and 0.95. The figure also shows that the implied volatility of the GaussianTGARCH prcing model outperforms both NIG-TGARCH with MCMM model and NIGTGARCH with conditional Esscher transform model for out of the money options. Furthermore, the NIG-TGARCH with MCMM model performs better than the NIG-TGARCH with
Esscher transform for in the money options and they do not have much difference for the
out of money options.
62
Figure 5.2: Black-Scholes implied volatilities for the three competing models
0.22
Market
Gaussian
NIG−Ess
NIG−MCMM
0.2
Implied Volatility
0.18
0.16
0.14
0.12
0.1
0.9
0.95
1
1.05
Moneyness
1.1
1.15
Table 5.10 reports the out-of-sample RMSE performance for the three competing models.
We use the in-sample estimation results in Table 5.7, Table 5.8 and Table 5.9 to compute
RMSE in Table 5.10. For example, we use the estimation results of week period from 1 to
26 for the Gaussian-TGARCH (Table 5.7) as the initial value, and then we use this initial
value to compute the RMSE of the Gaussian-TGARCH for week 27. We can also compute
the RMSE for NIG-MCMM and NIG-Esscher using the same approach. Moreover, we can
use the estimation results of week period from 2 to 27 as the initial value and compute the
RMSE for week 28 for the three competing models. Therefore, we can obtain the out-ofsample RMSE results from week 27 to 52 for the three competing models.
63
Table 5.10: Out-of-sample RMSE for the three competing models
Week
period
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
No. of No. of
weeks contract
1
24
1
24
1
27
1
26
1
29
1
22
1
29
1
28
1
18
1
22
1
20
1
28
1
18
1
22
1
16
1
24
1
25
1
29
1
25
1
24
1
20
1
33
1
23
1
10
1
26
1
20
Gaussian
1.3528
1.3559
1.3995
1.5415
2.3955
1.878
0.9865
0.9326
0.8025
0.8116
0.5165
1.6993
1.7106
1.2203
1.1356
1.2111
0.3441
0.5883
0.782
1.8002
0.6642
0.6643
0.8337
1.0515
0.8561
1.6257
64
NIG-MCMM
1.1014
1.4918
1.3164
1.7097
1.64
1.1696
0.8404
1.0361
0.4673
0.597
0.7026
1.8464
1.7704
1.5499
0.7995
1.2526
0.5677
1.1428
0.7104
1.4154
1.1989
1.6166
1.153
0.4593
1.0863
1.4362
NIG-Esscher
0.6292
1.3568
1.148
1.8047
1.8765
1.0328
0.7981
0.6884
0.6823
0.9328
0.7816
2.121
1.7478
1.4941
1.0399
1.2855
0.5486
1.0661
0.9342
1.7819
0.9152
1.1912
1.1512
1.247
1.5001
1.4119
Figure 5.3: Boxplot of out-of-sample RMSE for the three competing models
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
Gaussian
NIG−MCMM
NIG−Ess
Figure 5.3 presents the boxplot of out-of-sample RMSE for the three competing models.
From this boxplot, the Gaussian-TGARCH model has the smallest RMSE, but it still has
the largest RMSE. Thus the spread of the RMSE distribution is too wide. The out-of-sample
RMSE performance for the Guassian-TGARCH is not stable enough. For NIG-TGARCH
with MCMM and NIG-TGARCH with conditional Esscher transform models, the minimum
value of the NIG-TGARCH with MCMM is smaller and the maximum value of the NIGTGARCH with MCMM is also smaller. The spread of the RMSE distribution for the two
models is close. Therefore, the NIG-TGARCH with MCMM model performs better than
the NIG-TGARCH with conditional Esscher transform model in out-of-sample.
We would expect the approach that estimates the risk-neutral parameter directly from
option prices to work better than the approach based on the time series of asset returns for
several reasons. First, option prices contain forward-looking information over and beyond
historical returns, and thus using option price to find parameters can have an important
advantage simply from the perspective of the data used. Second, when using maximum
65
likelihood to estimate parameters under the physical measure, it is clear that the likelihood
function is quite different from the RMSE. Thus we compare the out-of-sample RMSE of
MLE and risk-neutral parameter for the three competing models.
These results are reported in Figure 5.4, Figure 5.5 and Figure 5.6. From these plots,
we can see that risk-neutral estimators perform better than MLE for the three competing
models. For all these three models, the minimum RMSE value of risk-neutral estimator is
smaller than that of MLE and the maximum RMSE value of risk-neural estimator is also
smaller than that of MLE. The distribution of RMSE using risk-neutral estimator is less
spread than that of RMSE using MLE.
Figure 5.4: Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for the
Gaussian-TGARCH model
3.5
3
2.5
2
1.5
1
0.5
risk−neutral esitmator
MLE
66
Figure 5.5: Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for the
NIG-TGARCH with MCMM model
4
3.5
3
2.5
2
1.5
1
0.5
risk−neutral esitmator
MLE
Figure 5.6: Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for the
NIG-TGARCH with Esscher model
4
3.5
3
2.5
2
1.5
1
0.5
risk−neutral esitmator
MLE
67
Table 5.11: Out-of-sample RMSE using MLE for the three competing models
Week
period
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
No. of No. of
weeks contract
1
24
1
24
1
27
1
26
1
29
1
22
1
29
1
28
1
18
1
22
1
20
1
28
1
18
1
22
1
16
1
24
1
25
1
29
1
25
1
24
1
20
1
33
1
23
1
10
1
26
1
20
Gaussian
2.2416
3.113
2.2388
3.4065
1.9155
1.1671
1.5924
2.1869
1.3153
2.4573
2.0647
1.5917
1.53
1.1077
0.5088
0.8803
0.7603
0.8167
0.6201
1.6481
0.5188
0.6779
1.0941
1.4682
1.2645
2.2932
68
NIG-MCMM
2.67
3.7391
2.6789
4.0246
1.9845
1.0938
1.6432
2.3513
1.681
2.7847
2.5547
1.5464
1.374
0.7724
0.7132
0.8933
1.0218
0.9393
0.9607
2.1811
1.0937
1.0772
1.5042
2.272
1.8035
3.0526
NIG-Esscher
2.3962
3.6244
2.5146
3.8937
1.5819
0.9877
1.586
2.0268
1.5994
2.1372
2.2362
1.1451
1.1295
0.7346
0.6802
0.6945
1.065
0.8794
1.044
2.0841
0.9699
0.9882
1.3138
1.963
1.3641
2.9119
Table 5.11 presents the RMSE results using MLE for the three competing models. We
use the same initial values for each week. For example, when we compute the Gaussian
RMSE for week 27, we use the MLE results in Table 5.6. We still use the same MLE results
for Gaussian RMSE in week 28. We just need to update the σt from the 1st week to the
27th week and use the volatility in the 27th week as our initial σ0 to compute the RMSE.
5.4 Conclusions
We conclude that the choice of the risk neutral measure is crucial for option pricing. There are
five risk neutral measures introduced in this thesis: Duan’s LRNVR, the mean correcting
martingale measure, the conditional Esscher transform method, the second order Esscher
transform method and the variance dependent pricing kernel. Three variance reduction
simulation methodologies are used: the Monte Carlo simulation, the Empirical Martingale
Simulation and the control variates. We examine two S&P 500 option data sets. Four
competing models are tested for S&P 500 options in 2002, we found that the GaussianTGARCH with second order Esscher transform model performs better than the other three
models for in-sample performance. Three competing models are further tested for S&P 500
options in 2004, and they show that the Gaussian-TGARCH model works better than the
other two models for in-sample performance and the NIG-TGARCH model with MCMM
works better than the other two models for the out-of-sample performance. For all these
three models, the RMSE of risk-neutral estimators performs better than that of MLE.
69
Appendix A
First Appendix
Martingales
Definition: Given a probability space (Ω, F , P ) with a filtration Ft , a cadlag process (Mt )t∈[0,T ]
is a martingale if M is adapted to Ft , E[|Mt|] is finite for t ∈ [0, T ] and
E[Ms |Ft ] = Mt , ∀s > t.
This is a crucial concept to establish the non-arbitrage pricing theory in mathematical finance. For Levey processes, different martingales can be constructed from their independent
increments property.
Brownian motion
Definition: Let (Ω, F , P ) be a probability space. For each ω ∈ Ω, suppose there is a continuous function Wt of t ≥ 0 that satisfies W0 = 0 and that depends on ω. Then Wt , t ≥ 0, is
a Brownian motion if for all 0 = t0 < t1 < · · · < tm the increments
Wt1 = Wt1 − Wt0 , Wt2 − Wt1 , . . . , Wtm − Wtm−1
are independent and each of these increments is normally distributed with
E[Wti+1 − Wti ] = 0
V ar[Wti+1 − Wti ] = ti+1 − ti
Itô’s process
Definition: Let Wt be a Brownian motion and Ft an associated filtration. An Itô’s process
is defined as:
Xt = X0 +
Z
t
µ(s)ds +
0
70
Z
t
σ(s)dWs ,
0
where µ(s), σ(s) is adapted stochastic processes.
Modified Bessel Function of the Third Kind with Index λ
The integral presentation of the modified Bessel function of the third kind with index λ can
be found in Barndorff-Nielsen et al. (1981),
1
Kλ (x) =
2
Z
0
∞
x
y λ−1 exp(− (y + y −1))dy,
2
Moreover, Kλ (x) = K−λ (x) and K−1/2 (x) = K1/2 (x) =
Normal distribution
pπ
2
x > 0.
x−1/2 e−x .
In probability theory, the normal (or Gaussian) distribution is a continuous probability
distribution that has a bell-shaped density function, known as the Gaussian function or
informally the bell curve
1
1
f (x) = √ e− 2
σ 2π
x−µ
σ
2
,
− ∞ < x < ∞.
The parameter µ is the mean and σ 2 is the variance, σ is known as the standard deviation.
The distribution with µ = 0 and σ 2 = 1 is called the standard normal distribution or the
unit normal distribution. The moment generating function of x is given by:
1
Mx (t) = exp(µt + σ 2 t2 ),
2
t ∈ R.
The cumulant function of x is:
1
κx (t) = logE(ext ) = logMx (t) = µt + σ 2 t2 ,
2
t ∈ R.
If x ∼ N(0, 1), the moment generating function and cumulant function are:
Mx (t) = exp
1
κx (t) = t2 ,
2
1 2
t ,
2
t ∈ R.
t ∈ R.
71
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