Risk Neutral Measures and GARCH Model Calibration

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2012-09-24 Risk neutral measures and GARCH model calibration

LI, SHENG

LI, SHENG. (2012). Risk neutral measures and GARCH model calibration (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/27669 http://hdl.handle.net/11023/221 master thesis

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Risk neutral measures and GARCH model calibration

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 cali-

bration” 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-of- sample 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 ...... ii Acknowledgements ...... iii TableofContents...... iv ListofTables ...... v ListofFigures...... vi 1 Introduction...... 1 2 Black-ScholesandGARCHModel ...... 7 2.1 Black-Scholesmodel ...... 7 2.2 GARCH-in-MeanModel ...... 10 2.3 GARCHModelExtensions...... 11 3 RiskNeutralMeasuresforGARCHModel ...... 14 3.1 Duan’s locally risk neutral valuation relationship ...... 15 3.2 MeanCorrectingMartingaleMeasure ...... 16 3.3 ConditionalEsscherTransformMethod ...... 18 3.4 A discrete version of the Girsanov change of measure ...... 22 3.5 Second order Esscher Transform Method ...... 26 3.6 VarianceDependentPricingKernel ...... 30 4 GeneralizedHyperbolicGARCHModel...... 35 4.1 Risk Neutral Dynamic from MCMM under GH-GARCH ...... 37 4.2 Risk Neutral Dynamic from Conditional Esshcer under GH-GARCH ..... 38 4.3 Special Case of Generalized Hyperbolic distribution ...... 41 4.3.1 Risk Neutral Dynamic from MCMM under NIG-GARCH ...... 44 4.3.2 Risk Neutral Dynamic from Conditional Esshcer under NIG-GARCH 44 5 EmpiricalAnalysisforGARCHModel ...... 46 5.1 Datadescription ...... 46 5.2 Simulationmethodology ...... 48 5.2.1 Monte Carlo Simulation ...... 49 5.2.2 Empirical Martingale Simulation ...... 50 5.2.3 ControlVariates ...... 51 5.3 SimulationResults ...... 52 5.4 Conclusions ...... 69 A FirstAppendix ...... 70

iv List of Tables

5.1 NumberofCalloptioncontractsin2004 ...... 48 5.2 Average Call option prices in 2004 ...... 48 5.3 MLE results for Gaussian-TGARCH and NIG-TGARCH ...... 54 5.4 In-sample estimation performance for four competing models based on RMSE 55 5.5 In-sample estimation performance for four competing models based on %RMSE 55 5.6 MLE results for Gaussian innovation ...... 56 5.7 In-sample estimation of the Guassian-TGARCH ...... 58 5.8 In-sample estimation of the NIG-TGARCH with MCMM ...... 59 5.9 In-sample estimation of the NIG-TGARCH with Esscher transform ..... 60 5.10 Out-of-sample RMSE for the three competing models ...... 64 5.11 Out-of-sample RMSE using MLE for the three competing models ...... 68

v List of Figures and Illustrations

5.1 Boxplot of in-sample RMSE for the three competing models ...... 62 5.2 Black-Scholes implied volatilities for the three competing models ...... 63 5.3 Boxplot of out-of-sample RMSE for the three competing models ...... 65 5.4 Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for theGaussian-TGARCHmodel...... 66 5.5 Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for theNIG-TGARCHwithMCMMmodel...... 67 5.6 Boxplot of out-of-sample RMSE between risk-neutral estimator and MLE for the NIG-TGARCH with Esscher model ...... 67

vi 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 Au- toregressive Conditional Heteroskedaticity (GARCH) models.

Stochastic volatility models were ﬁrst introduced by Hull and White (1987). The con- tribution 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 pos-

sible 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 Hes- ton 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 Exponen- tial 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), General- ized 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 ex-

ample, 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 in- stance, 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 in- vestigate 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 Gaussian-

TGARCH 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 Black-

Scholes 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 = µStdt + σStdWt. (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 S = S exp((µ σ2)t + σW ). (2.2) t 0 − 2 t

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)) = f (t, X(t))dt + f (t, X(t))dX(t)+ f (t, X(t))dX(t)dX(t). t x 2 xx

Setting f(t, X(t)) = log St and X(t)= St,

dSt 1 1 d log St = 2 dStdSt, St − 2 St

2 2 2 2 2 Since dtdt =0, dtdWt = 0 and dWtdWt = dt, dStdSt = µ St dtdt+2µσSt dtdWt+σ St dWtdWt =

2 2 σ St dt and substituting (2.1) into the above equation we have:

µStdt + σStdWt 1 2 d log St = σ dt St − 2

7 1 =(µ σ2)dt + σdW . − 2 t

Integration both sides it follows that

1 log S log S =(µ σ2)t + σW . t − 0 − 2 t

Therefore

1 S = S exp((µ σ2)t + σW ). t 0 − 2 t

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 W is a P -Brownian motion and X is an - t s F previsible process, then there exists a measure Q such that

1. Z = exp( t X dW 1 t X2ds), t − 0 s s − 2 0 s Q R t R 2. Wt = Wt + 0 Xsds. R Assume that

T EP [ X2Z2ds] < . s s ∞ Z0 Then Q deﬁned by

dQ = Z . dP T

Q is an equivalent probability measure w.r.t P under which Wt is also a Brownian motion.

From equation (2.2) and using the Girsanov Theorem

1 S = S exp((µ σ2)t + σW ) t 0 − 2 t

8 1 = S exp((r σ2 + µ r)t + σW ) 0 − 2 − t 1 (µ r) = S exp((r σ2)t + σ(W + − t)) 0 − 2 t σ 1 t (µ r) = S exp((r σ2)t + σ(W + − ds)) 0 − 2 t σ Z0 1 = S exp((r σ2)t + σW Q. 0 − 2 t

µ r Q Here Xs in the Girsanov theorem is the constant −σ . Moreover, Wt is the Brownian motion under the risk neutral measure. The call option price is computed by the formula:

r(T t) Q C(S, t)= e− − E [max(S K, 0)], T −

Thus we can obtain the famous Black-Scholes formula for European option:

r(T t) C(S, t)= SN(d ) Ke− − N(d ), 1 − 2 x 1 1 y2 N (x)= e− 2 dy, √2π −∞ Z 1 2 log(S/K)+(r + 2 σ )(T t) d1 = − , σ√T t −1 2 log(S/K)+(r 2 σ )(T t) d2 = − − , σ√T t − we may also write d = d σ√T t. The price of a corresponding put option based on 2 1 − − put-call parity is:

r(T t) P (S, t)= C(S, t)+ Ke− − 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, , ,P ) to model uncertainty. P is the historical F {Ft} (physical) measure and = , t = 0, 1, ...T (T < ), is a filtration, or a family F {Ft} ∞ 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 = . We assume the following general GARCH(p,q) model for the return {∅ } FT F

yt := log(St/St 1), where St is the stock price at time t. −

yt = mt + σtεt, (2.3) p q 2 2 2 σt = α0 + αiσt iϕ(εt i)+ biσt i. (2.4) − − − i=1 i=1 X X There are some assumptions we make:

εt 0 t T is a sequence of independent and identically distributed (i.i.d) ran- • { } ≤ ≤ dom variables with common distribution D(0, 1); the mean and variance for

the distribution is 0 and 1;

the conditional mean return m is assumed to be an -predictable process. • t Ft

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 ε on the • · t 2 conditional variance σt . This is called the news impact curve;

α , α and b are the coeﬃcients of the GARCH model, where α > 0 and • 0 1 1 0 α , b 0, and these parameters and function ϕ( ) are such that the conditional 1 1 ≥ · variance dynamics are covariance stationary.

10 Furthermore, throughout the thesis, we assume the conditional cumulant function of εt given

t 1 under P exists and is given by: F −

P P uεt κε (u) = log E [e t 1] < , u R. (2.5) t |F − ∞ ∈

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 + σtεt, (2.6)

2 2 2 σt = α0 + α1σt 1ϕ(εt 1)+ b1σt 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 t 1], (2.9) |F − 2 σt = V ar[yt t 1]. (2.10) |F −

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

2 consider p=q=1 and ϕ(εt 1) = εt 1 in equation (2.4), we can get the conditional variance − −

11 dynamic for the basic GARCH model proposed by Bollerslev (1986).

2 2 2 2 σt = α0 + α1σt 1εt 1 + b1σt 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:

εt 1 + γεt 1 ht = α0 + α1 | − | − + b1ht 1. (2.12) σt 1 − − 2 Here ht = log σt . 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 GJR-

GARCH 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 2 2 σt = α0 +(α1 + γI(εt 1 < 0))εt 1σt 1 + b1σt 1. (2.13) − − − −

Here γ is the positive parameter and I(εt 1 < 0) is the indicator function. The stationary − γ covariance property requires α1 + b1 + < 1. Depending on whether εt 1 is above or below 2 − 12 2 2 zero, εt 1 have different effects on the conditional variance σt . If there is good news, εt 1 0 − − ≥ 2 such that I(εt 1 < 0) = 0, the total effect is α1εt 1 on the next period conditional variance. − − 2 If there is bad news, εt 1 < 0 such that I(εt 1 < 0) = 1, the total effect is (α1 + γ)εt 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:

2 d d σt = α0 + α1( εt 1 + γεt 1) + b1σt 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

2 asymmetric behaviour in the volatility. When we set ϕ(εt 1)=(εt 1 γ) and p = q = 1 in − − − equation (2.4), we can get the conditional variance in the following form:

2 2 2 2 σt = α0 + α1σt 1(εt 1 γ) + b1σt 1. (2.15) − − − −

Here parameter γ is positive and the stationary covariance constraint for NGARCH model

2 is α1(1 + γ )+ 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 S˜t be the

rt discounted stock price at time t, so S˜t = e− St.

Deﬁnition 3.1. A probability measure Q is an equivalent martingale measure with respect to P if:

Q P ( B , Q(B)=0 P (B)=0); • ≈ ∀ ∈F ⇔ the discounted price process S˜ is a martingale under Q with respect to , that • t Ft Q is E S˜t t 1 = S˜t 1. |F − − h i Remark 3.1. The martingale condition for the discounted stock price can be rewritten as:

Q yt r E e t 1 = e . (3.1) |F − h i Proof. From second condition in Deﬁnition 3.1, we have

Q E S˜t t 1 = S˜t 1 |F − − Q rth i r(t 1) E e− St t 1 = e− − St 1 ⇔ |F − − h i Q St r E t 1 = e ⇔ "St 1 F − # − Q yt r E e t 1 = e . ⇔ |F − h i

14 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 t 1 is normally distributed under Q; • |F −

Q St r E t 1 = e ; • St−1 |F − h i Q St P St V ar log( ) t 1 = V ar log( ) t 1 almost surely with respect to • St−1 |F − St−1 |F − measureh P . i h i

From the previous proof, the second condition in LRNVR is the same martingale condi-

tion as for equation (3.1). In the deﬁnition of LRNVR, the conditional variance of log( St ) St−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

2 setup, we assume εt N(0, 1), and then yt t 1 N(mt, σt ). Therefore, the return dynamic ∼ |F − ∼ under P are equations from (2.6) to (2.8). We can specify as follows:

yt = mt + σtεt, (3.2)

15 2 2 2 σt = α0 + α1σt 1ϕ(εt 1)+ b1σt 1, (3.3) − − −

mt = r + λσt. (3.4)

Theorem 3.1. Suppose the asset return process y := yt t satisﬁes (3.2)-(3.4) under P . { } ∈T Under the risk neutral LRNVR QLRNV R the dynamics of the return are given by:

1 2 y = r σ + σ ε0 , t − 2 t t t

ε0 N(0, 1), t ∼ 2 2 1 2 σt = α0 + α1σt 1ϕ(εt0 1 σt 1 λ)+ b1σt 1. − − − 2 − − −

1 Here εt0 are i.i.d distributed and if ϕ(εt0 1 2 σt 1 λ) is given by: − − − − 1 1 1 γI(εt0 1 σt 1 λ< 0) 2 2 − ϕ(εt0 1 σt 1 λ)=(εt0 1 σt 1 λ) (1 + − − − ), − − 2 − − − − 2 − − α1 the conditional variance of return process follows a TGARCH model under the risk neutral

measure QLRNV R:

2 1 1 2 2 2 σt = α0 +(α1 + γI(εt0 1 σt 1 λ< 0))(εt0 1 σt 1 λ) σt 1 + b1σt 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

shift distributed. The main idea is to identify the quantity mt , which represents the shift to the conditional mean return m such that the discounted stock price is an -martingale t {Ft} under probability measure Qm. Consequently, under Qm, we have:

shift y = m + m + σ ε0 , ε0 D(0, 1). (3.5) t t t t t t ∼ 16 shift We implement MCMM in our approach, and mt will be determined in the following theorem.

Theorem 3.2. Suppose the asset return process y := yt t satisﬁes (3.2)-(3.4) under P . { } ∈T Under the risk neutral MCMM Qm the dynamics of the return are given by:

P y = r κ (σ )+ σ ε0 , t − εt t t t

εt0 D(0, 1), ∼ shift 2 2 mt 1 2 σt = α0 + α1σt 1ϕ(εt0 1 + − )+ b1σt 1. − − σt 1 − − Proof. Using martingale equation (3.1):

(m) Q yt r E [e t 1] = e |F − (m) shift 0 Q mt+mt +σtεt r E [e t 1] = e ⇔ |F − shift (m) 0 mt+mt Q σtεt r e E [e t 1] = e ⇔ |F − shift Q(m) mt+mt +κ 0 (σt) r e εt = e ⇔ m +mshift+κP (σ ) r e t t εt t = e ⇔ m + mshift + κP (σ ) = r. ⇔ t t εt t

shift Thus, the shift term mt is :

mshift = r m κP (σ ). (3.6) t − t − εt t

We use equation (3.5) and (3.6) to obtain the asset return dynamic in Theorem 3.2. Further- more, 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 εt0 :

shift mt εt0 = 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

shift shift shift mt−1 m m γI(εt0 1 + σ < 0) t 1 t 1 2 − t−1 ϕ(εt0 1 + − )=(εt0 1 + − ) (1 + ). − σt 1 − σt 1 α1 − − We can get the TGARCH conditional variance, that is :

shift shift 2 mt 1 mt 1 2 2 2 σt = α0 +(α1 + γI(εt0 1 + − < 0))(εt0 1 + − ) σt 1 + b1σt 1. − σt 1 − σt 1 − − − − If we assume ε N(0, 1) under historical measure P , then κP (σ ) = 1 σ2. The return t ∼ εt t 2 t dynamic under the risk neutral measure Qm is:

1 2 y = r σ + σ ε0 , ε0 N(0, 1). t − 2 t t t t ∼ shift mt εt0 = εt − σt 1 2 r mt 2 σt = εt − − − σt 1 2 r (r + λσt) 2 σt = εt − − − σt 1 2 λσt 2 σt = εt − − − σt 1 = ε + λ + σ . t 2 t

Then the conditional variance under risk neutral measure Qm is :

2 2 1 2 σt = α0 + α1σt 1ϕ(εt0 1 σt 1 λ)+ b1σt 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 eas-

ily, 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:

t P θkyk κy |F (θk) Zt = e − k k−1 , (3.7) Yk=1 where θ is -predictable, and we need to solve θ from: k Fk k

P P κy (1 + θk) κy (θk)= r, (3.8) k|Fk−1 − k|Fk−1

for all k 1, 2..., T . The conditional Esscher transform Qe with respect to P of the process ∈{ }

yt is deﬁned as:

dQe = ZT . (3.9) dP T F

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 -adapted stochastic process and deﬁne: { } ∈ F dQe m = Z = λ , dP Fm m k k=1 Y For any -measurable random variable h, n m: Fn ≥ P Qe E [hZn m] E [h m]= |F . |F Zm

P zyk Consider λk = exp(θkyk κy (θk)) and h = e . Setting n = k, m = k 1 and applying − k|Fk−1 − the Bayes’ Theorem. We have:

e e Q Q zyk κy (z) = log(E [e k 1]) k|Fk−1 |F − k exp(θ y κP (θ )) t t yt t−1 t P zyk t=1 − |F = log E e k 1 kQ1 F − " − exp(θ y κP (θ )) #! t t yt t−1 t − |F t=1 Q

P zyk P = log(E [e exp(θkyk κy (θk)) k 1]) − k|Fk−1 |F −

P zyk+θkyk P = log(E [e exp( κy (θk)) k 1]) − k|Fk−1 |F −

P zyk+θkyk P = log(E [e k 1]) + log(exp[( κy (θk)) k 1]) |F − − k|Fk−1 |F − P P = κy (z + θk) κy (θk). k|Fk−1 − k|Fk−1

Qe When z = 1, we can get r = κy (1). Then the martingale equation (3.8) holds. If k|Fk−1 ε N(0, 1), we will use conditional Esscher transform to derive the following theorem. t ∼

Theorem 3.3. Suppose the asset return process y := yt t satisﬁes (3.2)-(3.4) under P . { } ∈T Under the risk neutral conditional Esscher transform measure Qe the dynamics of the return

are given by:

1 2 y = r σ + σ ε0 , ε0 N(0, 1), t − 2 t t t t ∼ 2 2 1 2 σt = α0 + α1σt 1ϕ(εt0 1 σt 1 λ)+ b1σt 1. − − − 2 − − −

Proof. We start from the martingale equation in the previous proof. If ε N(0, 1), then t ∼ 2 yt t 1 N(mt, σt ). |F − ∼ 20 e κQ (z) = κP (z + θ ) κP (θ ) yt t−1 yt t−1 t yt t−1 t |F |F − |F 1 1 = (m (z + θ )) + (z + θ )2σ2) (m θ + θ2σ2) t t 2 t t − t t 2 t t 1 = m z + σ2z2 + θ σ2z t 2 t t t 1 = (m + θ σ2)z + σ2z2. t t t 2 t

This means

2 2 yt t 1 N(mt + θtσt , σt ), (3.10) |F − ∼

under Qe. Solving the above equation for z = 1.

e r = κQ (1) yt t−1 |F 1 = (m + θ σ2)+ σ2, t t t 2 t 1 2 r mt 2 σt θt = − 2− . ⇒ σt

Substituting θt into (3.10). We have:

1 2 2 yt t 1 N(r σt , σt ). |F − ∼ − 2

The relationship between εt and εt0 : 1 ε = ε0 σ λ. t t − 2 t −

Thus the risk neutral dynamic under Qe is:

1 2 y = r σ + σ ε0 , ε0 N(0, 1), t − 2 t t t t ∼ 2 2 1 2 σt = α0 + α1σt 1ϕ(εt0 1 σt 1 λ)+ b1σt 1, − − − 2 − − −

where εt0 are i.i.d distributed. These results are also consistent with Duan’s results for

1 Gaussian GARCH pricing models. Moreover, if ϕ(εt0 1 2 σt 1 λ) is given by: − − − − 1 1 1 γI(εt0 1 σt 1 λ< 0) 2 2 − ϕ(εt0 1 σt 1 λ)=(εt0 1 σt 1 λ) (1 + − − − ), − − 2 − − − − 2 − − α1

21 the conditional variance of return process follows a TGARCH model under the risk neutral

measure Qe:

2 1 1 2 2 2 σt = α0 +(α1 + γI(εt0 1 σt 1 λ< 0))(εt0 1 σt 1 λ) σt 1 + b1σt 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 W , 0 t T be a Brownian motion t ≤ ≤ on a probability space (Ω, ,P ) and let be the filtration for this Brownian motion. As F Ft 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 = µtStdt + σtStdWt. (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:

t 1 t S = S exp( (µ σ2)ds + σ dW ). (3.12) t 0 s − 2 s s s Z0 Z0 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 = µtStdt + σtStdWt

=(r + µ r)S dt + σ S dW t − t t t t

22 µt r = rStdt + σtSt( − dt + dWt) σt Q = rStdt + σtStdWt . (3.13)

µt r In this case, the stochastic process X = − . By analogy with this example for the con- t σt 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: t t

St = S0 exp( mk + σkεk). (3.14) Xk=1 Xk=1 Comparing (3.14) with its continuous counterpart (3.12) we notice that mt corresponds to µ 1 σ2. We assume that y follows the Normal-GARCH model given by (3.2)-(3.4), a t − 2 t t discrete version of the Girsanov theorem for the Normal-GARCH model can be deﬁned in

the following: dQ T 1 T = Z = exp( X ε X2) dP T − t t − 2 t t=1 t=1 X X T µ r 1 T µ r = exp( t − ε ( t − )2) − σ t − 2 σ t=1 t t=1 t X X T 1 2 T 1 2 mt + σ r 1 mt + σ r = exp( 2 t − ε ( 2 t − )2). (3.15) − σ t − 2 σ t=1 t t=1 t X X 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, ,P ) such that the following no-arbitrage condi- F tions hold:

P r E [Mt t 1]= e− , (3.16) |F −

P yt E [Mte t 1]=1. (3.17) |F − 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 M is given by: { t}

ΠM (h(S )) = EP [M , ..., M h(S ) ]. (3.18) t T t+1 T T |Ft

Here EP 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 dQ T r = ZT = e k=1 Mk, (3.19) dP P kY=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):

P r ft (εt + %t) Mt = e− P . (3.20) ft (εt)

Here

f P ( ) is the conditional probability density of the innovation ε at time t given • t · t

t 1 under the historical measure P ; F −

the market price of risk process, denoted as % , is an -predictable process • { t} {Ft} and is uniquely determined from condition (3.17) for any t 0 ; ∈T\{ }

it is clear by deﬁnition that the parametric form of the SDF from (3.20) satisﬁes • (3.16).

P mt+κεt (σt) r We let %t = − and use Deﬁnition 3.4. Under the assumption of εt N(0, 1), we σt ∼ can get:

T T k r ZT = e =1 Mk P kY=1 24 T P T r r ft (εk + %k) = e k=1 e− P f P (ε ) k=1 t k Y 2 T 1 exp (εk+%k) √2π − 2 = 2 1 exp (εk) kY=1 √2π − 2 T %2 = exp( % ε k ) − k k − 2 kY=1 T P P mk + κεk (σk) r 1 mk + κεk (σk) r 2 = exp( − εk ( − ) ) − σk − 2 σk kY=1 T 1 2 T 1 2 mk + 2 σk r 1 mk + 2 σk r 2 = exp( − εk ( − ) ). (3.21) − σk − 2 σk Xk=1 Xk=1

For conditional Esscher transform, the process ZT has already been deﬁned by equation

1 2 r mt 2 σt (3.7), (3.9) and θt = − −2 . Under the assumption of εt N(0, 1), we can derive ZT as σt ∼ in the following:

T P θkyk κy |F (θ ) ZT = e − k k−1 k Yk=1 T 1 = exp(θ (m + σ ε ) (θ m + θ2σ2)) k k k k − k k 2 k k Yk=1 T 1 = exp(θ σ ε θ2σ2) k k k − 2 k k Yk=1 T 1 2 1 2 r mk 2 σk 1 r mk 2 σk 2 2 = exp( − 2− σkεk ( − 2− ) σk) σk − 2 σk Yk=1 T 1 2 T 1 2 mk + 2 σk r 1 mk + 2 σk r 2 = exp( − εk ( − ) ). (3.22) − σk − 2 σk Xk=1 Xk=1

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 is given by: { } ∈T

2 P P θ1tyt+θ2tyt κ 2 (θ , θ ) = log E [e ], (3.23) (yt,y ) t−1 1t 2t t 1 t |F |F − where θ and θ are -predictable. 1t 2t Ft

2 P θ1tyt+θ2ty Here E [e t t 1] is the conditional second order laplace transform under the |F − historical measure P . In our GARCH setup, If we assume εt N(0, 1), then yt t 1 ∼ |F − ∼ 2 N(mt, σt ) under historical measure P . We can derive the following corollary.

Corollary 3.1. The conditional second order log laplace transform for Gaussian distribution

is:

2 2 P 1 2 mt 1 σt mt 2 κ 2 (θ , θ ) = log(1 2σ θ ) + ( )( + θ ) . (3.24) (yt,y ) t−1 1t 2t t 2t 2 2 2 1t t |F −2 − − 2σ 2 1 2σ θ σ t − t 2t t Proof. Starting from the second order laplace transform. For convenience, we assume y ∼ N(m, σ2). Thus we have:

2 P θ1y+θ2y P 2 E [e ] = f (y) exp(θ1y + θ2y )dy R Z 2 1 (y m) 2 = exp( − ) exp(θ1y + θ2y )dy σ√2π − 2σ2 ZR 26 2 1 (y m) 2 = exp( − 2 + θ1y + θ2y )dy R σ√2π − 2σ Z 2 2 1 y 2ym + m 2 = exp( − 2 + θ1y + θ2y )dy R σ√2π − 2σ Z 2 1 m 2 1 m = exp( 2 ) exp(y (θ2 2 )+ y(θ1 + 2 ))dy R σ√2π −2σ − 2σ σ Z 2 2 2 1 m 1 σ m 2 − 2 = (1 2σ θ2) exp 2 + ( 2 )( 2 + θ1) − − 2σ 2 1 2σ θ2 σ h −2 2 i 2 1 1 2σ θ2 σ m exp − y + θ1 dy 2 2 2 2 × R σ − 2σ − 1 2σ θ2 σ 2π 2 Z 1 2σ θ2 − − 2 2 q 1 m 1 σ m 2 2 2 = (1 2σ θ2)− exp 2 + ( 2 )( 2 + θ1) . − − 2σ 2 1 2σ θ2 σ h − i 2 2 2 1 1 2σ θ2 σ m Since exp − 2 y 2 2 + θ1 is the p.d.f of Normal distribu- σ2 2σ 1 2σ θ2 σ 2π 2 − − − r 1−2σ θ2 2 2 2 σ m σ 1 1 2σ θ2 tion with mean ( 2 )( 2 + θ1) and variance 2 , exp − 2 y 1 2σ θ2 σ 1 2σ θ2 R σ2 2σ − − 2π 2 − − r 1−2σ θ2 2 R σ2 m 2 2 + θ dy = 1 and taking the neutral logarithm we have: 1 2σ θ2 σ 1 − 2 2 2 1 m 1 σ m log EP [eθ1y+θ2y ] = log(1 2σ2θ ) + ( )( + θ )2. −2 − 2 − 2σ2 2 1 2σ2θ σ2 1 − 2

2 Given yt t 1 N(mt, σt ) under historical measure P , we can get equation (3.24). Based |F − ∼ 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:

t 2 P θ1kyk+θ2kyk κ 2 (θ1k,θ2k) − (yk,y )|Fk Zt = e k −1 , (3.25) Yk=1 θ1k and θ2k are the solution of the equation:

P P κ(y ,y2) (1 + θ1k, θ2k) κ(y ,y2) (θ1k, θ2k)= r, (3.26) k k |Fk−1 − k k |Fk−1 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 . (3.27) dP T F

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 -adapted stochastic process and deﬁne: { } ∈ F dQ2e m = Z = λ , dP Fm m k k=1 Y For any -measurable random variable h, n m: Fn ≥ P Q2e E [hZn m] E [h m]= |F . |F Zm

2 2 P z1yk+z2yk Consider λk = exp(θ1kyk + θ2kyk κ(y ,y2) (θ1k, θ2k)) and h = e . Setting n = k, − k k |Fk−1 m = k 1 and applying the Bayes’ Theorem. We have: −

2e 2e 2 Q Q z1yk+z2yk κ(y ,y2) (z1, z2) = log(E [e k 1]) k k |Fk−1 |F − k 2 P exp(θ y + θ y κ 2 (θ , θ )) 1t t 2t t (yt,y ) t−1 1t 2t 2 t P z1yk+z2y t=1 − |F = log E e k k 1 kQ1 − " − 2 P F #! exp(θ1tyt + θ2ty κ 2 (θ1t, θ2t)) t (yt,y ) t−1 − t |F t=1 Q 2 P z1yk+z2yk 2 P = log(E [e exp(θ1kyk + θ2kyk κ(y ,y2) (θ1k, θ2k)) k 1]) − k k |Fk−1 |F − 2 2 P z1yk+z2yk+θ1kyk+θ2kyk P = log(E [e exp( κ(y ,y2) (θ1k, θ2k)) k 1]) − k k |Fk−1 |F − 2 P (z1+θ1k)yk+(z2+θ2k)yk P = log(E [e k 1]) + log(exp[( κ(y ,y2) (θ1k, θ2k)) k 1]) |F − − k k |Fk−1 |F − P P = κ(y ,y2) (z1 + θ1k, z2 + θ2k) κ(y ,y2) (θ1k, θ2k). k k |Fk−1 − k k |Fk−1

Q2e When z1 = 1 and z2 = 0, we can get r = κ(y ,y2) (1, 0). Then the martingale equation k k |Fk−1 (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.

2 Corollary 3.2. Suppose yt t 1 N(mt, σt ) under the historical measure P . Under the |F − ∼ 2 2 mt+σt θ1t σt risk neutral measure Q, yt t 1 N( 2 , 2 ). 1 2θ2tσ 1 2θ2tσ |F − ∼ − t − t 28 Proof. Using distribution relationship between P and Q and without lots of generating.

Q P f (yt t 1) = f (yt t 1) Zt |F − |F − × 2 1 (yt mt) 2 P = exp − exp(θ1tyt + θ2ty κ 2 (θ1t, θ2t)) 2 2 t (yt,yt ) t−1 2πσ − 2σt − |F t h i 1 (y m )2 = p exp t − t 2 2 2πσ − 2σt t h i 1 m2 1 σ2 m pexp[θ y + θ y2 + log(1 2σ2θ )+ t ( t )( t + θ )2] × 1t t 2t t 2 − t 2t 2σ2 − 2 1 2σ2θ σ2 1t t − t 2t t 1 (y m )2 m2 1 σ2 m = exp ( t t + θ y + θ y2 + t ( t )( t + θ )2 2 − 2 1t t 2t t 2 2 2 1t σt − 2σt 2σt − 2 1 2σt θ2t σt 2π 2 1 2θ2tσ − − t h i q 1 1 2θ σ2 m + σ2θ = exp ( 2t t )(y t t 1t ) . 2 − 2 t 2 σt − 2σt − 1 2θ2tσt 2π 2 1 2θ2tσ − − t h i q Therefore under Q,

2 2 mt + σt θ1t σt yt t 1 N( , ). ⇒ |F − ∼ 1 2θ σ2 1 2θ σ2 − 2t t − 2t t

This property leads to the following theorem.

Theorem 3.4. Suppose the asset return process y := yt t satisﬁes (3.2)-(3.4) under P . { } ∈T Under the risk neutral second order Esscher transform measure Q2e the dynamics of the

return are given by:

1 Q 2 Q y = r (σ ) + σ ε0 , ε0 N(0, 1), t − 2 t t t t ∼ σ2 (σQ)2 = t . t 1 2θ σ2 − 2t t

2 Proof. If εt N(0, 1), then yt t 1 N(mt, σt ) under the historical measure P . Starting ∼ |F − ∼ from the equation (3.26).

P P r = κ 2 (1 + θ , θ ) κ 2 (θ , θ ) (yt,y ) t−1 1t 2t (yt,y ) t−1 1t 2t t |F − t |F

29 2 2 1 2 mt 1 σt mt 2 = log(1 2σt θ2t) 2 + ( 2 )( 2 +1+ θ1t) −2 − − 2σt 2 1 2σt θ2t σt 2 − 2 1 2 mt 1 σt mt 2 log(1 2σt θ2t) 2 + ( 2 )( 2 + θ1t) − − 2 − − 2σt 2 1 2σt θ2t σt h 2 − i 1 σt mt 2 mt 2 = 2 ( 2 +1+ θ1t) ( 2 + θ1t) 2 1 2σt θ2t σt − σt − 2 h i 1 σt 2mt = 2 2 +1+2θ1t 2 1 2σt θ2t σt − 2 2 1 σt 1 σt 2mt = 2 + 2 2 +2θ1t 2 1 2θ2tσt 2 1 2θ2tσt σt − 2 −2 2 1 σt σt mt + σt θ1t = 2 + 2 2 2 1 2θ2tσt 1 2θ2tσt σt − 2 − 2 1 σt mt + σt θ1t = 2 + 2 . 2 1 2θ2tσt 1 2θ2tσt − −

So we can get m + σ2θ 1 σ2 t t 1t = r t . 1 2θ σ2 − 2 1 2θ σ2 − 2t t − 2t t 2 2 2 mt+σt θ1t σt Q 2 σt 2 and 2 are the mean and variance under Q. We assume (σ ) = 2 and 1 2θ2tσ 1 2θ2tσ t 1 2θ2tσ − t − t − t 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,

2 2 1 σt mt + σt θ1t r = 2 + 2 2 1 2θ2tσt 1 2θ2tσt − 1 − r(1 2θ σ2)= σ2 +(m + θ σ2) ⇒ − 2t t 2 t t 1t t 1 r(1 2θ σ2) m σ2 = θ σ2 ⇒ − 2t t − t − 2 t 1t t 2 1 2 r(1 2θ2tσt ) mt 2 σt θ1t = − 2− − . ⇒ σt

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 volatil-

30 ity model. They showed that this general pricing kernel is non-monotonic, qualitatively ac-

counting 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:

t S φ N = N t exp(δt + η σ2 + ξ(σ2 σ2)), (3.28) t 0 S s t+1 − 1 0 s=1 X and therefore

φ Nt St 2 2 2 = exp(δ + ησt + ξ(σt+1 σt )). (3.29) Nt 1 St 1 − − − 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 + σtεt), St 1 − σ2 σ2 = α + α σ2ϕ(ε )+(b 1)σ2. t+1 − t 0 1 t t 1 − t

Substituting these into (3.29) gives

Nt 2 2 2 = exp(rφ + λσtφ + σtεtφ + δ + ησt + ξ(α0 + α1σt ϕ(εt)+(b1 1)σt )), Nt 1 − − Rearranging gives

Nt 2 2 2 = exp(rφ + λσtφ + δ + ησt + ξα0 + ξ(b1 1)σt + σtεtφ + ξα1σt 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(ε )=(ε γ)2. t t −

31 Then we have:

Nt 2 2 2 2 = exp(rφ + λσtφ + δ + ησt + ξα0 + ξ(b1 1)σt + σtεtφ + ξα1σt (εt γ) ). Nt 1 − − − Expanding the square and collecting terms gives

Nt 2 2 2 2 2 2 2 = exp(rφ + λσtφ + δ + ησt + ξα0 + ξ(b1 1)σt + σtεtφ + ξα1σt εt 2ξα1σt εtγ + ξα1σt γ ) Nt 1 − − − Nt 2 2 2 2 2 2 2 = exp(rφ + λσtφ + δ + ησt + ξα0 + ξ(b1 1)σt + ξα1σt γ +(σtφ 2ξα1σt γ)εt + ξα1σt εt ). 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.

Nt Et 1 = exp( r). (3.30) − Nt 1 − − Note that

Nt 2 2 2 2 Et 1 = exp(rφ + λσtφ + δ + ησt + ξα0 + ξ(b1 1)σt + ξα1σt γ ) − Nt 1 − − E(exp((σ φ 2ξα σ2γ)ε + ξα σ2ε2)) × t − 1 t t 1 t t Nt 2 2 2 2 Et 1 = exp(rφ + λσtφ + δ + ησt + ξα0 + ξ(b1 1)σt + ξα1σt γ ) − Nt 1 − − E(exp((φ 2ξα σ γ)σ ε + ξα σ2ε2)). × − 1 t t t 1 t t

We need the following result

1 2a2b2 E(exp(az2 +2abz)) = exp log(1 2a)+ . − 2 − 1 2a − For our application we have

a = ξα1, φ 2ξα σ γ b = − 1 t , 2ξα1

z = σtεt, and thus

2a2b2 (φ 2ξα σ γ)2 = − 1 t . 1 2a 2(1 2ξα ) − − 1 32 Therefore

2 2 2 1 (φ 2ξα1σtγ) E(exp((φ 2ξα1σtγ)σtεt + ξα1σt εt )) = exp log(1 2ξα1)+ − , − − 2 − 2(1 2ξα1) − and

Nt 2 2 2 2 Et 1 = exp rφ + λσtφ + δ + ησt + ξα0 + ξ(b1 1)σt + ξα1σt γ − Nt 1 − − 2 1 (φ 2ξα1σtγ) log(1 2ξα1)+ − − 2 − 2(1 2ξα1) − 2 Nt 1 φ Et 1 = exp rφ + δ + ξα0 log(1 2ξα1)+ − Nt 1 − 2 − 2(1 2ξα1) − − 2 2φξα1γ 2 2(ξα1γ) 2 +(λφ )σt +(η + ξ(b1 1) + ξα1γ + )σt . − 1 2ξα1 − 1 2ξα1 − − Using (3.30) we get

1 φ2 exp rφ + δ + ξα0 log(1 2ξα1)+ − 2 − 2(1 2ξα1) − 2 2φξα1γ 2 2(ξα1γ) 2 +(λφ )σt +(η + ξ(b1 1) + ξα1γ + )σt = exp( r) − 1 2ξα1 − 1 2ξα1 − − 2 − 1 φ 2φξα1γ rφ + δ + ξα0 log(1 2ξα1)+ +(λφ )σt ⇒ − 2 − 2(1 2ξα1) − 1 2ξα1 2 − − 2 2(ξα1γ) 2 +(η + ξ(b1 1) + ξα1γ + )σt = r − 1 2ξα1 − − 2 1 φ 2φξα1γ r(φ +1)+ δ + ξα0 log(1 2ξα1)+ +(λφ )σt ⇒ − 2 − 2(1 2ξα1) − 1 2ξα1 2 − − 2 2(ξα1γ) 2 +(η + ξ(b1 1) + ξα1γ + )σt =0. − 1 2ξα1 − Thus we must have

1 φ2 r(φ +1)+ δ + ξα log(1 2ξα )+ =0, 0 − 2 − 1 2(1 2ξα ) − 1 2φξα γ λφ 1 =0, − 1 2ξα − 1 2(ξα γ)2 η + ξ(b 1) + ξα γ2 + 1 =0. 1 − 1 1 2ξα − 1 We can obtain

λ ξ = , 2α1(λ + γ)

33 2(ξα γ)2 η = ξ(b 1) ξα γ2 1 , − 1 − − 1 − 1 2ξα − 1 1 φ2 δ = r(φ + 1) ξα + log(1 2ξα ) . − − 0 2 − 1 − 2(1 2ξα ) − 1 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) cap-

tures 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:

λ 1 2 1 2 γ λ x − δ x− + γ x fGIG(x,λ,δ,γ)=( ) exp . (4.1) δ 2Kλ(δγ) − 2 h i where x > 0, k ( ) is the modiﬁed Bessel function of the third kind associated with the λ · 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: ∼ 2 2 λ K 1 (α δ +(y µ) ) γ λ 2 fGH (y,λ,α,β,δ,µ)= − − exp[β(y µ)], y R. (4.2) λ 2 2 1 λ √2πδ Kλ(δγ) ( δ +(py µ) /α) 2 − − ∈ − Where α is the kurtosis, β is the skewness,p δ is the scale parameter and µ is the location

parameter. Moreover, γ2 = α2 β2. 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 λ α2 β2 κ (u)=µu + log − GH 2 α2 (β + u)2 − K (δ α2 (β + u)2) + log λ − , β + u < α. (4.3) K (δ α2 β2) | | λp − The mean and Variance of y area given by:p

δβ E[y] = µ + R (δγ), (4.4) γ λ δ β2δ2 V ar[y] = R (δγ)+ S (δγ). (4.5) γ λ γ2 λ

Here the functions R and S are deﬁned for all u R+ by: λ λ ∈

Kλ+1(u) Rλ(u) = , (4.6) Kλ(u) 2 Kλ+2(u)Kλ(u) Kλ+1(u) Sλ(u) = 2 − . (4.7) Kλ(u)

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 + σtεt, (4.8)

ε GH(λ,α,β,δ,µ), (4.9) t ∼ 2 2 2 σt = α0 + α1σt 1ϕ(εt 1)+ b1σt 1, (4.10) − − −

mt = r + λσt. (4.11)

The innovation process ε are i.i.d with common probability density function (4.2). To { t} 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 µ + R (δγ) = 0, (4.12) γ λ δ β2δ2 R (δγ)+ S (δγ) = 1. (4.13) γ λ γ2 λ

36 If ε GH(λ,α,β,δ,µ), then y = m + σ ε GH(λ,α/σ ,β/σ , δσ ,δµ + m ). The t ∼ t t t t ∼ t t t t conditional probability density function of yt given t 1 under P is given by: F −

λ 2 y mt 2 2 2 K 1 (α δ +( − µ) ) (α β ) 2 λ σt f P (y) = − − 2 − yt 1 √ λ 2 2 · 2 qy mt 2 λ 2πσtδ Kλ(δ α β ) ( δ +( − µ) /α) 2 − − σt − y mp q exp[β( − t µ)], y R. (4.14) · σt − ∈

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 satisﬁes equations (4.8)-(4.11) { } ∈T under P . Under the risk neutral MCMM Qm the dynamics of the return are given by:

λ α2 β2 y = r µσ log − t − t − 2 α2 (β + σ )2 − t 2 2 Kλ(δ α (β + σt) ) log − + σtεt0 , (4.15) − K (δ α2 β2) pλ − εt0 GH(λ,α,β,δ,µp), (4.16) ∼ shift 2 2 mt 1 2 σt = α0 + α1σt 1ϕ εt0 1 + − + b1σt 1, (4.17) − − σt 1 − − shift Where mt is given by:

λ α2 β2 mshift = r m µσ log − t − t − t − 2 α2 (β + σ )2 − t K (δ α2 (β + σ )2) log λ − t . (4.18) − K (δ α2 β2) pλ − p Proof. Using the cumulant generating function of εt.

λ α2 β2 κP (σ ) = µσ + log εt t t 2 − 2 2 α (β + σt) − K (δ α2 (β + σ )2) + log λ − t . K (δ α2 β2) pλ − p 37 By equation (3.6) and Theorem 3.2. We have:

mshift = r m κP (σ ) t − t − εt t λ α2 β2 = r mt µσt log 2 − 2 − − − 2 α (β + σt) − K (δ α2 (β + σ )2) log λ − t . 2 2 − Kpλ(δ α β ) P − y = r κ (σ )+ σ ε0 t − εt t pt t λ α2 β2 = r µσ log − − t − 2 α2 (β + σ )2 − t 2 2 Kλ(δ α (β + σt) ) log − + σtε0 . 2 2 t − Kpλ(δ α β ) shift − mt p εt0 = εt . − σt shift 2 2 mt 1 2 σt = α0 + α1σt 1ϕ εt0 1 + − + b1σt 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 satisﬁes equations (4.8)-(4.11) { } ∈T under P . Under the risk neutral conditional Esscher Qe the dynamics of the return are given by:

yt = mt + σt(µ + ν1t)+ σtν2tεt0 , (4.19)

δ ν1t εt0 t 1 GH λ,αν2t, β1tν2t, , , (4.20) |F − ∼ ν2t −ν2t Where β , ν and ν are some -predictable processes given by: { 1t} { 1t} { 2t} {Ft}

β1t = β + θtσt, (4.21)

38 δβ1t 2 2 ν1t = Rλ δ α β , (4.22) 2 2 1t α β1t − − q 2 2 1 δ 2 2 δ β1t 2 2 2 ν2t = p Rλ(δ α β )+ Sλ(δ α β ) . (4.23) 2 2 1t 2 2 1t α β − α β1t − − 1t q − q p such that for each t T . ε0 has zero conditional mean and unit conditional variance given ∈ t

t 1 and θt is the unique predictable solution of the following martingale equation: F − 2 2 2 2 λ α (β +(1+ θt)σt) Kλ(δ α (β + θtσt) ) log −2 2 + log − = µσt + mt r. (4.24) 2 α (β + θtσt) K (δ α2 (β +(1+ θ )σ )2) − − λ p− t t Proof. We know that if ε GH(λ,α,β,δ,µp), then y = m +σ ε GH(λ,α/σ ,β/σ , δσ ,δµ+ t ∼ t t t t ∼ t t t

mt), and using (4.3) we can get the cumulant function of the returns under the physical measure P in the following:

2 2 2 2 P λ α β Kλ(δ α (β + uσt) ) k (u) =(mt + µσt)u + log − + log − , yt 2 2 2 2 2 α (β + uσt) Kλ(δ α β ) − p − β + uσ < α. p | t|

We start by evaluating the conditional cumulant generating function of yt given t 1 under F − Qe.

e kQ (u)= kP (u + θ ) kP (θ ) yt yt t − yt t λ α2 β2 =(m + µσ )(u + θ )+ log − t t t 2 α2 (β +(u + θ )σ )2 − t t 2 2 Kλ(δ α (β +(u + θt)σt) ) + log − (mt + µσt)θt K (δ α2 β2) − p λ − λ α2 β2 K (δ α2 (β + θ σ )2) log −p log λ − t t 2 2 2 2 − 2 α (β + θtσt) − Kλ(δ α β ) − p − λ α2 (β + θ σ )2 =(m + µσ )u + log − t t p t t 2 α2 (β + θ σ + uσ )2 − t t t K (δ α2 (β + θ σ + uσ )2) + log λ − t t t . K (δ α2 (β + θ σ )2) pλ − t t Qe p When u = 1, kyt (1) = r. We can get the martingale equation (4.24). This expression is well deﬁned provided that β + θ σ + uσ <α. From the above expression we can see that | t t t|

39 e under Q , the conditional distribution of the return yt given t 1 is again a GH distribution F − as follows:

yt t 1 GH(λ,α/σt, β1t/σt, δσt, mt + σtµ), |F − ∼

where β1t = β + θtσt. 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

e with θt. Therefore, the return dynamics under Q are:

yt = mt + σtηt,

ηt t 1 GH(λ,α,β + θtσt,δ,µ). |F − ∼

P In order to represent yt in the form given by (4.19)-(4.20) we denote by ν1t = E [ηt t 1] µ |F − −

and ν the conditional standard deviation of η and we let ε0 = η/ν (ν +µ)/ν . Moreover, 2t t t 2t− 1t 2t we conclude that, under the risk neutral conditional Esscher transform Qe, the conditional

mean and standard deviation of the returns, mQe and σQe , are given by:

e Q σtδ(β + θtσt) 2 2 m = mt + σtµ + Rλ(δ α (β + θtσt) ), t 2 2 α (β + θtσt) − − p e Q δ p 2 2 σ =σt Rλ(δ α (β + θtσt) ) t 2 2 α (β + θtσt) − − p 1 p δ2(β + θ σ )2 2 + t t . 2 2 2 2 α (β + θtσt) Sλ(δ α (β + θtσt) )! − − p

We remark that the risk neutral return dynamics have no longer a linear GARCH form since the innovations εt0 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 y under this new probability measure using { t} (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:

1/2 2 2 1 γ− K 1(α δ +(y µ) ) fNIG y, ,α,β,δ,µ = − − 2 √ 1/2 1 2 2 − 2πδ− K (δγ) ( δ p+(y µ) /α) − 2 − exp[β(y µ)], y pR. (4.25) · − ∈

Where γ2 = α2 β2. −

There are various scale and location invariant parametrizations proposed in the literature.

For numerical purposes we use the following parametrization:

Parametrizationα ˜ = αδ, β˜ = βδ. •

Corollary 4.2. If we apply parametrization α˜ = αδ, β˜ = 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:

x µ˜ 2 K1 α˜ 1+( −˜ ) α˜ 2 2 x µ˜ δ fNIG(x)= exp α˜ β˜ + β˜ − , (4.26) ˜ − ˜ q x µ˜ πδ δ 1+( − )2 hq i δ˜ q δ˜ and µ˜ are given by:

3/2 α˜2 β˜2 − ! δ˜ = q , (4.27) α˜

41 1/2 β˜ µ˜ = α˜2 β˜2 . (4.28) −α˜ − q ! Proof. Starting from the equation (4.25).

1/2 2 2 γ− K 1(α δ +(x µ) ) fNIG(x)= − − exp[β(x µ)] 1/2 2 2 √2πδ− K 1 (δγ) ( δ +(x µ) /α) − − 2 p − 1/2 2 2 − p α β 2 2 K 1(α δ +(x µ) ) = − − − exp[β(x µ)] 1/2 2 2 2 2 √2πδ−p K 1 (δ α β ) ( δ p+(x µ) /α) − − 2 − − 1/2 α˜2 β˜2p− p 2 2 α˜ 2 2 ˜ δ − δ K 1( δ δ +(x µ) ) β = q − − exp[ (x µ)] α˜2 β˜2 ( δ2 +(x µ)2/(˜α/δ)) δ − √ 1/2 1 p 2πδ− K δ δ2 δ2 − − 2 − q p 1/2 2 − x µ δ α˜2 β˜2 K 1 α˜ 1+ − − δ β˜ = − r exp[ (x µ)] q 2 δ − √ 1 2 ˜2 x µ 2πK α˜ β δ2 1+ − α˜ − 2 − δ q r 1/2 . 2 2 ˜2 − x µ α˜ α˜ β K 1 α˜ 1+ −δ ˜ − − β = q r exp[ (x µ)] 1/2 2 δ π − x µ − √2πδ α˜2 β˜2 exp α˜2 β˜2 1+ − 2 − − − δ q q r 2 p x µ K α˜ 1+ − α˜ x µ 1 δ = exp α˜2 β˜2 + β˜( − ) r . πδ − δ 2 q x µ h i 1+ −δ r By equation (4.12).

δβ µ + R (δγ)=0 γ λ δβ˜ α˜2 β˜2 1 µ + R δ 2 2 =0 ⇒ 2 β˜2 − 2 s δ − δ δ α˜ δ2 − δ2 q ˜ δβ 2 2 µ + R 1 ( α˜ β˜ )=0 ⇒ − 2 − α˜2 β˜2 q − q 2 ˜2 δβ˜ K 1 ( α˜ β ) µ + + 2 − =0 ⇒ 2 2 q 2 2 α˜ β˜ K 1 ( α˜ β˜ ) − − 2 − q q

42 2 ˜2 δβ˜ K 1 ( α˜ β ) µ + + 2 − =0 ⇒ 2 2 q 2 2 α˜ β˜ K 1 ( α˜ β˜ ) − 2 − q δβ˜ q µ + =0. (4.29) ⇒ α˜2 β˜2 − q By equation (4.13).

δ β2δ2 R (δγ)+ S (δγ)=1 γ λ γ2 λ ˜2 2 ˜2 β 2 2 ˜2 δ α˜ β δ2 δ α˜ β R 1 δ + S 1 δ =1 2 2 2 ˜2 2 2 ⇒ α˜2 β˜2 − 2 s δ − δ α˜ β − 2 s δ − δ δ2 δ2 δ2 − δ2 − q 2 2 ˜2 δ 2 2 δ β 2 2 R 1 ( α˜ β˜ )+ S 1 ( α˜ β˜ )=1 ⇒ − 2 − α˜2 β˜2 − 2 − α˜2 β˜2 q q − − q 2 ˜2 2 ˜2 2 2 ˜2 δ2 δ2β˜2 K 3 ( α˜ β )K 1 ( α˜ β ) K 1 ( α˜ β ) + − 2 − − 2 − − 2 − =1 2 ˜2 q q q ⇒ 2 ˜2 α˜ β 2 2 ˜2 α˜ β − K 1 ( α˜ β ) − 2 − q − q 2 ˜2 1 2 ˜2 2 2 ˜2 2 2 ˜2 K 1 ( α˜ β )[1 + ]K 1 ( α˜ β ) K 1 ( α˜ β ) δ δ β 2 − √α˜2 β˜2 2 − − 2 − + − =1 2 ˜2 q q q ⇒ 2 ˜2 α˜ β 2 2 ˜2 α˜ β − K 1 ( α˜ β ) − 2 − q q δ2 δ2β˜2 1 + =1 ⇒ 2 ˜2 α˜2 β˜2 α˜ β α˜2 β˜2 − − − q ˜q2 2 1 β δ + 2 =1 ⇒ 2 ˜2 3 α˜2 β˜2 (˜α β ) − − q 3/2 α˜2 β˜2 − ! δ˜ = q . (4.30) ⇒ α˜

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 trans-

form 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 + σtεt, (4.31)

43 ε NIG(˜α, β,˜ δ,˜ µ˜), (4.32) t ∼ 2 2 2 σt = α0 + α1σt 1ϕ(εt 1)+ b1σt 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 satisﬁes equations (4.31)- { } ∈T (4.34) under P . Under the risk neutral MCMM Qm the dynamics of the return are given

by:

2 2 2 2 y = r µσ˜ α˜ β˜ + α˜ (β˜ + σ δ˜) + σ ε0 , (4.35) t − t − − − t t t q q ε0 NIG(˜α, β,˜ δ,˜ µ˜), (4.36) t ∼ shift 2 2 mt 1 2 σt = α0 + α1σt 1ϕ εt0 1 + − + b1σt 1, (4.37) − − σt 1 − − shift where mt is given by:

mshift = r m µσ˜ α˜2 β˜2 + α˜2 (β˜ + σ δ˜)2. (4.38) t − t − t − − − t q q

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 satisﬁes equations (4.31)- { } ∈T (4.34) under P . Under the risk neutral conditional Esscher Qe the dynamics of the return are given by:

˜ ˜ yt = mt + σt(˜µ + δν1t)+ σtδν2tεt0 , (4.39)

44 ˜ 1 ν1t εt0 NIG(˜α, β1t, , ), (4.40) ∼ ν2t −ν2t where β˜ , ν , ν are some -predictable processes given by: { 1t} { 1t} { 2t} {Ft}

β˜1t = β˜ + θtδσ˜ t, (4.41)

β˜1t ν1t = , (4.42) α˜2 β˜2 − 1t q α˜ ν2t = 3 . (4.43) 2 α˜2 β˜2 − 1t q Such that εt0 has zero conditional mean and unit conditional variance given t 1, and θt is F − the unique predictable solution of the following martingale equation:

α˜2 (β˜ + δσ˜ + θ δσ˜ )2 α˜2 (β˜ + θ δσ˜ )2 =µσ ˜ + m r, t 0 . (4.44) − t t t − − t t t t − ∈T\{ } q q

In this case we can determine an analytical form for the risk neutral Esscher parameter θt:

2 2 1 β˜ 1 (r mt σtµ˜) 4˜α θt = − − 1 . (4.45) −2 − ˜ − 2v 2 ˜2 2 ˜2 2 − σtδ u σt δ σt δ +(r mt σtµ˜) ! u − − t

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 out- of-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(M 1) > 0.1 for | 0 − | 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, ac-

cording 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

M 0 < DTM 40 40 < DTM 80 80 < DTM 180 180 < DTM 200 All o ≤ ≤ ≤ ≤ 1 [0.9, 0.95] 25 5 3 0 33 2 [0.95, 0.975] 27 6 2 1 36 3 [0.975, 0.99] 62 19 8 4 93 4 [0.99, 1.01] 201 75 37 7 320 5 [1.01, 1.025] 151 37 17 2 207 6 [1.025, 1.05] 171 57 32 0 260 7 [1.05, 1.1] 68 76 44 8 196 All 705 275 143 22 1145

Table 5.2: Average Call option prices in 2004

M 0 < DTM 40 40 < DTM 80 80 < DTM 180 180 < DTM 200 All o ≤ ≤ ≤ ≤ 1 [0.9, 0.95] 82.69 88.22 106.3 0 85.67 2 [0.95, 0.975] 48.88 52.75 67.4 93.6 51.8 3 [0.975, 0.99] 27.3 40.99 61.25 71.98 34.94 4 [0.99, 1.01] 15.73 30.94 47.69 64.11 24.05 5 [1.01, 1.025] 7.20 19.95 38.21 56.25 12.5 6 [1.025, 1.05] 3.56 11.82 28.79 0 8.47 7 [1.05, 1.1] 1.34 4.51 15.84 27.11 6.88 All 14.22 20.41 34.8 52.71 19.02

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 1 r(T t) (Q) method(i) : Cˆ = e− − max(0,S K), (5.1) t n T,i − i=1 Xn 1 r(T t) (P ) dQ method(ii) : Cˆ = e− − max(0,S K) (i), (5.2) t n T,i − dP i=1 X 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:

Zi(tj, n) Si∗(tj, n)= S0 , (5.3) Z0(tj, n) where

Sˆi(tj) Zi(tj, n)= Si∗(tj 1, n) , (5.4) − Sˆi(tj 1) − n 1 rtj Z (t , n)= e− Z (t , n). (5.5) 0 j n i j i=1 X

Note that Sˆi(t) is the ith simulated asset price at time t prior ro the EMS adjustment, and ˆ Si(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., Sˆi(tj)/Sˆi(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. Sup- pose 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

Y (b)= Y b(X E(X)), (5.6) i i − i −

from the ith replication and then compute the sample mean

1 n Y¯ (b)= Y¯ b(X¯ E(X)) = (Y b(X E(X))). (5.7) − − n i − i − i=1 X This is a control variate estimator and the observed error X¯ E(X) serves as a control in −

estimating E(Y ). The optimal coeﬃcient b∗ is given by

σY Cov(X,Y ) b∗ = ρXY = , (5.8) σX V ar(X)

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 ˆbn, where the latter is given by

n (X X¯)(Y Y¯ ) ˆb = i=1 i − i − . (5.9) n n (X X¯)2 P i=1 i − P 51 Then, we can rewrite equation (5.8) and (5.9) as:

Y (ˆb )= Y ˆb (X E(X)), (5.10) i n i − n i − 1 n Y¯ (ˆb )= Y¯ ˆb (X¯ E(X)) = (Y ˆb (X E(X))). (5.11) n − n − n i − n i − i=1 X 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 measure in Black-Scholes model by S(i) = S exp((r 1 σ2)T + σ√TZ ), i = 1, ..., n, Z T 0 − 2 i i ∼ N(0, 1). Then we can calculate the discounted payoﬀ for each sample path in the Black-

Scholes 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 Black-

Scholes option price and terminal stock price are computed using the stationary variance

1 α (1 α b 0.5γ)− of the TGARCH model in the Black-Scholes closed-form formula. 0 − 1 − 1 − 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(ˆbn) 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 cor-

recting 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 Ess-

cher 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 proba-

bility 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:

1 N RMSE = (Cmarket Cˆmodel(θ))2, θ vN i − i u i=1 u X t 2 1 N Cmarket Cˆmodel(θ) %RMSE = i − i . θ vN Cmarket u i=1 i ! u X t Where N is the number of option contracts, Cmarket is the option price observed from the market and Cˆmodel is the simulated price for the model considered. θ represents the parameter

set, θ = α ,α , b , λ, γ, α,˜ β,˜ θ . α ,α , b ,λ,γ are TGARCH parameters.α, ˜ β˜ are NIG- { 0 1 1 2} 0 1 1

distributed parameters. θ2 is the parameter from the second order Esscher transform. In models 1 and 2, Cˆmodel is derived by Monte Carlo method (i) (equation(5.1)), EMS method

and control variates technique. In models 3 and 4, Cˆmodel 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.α, ˜ β 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(θˆ) is the log likelihood evaluated at the MLE θˆ. 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 NIG 6 7 α0 1.11 10− 8.4 10− ∗ 7 ∗ 7 (3.1 10− ) (2.5 10− ) ∗ ∗

α1 0.0076 0.0093 (0.0057) (0.0068)

b1 0.9428 0.944 (0.0097) (0.0099)

λ 0.0443 0.042 (0.0165) (0.0167)

γ 0.0721 0.0738 (0.0141) (0.0160)

α˜ 1.6893 (0.2375)

β˜ -0.1916 (0.0682)

Skw 0 -0.263 Kts 0 1.879 l(θˆ) -4697.3 -4551.5 AIC 9404.6 9117.0 BIC 9435.6 9160.3

54 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

Gaussian NIG-MCMM NIG-Esscher Gaussian-2nd Esscher 6 7 6 7 α 1.134 10− 8.458 10− 1.202 10− 3.175 10− 0 ∗ ∗ ∗ ∗ α1 0.0148 0.009 0.0071 0.000067

b1 0.9169 0.944 0.9416 0.9425 λ 0.0029 0.0623 0.0028 0.1885 γ 0.1238 0.074 0.0787 0.1026 α˜ 1.5822 1.5318 β˜ -0.2404 -0.375

θ2 -507.47 RMSE 1.495 1.455 1.153 0.569

Table 5.5: In-sample estimation performance for four competing models based on %RMSE

Gaussian NIG-MCMM NIG-Esscher Gaussian-2nd Esscher 7 6 6 7 α 7.139 10− 1.442 10− 1.424 10− 3.7 10− 0 ∗ ∗ ∗ ∗ α1 0.0292 0.0134 0.0025 0.0023

b1 0.895 0.942 0.9435 0.9427 5 λ 0.028 0.0145 2.464 10− 0.2038 ∗ γ 0.1437 0.055 0.0736 0.098 α˜ 2.096 2.1535 β˜ -1.547 -1.1311

θ2 -634.82 %RMSE 0.0524 0.0434 0.0364 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 cor-

recting 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

7 α0 10.7404 10− γ 0.0761998 ∗ 7 (2 10− ) (0.013) ∗ α1 0.00631494 Skw 0 (0.005) Kts 0

b1 0.94328 l(θˆ) -5409.55 (0.008) AIC 10829.1 λ 0.0435936 BIC 10860.62 (0.016)

56 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, Cˆmodel is derived by Monte Carlo method (i) (equation(5.1)), EMS

method and control variates technique. In model 3, Cˆmodel is derived by Monte Carlo method

(ii) (equation(5.2)) and control variates technique. The number of the Monte Carlo simula-

tion 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 No. of No.of period weeks contract α0 α1 b1 λ γ RMSE 1-26 26 532 2.685 10−6 0.00077 0.8326 1.2409 0.0576 1.5931 ∗ 2-27 26 529 2.374 10−6 0.0024 0.8477 1.2468 0.0508 1.5416 ∗ 3-28 26 534 2.141 10−6 0.0507 0.8521 1.2598 0.0003 1.4715 ∗ 4-29 26 540 2.406 10−6 0.0593 0.8493 1.0712 0.0025 1.3746 ∗ 5-30 26 550 2.08 10−6 0.0074 0.8655 1.1592 0.0445 1.3349 ∗ 6-31 26 559 1.7506 10−6 0.0152 0.8815 1.1193 0.0328 1.3751 ∗ 7-32 26 567 1.833 10−6 0.0193 0.8889 0.9363 0.0343 1.3969 ∗ 8-33 26 586 1.797 10−6 0.049 0.8773 1.1164 5.605 10−12 1.3756 ∗ ∗ 9-34 26 597 1.8834 10−6 0.0506 0.8724 1.1211 4.245 10−13 1.3685 ∗ ∗ 10-35 26 597 1.96 10−6 0.049 0.8723 1.079 0.0036 1.3729 ∗ 11-36 26 602 2.072 10−6 0.0037 0.8833 0.7733 0.0647 1.3098 ∗ 12-37 26 597 1.517 10−6 0.0322 0.8854 1.1312 0.0144 1.2312 ∗ 13-38 26 599 1.251 10−6 0.0371 0.8941 1.2797 0.0002 1.2482 ∗ 14-39 26 599 1.183 10−6 0.0192 0.9008 1.1626 0.0207 1.2813 ∗ 15-40 26 605 9.233 10−7 0.0097 0.9052 1.3037 0.0242 1.256 ∗ 16-41 26 608 8.241 10−7 0.0316 0.9102 1.3067 0.0004 1.2453 ∗ 17-42 26 605 8.97 10−7 0.013 0.9088 1.207 0.0231 1.2144 ∗ 18-43 26 605 9.356 10−7 0.0344 0.9073 1.1921 0.0023 1.1995 ∗ 19-44 26 607 8.794 10−7 0.0275 0.9075 1.2444 0.0077 1.1783 ∗ 20-45 26 605 7.774 10−7 0.0189 0.9179 1.1588 0.0155 1.165 ∗ 21-46 26 615 1.185 10−6 0.0355 0.9081 0.7873 0.0189 1.253 ∗ 22-47 26 614 8.09 10−7 0.0181 0.922 1.093 0.0165 1.1954 ∗ 23-48 26 620 7.921 10−7 0.0151 0.9235 1.0673 0.02 1.178 ∗ 24-49 26 627 8.143 10−7 0.0311 0.9215 1.102 0.003 1.1664 ∗ 25-50 26 616 8.306 10−7 0.0328 0.922 1.1276 1.305 10−11 1.1532 ∗ ∗ 26-51 26 620 8.929 10−7 0.01 0.9238 1.055 0.0249 1.159 ∗

58 Table 5.8: In-sample estimation of the NIG-TGARCH with MCMM

Week No. of No.of period weeks contractα ˜ β˜ α0 α1 b1 λ γ RMSE 1-26 26 532 2.4616 -0.2561 1.262 10−6 0.011 0.9431 0.0049 0.073 1.8823 ∗ 2-27 26 529 2.6254 -0.2813 2.436 10−6 0.0095 0.8946 0.0058 0.1501 1.6294 ∗ 3-28 26 534 1.2692 -0.0628 2.369 10−6 0.0029 0.898 0.0096 0.1615 1.5771 ∗ 4-29 26 540 3.5049 -0.6384 2.469 10−6 0.0079 0.8989 0.0782 0.127 1.432 ∗ 5-30 26 550 6.9424 -2.4878 1.319 10−6 0.0091 0.9419 0.0069 0.0754 1.5656 ∗ 6-31 26 559 4.9366 -2.3388 1.828 10−6 0.0009 0.9338 0.0088 0.0918 1.4419 ∗ 7-32 26 567 4.6376 -1.9117 1.46 10−6 0.0061 0.939 0.0031 0.0816 1.4894 ∗ 8-33 26 586 5.9588 -2.0589 1.174 10−6 0.0095 0.9446 0.0014 0.0732 1.5573 ∗ 9-34 26 597 9.785 -6.1269 2.375 10−6 0.0095 0.9051 0.1994 0.091 1.3703 ∗ 10-35 26 597 9.2537 -6.0745 2.521 10−6 0.0111 0.9013 0.1752 0.0945 1.3695 ∗ 11-36 26 602 4.8491 -3.206 1.776 10−6 0.0102 0.929 0.0317 0.08 1.2697 ∗ 12-37 26 597 4.1035 -2.8509 2.014 10−6 0.0077 0.923 0.0446 0.0876 1.2214 ∗ 13-38 26 599 0.7653 -0.2447 1.542 10−6 0.0097 0.9349 0.0399 0.0781 1.2036 ∗ 14-39 26 599 5.4575 -2.5972 1.2602 10−6 0.0097 0.9418 0.0001 0.0736 1.3092 ∗ 15-40 26 605 5.4087 -4.0857 1.666 10−6 0.0079 0.9338 0.0374 0.0747 1.2368 ∗ 16-41 26 608 2.3136 -1.339 1.883 10−6 0.0088 0.929 0.0025 0.0818 1.2396 ∗ 17-42 26 605 3.3775 -2.01 1.652 10−6 0.0104 0.9317 0.0049 0.0793 1.2248 ∗ 18-43 26 605 4.4187 -2.6642 1.361 10−6 0.1001 0.9387 0.0025 0.0742 1.2315 ∗ 19-44 26 607 1.9672 -0.9244 1.519 10−6 0.0072 0.9364 0.0044 0.0803 1.2251 ∗ 20-45 26 605 2.0477 -1.0852 1.661 10−6 0.0091 0.9332 0.000011 0.079 1.1922 ∗ 21-46 26 615 0.665 -0.207 1.534 10−6 0.0095 0.9321 0.0883 0.0789 1.1921 ∗ 22-47 26 614 3.7469 -2.4216 1.737 10−6 0.0088 0.9281 0.0164 0.084 1.2169 ∗ 23-48 26 620 2.4137 -1.0405 1.194 10−6 0.0079 0.9415 0.0013 0.0778 1.2277 ∗ 24-49 26 627 2.1434 -0.8117 9.963 10−7 0.0036 0.9509 0.0462 0.0681 1.1992 ∗ 25-50 26 616 2.319 -0.7941 1.099 10−6 0.0047 0.9431 0.06 0.0781 1.2092 ∗ 26-51 26 620 2.3341 -0.7761 1.058 10−6 0.0082 0.9437 0.0342 0.0746 1.2301 ∗

59 Table 5.9: In-sample estimation of the NIG-TGARCH with Esscher transform

Week No. of No.of period weeks contractα ˜ β˜ α0 α1 b1 λ γ RMSE 1-26 26 532 0.5392 0.0313 2.125 10−6 0.0104 0.9118 0.234 0.0765 1.7343 ∗ 2-27 26 529 1.6635 0.1385 1.806 10−6 0.0084 0.9243 0.2319 0.0752 1.6917 ∗ 3-28 26 534 1.2752 -0.0841 2.537 10−6 0.0082 0.9003 0.1771 0.0967 1.6005 ∗ 4-29 26 540 0.6851 0.1092 1.771 10−6 0.0083 0.93 0.2184 0.0736 1.5354 ∗ 5-30 26 550 2.3741 -0.5858 2.891 10−6 0.0035 0.901 0.0704 0.1152 1.4461 ∗ 6-31 26 559 0.5641 -0.0201 2.535 10−6 0.0081 0.908 0.1475 0.0899 1.4541 ∗ 7-32 26 567 0.4407 -0.0683 2.276 10−6 0.0094 0.9262 0.045 0.0784 1.4588 ∗ 8-33 26 586 0.6355 -0.1605 2.953 10−6 0.0083 0.9073 0.0534 0.0937 1.4143 ∗ 9-34 26 597 0.2764 0.0398 1.607 10−6 0.0085 0.9252 0.2107 0.0785 1.3665 ∗ 10-35 26 597 0.4224 0.0648 1.601 10−6 0.0095 0.9294 0.2217 0.0727 1.4037 ∗ 11-36 26 602 0.3561 -0.0101 1.596 10−6 0.0079 0.9355 0.0854 0.0759 1.3441 ∗ 12-37 26 597 1.0718 -0.2108 2.388 10−6 0.0057 0.9049 0.1051 0.1043 1.2611 ∗ 13-38 26 599 0.3494 0.0108 1.55 10−6 0.0074 0.9297 0.1421 0.0791 1.2692 ∗ 14-39 26 599 0.6523 0.0837 1.415 10−6 0.0065 0.9314 0.206 0.0795 1.3184 ∗ 15-40 26 605 1.4011 -0.1818 1.973 10−6 0.0065 0.9096 0.1432 0.0992 1.3084 ∗ 16-41 26 608 1.0055 -0.1643 2.114 10−6 0.0083 0.9072 0.1358 0.0942 1.289 ∗ 17-42 26 605 1.4482 -0.252 1.905 10−6 0.0069 0.9131 0.1209 0.0967 1.2752 ∗ 18-43 26 605 1.9829 -0.2833 1.777 10−6 0.0062 0.9169 0.1143 0.0989 1.2877 ∗ 19-44 26 607 0.8572 -0.0888 1.618 10−6 0.0073 0.9258 0.1048 0.0858 1.2781 ∗ 20-45 26 605 1.9653 -0.5268 1.932 10−6 0.0089 0.9169 0.0546 0.098 1.254 ∗ 21-46 26 615 1.7533 -0.1565 1.096 10−6 0.0093 0.9427 0.0609 0.0735 1.323 ∗ 22-47 26 614 1.7371 0.1005 1.151 10−6 0.0082 0.9329 0.2239 0.0741 1.277 ∗ 23-48 26 620 2.0898 -0.4327 1.856 10−6 0.0088 0.9108 0.1147 0.0985 1.2673 ∗ 24-49 26 627 1.7909 -0.5277 2.045 10−6 0.0081 0.9122 0.0586 0.1012 1.2535 ∗ 25-50 26 616 1.5252 -0.3409 1.84 10−6 0.0085 0.9181 0.0775 0.0925 1.2538 ∗ 26-51 26 620 1.8137 -0.3216 1.498 10−6 0.0091 0.9284 0.0712 0.0849 1.2777 ∗

60 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 ofα ˜ and β˜ are quite diﬀerent from the MLE in Table 5.3, but the

estimates of β˜ 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 ofα ˜ and β˜ are not much diﬀerent from the MLE, but some estimates

of β˜ 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 NIG-

TGARCH 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 Gaussian-

TGARCH prcing model outperforms both NIG-TGARCH with MCMM model and NIG-

TGARCH with conditional Esscher transform model for out of the money options. Further- more, 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 0.2 NIG−MCMM

0.18

0.16 Implied Volatility 0.14

0.12

0.1 0.9 0.95 1 1.05 1.1 1.15 Moneyness

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-of- sample 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 No. of No. of period weeks contract Gaussian NIG-MCMM NIG-Esscher 27 1 24 1.3528 1.1014 0.6292 28 1 24 1.3559 1.4918 1.3568 29 1 27 1.3995 1.3164 1.148 30 1 26 1.5415 1.7097 1.8047 31 1 29 2.3955 1.64 1.8765 32 1 22 1.878 1.1696 1.0328 33 1 29 0.9865 0.8404 0.7981 34 1 28 0.9326 1.0361 0.6884 35 1 18 0.8025 0.4673 0.6823 36 1 22 0.8116 0.597 0.9328 37 1 20 0.5165 0.7026 0.7816 38 1 28 1.6993 1.8464 2.121 39 1 18 1.7106 1.7704 1.7478 40 1 22 1.2203 1.5499 1.4941 41 1 16 1.1356 0.7995 1.0399 42 1 24 1.2111 1.2526 1.2855 43 1 25 0.3441 0.5677 0.5486 44 1 29 0.5883 1.1428 1.0661 45 1 25 0.782 0.7104 0.9342 46 1 24 1.8002 1.4154 1.7819 47 1 20 0.6642 1.1989 0.9152 48 1 33 0.6643 1.6166 1.1912 49 1 23 0.8337 1.153 1.1512 50 1 10 1.0515 0.4593 1.247 51 1 26 0.8561 1.0863 1.5001 52 1 20 1.6257 1.4362 1.4119

64 Figure 5.3: Boxplot of out-of-sample RMSE for the three competing models

2.4

2.2

1.8

1.6

1.4

1.2

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 NIG-

TGARCH 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

2.5

1.5

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

3.5

2.5

1.5

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

3.5

2.5

1.5

0.5 risk−neutral esitmator MLE

67 Table 5.11: Out-of-sample RMSE using MLE for the three competing models

Week No. of No. of period weeks contract Gaussian NIG-MCMM NIG-Esscher 27 1 24 2.2416 2.67 2.3962 28 1 24 3.113 3.7391 3.6244 29 1 27 2.2388 2.6789 2.5146 30 1 26 3.4065 4.0246 3.8937 31 1 29 1.9155 1.9845 1.5819 32 1 22 1.1671 1.0938 0.9877 33 1 29 1.5924 1.6432 1.586 34 1 28 2.1869 2.3513 2.0268 35 1 18 1.3153 1.681 1.5994 36 1 22 2.4573 2.7847 2.1372 37 1 20 2.0647 2.5547 2.2362 38 1 28 1.5917 1.5464 1.1451 39 1 18 1.53 1.374 1.1295 40 1 22 1.1077 0.7724 0.7346 41 1 16 0.5088 0.7132 0.6802 42 1 24 0.8803 0.8933 0.6945 43 1 25 0.7603 1.0218 1.065 44 1 29 0.8167 0.9393 0.8794 45 1 25 0.6201 0.9607 1.044 46 1 24 1.6481 2.1811 2.0841 47 1 20 0.5188 1.0937 0.9699 48 1 33 0.6779 1.0772 0.9882 49 1 23 1.0941 1.5042 1.3138 50 1 10 1.4682 2.272 1.963 51 1 26 1.2645 1.8035 1.3641 52 1 20 2.2932 3.0526 2.9119

68 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 Gaussian-

TGARCH 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, ,P ) with a filtration t, a cadlag process (Mt)t [0,T ] F F ∈ is a martingale if M is adapted to , E[ Mt ] is finite for t [0, T ] and Ft | | ∈

E[M ]= M , s > t. s|Ft t ∀

This is a crucial concept to establish the non-arbitrage pricing theory in mathematical ﬁ- nance. For Levey processes, diﬀerent martingales can be constructed from their independent increments property. Brownian motion

Deﬁnition: Let (Ω, ,P ) be a probability space. For each ω Ω, suppose there is a contin- F ∈ uous function W of t 0 that satisﬁes W = 0 and that depends on ω. Then W , t 0, is t ≥ 0 t ≥ a Brownian motion if for all 0 = t < t < < t the increments 0 1 ··· m

W = W W , W W ,...,W W t1 t1 − t0 t2 − t1 tm − tm−1 are independent and each of these increments is normally distributed with

E[W W ]=0 ti+1 − ti V ar[W W ]= t t ti+1 − ti i+1 − i

Itoˆ’s process

Definition: Let W be a Brownian motion and an associated filtration. An Itô’s process t Ft is defined as:

t t Xt = X0 + µ(s)ds + σ(s)dWs, Z0 Z0 70 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 ∞ λ 1 x 1 K (x)= y − exp( (y + y− ))dy, x> 0. λ 2 − 2 Z0 π 1/2 x Moreover, Kλ(x)= K λ(x) and K 1/2(x)= K1/2(x)= x− e− . − − 2 Normal distribution p

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

2 1 1 x−µ f(x)= e− 2 σ ,

The distribution with µ = 0 and σ2 = 1 is called the standard normal distribution or the

unit normal distribution. The moment generating function of x is given by:

1 M (t) = exp(µt + σ2t2), t R. x 2 ∈

The cumulant function of x is:

1 κ (t)= logE(ext)= logM (t)= µt + σ2t2, t R. x x 2 ∈

If x N(0, 1), the moment generating function and cumulant function are: ∼ 1 M (t) = exp t2 , t R. x 2 ∈ 1 κ (t)= t2, t R. x 2 ∈

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