Parameter Estimation in Stochastic Volatility Models Via Approximate Bayesian Computing
A Thesis
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University
By
Achal Awasthi, B.S.
Graduate Program in Department of Statistics
The Ohio State University
2018
Master’s Examination Committee:
Radu Herbei,Ph.D., Advisor Laura S. Kubatko, Ph.D. c Copyright by
Achal Awasthi
2018 Abstract
In this thesis, we propose a generalized Heston model as a tool to estimate volatil- ity. We have used Approximate Bayesian Computing to estimate the parameters of the generalized Heston model. This model was used to examine the daily closing prices of the Shanghai Stock Exchange and the NIKKEI 225 indices. We found that this model was a good fit for shorter time periods around financial crisis. For longer time periods, this model failed to capture the volatility in detail.
ii This is dedicated to my grandmothers, Radhika and Prabha, who have had a
significant impact in my life.
iii Acknowledgments
I would like to thank my thesis supervisor, Dr. Radu Herbei, for his help and his availability all along the development of this project. I am also grateful to Dr. Laura
Kubatko for accepting to be part of the defense committee. My gratitude goes to my parents, without their support and education I would not have had the chance to study worldwide. I would also like to express my gratitude towards my uncles,
Kuldeep and Tapan, and Mr. Richard Rose for helping me transition smoothly to life in a different country. In addition, my deepest appreciation goes to my friends at the department of Statistics who have been there for me since my first day of class at the Ohio State University. Finally, I am extremely thankful to my housemates for bearing with me during the past one year.
iv Vita
2016 ...... B.S. Physics
2016-present ...... Graduate Teaching Associate, The Ohio State University.
Publications
Fields of Study
Major Field: Department of Statistics
v Contents
Page
Abstract...... ii
Dedication...... iii
Acknowledgments...... iv
Vita...... v
List of Tables...... viii
List of Figures...... xii
1. Introduction...... 1
1.1 Motivation...... 1 1.2 Emerging Markets during Financial Crisis...... 2 1.3 Structure of Thesis...... 5
2. Background...... 6
2.1 Introduction...... 6 2.2 Brownian Motion...... 8 2.3 Geometric Brownian Motion (GBM)...... 10 2.3.1 Parameter Estimation for the GBM process using Maximum Likelihood Estimation...... 14 2.4 The Ornstein-Uhlenbeck Process...... 17 2.4.1 Simulation of the OU Process...... 18 2.4.2 Parameter Estimation for OU Process using Maximum Like- lihood...... 20
vi 2.4.3 Parameter Estimation for OU Process using Ordinary Least Squares...... 23 2.5 Cox-Ingersoll-Ross Process...... 26 2.5.1 Simulation of CIR process...... 27 2.5.2 Parameter Estimation for CIR Process using Maximum Like- lihood...... 32 2.6 Generalized Cox-Ingersoll-Ross model...... 35 2.6.1 Parameter Estimation for generalized CIR Process using Max- imum Likelihood...... 37 2.6.2 Distribution of R t2 W (s) ds ...... 40 t1
3. Approximate Bayesian Computing for Stochastic Volatility Models... 43
3.1 Heston Model...... 43 3.1.1 Simulation of sample paths of the Heston Model...... 45 3.1.2 Euler-Maruyama (EM) Approximation...... 46 3.1.3 Euler-Maruyama scheme with Lord et al’.s modification.. 47 3.1.4 Milstein scheme...... 47 3.1.5 Broadie and Kaya’s Exact Algorithm...... 48 3.2 A generalized Heston Model...... 51 3.2.1 Simulation of sample paths of the generalized Heston model 53 3.3 Approximate Bayesian Computing (ABC)...... 60 3.3.1 ABC for Heston Model...... 61 3.3.2 ABC for generalized Heston Model...... 83
4. Application: Modeling Volatility in Financial Markets...... 100
4.1 Introduction...... 100 4.1.1 Stock Index...... 100 4.2 Exploratory Data Analysis...... 104 4.3 Parameter estimation of the Generalized Heston model using ABC 107 4.3.1 Parameter estimation using ABC for SSE...... 107 4.3.2 Parameter estimation using ABC for NIKKEI 225..... 134
5. Contributions and Future Work...... 142
5.1 Results Overview...... 142 5.2 Future Work...... 144 5.2.1 Moments of generalized Heston model...... 145
Bibliography...... 147
vii List of Tables
Table Page
3.1 Table showing the number of simulations vs number of accepted pa-
rameters for different = 100...... 62
3.2 Table showing the number of simulations vs number of accepted pa-
rameters for different = 200...... 62
3.3 Table showing the number of simulations vs number of accepted pa-
rameters for different = 500...... 63
3.4 Table showing the number of simulations vs number of accepted pa-
rameters for different = 800...... 63
3.5 Table showing the number of simulations vs number of accepted pa-
rameters for different = 1000...... 63
viii 3.6 Table showing the number of simulations vs number of accepted pa-
rameters for different = 1500...... 64
3.7 Table showing the number of simulations vs number of accepted pa-
rameters for = 100...... 84
3.8 Table showing the number of simulations vs number of accepted pa-
rameters for = 200...... 84
3.9 Table showing the number of simulations vs number of accepted pa-
rameters for = 500...... 85
3.10 Table showing the number of simulations vs number of accepted pa-
rameters for = 800...... 85
3.11 Table showing the number of simulations vs number of accepted pa-
rameters for = 1, 000...... 85
3.12 Table showing the number of simulations vs number of accepted pa-
rameters for = 1, 500...... 86
4.1 Table showing the number of simulations vs number of accepted pa-
rameters for different = 10, 000...... 108
ix 4.2 Table showing the number of simulations vs number of accepted pa-
rameters for different levels...... 111
4.3 Table showing the estimated parameters for different levels (100 sim-
ulations)...... 114
4.4 Table showing the number of simulations vs number of accepted pa-
rameters for different levels...... 116
4.5 Table showing the estimated parameters for different levels (100 sim-
ulations)...... 120
4.6 Table showing the number of simulations vs number of accepted pa-
rameters for different levels...... 122
4.7 Table showing the estimated parameters for different levels (100 sim-
ulations)...... 126
4.8 Table showing the number of simulations vs number of accepted pa-
rameters for different levels...... 128
4.9 Table showing the estimated parameters for different levels...... 132
x 4.10 Table showing the number of simulations vs number of accepted pa-
rameters for different levels...... 135
4.11 Table showing the estimated parameters for different levels...... 139
xi List of Figures
Figure Page
2.1 Simulated paths of the GBM process with parameters as described in
algorithm1...... 13
2.2 Histogram of log of GBM at the 50th time-step. The orange curve repre-
sents the superimposed normal density curve with parameters obtained
from simulated data at the 50th time-step...... 14
2.3 Histogram of estimated values of µ of the GBM as simulated above.
The dashed red line represents the true value of the parameter..... 16
2.4 Histogram of estimated values of σ of the GBM as simulated above.
The dashed red line represents the true value of the parameter..... 16
2.5 Simulated paths of the OU process with parameters as described above 19
xii 2.6 Histogram of estimated values of β of the OU process as simulated
above. The dashed red line represents the true value of the parameter. 21
2.7 Histogram of estimated values of θ of the OU process as simulated
above. The dashed red line represents the true value of the parameter. 22
2.8 Histogram of estimated values of σ of the OU process as simulated
above. The dashed red line represents the true value of the parameter. 22
2.9 Histogram of estimated values of β of the OU process using least
squares approximation. The dashed red line represents the true value
of the parameter...... 24
2.10 Histogram of estimated values of θ of the OU process using least squares
approximation. The dashed red line represents the true value of the
parameter...... 25
2.11 Histogram of estimated values of σ of the OU process using least
squares approximation. The dashed red line represents the true value
of the parameter...... 25
2.12 Simulated paths of the CIR process with parameters as described above. 32
xiii 2.13 Histogram of estimated values of α of the CIR process. The dashed
red line represents the true value of the parameter...... 34
2.14 Histogram of estimated values of β of the CIR process. The dashed
red line represents the true value of the parameter...... 34
2.15 Histogram of estimated values of σ of the CIR process. The dashed
red line represents the true value of the parameter...... 35
2.16 Simulated paths of the generalized CIR process with parameters as
described above...... 37
2.17 Histogram of estimated values of α of the generalized CIR process using
normal approximation. The dashed red line represents the true value
of the parameter...... 38
2.18 Histogram of estimated values of β of the generalized CIR process using
normal approximation. The dashed red line represents the true value
of the parameter...... 39
2.19 Histogram of estimated values of σ of the generalized CIR process using
normal approximation. The dashed red line represents the true value
of the parameter...... 39
xiv 2.20 Histogram of estimated values of γ of the generalized CIR process using
normal approximation. The dashed red line represents the true value
of the parameter...... 40
3.1 Simulation of a path of CIR process with N = 252, α = 0.09, β = 0.145
and σ = 0.055...... 50
3.2 Simulation of a path of Heston process with N = 252, α = 0.09, β =
0.145, µ = 0.009 and σ = 0.055...... 51
3.3 s=4 intermediate points between ti and ti+1...... 53
3.4 Simulated path of the CIR process with parameters α2 = 0.221, β2 =
th 0.601, σ2 = 0.055. Every (s + 1) value has been chosen for the plot,
where s has been defined in step-I...... 54
3.5 Simulated path of the OU process with parameters α1 = 0.14, β1 =
th 0.861, σ1 = 0.009. Every (s + 1) value has been chosen for the plot,
where s has been defined in step-I...... 55
3.6 Simulated path of the estimates of R t2 ν(s) ds at different time points. 56 t1
3.7 Simulated path of the estimate of R t2 µ(s) ds...... 57 t1
xv 3.8 Simulated path of the estimate of R t2 pν(s) dW ν(s)...... 58 t1
3.9 Simulated path of the estimate of R t2 pν(s) dW Z ...... 59 t1
3.10 Simulated sample path of the generalized Heston model...... 60
3.11 Histograms of accepted values of the parameters of the Heston Model
for = 100 and 1000 simulations. The dashed red lines represent the
true values of the parameters...... 65
3.12 Histograms of accepted values of the parameters of the Heston Model
for = 100 and 10, 000 simulations. The dashed red lines represent
the true values of the parameters...... 66
3.13 Histograms of accepted values of the parameters of the Heston Model
for = 100 and 100, 000 simulations. The dashed red lines represent
the true values of the parameters...... 67
3.14 Histograms of accepted values of the parameters of the Heston Model
for = 200 and 1, 000 simulations. The dashed red lines represent the
true values of the parameters...... 68
xvi 3.15 Histograms of accepted values of the parameters of the Heston Model
for = 200 and 10, 000 simulations. The dashed red lines represent
the true values of the parameters...... 69
3.16 Histograms of accepted values of the parameters of the Heston Model
for = 200 and 100, 000 simulations. The dashed red lines represent
the true values of the parameters...... 70
3.17 Histograms of accepted values of the parameters of the Heston Model
for = 500 and 1, 000 simulations. The dashed red lines represent the
true values of the parameters...... 71
3.18 Histograms of accepted values of the parameters of the Heston Model
for = 500 and 10, 000 simulations. The dashed red lines represent
the true values of the parameters...... 72
3.19 Histograms of accepted values of the parameters of the Heston Model
for = 500 and 100, 000 simulations. The dashed red lines represent
the true values of the parameters...... 73
3.20 Histograms of accepted values of the parameters of the Heston Model
for = 800 and 1, 000 simulations. The dashed red lines represent the
true values of the parameters...... 74
xvii 3.21 Histograms of accepted values of the parameters of the Heston Model
for = 800 and 10, 000 simulations. The dashed red lines represent
the true values of the parameters...... 75
3.22 Histograms of accepted values of the parameters of the Heston Model
for = 800 and 100, 000 simulations. The dashed red lines represent
the true values of the parameters...... 76
3.23 Histograms of accepted values of the parameters of the Heston Model
for = 1000 and 1, 000 simulations. The dashed red lines represent
the true values of the parameters...... 77
3.24 Histograms of accepted values of the parameters of the Heston Model
for = 1000 and 10, 000 simulations. The dashed red lines represent
the true values of the parameters...... 78
3.25 Histograms of accepted values of the parameters of the Heston Model
for = 1000 and 100, 000 simulations. The dashed red lines represent
the true values of the parameters...... 79
3.26 Histograms of accepted values of the parameters of the Heston Model
for = 1500 and 1, 000 simulations. The dashed red lines represent
the true values of the parameters...... 80
xviii 3.27 Histograms of accepted values of the parameters of the Heston Model
for = 1500 and 10, 000 simulations. The dashed red lines represent
the true values of the parameters...... 81
3.28 Histograms of accepted values of the parameters of the Heston Model
for = 1500 and 100, 000 simulations. The dashed red lines represent
the true values of the parameters...... 82
3.29 Histograms of estimated values of the parameters of the generalized
Heston Model for = 100 and 1000 simulations. The dashed red lines
represent the true values of the parameters...... 87
3.30 Histograms of estimated values of the parameters of the generalized
Heston Model for = 100 and 10000 simulations. The dashed red lines
represent the true values of the parameters...... 88
3.31 Histograms of estimated values of the parameters of the generalized
Heston Model for = 200 and 1000 simulations. The dashed red lines
represent the true values of the parameters...... 89
3.32 Histograms of estimated values of the parameters of the generalized
Heston Model for = 200 and 10, 000 simulations. The dashed red
lines represent the true values of the parameters...... 90
xix 3.33 Histograms of estimated values of the parameters of the generalized
Heston Model for = 500 and 1, 000 simulations. The dashed red lines
represent the true values of the parameters...... 91
3.34 Histograms of estimated values of the parameters of the generalized
Heston Model for = 500 and 10, 000 simulations. The dashed red
lines represent the true values of the parameters...... 92
3.35 Histograms of estimated values of the parameters of the generalized
Heston Model for = 800 and 1, 000 simulations. The dashed red lines
represent the true values of the parameters...... 93
3.36 Histograms of estimated values of the parameters of the generalized
Heston Model for = 800 and 10, 000 simulations. The dashed red
lines represent the true values of the parameters...... 94
3.37 Histograms of estimated values of the parameters of the generalized
Heston Model for = 1, 000 and 1, 000 simulations. The dashed red
lines represent the true values of the parameters...... 95
3.38 Histograms of estimated values of the parameters of the generalized
Heston Model for = 1, 000 and 10, 000 simulations. The dashed red
lines represent the true values of the parameters...... 96
xx 3.39 Histograms of estimated values of the parameters of the generalized
Heston Model for = 1, 500 and 1, 000 simulations. The dashed red
lines represent the true values of the parameters...... 97
3.40 Histograms of estimated values of the parameters of the generalized
Heston Model for = 1, 500 and 10, 000 simulations. The dashed red
lines represent the true values of the parameters...... 98
4.1 Daily Adjusted Closing Price of SSE from 01/01/96 to 04/08/16... 105
4.2 Daily Log Adjusted Closing Price of SSE from 01/01/96 to 04/08/16. 106
4.3 Daily Adjusted Closing Price of NIKKEI 225 from 01/05/15 to 07/24/18.106
4.4 Daily Log Returns of NIKKEI 225 from 01/05/15 to 07/24/18..... 107
4.5 Histograms of accepted values of the parameters for = 10, 000 and
100 simulations...... 109
4.6 Comparison between simulated dataset and testing dataset...... 110
4.7 Histograms of accepted values of the parameters for = 10, 000 and
100 simulations...... 112
xxi 4.8 Histograms of accepted values of the parameters for = 5, 000 and 100
simulations...... 113
4.9 Comparison between simulated dataset and testing dataset for =
5, 000 for the first period...... 115
4.10 Comparison between simulated dataset and testing dataset for =
10, 000 for the first period...... 115
4.11 Histograms of accepted values of the parameters for = 1, 000 and 100
simulations...... 117
4.12 Histograms of accepted values of the parameters for = 5, 000 and 100
simulations...... 118
4.13 Histograms of accepted values of the parameters for = 10, 000 and
100 simulations...... 119
4.14 Comparison between simulated dataset and testing dataset for =
1, 000 for the second period...... 121
4.15 Comparison between simulated dataset and testing dataset for =
5, 000 for the second period...... 121
xxii 4.16 Comparison between simulated dataset and testing dataset for =
10, 000 for the second period...... 122
4.17 Histograms of accepted values of the parameters for = 10, 000 and
100 simulations...... 123
4.18 Histograms of accepted values of the parameters for = 5, 000 and 100
simulations...... 124
4.19 Histograms of accepted values of the parameters for = 1, 000 and 100
simulations...... 125
4.20 Comparison between simulated dataset and testing dataset for =
1, 000 for the third period...... 127
4.21 Comparison between simulated dataset and testing dataset for =
5, 000 for the third period...... 127
4.22 Comparison between simulated dataset and testing dataset for =
10, 000 for the third period...... 128
4.23 Histograms of accepted values of the parameters for = 10, 000 and
100 simulations...... 129
xxiii 4.24 Histograms of accepted values of the parameters for = 5, 000 and 100
simulations...... 130
4.25 Histograms of estimated values of the parameters for = 1, 000 and
100 simulations...... 131
4.26 Comparison between simulated dataset and testing dataset for =
1, 000 for the fourth period...... 133
4.27 Comparison between simulated dataset and testing dataset for =
5, 000 for the fourth period...... 133
4.28 Comparison between simulated dataset and testing dataset for =
10, 000 for the fourth period...... 134
4.29 Histograms of accepted values of the parameters for = 10, 000 and
100 simulations...... 136
4.30 Histograms of accepted values of the parameters for = 5, 000 and 100
simulations...... 137
4.31 Histograms of accepted values of the parameters for = 1, 000 and 100
simulations...... 138
xxiv 4.32 Comparison between simulated dataset and testing dataset for = 1, 000.140
4.33 Comparison between simulated dataset and testing dataset for = 5, 000.140
4.34 Comparison between simulated dataset and testing dataset for =
10, 000...... 141
xxv Chapter 1: Introduction
1.1 Motivation
Physicists, statisticians and mathematicians have long been interested in theories related to finance. The tools developed in statistical physics, statistics and theoretical mathematics can be used to model complex financial systems. Many changes have taken place in the world of finance in the later half of the last century. For exam- ple, in 1973 currencies began to be traded in financial markets. The values of these were determined by the foreign exchange markets that are active 24 hours a day all over the world. Among other changes are new models that have come up for esti- mating volatility which is an inherent framework for pricing European options. The
Black-Scholes model (BSM) was among the first successful models to price options.
However, this model is based on several assumptions that are not representative of the real world. In particular, the BSM assumes that volatility is deterministic and remains constant through the option’s life, which clearly contradicts the behavior observed in financial markets. While the BSM framework can be adapted to obtain reasonable prices for plain vanilla options, the constant volatility assumption may lead to significant mispricings when used to evaluate options with non-conventional or exotics features.
1 During the last decades several alternatives have been proposed to improve volatility modeling in the context of derivatives pricing. One such approach is to model volatil- ity as a stochastic quantity. By introducing uncertainty in the behavior of volatility, the evolution of financial assets can be estimated more realistically. In addition, using appropriate parameters, stochastic volatility models can be calibrated to reproduce the market prices of liquid options and other derivatives contracts. One of the most widely used stochastic volatility models was proposed by Heston in 1993. The Heston model introduces a dynamic for the underlying asset which can take into account the asymmetry and excess kurtosis that are typically observed in financial assets returns.
It also provides a closed-form valuation formula that can be used to efficiently price plain vanilla options. This will be particularly useful in the calibration process, where many option pricings are usually required in order to find the optimal parameters that reproduce market prices.
1.2 Emerging Markets during Financial Crisis
Previous research shows us that the strong functioning of stock markets has con- siderable effect on the growth of an economy, especially so in a developing one. Over the past few decades, studies have been conducted around the globe by many re- searchers on the subject of stock market efficiency, and the conflicting results have made it difficult to comment on the status of stock market of a particular country. So, we focus our attention on the stock market behavior in developing countries which aren’t considered to be as stable as the developed ones. They are unlikely to be fully information-efficient, partly due to institutional barriers restricting information flows to the market and partly due to lack of experience of market participants to rapidly
2 lock up new information into security prices. Therefore, it would be interesting to investigate this period of last 20 years studying both the Global Financial and the
Chinese crisis and its effects on fastest emerging economies of India and China. Re- cession had crumpled economies worldwide but these two were relatively unaffected and hence are of particular interest.
The the current fastest growing economies BRICS (Brazil, Russia, India, China, South
Africa) were affected primarily through four channels of trade, finance, commodity, and confidence. The slump in export demand and firmer trade credit caused a slow- down in aggregate demand. The global financial crisis inflicted significant loss in output in all these countries. However, the real GDP growth in India and China remained impressive even though they witnessed some moderation due to weakening global demand. The crisis also exposed the structural weakness of the global financial and real sectors. The BRICS were able to recover quickly with the support of domes- tic demand. The reversal of capital flows led to equity market losses and currency depreciations, resulting in lower external credit flows. The banking sectors of the
BRICS economies performed relatively well [20].
Since our analysis revolves around the two recent financial crises, we need to under- stand its effects as well. A financial crisis is a disruption to financial markets in which adverse selection and moral hazard problems become much worse, so that financial markets are unable to efficiently channel funds to those who have the most produc- tive investment opportunities. As a result, a financial crisis can drive the economy away from an equilibrium with high output in which financial markets perform well to one in which output declines sharply [19]. The end of 2007 and beginning of 2008 observed that the onset of global financial crisis had brought disorder to the financial
3 markets around the world and it is the first crisis in consideration for our study. The
instability in the global stock market scenario began with a shortfall of liquid assets in
US banking system and the continual fall in stock prices on information that Lehman
Brothers, Merill Lynch and many other investment banks and companies were col-
lapsing. The stock markets around the globe suffered huge losses and Indian stock
market was no exception. The SENSEX which had reached historically high levels in
the beginning of 2008, turned down to its level about three years back and the S&P
CNX NIFTY also followed a similar trend. Economic growth decelerated in 2008-09 to 6.7 percent. This represented a decline of 2.1 percent from the average growth rate of 8.8 percent in the previous five years. China was not one of the countries hardest hit by the crisis, neither was it as insulated as many had assumed. This can be seen from the fact that China continued to have one of the highest rates of economic growth across the globe, recording 9.6% in 2008 and 9.2% in 2009.
While most countries would be delighted to have such growth rates, the point to be considered is that these rates reflected a substantial drop from the 14.2% growth in
2007. In terms of short term impact on China, the most visible damage was inflicted on its export-oriented light industry in southern China. Thousands of companies went bust, tens of thousands of workers have been laid-off and official statistics revealed that 10 million migrant workers had returned back to their home provinces. In the
financial sector the stock market crash that started in late 2007 had wiped out more than two thirds of market value although this dramatic collapse was not without any home-made reasons [16]. The Chinese banks for all their profitability witnessed the sudden pull-out of many of their Western partners which (Bank of America, UBS,
RBS) sold their minority stakes in order to retrieve capital. Another massive blow
4 was to the China’s fledgling sovereign wealth fund, China Investment Corporation.
The second crisis in consideration for our study is the Chinese stock market crash which began with the popping of the stock market bubble on 12 June 2015. A third of the value of A-shares on the Shanghai Stock Exchange was lost within one month of the event since mid-June. By 89 July 2015, the Shanghai stock market had fallen
30 percent over three weeks as 1,400 companies, or more than half listed, filed for a trading halt in an attempt to prevent further losses. This crisis was inevitable because over major part of 2014-15, investors kept investing more and more into
Chinese stocks, encouraged by falling borrowing costs as the central bank loosened monetary policy even though economic growth and company profits were weak with retail investors being the one leading this.
1.3 Structure of Thesis
This thesis is organized as follows: in chapter2, we present the most commonly en- countered stochastic models in finance, their simulations and parameter estimations.
Section 3 is devoted to a complete analysis of estimation of parameters of the Heston model using Approximate Bayesian Computing. In chapter3, we also present a new model namely, the generalized Heston model for estimating volatility. In chapter4, we fit the generalized Heston model to the data from the Shanghai Stock Exchange and NIKKEI 225. In chapter5 we discuss some of the results and talk about future work.
5 Chapter 2: Background
2.1 Introduction
In this chapter, we introduce the basic concepts from probability theory and its applications in the field of finance. We also introduce several important and widely used stochastic processes. In addition to their definitions, we describe a statistical approach to estimating the parameters defining these processes.
Definition 1. Let Ω be a non-empty set, and let F be a collection of subsets of Ω.
F is a σ−algebra if it satisfies,
1. ∅ ∈ F,
2. If a set A ∈ F, then Ac ∈ F,
∞ 3. If a sequence of sets A1,A2, · · · ∈ F, then ∪n=1An ∈ F.
Definition 2. Let Ω be a non-empty set, and let F be a σ−algebra over Ω.A probability measure P is a function that, to every set A ∈ F assigns a number in
[0, 1]. This number is called the probability of A and is represented as P(A).
The measure P should satisfy the following properties,
1. P(Ω) = 1, and
6 2. If A1,A2,... is a sequence of disjoint sets such that An ∈ F for all n ≥ 1, then
! ∞ ∞ X P ∪n=1 An = P(An) (2.1) n=1
The triple (Ω, F, P) is called a probability space.
Definition 3. Let F be a σ−algebra and Ω the space of outcomes which are specific to an experiment. A function X : (Ω, F) → R is a random variable if for every subset
Fr = {ω: X(ω) ≤ r} r ∈ R, the condition Fr ∈ F is satisfied.
A random variable X is called a discrete random variable if its range {X(ω): ω
∈ Ω} is countable. A random variable X is called a continuous random variable if its range is a continuous subset of R. A continuous random variable has a cumulative distribution function (CDF) which is absolutely continuous. On the other hand, the
CDF of a discrete random variable is a step function with discontinuities at the values taken on by the random variable.
Definition 4. Let T ⊆ [0, ∞). A family of random variables {Xt}t∈T is called a stochastic process. If T ⊆ N, then the stochastic process is discrete and if T ⊆ [0, ∞), the stochastic process is continuous.
For example, let {X(t): t = 0, 1, 2,...} be a stochastic process that evolves according to the following rule: X(0) = 0 and, for t ≥ 0,
X(t + 1) = X(t) + 1 with probability p X(t + 1) = X(t) − 1 with probability 1 − p,
Then, the stochastic process {X(t): t > 0} is called a random walk. If p = 1/2 i.e., we are equally likely to move forward or backward, then the random walk is called a symmetric random walk. If p 6= 1/2, i.e. we have a preferred direction, then
7 the random walk is called a biased random walk. The random walk process has the following properties,
• If p = 1/2 all states of a random walk are recurrent. If p 6= 1/2 all states are
transient.
• Each state of a random walk has period 2 except for the first and last states, if
the process is assumed to live in 1, 2, . . . , k for some positive integer k.
2.2 Brownian Motion
Brownian Motion (BM) was first observed by biologist Robert Brown [9] in 1827 while studying pollen particles. He observed that when seen under a microscope, the pollen particles floating in water exhibited a zig-zag jittery motion. He repeated the experiment with particles of dust and concluded that the motion was due to the pollen being alive. But, he could not explain the source of this random motion. The theory of
BM was first given by French mathematician Louis Bachelier in his PhD thesis titled
”Theory of Speculation” [7]. It was in 1905 when renowned physicist Albert Einstein using probabilistic arguments was able to explain the theory of BM. He observed that under the right kinetic energy, molecules of water would move randomly. This is how
Robert Brown described the movement of pollens.
The theory of BM has been applied to a variety of fields ranging from biology, physics, economics, mathematics to finance. Stock market researchers were battling with a problem similar to what Robert Brown had encountered in 1827. They were able to
figure out the path of market price but they did not know the reason behind it. They could not determine who was buying, who was selling and how demand and supply were affecting price movements.
8 Definition 5. Let (Ω, F, P) be a probability space. A stochastic process {W (t): t ≥ 0} is said to be a standard Brownian motion process if,
• W (0) = 0 almost surely;
• The increments for non-overlapping time intervals are independent.
• W (t) − W (s) ∼ N(0, t − s) for s < t,
• cov(W (s),W (t)) = min(s, t).
Next, we briefly introduce the concept of a stochastic differential equation (SDE).
Let {X(t): t ≥ 0} be a stochastic process and assume that the process satisfies the following equation,
Z t Z t X(t) = X(0) + a(X(s), s) ds + B(X(s), s) dW (s), (2.2) 0 0 where a(·, ·) and b(·, ·) are known functions and {W (t): t ≥ 0} is a standard Brownian motion. In the equation above, the integral
Z t a(X(s), s) ds 0 is a Riemann integral whereas the integral
Z t B(X(s), s)dW (s) 0 is an Itˆo integral. Throughout this dissertation, we will assume that the functions a(·, ·) and b(·, ·) satisfy sufficient conditions for such integrals to exist and to be finite almost surely. Such conditions can be found in [15]. If a process X(t) satisfies equation
(2.2), we say that X(t) is a diffusion process. Equation (2.2) can be briefly written as,
dX(t) = a(X(t), t) dt + b(X(t), t) dW (t) (2.3)
9 The term a(·, ·) is called the drift term while the function b(·, ·) is called the diffusion
coefficient. In this dissertation we only briefly review some of the necessary tools and
processes from this area. The equation (2.3) is referred to as a stochastic differential
equation (SDE).
Proposition 1. Itoˆ’s Lemma - Let X(t) be a stochastic process which satisfies the
following stochastic differential equation,
dX(t) = a(X(t), t) dt + b(X(t), t) dW (t)
and let f(x,t) be any twice differentiable scalar function of two real variables x and t,
then Itoˆ’s lemma states that, " # ∂f(X, t) ∂f(X, t) b2(X, t) ∂2f(X, t) ∂f(X, t) df(X(t), t) = +a(X, t) + dt+b(X, t) dW (t). ∂t ∂x 2 ∂x2 ∂x
A proof of this lemma can be found in [15].
2.3 Geometric Brownian Motion (GBM)
Definition 6. Let {W (t): t ≥ 0} be a stochastic process that describes a Brownian
Motion. Let S(0) > 0 and µ ∈ R and σ ∈ R+ be constants. If S(t) satisfies the following stochastic differential equation,
dS(t) = µS(t)dt + σS(t)dW (t) (2.4)
then it is said to be a Geometric Brownian Motion (GBM).
The solution of (2.4) is,
n o S(t) = S(0) · exp (µ − 0.5σ2)t + σW (t)
10 For, a small increase in time from t to t + ∆t, the ratio of S(t + ∆t)/S(t) is
S(t + ∆t) n o = exp (µ − 0.5σ2)∆t + σ(W (t + ∆t) − W (t)) S(t) where, W (t+∆t)−W (t) ∼ N(0, ∆t). From this definition, it follows that S(t) cannot be zero at any point of time. If σ (the volatility) equals zero, then equation (2.4) reduces to
S(t) = S(0) exp (µt) .
This implies that given S(0) > 0, S(t) is an increasing function of time t. As noted, for any particular time interval ∆t,
n o S(t + ∆t) = S(t) · exp (µ − 0.5σ2)∆t + σ(W (t + ∆t) − W (t)) (2.5)
If we take logarithms on both sides, we obtain the following equation,
log(S(t + ∆t)) − log(S(t)) = (µ − 0.5σ2)∆t + σ[W (t + ∆t) − W (t)] where, W (t + ∆t) − W (t) ∼ N(0, ∆t). So, σ[W (t + ∆t) − W (t)] ∼ N(0, σ2∆t).
It follows that, (µ − 0.5σ2)∆t + σ[W (t + ∆t) − W (t)] ∼ N[(µ − 0.5σ2)∆t, σ2∆t].
Consequently, conditionally on log(S(t)),
log(S(t + ∆t)) ∼ N[log(S(t)) + (µ − 0.5σ2)∆t, σ2∆t].
The expectation of this process is,
h n o i 2 E(S(t)|S(0)) = E S(0) · exp σW (t) + (µ − 0.5σ )t S(0)
n 2 o = S(0) · exp (µ − 0.5σ )t · E[exp (σW (t))] n o = S(0) · exp (µ − 0.5σ2)t · exp{0.5σ2 · t}
= S(0) · exp (µt)
11 h i Here, we have used the fact that E exp{cW (t)} = exp(c2t/2), where c ∈ R. Simi- larly, the variance of S(t) is,
V ar(S(t)|S(0)) = S(0)2 · exp(2µt) · exp(σ2t − 1).
This stochastic process has been used to model quantities that must be positive. In
figure 2.1, we show 500 simulated paths of a GBM process, which have been obtained according to algorithm1.
Algorithm 1. (Simulation of the GBM process)
• Set the process parameters i.e. total time period (T) = 10, number of steps (N)
= 1000, number of simulations (n) = 500, β = 1.5, θ = 0.15, σ = 0.1.
• Let ∆t = T/N and initialize the process by setting S(0).
• Recursively simulate S(t + ∆t) using (2.5), where W (t + ∆t) − W (t) ∼ N(0,∆t)
is independent of everything else.
12 160
140
120
100
S(t) 80
60
40
20
0 2 4 6 8 10 t
Figure 2.1: Simulated paths of the GBM process with parameters as described in algorithm1
13 Histogram of ln(S(t=50dt))
6
5
4
3 Frequency 2
1
0 2.8 2.9 3.0 3.1 3.2 3.3 Value
Figure 2.2: Histogram of log of GBM at the 50th time-step. The orange curve represents the superimposed normal density curve with parameters obtained from simulated data at the 50th time-step.
2.3.1 Parameter Estimation for the GBM process using Max- imum Likelihood Estimation
Let {X(t): t ≥ 0} be a stochastic process that satisfies the Markov’s property. As- sume that we observe this process at a discrete collection of time points {t0, t1, . . . , tn} where, t0 = 0, ti = iT/n for i = 1, 2, . . . , n. Let X = {X(t0),X(t1),...,X(tn)} be the available data. For simplicity, we use Xi = X(ti). Let θ be the parameters defining the process {X(t): t ≥ 0}. The likelihood function is defined as,
n Y L(θ|X1,X2,...,Xn) = fθ (Xi|Xi−1) i=1 where fθ (Xi|Xi−1) is called the transition density, and X0 is assumed to be fixed. We make this assumption throughout this document. For the GBM process the transition
14 density is, ! 1 (log(X /X ) − ντ)2 √ i i−1 f(Xi|Xi−1) = exp − 2 σXi 2πτ 2σ τ where ν = µ − σ2/2 and τ = T/n. Thus, the likelihood function is,
t 2 ! Y 1 (log(Xi/Xi−1) − ντ) L(µ, σ|X) = √ exp − 2σ2τ i=1 σXi 2πτ
Instead of maximizing the likelihood function, we maximize the log likelihood func- tion l(µ, σ|X).
For a simulation study, we generated a data set according to algorithm1 using the same parameter values as above. Based on such data, we used the built in mini- mization function from Python to estimate the parameter values by minimizing the negative of log-likelihood. This process is repeated 500 times and the histogram of all estimates of the parameter µ is presented in Figure 2.3. The dashed red line repre- sents the true value of the parameter µ. Similarly, Figure 2.4 displays the histogram of all estimates of the parameter σ and the dashed red line represents the true value of the parameter σ.
15 Histogram of est_mu
250
200
150
Frequency 100
50
0 0.05 0.10 0.15 0.20 0.25 Value
Figure 2.3: Histogram of estimated values of µ of the GBM as simulated above. The dashed red line represents the true value of the parameter.
Histogram of est_sigma
200
150
100 Frequency
50
0 0.094 0.096 0.098 0.100 0.102 0.104 0.106 Value
Figure 2.4: Histogram of estimated values of σ of the GBM as simulated above. The dashed red line represents the true value of the parameter.
16 2.4 The Ornstein-Uhlenbeck Process
The Ornstein-Uhlenbeck (OU) process is a stochastic process that was introduced
to model the velocity of a particle that is undergoing a Brownian Motion [22]. The
OU process was an attempt to model the velocity of a particle directly. This was
particularly important because if the position of a particle is given by Brownian
Motion, then its time derivative would not exist. This difficulty was overcome by
using the OU process to model the velocity of a particle.
In addition, the OU process was one of the first models used to model no arbitrage
interest rates as it had favorable properties, like mean reversion. Later, better models
were developed because this model could assume negative values with a positive
probability whereas the quantities it was used to model, like the no arbitrage interest
rates, could never take negative values. In the financial literature, it is also known as
the Vasicek model
Definition 7. Let {X(t): t ≥ 0} be a stochastic process and θ ∈ R and β, σ ∈ R+ be constants. If {X(t): t ≥ 0} satisfies the following stochastic differential equation,
+ dX(t) = −β(X(t) − θ)dt + σdW (t), β, σ ∈ R , θ ∈ R (2.6)
then X(t) is said to be an OU process.
In (2.6) above, the term dX(t) is called the infinitesimal change in X(t), β > 0 is
called the rate of mean reversion and θ is the long term mean of the OU process. The
parameter σ > 0 is called the volatility and dW (t) is Gaussian Noise. In (2.6) the
−β(X(t) − θ)dt term is known as the drift term and the term σdW (t) is known as the diffusion term.
17 The OU process is a mean reverting process, i.e., even though the process is stochastic, it has a tendency to revert to an equilibrium value. The OU process is very helpful in modeling the interest rates or volatility as these quantities are assumed to fluctuate around an equilibrium quantity. As can be seen from (2.6), if σ = 0, we get an ordinary differential equation. Let X(0) = 0, when σ = 0, (2.6) reduces to,
dX(t) = −β(X(t) − θ)dt which can be solved to get
X(t) = θ − θ exp(−βt)
As t → ∞, the general solution converges to θ. So, with the addition of the term
σdW (t), we are merely adding random fluctuations about the equilibrium position θ.
If X(t) is very far from the equilibrium position θ, then the mean reversion term
−β(X(t) − θ)dt becomes larger and pushes X(t) towards the equilibrium position θ.
2.4.1 Simulation of the OU Process
Euler-Maruyama Approximation for OU Process - Let h > 0 be the step size.
The Euler-Maruyama (EM) approximation for OU process is,
+ X(t + h) − X(t) ≈ −β(X(t) − θ)h + σ(W (t + h) − W (t)), β, σ ∈ R , θ ∈ R (2.7)
This approximation leads to the following transition distribution,
[X(t + h)|X(t)] ∼ N(X(t) − β(X(t) − θ), σ2h).
It can be shown that the exact transition density for an OU process is, ! σ2(1 − exp(−2βh)) [X(t + h)|X(t)] ∼ N θ + (X(t) − θ) exp(−βh), (2.8) 2β
18 For a fixed t and a large h > 0, [X(t + h)|X(t)] follows a normal distribution with mean θ and variance σ2/2β.
In Figure 2.5, we show 50 simulated paths according to algorithm2.
Algorithm 2. (Simulation of the OU process)
• Set the process parameters i.e. total time period (T) = 10, number of steps (N)
= 100, number of simulations (n) = 1000, β = 3.5, θ = 0.7, σ = 0.1.
• Let ∆t = T/N and initialize the process by setting X(0) = 0.7.
• Recursively simulate X(t + ∆t) using the distribution given in (2.8).
0.80
0.75
0.70 X(t)
0.65
0.60
0 2 4 6 8 10 t
Figure 2.5: Simulated paths of the OU process with parameters as described above
19 2.4.2 Parameter Estimation for OU Process using Maximum Likelihood
Let {X(t): t ≥ 0} be an OU stochastic process as defined in (2.6). Assume that
we observe this process at a discrete collection of time points {t0, t1, . . . , tn} where,
t0 = 0, ti = iT/n for i = 1, 2, . . . , n. Let X = {X(t0),X(t1),...,X(tn)} be the data.
For simplicity, we use Xi = X(ti). Let θ = (β, θ, σ). Given that this process satisfies
Markov’s property, the likelihood function is defined as,
n Y L(θ|X) = f(Xi|Xi−1) i=1
where f(Xi|Xi−1) is the transition density. For the OU process the transition density is, ! 1 −(X − α )2 f(X |X ) = √ · exp i i−1 i i−1 2πη 2η2 ! 2 where αi−1 = θ + (Xi−1 − θ) · exp (−βh) and η = σ /2β · 1 − exp(−2βh) . Thus,
the likelihood function can be written as,
t 2 ! Y 1 −(Xi − αi−1) L(θ, β, σ|X) = √ · exp (2.9) 2πη 2η i=1 The log likelihood function is,
t 2 ! −t X −(Xi − αi−1) l(θ, β, σ|X) = log(2πη) − (2.10) 2 2η i=1 For a simulation study, we generated a data set according to algorithm2 using the
same parameter values as above. Based on such data, we used the built in mini-
mization function from Python to estimate the parameter values by minimizing the
negative of log-likelihood. This process is repeated 500 times and the histogram of
all estimates of the parameter β is presented in Figure 2.6. The dashed red line rep-
resents the true value of the parameter β. Similarly, Figures 2.7 and 2.8 display the
20 histograms of all estimates of the parameters θ and σ, respectively. The dashed red
lines represent the true value of the parameters θ and σ.
Histogram of est_beta
250
200
150
Frequency 100
50
0 2 3 4 5 6 7 8 9 Value
Figure 2.6: Histogram of estimated values of β of the OU process as simulated above. The dashed red line represents the true value of the parameter.
21 Histogram of est_theta
250
200
150
Frequency 100
50
0 0.68 0.69 0.70 0.71 0.72 0.73 Value
Figure 2.7: Histogram of estimated values of θ of the OU process as simulated above. The dashed red line represents the true value of the parameter.
Histogram of est_sigma 250
200
150
Frequency 100
50
0 0.07 0.08 0.09 0.10 0.11 0.12 0.13 Value
Figure 2.8: Histogram of estimated values of σ of the OU process as simulated above. The dashed red line represents the true value of the parameter.
22 2.4.3 Parameter Estimation for OU Process using Ordinary Least Squares
We consider an OU process as represented by (2.6). Using the EM discretization procedure, we can approximate the OU process as (2.7). This can be further simplified as, √ Xt+dt = Xt(1 − βdt) + βθdt + σ dtZ (2.11) where, Z ∼ N(0, 1) represents the standard normal distribution. When represented this way, equation (2.11) can be thought of as a normal linear model with independent errors. This normal linear model is of the form Y = βX +, where Y is a N×1 vector of Xt+dt values. Thus, we can estimate the coefficient vector β and then use that to estimate the parameters of the OU process. If we compare (2.11) to an AR(1) model whose equation is of the form Xi+1 = β0 + β1Xi + , then we get βθdt = β0 and
β1 = (1 − βdt). It so happens that in this case, we would get the same estimates as we would be get from using the maximum likelihood procedure. This is true because we have a normal linear model and in the case of a normal linear model, ˆ ˆ βols = βmle i.e. the estimator obtained using ordinary least squares is the same as the estimate obtained using maximum likelihood estimation. However, we would lose some information as the least square estimates only use information from the second observation onwards where as the maximum likelihood estimates use information from the first observations itself.
th Let ˆ = Xi+1 − (β0 + β1Xi) be the i residual. The sum of squares of residuals (SSE) is defined as,
N N N 2 X 2 X 2 X SSE = ˆ = Xi+1 + (β0 + β1Xi) − 2 Xi+1(β0 + β1Xi) (2.12) i=1 i=1 i=1
23 Now, we maximize equation 2.11 with respect to the parameters β0 and β1. To do
this we differentiate SSE with respect to the parameters and set them equal to zero.
On doing the aforementioned, we obtain,
PN X − PN βˆ X βˆ = i=1 i+1 i=1 1 i (2.13) 0 n PN PN PN ˆ (N i=1 Xi+1Xi) − ( i=1 Xi i=1 Xi+1) β1 = (2.14) PN 2 PN 2 N i=1 Xi − ( i=1 Xi)
The data generation process and the true parameter values used to generate data were identical to the processes in the previous section. After getting the least square ˆ ˆ estimates, the estimates of the OU process were obtained as follows:- β = (1−β1)/dt, ˆ ˆ ˆ ˆ θ = β0/(1 − β1),σ ˆ = se().
Histogram of est_beta
250
200
150
Frequency 100
50
0 2 3 4 5 6 Value
Figure 2.9: Histogram of estimated values of β of the OU process using least squares approximation. The dashed red line represents the true value of the parameter.
24 Histogram of est_theta 250
200
150
Frequency 100
50
0 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 Value
Figure 2.10: Histogram of estimated values of θ of the OU process using least squares approximation. The dashed red line represents the true value of the parameter.
Histogram of est_sigma
250
200
150
Frequency 100
50
0 0.05 0.06 0.07 0.08 0.09 0.10 Value
Figure 2.11: Histogram of estimated values of σ of the OU process using least squares approximation. The dashed red line represents the true value of the parameter.
25 2.5 Cox-Ingersoll-Ross Process
The Cox-Ingersoll-Ross (CIR) model [11] was introduced in 1985 by John C. Cox,
Jonathan E. Ingersoll and Stephen A. Ross in order to improve the existing Vasicek
model which allowed for negative interest rates. Earlier, the OU model was used to
model interest rates rt. But, the fundamental problem with that approach was that
the change in rt assumed a constant volatility σ regardless of what happened in the
economy. There is empirical evidence that suggests that ∆rt is more volatile, if rt is
high and it is not so volatile if rt is low, i.e. the change in interest rates would be more volatile if the interest rates themselves are very high and that change is relatively less volatile if the interest rates are relatively lower. Also, the interest rates can never be negative but if modeled using an OU process, they can assume negative values with some positive probability. With regards to this, the CIR model was used to model interest rates as it was more efficient and violated fewer assumptions than the OU model used to model the same interest rates.
Definition 8. Let X(t) be a stochastic process and β, σ ∈ R+, and θ ∈ R be constants. If X(t) satisfies the following stochastic differential equation,
p + dX(t) = α(β − X(t))dt + σ X(t)dW (t), β, σ ∈ R , θ ∈ R (2.15)
then X(t) is said to be a CIR process.
In equation (2.15), dX(t) is the infinitesimal change in X(t), α is the rate of mean
reversion, β is the long term mean of the process which is also known as the asymptotic
mean, σ > 0 is the volatility and dW (t) Gaussian Noise. The drift function is linear
and has a mean reverting tendency because of which the CIR process is also a mean
26 reverting process. The diffusion function is proportional to X(t) and thus helps in
ensuring that the process never becomes negative. If all the process parameters, i.e.,
σ, α and β, are positive and 2αβ ≥ σ2 (Feller’s condition), then the CIR process is
well-defined.
The transition density of X(t) given X(s) is,
q ! 2 u √ f(X(t)|X(s)) = c exp(−v − u) I (2 uv) s < t (2.16) v q
where,
2α c = , σ2[1 − exp(−α(t − s))] u = cX(s)e(−α(t−s)),
v = cX(t), 2αβ q = − 1, σ2 √ and Iq(2 uv) is the modified Bessel function of the first kind and of order q. We use
the transformation S(t) = 2cX(t). Thus, the transition density of S(t) given S(s) is,
1 f(S(t)|S(s)) = f(X(t)|X(s)), s < t. 2c
Here, f(S(t)|S(s)) is a non-central χ2 distribution with 2u as the non-centrality pa- rameter and 2q + 2 degrees of freedom.
2.5.1 Simulation of CIR process
Proposition 2. Let Z1,Z2,...,Zk ∼ N(0, 1) be independent random variables, then
2 2 2 2 2 U = Z1 + Z2 + ... + Zk ∼ χk(0), where χk(0) is a (central) chi-squared distribution with k degrees of freedom.
27 2 Let U ∼ χk(0). Then, the probability density function of the random variable U is, uk/2−1 exp(−u/2) f (u) = , u > 0 U 2k/2Γ(k/2)
R ∞ x−1 where, Γ(x) = 0 t exp(−t)dt is the gamma function. It is known that Γ(n) = (n − 1)! for an integer n > 0. The moment generating function of U is,
−k/2 MU (t) = E(exp(tU)) = (1 − 2t) , |t| < 1/2.
Proposition 3. Let Z1,Z2,...,Zk ∼ N(µj, 1) for j = 1, 2, . . . , k be independent
2 2 2 2 2 random variables, then U = Z1 + Z2 + ... + Zk ∼ χk(λ), where χk(λ) is a non-central chi-squared distribution with k degrees of freedom with non-centrality parameter λ
1 Pk 2 where, λ = 2 j=1 µj .
2 Let V ∼ χk(λ). Then, the probability density function of the random variable V is, ∞ X hexp(λ)λjv(j+k/2)−1 exp(−v/2)i f (v) = , v > 0 (2.17) V j!2j+k/2Γ(j + k/2) j=0 The moment generating function of V is, ! λt M (t) = (exp(tV )) = exp (1 − 2t)−k/2, |t| < 1/2 V E 1 − 2t
We note that equation (2.17) is a mixture of Poisson and Gamma distributions. The
non-centrality parameter λ is equal to 0 if and only if µj = 0 for all j = 1, 2, . . . , k.
2 Note that a random variable V ∼ χk(λ) can be simulated using the following hierar- chy:
2 V |Y ∼ χk+2Y (0)
Y ∼ P oisson(λ)
28 We can use the law of iterated expectations to calculate E(V ) and V ar(V ).
E(V ) = E[E(V |Y )] = E(k + 2Y ) = k + 2E(Y ) = k + 2λ
Similarly, the variance of V is,
V ar(V ) = V ar(E(V |Y )) + E(V ar(V |Y ))
= V ar(k + 2Y ) + E(2(k + 2Y ))
= 4λ + 2k + 4λ = 2(k + 4λ)
The characteristic function of V is,
exp{λ2it/(1 − 2it)} φ(t) = (exp{itV }) = , |2it| < 1 (2.18) E (1 − 2it)k/2
It can be shown using equation (2.18) that if we have two independent random vari- ables V ∼ χ2 (λ ) and V ∼ χ2 (λ ) then, 1 k1 1 2 k2 2
d 2 V1 + V2 = χk1+k2 (λ1 + λ2) (2.19)
The above also holds true for any finite number of independent non-central chi- squared distributions. Equation (2.19) implies that the sum random variables which follow a non-central chi-squared distribution is equal in distribution to another ran- dom variable which follows a non-central chi-squared distribution. In particular, if
2 we have a random variable V ∼ χk(λ) then,
d 2 2 V = χ1(λ) + χk−1(0) d > 1 (2.20)
It is important to understand that,
√ √ 2 d 2 d 2 χ1(λ) = [N( λ, 1)] = (N(0, 1) + λ)
29 Equation (2.20) implies that a random variable which follows a non-central chi- squared distribution is equal in distribution to the sum of two independent random variables following a central chi-squared distribution and a standard normal distribu- tion.
Proposition 4. Assume that k > 1. Then, it is true that,
√ 2 d 2 2 χk(λ) = (Z + λ) + χk−1(0) .
Therefore, when the degrees of freedom k > 1, sampling from a non-central chi- squared distribution is equivalent to sampling from an central chi-squared distribu- tion and an independent normal distribution. This sampling method is not compu- tationally intensive and is generally efficient. When 0 < k < 1, we cannot use the above mentioned method to sample from a non-central chi-squared distribution. If
0 < k < 1, a non-central chi-squared distribution can be sampled using a central chi-squared distribution with random degrees of freedom.
Let Y ∼ P oisson(λ/2) random variable. The probability mass function (pmf) of Y is, (λ/2)y {Y = y} = exp(−λ/2) y = 1, 2,... P y!
2 Let U ∼ χk+2N (0). Conditional on the value of Y = y, let U follow a central chi- squared distribution with k + 2y degrees of freedom whose CDF is,
Z x 1 (k/2)+y−1 P{U ≤ u|Y = y} = (k/2)+y exp{−z/2}z dz (2.21) 2 Γ[(k/2) + y] 0
The unconditional cumulative distribution of U is,
∞ ∞ X X (λ/2)y {Y = y} {U ≤ u|Y = y} = exp(−λ/2) {U ≤ y} (2.22) P P y! P 0 0
30 Equation (2.22) is the CDF of a non-central chi-squared distribution with k degrees of freedom and non-centrality parameter λ.
Proposition 5. Assume that k < 1 and Y ∼ P oisson(λ). Then, it is true that,
2 d 2 χk(λ) = χk+2Y (0) .
Therefore, when the degrees of freedom are less than 1, we can sample from a non- central chi-squared distribution by first generating a Poisson random variable Y with parameter λ/2 and then sampling from a central chi-squared distribution with k +2Y degrees of freedom. Even though this hierarchical model to sample from a non-central chi-squared distribution produces unbiased results, it is usually computationally in- tensive.
Algorithm 3. (Simulation of the CIR process)
• Set the process parameters i.e. total time period (T) = 10, number of steps (N)
= 1000, number of simulations (n) = 500, α = 0.9, β = 4.0, σ = 1.5.
• Let ∆t = T/N and initialize the process by setting X(0) = 4.0.
• Recursively simulate X(t + ∆t) using the distribution given in (2.15).
31 14
12
10
8
X(t) 6
4
2
0 0 2 4 6 8 10 t
Figure 2.12: Simulated paths of the CIR process with parameters as described above.
2.5.2 Parameter Estimation for CIR Process using Maxi- mum Likelihood
Let {X(t): t ≥ 0} be a CIR process as defined in (2.15). Assume that we
observe this process at a discrete collection of time points {t0, t1, . . . , tn} where, t0 =
0, ti = iT/n for i = 1, 2, . . . , n. Let X = {X(t0),X(t1),...,X(tn)} be the data. For
simplicity, we use Xi = X(ti). Let θ = (α, β, σ). Given that this process is Markovian, the likelihood function is,
n Y L(θ|X1,X2,...,Xn) = f(Xi|Xi−1) i=1
32 where f(Xi|Xi−1) is the transition density. The transition density for a CIR process is, q ! 2 u √ f(X |X ) = c exp(−v − u) I (2 uv) (2.23) i i−1 v q
where,
2α c = , σ2[1 − exp(−αdt)] (−αdt) u = cXi−1e ,
v = cXi, 2αβ q = − 1, σ2 √ and Iq(2 uv) is the modified Bessel function of the first kind and of order q. The log
likelihood function is,
t X l(α, β, σ|X) = f(Xi|Xi−1) i=1 t X vi √ = t log(c) + [−u − v + q/2 log + log(I (2 u v ))] i−1 i u q i−1 i i=1 i−1 (2.24)
where, c, ui−1, vi = cXi, q and Iq have the usual meaning. For a simulation study, we
generated a data set according to algorithm3 using θ = (0.9, 4.0, 1.5). Based on such
data, we used the built in minimization function from Python to estimate the param-
eter values by minimizing the negative of log-likelihood. This process is repeated 500
times and the histogram of all estimates of the parameter α is presented in Figure
2.13. The dashed red line represents the true value of the parameter β. Similarly,
Figures 2.14 and 2.15 display the histograms of all estimates of the parameters α and
σ respectively. The dashed red lines represent the true value of the parameters α and
σ.
33 Histogram of est_alpha
120
100
80
60 Frequency 40
20
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Value
Figure 2.13: Histogram of estimated values of α of the CIR process. The dashed red line represents the true value of the parameter.
Histogram of est_beta
160
140
120
100
80 Frequency 60
40
20
0 2 3 4 5 6 7 8 9 Value
Figure 2.14: Histogram of estimated values of β of the CIR process. The dashed red line represents the true value of the parameter.
34 Histogram of est_sigma
100
80
60
Frequency 40
20
0 1.425 1.450 1.475 1.500 1.525 1.550 1.575 1.600 Value
Figure 2.15: Histogram of estimated values of σ of the CIR process. The dashed red line represents the true value of the parameter.
2.6 Generalized Cox-Ingersoll-Ross model
+ Definition 9. Let Xt be a stochastic process and β, σ ∈ R , γ ∈ (0, 1), and θ ∈ R be constants. If Xt satisfies the following stochastic differential equation,
γ + dXt = α(β − Xt)dt + σXt dWt, β, σ ∈ R , θ ∈ R, γ ∈ (0, 1) (2.25) then X(t) is said to be a generalized CIR [10] process.
Let {X(t): t ≥ 0} be a stochastic process. Assume that we observe this process at a discrete collection of time points {t0, t1, . . . , tn} where, t0 = 0, ti = iT/n for i = 1, 2, . . . , n. Let X = {X(t0),X(t1),...,X(tn)} be the data. For simplicity, we use Xi = X(ti). Let θ = (α, β, γ, σ). The likelihood function is,
n Y L(θ|X) = f(Xi|Xi−1) i=1
35 where f(Xi|Xi−1) is the transition density.
Even though the exact likelihood function does not have a closed form solution, we use a Gaussian approximation which works relatively well for smaller intervals of time
(∆t). In order to get accurate results, we would like to have the change in the time interval (∆t) as small as possible.
So, using the Gaussian approximation we have,
2 2γ Xi+1|Xi ∼ N(Xi + α(β − Xi−1)dt, σ Xi−1dt). (2.26)
This is true because we assume that W (t + dt) ∼ N(0, dt).
Algorithm 4. (Simulation of the generalized CIR process)
• Set the process parameters i.e. total time period (T) = 10, number of steps (N)
= 1000, number of simulations (n) = 1000, α = 0.5, β = 3, σ = 0.1, γ = 0.2.
• Let ∆t = T/N and initialize the process by setting X(0) = 0.3.
• Recursively calculate X(t + ∆t) using the distribution given in (2.26).
In Figure 2.16, we show 50 simulated paths according to algorithm4.
36 0.80
0.75
0.70 X(t)
0.65
0.60
0 2 4 6 8 10 t
Figure 2.16: Simulated paths of the generalized CIR process with parameters as described above
2.6.1 Parameter Estimation for generalized CIR Process us- ing Maximum Likelihood
We calculate the likelihood function using this Gaussian approximation. Thus,
the likelihood function can be written as,
N 2 ! Y 1 −(Xi+1 − Xi − α(β − Xt)dt) L(θ, β, σ, γ|X) = q · exp 2γ (2.27) 2γ ηXi i=1 πηXi
where, η = 2π2dt. But instead of maximizing the likelihood function, we maximize the log likelihood function l(θ, β, σ|X). The log likelihood function is,
N 2 ! −N X −(Xi+1 − Xi − α(β − Xt)dt) l(θ, β, σ|X) = log(2πη) − + γlog(X ) 2 2γ i i=1 ηXi (2.28)
37 We maximize l(α, β, γ, σ|X) in equation (2.28) to get estimates for the parameters.
Figure 2.17: Histogram of estimated values of α of the generalized CIR process using normal approximation. The dashed red line represents the true value of the parameter.
38 Figure 2.18: Histogram of estimated values of β of the generalized CIR process using normal approximation. The dashed red line represents the true value of the parameter.
Figure 2.19: Histogram of estimated values of σ of the generalized CIR process using normal approximation. The dashed red line represents the true value of the parameter.
39 Figure 2.20: Histogram of estimated values of γ of the generalized CIR process using normal approximation. The dashed red line represents the true value of the parameter.
2.6.2 Distribution of R t2 W (s) ds t1
In this subsection, we derive the distribution of the quantity R t2 W (s) ds, where t1 {W(t)} is a standard BM process. This distribution will play a significant role later
R t in this thesis. The distribution of 0 W (s) ds is the special case of the distribution of R t2 W (s) ds. Let us start by finding the mean and variance of R t W (s) ds. Let t1 0 f(x) = x3 and applying Ito’s Lemma (proposition1) we get,
Z t Z t W 3(t) = 3 W 2(s) dW (s) + 3 W (s) ds 0 0 Z t 1 Z t W (s) ds = W 3(t) − W 2(s) dW (s). 0 3 0
R t Thus, the mean of 0 W (s) ds is,
Z t E W (s) ds = 0. 0
40 R t The variance of 0 W (s) ds is, Z t "Z t 2# Z t Z t V ar W (s) ds = E W (s) ds = E W (s)W (u) du ds 0 0 0 0 Z t Z t Z t Z t = E[W (s)W (u)] du ds = min(s, u) du ds 0 0 0 0 Z t Z s Z t Z t = u du ds + s du ds = t3/3. 0 0 0 s
R t t3 R t Thus, 0 W (s) ds is a random variable that has a mean 0 and variance 3 . 0 W (s) ds is a random variable that has a normal distribution so,
Z t t3 W (s) ds ∼ N 0, . 0 3 R t Once we’ve found the mean and variance of 0 W (s) ds, we move on to the more general case of finding the mean and variance of R t2 W (s) ds. The mean of R t2 W (s) ds t1 t1 is, Z t2 Z t2 E W (s) ds = E[W (s)] ds = 0. t1 t1 The variance of R t2 W (s) ds is, t1 " 2# Z t2 Z t2 V ar W (s) ds = E W (s) ds t1 t1 Z t2 Z t2 Z t2 Z t2 = E[W (s)W (u)] du ds = min(s, u) du ds t1 t1 t1 t1 Z t2 Z s Z t2 Z t2 = u du ds + s du ds t1 t1 t1 s Z t2 2 2 Z t2 s t1 = − ds + s(t2 − s) ds t1 2 2 t1 2t3 t3 = 1 + 2 − t2t 3 3 1 2
Thus, the variance of R t2 W (s) ds is t1 2t3 t3 1 + 2 − t2t . (2.29) 3 3 1 2
We note that,
41 • If we let t1 = 0 and t2 = t then equation (2.29) gets reduced to the variance of
t3 the special case described earlier i.e 3 .
• If we let t1 = t and t2 = t then equation (2.29) gets reduced to 0.
Thus, Z t2 3 3 2t1 t2 2 W (s) ds ∼ N 0, + − t1t2 . t1 3 3
42 Chapter 3: Approximate Bayesian Computing for Stochastic Volatility Models
3.1 Heston Model
In his 1993 paper “A Closed-Form Solution for Options with Stochastic Volatil- ity with Applications to Bond and Currency Option” [13] Heston proposed a new stochastic volatility model, which now carries his name. The Heston model is used extensively in estimating the volatility of financial assets or derivatives. This model is an extension of the Black-Scholes model, i.e., the assumption is that underlying asset price still evolves according to the Black-Scholes model but it also introduces a stochastic behavior for the volatility component. That is, the model assumes that the volatility component in the Black-Scholes model is not fixed but rather is governed by another stochastic differential equation. In particular, the Heston model uses a mean reverting CIR model to describe the evolution of the volatility.
Definition 10. Let S(t): t ≥ 0 to be the price of the asset and ν(t); t ≥ 0 be the variance process. The equations governing the Heston model are,
dS(t) = µS(t)dt + pν(t)S(t) dW S(t) (3.1)
dν(t) = α(β − ν(t))dt + σpν(t) dW ν(t) (3.2)
43 where W S(t) and W ν(t) are correlated standard BM processes with the correlation between them given by ρ ∈ [−1, 1], µ is called the risk-free rate, dS(t) is the infinites-
imal change in S(t), the price of the underlying asset, α is the rate of mean reversion,
β is the long term mean of the CIR process which is also known as the asymptotic
mean, σ > 0 is the volatility of the CIR process.
The Heston model has certain desirable properties which make it a useful model.
Under the Heston model, volatility is modeled as a mean reverting process. This
assumption of the Heston model is also corroborated by observing its behavior in the
financial markets. If the volatility of an asset was not mean reverting, there would be
many assets whose volatility would be close to zero or very high. However, in practice
the probability of occurrence of these cases is very low and short lived.
The Heston model also associates asset prices with volatility by introducing correlated
shocks between the two. This assumption is particularly useful as it helps us to model
the statistical dependence between an asset and its volatility. Empirical evidence [21]
and [14] shows that in an equity market, the volatility and change in price of an asset
are inversely related, i.e., high changes in asset prices result in an increased volatility.
However, the flexibility that the Heston framework provides comes at the expense of
increased model complexity. It is generally difficult to implement the Heston model
as compared to the Black-Scholes model and there is always a tradeoff between the
two models in terms of complexity and accuracy. The Heston model is generally more
complex but also more accurate.
Proposition 6. Let dW ν(t) ∼ N(0, dt) and dW S(t) = ρdW ν(t) + p1 − ρ2dZ(t),
S where dZ(t) ∼ N(0, dt) is independent of dW (t).
44 Then,
V ar[dW ν(t)] = dt p V ar[dW S(t)] = ρCov[dW ν(t), dW ν(t)] + 1 − ρ2Cov[dW ν(t), dZ(t)]
= ρV ar[dW ν(t)]
= ρdt
The correlation between dW S(t) and dW ν(t) is equal to ρ. Let X(t) = log(S(t)).
Using Itˆo’s Lemma (proposition1) we can rewrite the equation (3.1) as, " ! !# 1 ν(t)S2(t) −1 1 dX(t) = µS(t) · + · dt + pν(t) · S(t) · dW X (t) S(t) 2 S2(t) S(t) ! ν(t) dX(t) = µ − dt + pν(t) dW X (t) 2
Thus, after using Itˆo’s lemma we get the following set of equations, ! ν(t) dX(t) = µ − dt + pν(t) dW X (t) (3.3) 2 dν(t) = α(β − ν(t))dt + σpν(t) dW ν(t) (3.4) where, dW X (t) = dW S(t) and all the other parameters have the usual meanings.
Feller’s Condition - It can be seen from equations (3.3) and (3.4) that ν(t) is under the square root sign. Thus, we require ν(t) to be non-negative. Feller proposed a condition which guarantees that ν(t) would be non-negative. If 2αβ ≥ σ2, then ν(t) takes non-negative values.
3.1.1 Simulation of sample paths of the Heston Model
There have been extensive studies on how to simulate sample paths of a Heston model. The basic idea is to partition a time interval into equally spaced intervals and
45 then simulate asset price paths for a given partition. Apart from the generic E-M discretization and Miller’s algorithm, Broadie and Kaya’s [8] algorithm is also popu- lar. There have been several modifications to Broadie and Kaya’s algorithm such as
Smith’s Approximation [23], Broadie and Kaya’s drift interpolation [25], Anderson’s quadratic exponential [6], and Tse and Wan’s Inverse Gaussian [24]. In this project, we use the exact scheme by Broadie and Kaya [8] but we estimate the integrals using
Riemann sums. This is slightly different from the work done by A. Van Haastrecht and A. Pelsser [25] who use the trapezoidal rule to estimate the integrals.
3.1.2 Euler-Maruyama (EM) Approximation
The Euler-Maruyama(EM) algorithm is an easily implementable approximation which can be used to approximate any SDE. The original process X(t) is approximated by another process X˜(t) which is defined in the following way, " # 1 X˜(t + ∆t) = X˜(t) + µ − ν˜(t) ∆t + pν˜(t)∆tZ 2 X
" # p ν˜(t + ∆t) =ν ˜(t) + α β − ν˜(t) ∆t + σ ν˜(t)∆tZν whereν ˜(t) is another process approximating the process ν(t). In between any two time points t, t+∆t, the processes X˜(·) andν ˜(·) are defined via a linear-interpolation of the values defined through the above equations. Above, ZX and Zν are standard normal random variables such that the correlation between them is ρ i.e. Corr(ZX ,Zν) = ρ.
In practice, this algorithm is not robust. When Feller’s condition is violated, the un- derlying variance process does not remain non-negative and has a positive probability of becoming negative. In addition, the Gaussian approximation above is valid only
46 when ∆t is very small. To circumvent this problem, Lord, Koekkoek and van Dijk
[17] propose a modification to the EM algorithm.
3.1.3 Euler-Maruyama scheme with Lord et al’.s modifica- tion
The equations of the modified EM algorithm are, " # 1 X˜(t + ∆t) = X˜(t) + µ − (˜ν(t)) ∆t + pν˜(t)∆tZ 2 X " # p ν˜(t + ∆t) =ν ˜(t) + α β − f(˜ν(t)) ∆t + σ ν˜(t)∆tZν
where, f(z) = max(0, z). If the variance process V˜ becomes negative, it corrects itself
with a deterministic upward drift of αβ.
3.1.4 Milstein scheme
The Milstein scheme is very similar to the EM algorithm. However, the Milstein scheme uses a second-order approximation to the SDE whereas the EM algorithm uses a first-order approximation or linear approximation to the SDE.
The algorithm under the Milstein scheme is, " # 1 X˜(t + ∆t) = X˜(t) + µ − (˜ν(t)) ∆t + pν˜(t)∆tZ , 2 X " # σ2 ν˜(t + ∆t) =ν ˜(t) + α β − f(˜ν(t)) ∆t + σpν˜(t)∆tZ + Z2h, ν 4 ν where, f(z) = max(0, z). It is important to know that ν(t + ∆t) > 0 if ν(t) > 0 and 4αβ ≥ σ2. This fact was stated by Gartner in [12]. When this inequality is not satisfied, it can still be shown that the occurrence of negative realizations ofν ˜ is greatly reduced as compared to the EM algorithm.
47 3.1.5 Broadie and Kaya’s Exact Algorithm
An exact simulation algorithm to simulate the Heston model is proposed by
Broadie and Kaya [8]. However, this algorithm is rarely used in practice as it is
computationally intensive. The solution to (3.1) can be written as, ! 1 Z t+∆t Z t+∆t p S(t + ∆t) = S(t) exp µ∆t − ν(u)du + ν(u)dWS(u) 2 t t
Using this and the transformation X = log(S), we get the following explicit solution
for X(t),
1 Z t+∆t X(t + ∆t) = X(t) + µ∆t − ν(u) du 2 t Z t+∆t Z t+∆t p p 2 p + ρ ν(u) dWν(u) + 1 − ρ ν(u) dWX (u) (3.5) t t
where, W ν(u) and W X (u) are values from two independent Brownian motions at time u. If we integrate (3.4), we get,
Z t+∆t Z t+∆t p ν(t + ∆t) = ν(t) + [α(β − ν(u))]du + σ ν(u)dWν(u) (3.6) t t
Equation (3.6) can be re-written as,
Z t+∆t " Z t+∆t # p −1 ν(u)dWν(u) = σ ν(t + ∆t) − ν(t) − αβ∆t + α ν(u)du t t
R t+∆t p and then if we substitute the value of t ν(u)dWν(u) into equation (3.5), we get,
1 Z t+∆t ρ X(t + ∆t) = X(t) + µ∆t − ν(u)du [ν(t + ∆t) − ν(t) − αβ∆t] 2 t σ Z t+∆t Z t+∆t αρ p 2 p + ν(u)du + 1 − ρ ν(u)dWX (u) σ t t
Thus, we have to sample the following quantities in the required order,
1. ν(t + ∆t) given ν(t)
48 R t+∆t 2. t ν(u)du given ν(t + ∆t), ν(t)
R t+∆t p R t+∆t 3. t ν(u)dWν(u) given t ν(u)du
2 We know that a transformation of νt+dt follows a scaled χ distribution. So,
n(dt) ν(t + dt) exp{−αdt}
has a χ2 distribution with λ(t) as the non-centrality parameter and
4αβ d = σ2
degrees of freedom. Here,
λ = ndtν(t), 4α exp{−αdt} n(dt) = . σ2(1 − exp{−αdt})
To get a value for a future time step (t + dt), we sample from a non-central χ2 distribution with λ(t) as the non-central parameter and d as the degrees of freedom.
We use an built in random number generator in the numpy module to achieve this.
Algorithm 5. The sample paths of Heston Model can be simulated using the following algorithm,
1. Sample νˆ(t + ∆t) given νˆ(t) from a non-central χ2 distribution.
R t+∆t 2. Given νˆ(t + ∆t) and νˆ(t), we estimate t ν(u)du. For this we use the trape- zoidal rule and estimate the integrated variance as,
νˆ(t + ∆t) +ν ˆ(t) IVˆ (t, t + ∆t) ≈ . 2
3. Generate a random observation Zx from an independent standard Gaussian ran-
dom variable.
49 4. Use the following exact scheme to get the different values of a sample path.
αρ IVˆ (t, t + ∆t) Xˆ(t + ∆t) = Xˆ(t) + µ∆t + IVˆ (t, t + ∆t) − σ 2 ρ p q + [ˆν(t + ∆t) − νˆ(t) − αβ∆t] + 1 + ρ2Z IVˆ (t, t + ∆t) (3.7) σ x
V(t) vs time
0.30
0.29
0.28 V(t)
0.27
0.26
0 50 100 150 200 250 t
Figure 3.1: Simulation of a path of CIR process with N = 252, α = 0.09, β = 0.145 and σ = 0.055.
50 X(t) vs time
2.3
2.2
2.1
2.0 X(t)
1.9
1.8
1.7
1.6 0 50 100 150 200 250 t
Figure 3.2: Simulation of a path of Heston process with N = 252, α = 0.09, β = 0.145, µ = 0.009 and σ = 0.055.
3.2 A generalized Heston Model
In this section, we propose a generalization of the Heston model. We extend the Heston model (10) by allowing the drift µ to be governed by another stochastic process. The rationale behind this idea is that there are some local variations in the drift component which we feel might be captured by the generalized Heston model.
As far as we know, all the models that have been proposed in the literature assume the interest rates to be a strictly positive quantity. But, there have been instances when the interest rates have been negative [5]. We feel the generalized Heston model would be more appropriate to estimate the volatility in these markets.
51 Definition 11. Let S(t): t ≥ 0 to be the price of the asset and ν(t); t ≥ 0 be the variance process. The equations governing the generalized Heston model are as follows:
dS(t) = µ(t)S(t)dt + pν(t)S(t) dW S(t) (3.8)
µ dµ(t) = α1(β1 − µ(t))dt + σ1 dW (t) (3.9)
p ν dν(t) = α2(β2 − ν(t))dt + σ2 ν(t) dW (t) (3.10) where dW µ(t) is uncorrelated with both dW S(t) and dW ν(t) by construction. The rest of the parameters have their usual meanings as defined earlier. From proposition
(2.2) we know that equation (3.9) can be written as,
Z t Z t µ µ(t) = µ(0) + α1(β1 − µ(s))ds + σ1 dW (s). (3.11) 0 0 Using the transformation f(X, t) = X(t) = log(S(t)) and Itˆo’s lemma (proposition
1), equation (3.8) can be written as, " ! !# 1 ν(t)S2(t) −1 1 dX(t) = µ(t)S(t) · + · dt + pν(t) · S(t) · dW X (t) S(t) 2 S2(t) S(t) ν(t) dX(t) = µ(t) − dt + pν(t) dW X (t), 2 where µ(t) follows a mean reverting OU process given by equation (3.9) and ν(t) follows a CIR process given by equation (3.10).
Using equation (2.2), X(t) can be written as,
Z t ν(s) Z t X(t) = X(0) + µ(s) − ds + pν(s) dW X (s). (3.12) 0 2 0
For any two times t1, t2 such that t2 > t1, equation (3.12) translates to,
Z t2 Z t2 ν(s) p X X(t2) = X(t1) + µ(s) − ds + ν(s) dW (s). t1 2 t1 52 This can be further simplified as,
Z t2 Z t2 Z t2 ν(s) p X X(t2) = X(t1) + µ(s)ds − ds + ν(s) dW (s). t1 t1 2 t1
Using proposition6 we get,
Z t2 Z t2 Z t2 ν(s) p ν X(t2) = X(t1) + µ(s)ds − ds + ρ ν(s) dW (s) t1 t1 2 t1 p Z t2 + 1 − ρ2 pν(s) dW Z (s), (3.13) t1 where dW µ(s) and dW Z (s) are independent of each other.
3.2.1 Simulation of sample paths of the generalized Heston model
The sample paths of the modified Heston model can be simulated using the fol- lowing multistep procedure.
Step-I
Set the process parameters, i.e., total time period (T)= 1.0, number of steps (N) =
100, ρ = −0.6. Let s be the number of intermediate points between ti and ti+1.
Figure 3.3 illustrates this with s=4. For our simulations, we choose s as 100.
Figure 3.3: s=4 intermediate points between ti and ti+1.
53 We need to simulate both the CIR process and the OU process in order to simulate a path of the generalized Heston model. We simulate the OU process using algorithm
2. Similarly, the CIR process is simulated using algorithm described in Chapter 2.
0.75
0.70
0.65
0.60 V(t)
0.55
0.50
0.45
0.0 0.2 0.4 0.6 0.8 1.0 t
Figure 3.4: Simulated path of the CIR process with parameters α2 = 0.221, β2 = th 0.601, σ2 = 0.055. Every (s + 1) value has been chosen for the plot, where s has been defined in step-I.
54 0.76
0.74
0.72
0.70 M(t)
0.68
0.66
0.0 0.2 0.4 0.6 0.8 1.0 t
Figure 3.5: Simulated path of the OU process with parameters α1 = 0.14, β1 = th 0.861, σ1 = 0.009. Every (s + 1) value has been chosen for the plot, where s has been defined in step-I.
Step-II
We estimate the integral R t2 ν(s) ds using the Riemann sum. t1
s Z t2 X ν(s) ds ≈ IVc = (ν(t1) + ν(si))∆, t1 i=1 where s = 100 is the number of divisions between t1 and t2 and
t − t ∆ = 2 1 . s
ν(t1) and ν(si) have already been simulated in Step - I as part of simulating the CIR process. In this step, we just add the product of all the simulated values of the CIR process between the time points t1 and t2 and ∆.
55 0.00060
0.00055
0.00050
Value 0.00045
0.00040
0.00035
0 20 40 60 80 100 N
Figure 3.6: Simulated path of the estimates of R t2 ν(s) ds at different time points. t1
The X axis here represents the number of divisions between 0 and the total time period T . If N = 100, then there would be 99 estimated values of the integral at
R t2 R t3 R t100 different times, i.e., ν(s) ds, ν(s) ds, . . . , ν(s) ds and first value is just νt1. t1 t2 t99
Step-III
We estimate the integral R t2 µ(s) ds using Riemann sum. t1
s Z t2 X µ(s) ds ≈ Iµc = (µ(t1) + µ(si))∆, t1 i=1 where s and ∆ have their usual meanings.
56 0.000375
0.000350
0.000325
0.000300
Value 0.000275
0.000250
0.000225
0 20 40 60 80 100 N
Figure 3.7: Simulated path of the estimate of R t2 µ(s) ds. t1
The X axis here represents the number of divisions between 0 and the total time period T . If N = 100, then there would be 99 estimated values of the integral at
R t2 R t3 R t100 different times, i.e., µ(s) ds, µ(s) ds, . . . , µ(s) ds, and first value is just µt1. t1 t2 t99
Step-IV
The solution to the CIR process simulated in Step-I is given as,
Z t2 Z t2 p ν(t2) = ν(t1) + α2(β2 − ν(u))du + σ2 ν(u)dW (u). t1 t1
We estimate the integral R t2 pν(s) dW ν(s) as follows, t1
Z t2 Z t2 p −1 ν(u)dW (u) = [ν(t2) − ν(t1) + α2(β2 − ν(u))du]σ2 . t1 t1
We have already simulated all the terms on the right hand side and thus, we know the value of Z t2 pν(u)dW ν(u) t1 57 .
0
0
0 0
0
0
0 0 0 0 0 00
Figure 3.8: Simulated path of the estimate of R t2 pν(s) dW ν(s). t1
The interpretation of the X-axis is similar to the one described above.
Step-V
We estimate the integral R t2 pν(s) dW Z (s) as, t1
Z t2 p pν(s) dW Z (s) ≈ Z IV.ˆ t1 where, Z is a value from a standard normal random variable and IVc has already been explained in Step-II.
58 0 0
0 0
0 0
0 0
0 00
0 0
0 0
0 0
0 0
0 0 0 0 0 00
Figure 3.9: Simulated path of the estimate of R t2 pν(s) dW Z . t1
The interpretation of the X-axis is similar to the one described above. We now have all the estimated integrals. We assume that the initial value X(0) ∼ N(1, 1). Using
X(0) and (3.13) we can now simulated a sample path of the generalized Heston model.
59 5
4
3 X(t)
2
1
0 20 40 60 80 100 t
Figure 3.10: Simulated sample path of the generalized Heston model.
3.3 Approximate Bayesian Computing (ABC)
Approximate Bayesian Computing (ABC) is a computational technique used when an analytical formula for the likelihood function is difficult to derive or is computa- tionally costly to evaluate. Assume we want to perform Bayesian inference and wish to explore an intractable posterior density P (θ|D0) where θ is the parameter of interest and D0 is a generic notation for ”observed data”.
Algorithm 6. The ABC algorithm performs the following steps,
1. Sample a new value of the parameter θ* from the prior distribution P (·).
2. Simulate a data set D* from the likelihood model f(·|θ*).
60 3. Compare the newly simulated data D* to the observed data D0 using a well
defined distance function d and tolerance ≥ 0. The tolerance is the desired
level of closeness or agreement between D* and D0.
4. If d(D*,D0) ≤ , we accept θ* else we reject θ*.
Repeat the process until R such parameters sampled from P (·) have been accepted.
The accepted parameters represent a sample from P (θ|d(D*,D0) ≤ ). For a suffi-
ciently small , the distribution P (θ|d(D*,D0) ≤ ) would be a very good approxi-
mation to the “true” distribution P (θ|D0).
3.3.1 ABC for Heston Model
In this section, we estimate the parameters of the Heston Model using ABC. The
observed data D0 is simulated using algorithm5. We use Gaussian distribution priors
for parameters that have no restrictions and Gamma priors for parameters that are
restricted to be positive. The parameters used to generate the observed data are,
α = 0.290, β = 0.445, σ = 0.055, µ = 0.1, ρ = −0.2,T = 1.0,N = 100.
Here, the parameters have their usual meanings as described in section 3.1.
Algorithm 7. The ABC algorithm for the Heston model performs the following steps,
1. Let θ* = (α*, β*, σ*, µ*, ρ*). Sample α*, β* from a Gamma(1, 1) distribu-
tion, σ* and µ* from a Gamma(0.45, 0.45) distribution. We sample ρ* from a
Gamma(0.3, 0.3) distribution and then multiply the selected value by −1 because
of the additional constraints on ρ as described above.
2. Simulate a data set D* in accordance to a simulation framework f(D|θ*) which
is described in5.
61 3. Let the distance function be,
N X d = |D0,i − D*i| i=1
Where D0,i = X(ti) is the observed value of the process at time ti.
4. If d(D*,D0) ≤ , we accept θ* else we reject θ*.
Table 3.1: Table showing the number of simulations vs number of accepted parameters for different = 100.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 54 2) 1,000 656 3) 10,000 5488 4) 100,000 65766
Table 3.2: Table showing the number of simulations vs number of accepted parameters for different = 200.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 70 2) 1,000 740 3) 10,000 7321 4) 100,000 74217
62 Table 3.3: Table showing the number of simulations vs number of accepted parameters for different = 500.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 84 2) 1,000 767 3) 10,000 7873 4) 100,000 79502
Table 3.4: Table showing the number of simulations vs number of accepted parameters for different = 800.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 85 2) 1,000 816 3) 10,000 8129 4) 100,000 81515
Table 3.5: Table showing the number of simulations vs number of accepted parameters for different = 1000.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 86 2) 1,000 845 3) 10,000 8303 4) 100,000 82981
63 Table 3.6: Table showing the number of simulations vs number of accepted parameters for different = 1500.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 88 2) 1,000 858 3) 10,000 8410 4) 100,000 84712
For the same number of simulations, we observe that the number of accepted parame- ters increases with the tolerance () level. This is in accordance with our expectations as having a higher tolerance level () corresponds to a weaker constraint which al- lows more parameters to be accepted. Similarly, for the same tolerance () level, the number of accepted parameters increases with the number of simulations. Below, we show the histograms of accepted values of the parameters of the Heston Model for different number of simulations and different tolerance () levels.
64 Histogram of est_alpha Histogram of est_beta
350 250
300 200 250
200 150
150 Frequency Frequency 100
100 50 50
0 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
400
350 400
300
250 300
200 200 Frequency 150 Frequency
100 100 50
0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
of est_rho
− − − − − − − − (e) Histogram of accepted values of ρ.
Figure 3.11: Histograms of accepted values of the parameters of the Heston Model for = 100 and 1000 simulations. The dashed red lines represent the true values of the parameters.
65 Histogram of est_alpha Histogram of est_beta 3000 3000 2500 2500
2000 2000
1500 1500 Frequency Frequency 1000 1000
500 500
0 0 0 2 4 6 8 0 1 2 3 4 5 6 7 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 4000 4500 4000 3500 3500 3000 3000 2500 2500 2000 2000
Frequency 1500 Frequency 1500
1000 1000
500 500
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.12: Histograms of accepted values of the parameters of the Heston Model for = 100 and 10, 000 simulations. The dashed red lines represent the true values of the parameters.
66 Histogram of est_alpha Histogram of est_beta
40000 40000
30000 30000
20000 20000 Frequency Frequency
10000 10000
0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
50000 50000
40000 40000
30000 30000
Frequency 20000 Frequency 20000
10000 10000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.13: Histograms of accepted values of the parameters of the Heston Model for = 100 and 100, 000 simulations. The dashed red lines represent the true values of the parameters.
67 Histogram of est_alpha Histogram of est_beta
350 350
300 300
250 250
200 200
Frequency 150 Frequency 150
100 100
50 50
0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
500 500
400 400
300 300
Frequency 200 Frequency 200
100 100
0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.14: Histograms of accepted values of the parameters of the Heston Model for = 200 and 1, 000 simulations. The dashed red lines represent the true values of the parameters.
68 Histogram of est_alpha Histogram of est_beta
5000 4000
4000
3000 3000
2000 Frequency Frequency 2000
1000 1000
0 0 0 2 4 6 8 0 2 4 6 8 10 12 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
5000 5000
4000 4000
3000 3000 Frequency Frequency 2000 2000
1000 1000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.15: Histograms of accepted values of the parameters of the Heston Model for = 200 and 10, 000 simulations. The dashed red lines represent the true values of the parameters.
69 Histogram of est_alpha Histogram of est_beta
50000
40000 40000
30000 30000
20000 Frequency Frequency 20000
10000 10000
0 0 0 2 4 6 8 10 0 2 4 6 8 10 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 60000 60000
50000 50000
40000 40000
30000 30000 Frequency Frequency 20000 20000
10000 10000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.16: Histograms of accepted values of the parameters of the Heston Model for = 200 and 100, 000 simulations. The dashed red lines represent the true values of the parameters.
70 Histogram of est_alpha Histogram of est_beta 350
400 300
250 300 200
200 150 Frequency Frequency
100 100 50
0 0 0 2 4 6 8 0 1 2 3 4 5 6 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 500
500 400
400 300 300
200 Frequency Frequency 200
100 100
0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.0 0.5 1.0 1.5 2.0 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − (e) Histogram of accepted values of ρ.
Figure 3.17: Histograms of accepted values of the parameters of the Heston Model for = 500 and 1, 000 simulations. The dashed red lines represent the true values of the parameters.
71 Histogram of est_alpha Histogram of est_beta 6000
4000 5000
4000 3000
3000 2000 Frequency Frequency 2000
1000 1000
0 0 0 2 4 6 8 0 2 4 6 8 10 12 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
6000 6000
5000 5000
4000 4000
3000 3000 Frequency Frequency 2000 2000
1000 1000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.18: Histograms of accepted values of the parameters of the Heston Model for = 500 and 10, 000 simulations. The dashed red lines represent the true values of the parameters.
72 Histogram of est_alpha Histogram of est_beta 50000
50000 40000
40000
30000 30000
20000 Frequency Frequency 20000
10000 10000
0 0 0 2 4 6 8 10 0 2 4 6 8 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 70000 60000 60000 50000 50000
40000 40000
30000 30000 Frequency Frequency
20000 20000
10000 10000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.19: Histograms of accepted values of the parameters of the Heston Model for = 500 and 100, 000 simulations. The dashed red lines represent the true values of the parameters.
73 Histogram of est_alpha Histogram of est_beta 400 400
350 350
300 300
250 250
200 200
Frequency 150 Frequency 150
100 100
50 50
0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 600
500 500
400 400
300 300 Frequency Frequency 200 200
100 100
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − (e) Histogram of accepted values of ρ.
Figure 3.20: Histograms of accepted values of the parameters of the Heston Model for = 800 and 1, 000 simulations. The dashed red lines represent the true values of the parameters.
74 Histogram of est_alpha Histogram of est_beta
5000 5000
4000 4000
3000 3000 Frequency Frequency 2000 2000
1000 1000
0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 6000
6000 5000 5000 4000 4000
3000 3000 Frequency Frequency 2000 2000
1000 1000
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.21: Histograms of accepted values of the parameters of the Heston Model for = 800 and 10, 000 simulations. The dashed red lines represent the true values of the parameters.
75 Histogram of est_alpha Histogram of est_beta
50000 50000
40000 40000
30000 30000 Frequency Frequency 20000 20000
10000 10000
0 0 0 2 4 6 8 10 0 2 4 6 8 10 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
70000 70000
60000 60000
50000 50000
40000 40000
30000 30000 Frequency Frequency
20000 20000
10000 10000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.22: Histograms of accepted values of the parameters of the Heston Model for = 800 and 100, 000 simulations. The dashed red lines represent the true values of the parameters.
76 Histogram of est_alpha Histogram of est_beta 500 400
400
300 300
200
Frequency Frequency 200
100 100
0 0 0 1 2 3 4 5 6 7 0 2 4 6 8 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
600 500 500
400 400
300 300 Frequency Frequency 200 200
100 100
0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0 0.5 1.0 1.5 2.0 2.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.23: Histograms of accepted values of the parameters of the Heston Model for = 1000 and 1, 000 simulations. The dashed red lines represent the true values of the parameters.
77 Histogram of est_alpha Histogram of est_beta
5000 4000
4000
3000 3000
2000
Frequency Frequency 2000
1000 1000
0 0 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
6000 6000
5000 5000
4000 4000
3000 3000 Frequency Frequency
2000 2000
1000 1000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.24: Histograms of accepted values of the parameters of the Heston Model for = 1000 and 10, 000 simulations. The dashed red lines represent the true values of the parameters.
78 Histogram of est_alpha Histogram of est_beta 60000
50000 50000
40000 40000
30000 30000 Frequency Frequency 20000 20000
10000 10000
0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 70000 70000 60000 60000 50000 50000
40000 40000
30000 30000 Frequency Frequency
20000 20000
10000 10000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.25: Histograms of accepted values of the parameters of the Heston Model for = 1000 and 100, 000 simulations. The dashed red lines represent the true values of the parameters.
79 Histogram of est_alpha Histogram of est_beta 350 400 300 350
300 250
250 200
200 150 Frequency Frequency 150 100 100 50 50
0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu 600 600
500 500
400 400
300 300 Frequency Frequency 200 200
100 100
0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.0 0.5 1.0 1.5 2.0 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
of est_rho
− − − − (e) Histogram of accepted values of ρ.
Figure 3.26: Histograms of accepted values of the parameters of the Heston Model for = 1500 and 1, 000 simulations. The dashed red lines represent the true values of the parameters.
80 Histogram of est_alpha Histogram of est_beta 5000 5000
4000 4000
3000 3000
Frequency 2000 Frequency 2000
1000 1000
0 0 0 2 4 6 8 0 2 4 6 8 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
6000 6000
5000 5000
4000 4000
3000 3000 Frequency Frequency
2000 2000
1000 1000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.27: Histograms of accepted values of the parameters of the Heston Model for = 1500 and 10, 000 simulations. The dashed red lines represent the true values of the parameters.
81 Histogram of est_alpha Histogram of est_beta 60000 60000
50000 50000
40000 40000
30000 30000 Frequency Frequency 20000 20000
10000 10000
0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Value Value (a) Histogram of accepted values of α. (b) Histogram of accepted values of β.
Histogram of est_sigma Histogram of est_mu
70000 70000
60000 60000
50000 50000
40000 40000
30000 30000 Frequency Frequency
20000 20000
10000 10000
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Value Value (c) Histogram of accepted values of σ. (d) Histogram of accepted values of µ.
− − − − − (e) Histogram of accepted values of ρ.
Figure 3.28: Histograms of accepted values of the parameters of the Heston Model for = 1500 and 100, 000 simulations. The dashed red lines represent the true values of the parameters.
82 We observe that in almost all the histograms, the slab of the highest frequency
contains the dashed red lines, i.e., the highest number of accepted values lie very close
to the true parameter value for α, β, σ and µ. This is not the case for ρ. Next, we
look at the implementation of the ABC algorithm for the generalized Heston model.
3.3.2 ABC for generalized Heston Model
In this section, we estimate the parameters of the generalized Heston Model using
ABC. The observed data D0 is simulated using the process described in section 3.2.
We use Gaussian distribution priors for parameters that have no restrictions and
Gamma priors for parameters that are restricted to be positive. The parameters used to generate the observed data are,
α1 = 0.283, β1 = 0.661, σ1 = 0.009, α2 = 0.221, β2 = 0.601, σ2 = 0.055,
ρ = −0.6,T = 1.0,N = 100.
Here, the parameters have their usual meanings as described in section 3.2.
Algorithm 8. The ABC algorithm for the generalized Heston model performs the
following steps,
1. Let θ* = (α1*, β1*, σ1*, α1*, β1*, σ1*), ρ*. Sample α1*, α2*, β2* from a Gamma(1, 1)
distribution, σ1* and σ2* from a Gamma(0.25, 0.25) distribution, β2* from a
Normal(1, 1) distribution. We sample ρ* from a Gamma(1, 1) distribution and
then multiply the selected value by −1 because of the additional constraints on
rho as described above.
2. Simulate a data set D* in accordance to a simulation framework f(D|θ*) which
is described is section 3.2.
83 3. Let the distance function be,
N X d = |D0,i − D*i| (3.14) i=1
Where D0,i = X(ti) is the observed value of the process at time ti. Compare
the simulated data D* to the observed data D0 using a well-defined distance
function d and tolerance ≥ 0.
4. If d(D*,D0) ≤ , we accept θ* else we reject θ*.
Table 3.7: Table showing the number of simulations vs number of accepted parameters for = 100.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 1 2) 1,000 14 3) 10,000 148
Table 3.8: Table showing the number of simulations vs number of accepted parameters for = 200.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 2 2) 1,000 53 3) 10,000 440
84 Table 3.9: Table showing the number of simulations vs number of accepted parameters for = 500.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 20 2) 1,000 240 3) 10,000 2236
Table 3.10: Table showing the number of simulations vs number of accepted param- eters for = 800.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 36 2) 1,000 246 3) 10,000 2671
Table 3.11: Table showing the number of simulations vs number of accepted param- eters for = 1, 000.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 31 2) 1,000 302 3) 10,000 2889
85 Table 3.12: Table showing the number of simulations vs number of accepted param- eters for = 1, 500.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 43 2) 1,000 284 3) 10,000 3189
We observe that for the same number of simulations, the number of accepted parameters increases with the tolerance () level. Similarly, for the same tolerance () level, the number of accepted parameters increases with the number of simulations.
For the same number of simulations and same tolerance () level, the number of accepted parameters for the Heston model is greater than the number of accepted parameters for the generalized Heston model. This is due to the fact that the number of parameters in the Heston model are less than the number of parameters in the generalized Heston model. As we increase the complexity of the model, it becomes harder to find the right set of parameters that satisfy the constraints of ABC. Below, we show the histograms of accepted values of the parameters of the generalized Heston
Model for different number of simulations and different tolerance () levels.
86 Histogram of est_alpha1 Histogram of est_beta1
4.0 4.0
3.5 3.5
3.0 3.0
2.5 2.5
2.0 2.0
Frequency 1.5 Frequency 1.5
1.0 1.0
0.5 0.5
0.0 0.0 0 1 2 3 4 5 0.5 1.0 1.5 2.0 2.5 3.0 Value Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
6
8 5
6 4
3 4 Frequency Frequency 2
2 1
0 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
5 5
4 4
3 3
Frequency 2 Frequency 2
1 1
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.29: Histograms of estimated values of the parameters of the generalized Heston Model for = 100 and 1000 simulations. The dashed red lines represent the true values of the parameters. 87 Histogram of est_alpha1 50
40
30
20 Frequency
10
0 0 1 2 3 4 5 − − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
120 50 100 40 80
30 60 Frequency Frequency 20 40
20 10
0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
100 60
50 80
40 60
30
Frequency Frequency 40 20
20 10
0 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.30: Histograms of estimated values of the parameters of the generalized Heston Model for = 100 and 10000 simulations. The dashed red lines represent the true values of the parameters. 88 Histogram of est_alpha1
14
12
10 8 6 Frequency 4
2
0 0.0 0.5 1.0 1.5 2.0 2.5 − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
40 20
30 15
20 10 Frequency Frequency
10 5
0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 4 5 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
25 20
20 15
15
10 Frequency Frequency 10
5 5
0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 0.5 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.31: Histograms of estimated values of the parameters of the generalized Heston Model for = 200 and 1000 simulations. The dashed red lines represent the true values of the parameters. 89 Histogram of est_alpha1
200 175
150
125 100 Frequency 75
50 25
0 0 1 2 3 4 5 6 − − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
350 175
300 150
250 125
200 100
150 Frequency Frequency 75
100 50
50 25
0 0 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 100
200 80
150 60
40 100 Frequency Frequency
20 50
0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.32: Histograms of estimated values of the parameters of the generalized Heston Model for = 200 and 10, 000 simulations. The dashed red lines represent the true values of the parameters. 90 Histogram of est_alpha1 120
100
80
60
Frequency 40
20
0 0 1 2 3 4 5 6 − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2 100 175
150 80
125 60 100
75 Frequency Frequency 40
50 20 25
0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 120 80
70 100
60 80 50 60 40 Frequency Frequency 30 40
20 20 10
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.33: Histograms of estimated values of the parameters of the generalized Heston Model for = 500 and 1, 000 simulations. The dashed red lines represent the true values of the parameters. 91 Histogram of est_alpha1 1200
1000
800
600 Frequency 400
200
0 0 1 2 3 4 5 6 7 − − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
1750 1000 1500
800 1250
1000 600
Frequency 750 Frequency 400 500 200 250
0 0 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 7 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
1200 1400
1200 1000
1000 800
800 600
Frequency 600 Frequency 400 400
200 200
0 0 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.34: Histograms of estimated values of the parameters of the generalized Heston Model for = 500 and 10, 000 simulations. The dashed red lines represent the true values of the parameters. 92 Histogram of est_alpha1 80
70
60 50
40
Frequency 30
20 10
0 0 1 2 3 4 − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2 200
175 80
150
125 60
100 40 Frequency 75 Frequency
50 20 25
0 0 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 100 140
80 120
100 60 80
Frequency 40 Frequency 60
40 20 20
0 0 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.35: Histograms of estimated values of the parameters of the generalized Heston Model for = 800 and 1, 000 simulations. The dashed red lines represent the true values of the parameters. 93 Histogram of est_alpha1
1400
1200
1000
800
Frequency 600
400
200
0 0 1 2 3 4 5 6 7 8 − − − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
2000 1400
1200
1500 1000
800 1000
Frequency Frequency 600
400 500 200
0 0 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 1600 1600
1400 1400
1200 1200
1000 1000
800 800
Frequency 600 Frequency 600
400 400
200 200
0 0 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.36: Histograms of estimated values of the parameters of the generalized Heston Model for = 800 and 10, 000 simulations. The dashed red lines represent the true values of the parameters. 94 Histogram of est_alpha1
120
100
80
60
Frequency 40
20
0 0 1 2 3 4 5 − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
120 200 100
150 80
60 100 Frequency Frequency
40
50 20
0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 4 5 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 160 100
140
80 120
100 60 80
Frequency 40 Frequency 60
40 20 20
0 0 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
of est_rho
− − − − − (g) Histogram of estimated values of ρ. Figure 3.37: Histograms of estimated values of the parameters of the generalized Heston Model for = 1, 000 and 1, 000 simulations. The dashed red lines represent the true values of the parameters. 95 Histogram of est_alpha1
1400 1200
1000
800
Frequency 600
400
200
0 0 1 2 3 4 5 6 7 − − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
1600
2000 1400
1200 1500 1000
800 1000 Frequency Frequency 600
500 400 200
0 0 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
1600 1750
1400 1500
1200 1250 1000 1000 800 750 Frequency Frequency 600 500 400
200 250
0 0 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.38: Histograms of estimated values of the parameters of the generalized Heston Model for = 1, 000 and 10, 000 simulations. The dashed red lines represent the true values of the parameters. 96 Histogram of est_alpha1
100
80
60 Frequency 40
20
0 0 1 2 3 4 5 6 − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
120 200 100
150 80
60 100 Frequency Frequency 40
50 20
0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 4 5 6 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
175 120
150 100 125 80 100 60 75 Frequency Frequency
40 50
20 25
0 0 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
− − − − − (g) Histogram of estimated values of ρ. Figure 3.39: Histograms of estimated values of the parameters of the generalized Heston Model for = 1, 500 and 1, 000 simulations. The dashed red lines represent the true values of the parameters. 97 Histogram of est_alpha1
2000
1500
1000 Frequency
500
0 0 2 4 6 8 10 12 − − − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2 2000 2500 1750
2000 1500
1250 1500 1000
Frequency 1000 Frequency 750
500 500 250
0 0 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 Value Value (c) Histogram of estimated values (d) Histogram of estimated values of σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
1750 2000
1500 1750
1250 1500
1250 1000 1000 750 Frequency Frequency 750 500 500
250 250
0 0 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 Value Value (e) Histogram of estimated values (f) Histogram of estimated values of β2. of σ2.
of est_rho
− − − − − (g) Histogram of estimated values of ρ. Figure 3.40: Histograms of estimated values of the parameters of the generalized Heston Model for = 1, 500 and 10, 000 simulations. The dashed red lines represent the true values of the parameters. 98 We observe that in almost all the histograms, the slab of the highest frequency contains the dashed red lines, i.e., the highest number of accepted values lie very close to the true parameter value for α1, β1, σ1, σ2 and α2. This is not the case for ρ and
β2.
99 Chapter 4: Application: Modeling Volatility in Financial Markets
4.1 Introduction
In this section, we define a stock index and its purpose, and describe different stock indices of the world. We also describe the importance of volatility in this section.
4.1.1 Stock Index
A stock index is defined as a collection of stocks that are grouped together using a specific criteria so that a particular sector, market, commodity, bond, currency or any other asset could be monitored [3]. It is difficult to track every asset so a statistical measuring tool like an index is really useful.
Standard & Poor’s 500
The Standard & Poor’s 500 or better abbreviated as S&P 500 is an index based in the American Stock market. It consists of the largest 500 companies listed on the
New York Stock Exchange (NYSE) based on market capitalization [4]. The S&P 500 is one of the most followed indices globally and many economists believe it to be a fair and apt representation of the US stock market and an index that acts as a bellwether for the economy of United States.
100 S&P BSE 200
S&P BSE 200 Index is a free float weighted index of 200 companies selected from Specified and Non-Specified lists of BSE India Exchange, selected based on their market capitalization. It started as a cap-weighted index with a base value of
100, and base year 1989-90. Effective from 8/16/05, it was changed to a free float index. Though S&P BSE SENSEX was serving the purpose of quantifying the price movements as also reflecting the sensitivity of the market in an effective manner, the rapid growth of the market necessitated compilation of a new broad-based index series reflecting the market trends in a more effective manner and providing a better representation of the increased equity stocks, market capitalization as also to the new industry groups. As such BSE launched on 27th May 1994, two new index series S&P
BSE 200 and S&P Dollex 200. The equity shares of 200 selected companies from the specified and non-specified lists of BSE were considered for inclusion in the sample for ‘S&P BSE 200’. The selection of companies was primarily done on the basis of current market capitalization of the listed scrips. Moreover, the market activity of the companies as reflected by the volumes of turnover and certain fundamental factors were considered for the final selection of the 200 companies [1].
Shanghai Stock Exchange (SSE)
The Chinese stock market has been one of the fastest growing emerging capital markets, and is now the second largest in Asia, only behind Japan. The Shang- hai Stock Exchange Composite Index is a capitalization-weighted index. The index tracks the daily price performance of all A-shares and B-shares listed on the Shanghai
Stock Exchange. The index was developed on December 19, 1990 with a base value
101 of 100. The first day of reporting was July 15, 1991 [18]. A-shares are shares of the Renminbi currency that are purchased and traded on the Shanghai and Shen- zhen stock exchanges. This is contrast to Renminbi B-shares which are owned by foreigners who cannot purchase A-shares due to Chinese government restrictions. B shares (officially Domestically Listed Foreign Investment Shares) on the Shanghai and Shenzhen stock exchanges refers to those that are traded in foreign currencies.
The composite figure can be calculated by using the formula:
Market Cap of Composite Numbers Current Index = × Base Value Base Period
The B-share stocks are generally denominated in US dollars for calculation purposes.
For calculation of other indices, B share stock prices are converted to RMB at the applicable exchange rate (the middle price of US dollar on the last trading day of each week) at China Foreign Exchange Trading Center and then published by the exchange.
Nikkei 225
The Nikkei 225 is Japan’s top stock index which consists of the top 225 blue chip companies that are listed in the Tokyo Stock Exchange [2]. The Nikkei 225 is the oldest index in Asia.
According to the US National Bureau of Economic Research (the official arbiter of US recessions) the US recession began in December 2007 and ended in June 2009, and thus extended over 19 months. In order to see the impact of recent financial crisis and have time varying results, the total data is divided into four sub-periods, i.e. before financial crisis period (period-I, January, 1996- November, 2007), during recession (period-II, December, 2007- June, 2009), after recession and before Chinese
102 Crisis (period- III, July, 2009- May, 2015) and from the start of Chinese crisis till date(period- IV, June, 2015 - April, 2016). We assume the sample period is sufficient to evaluate the information asymmetry especially after the huge Foreign Institutional
Investors investments in stock markets, sub-prime crisis disorder and the recent fi- nancial crisis.
Most classic equations in finance like the Black-Scholes equation that is used to price options consider volatility to be a fixed quantity. But empirical studies have shown that volatility varies over time. Capturing volatility is very important to predict the price of stocks and commodities. Having some information about volatility also helps an investor make informed decisions. There have been some studies aimed at this but very few of them pertain to emerging markets. The goal of this project is to be able to predict volatility for emerging markets so that investors can make informed decisions. Volatility estimation is of utmost importance for option valuation.
Volatility is defined as the uncertainty or dispersion in stock price movements or variability in the returns. Regulated utilities and blue chip companies that are expected to grow slowly but steadily over time have usually been associated with a low volatility. Investing in the stocks of these companies turns out to be a viable investment in the long run. On the other hand, the stock prices of companies that have a higher volatility associated with them vary rapidly in a short period of time.
Start-ups are a prime example of these type of companies that have a higher volatility associated with them. A very recent example is the prices of cryptocurrencies espe- cially Bitcoin. The price movement of Bitcoin over the past one year indicates that it has a high volatility. Investing in the stocks of companies that have a higher volatility
103 associated with them results in short term gains but it is not a feasible option to go long on the stocks of these companies.
Volatility is an important indicator and most companies estimate their volatility by means three measures. The first one is historic volatility followed by implied volatility and the last one is historical or implied volatility using a subset of peer companies. Historical volatility is the actual variability that was observed in the past during a specific time period. The disadvantage of historical volatility is that it is often calculated using the past stock prices and is of little or no use for future use. Even though it does provide us with a rough estimate of volatility, it would be beneficial if we had a better estimate of the volatility. Implied volatility, on the other hand, is the volatility that gives the theoretical trading price of an option that is traded in the market. When the value of implied volatility is plugged in the famous
Black-Scholes equation, we get the theoretical trading price. The only problem is that implied volatility is rarely available for all time periods or for all companies. It is also subject to short-term market fluctuations. To circumvent this problem, the companies that have usable option data rely make use of a combination of historical volatility and implied volatility. But many companies have to exclusively make use of historical volatility because of several reasons. Not having usable option data is the primary reason why companies have to exclusively use historical volatility.
4.2 Exploratory Data Analysis
Daily closing prices of Shanghai Stock Exchange (SSE) composite index for the period Jan 1, 1996 to April 8, 2016 are considered for the study. The data for
SSE was retrieved from Yahoo! Finance. For this purpose, we have used the daily
104 adjusted closing prices for the SSE Composite Index. We have also considered the
daily adjusted closing price of Nikkei 225 from January 5, 2015 to July 24, 2018 which
corresponds to 927 days. We have downloaded the data from the Federal Reserve
Economic Data (FRED) database which is maintained by the Research division of
the Federal Reserve Bank of St. Louis. These indices are considered because of their
popularity around the world so as to represent these markets. In order to apply
our model, we use the log adjusted closing prices. Figures 4.1- 4.4 represent the
daily adjusted closing prices and daily log adjusted closing prices for the desired time
periods for different indices. The daily log closing prices are calculated as,
X(t) = log S(t).
If the adjusted closing price is missing for a particular day, the price of the preceding day is taken as the adjusted closing price of the current day (for which the closing price was missing) [26].
Daily Adjusted Closing Price of SSE from 01/01/96 to 04/08/16
6000
5000
4000
3000
2000
1000
Adjusted Closing Price (in Renminbi) (in Price Closing Adjusted 0 1000 2000 3000 4000 5000 time (no. of days)
Figure 4.1: Daily Adjusted Closing Price of SSE from 01/01/96 to 04/08/16.
105 Daily Log Closing Price of SSE from 01/01/96 to 04/08/16
8.5
8.0
7.5
7.0
6.5
0 1000 2000 3000 4000 5000
Log Adjusted Closing Price (in Renminbi) (in Price Closing Adjusted Log time (no. of days)
Figure 4.2: Daily Log Adjusted Closing Price of SSE from 01/01/96 to 04/08/16.
Daily Adjusted Closing Price of NIKKEI 225 from 01/05/15 to 07/24/18
24000
22000
20000
18000
16000
Adjusted Closing Price (in Yen) (in Price Closing Adjusted 0 200 400 600 800 time (no. of days)
Figure 4.3: Daily Adjusted Closing Price of NIKKEI 225 from 01/05/15 to 07/24/18.
106 Daily Log Closing Price of NIKKEI 225 from 01/05/15 to 07/24/18 10.1
10.0
9.9
9.8
9.7
9.6 0 200 400 600 800
Log Adjusted Closing Price (in Yen) (in Price Closing Adjusted Log time (no. of days)
Figure 4.4: Daily Log Returns of NIKKEI 225 from 01/05/15 to 07/24/18.
4.3 Parameter estimation of the Generalized Heston model using ABC
4.3.1 Parameter estimation using ABC for SSE
We fit the generalized Heston model to the data from SSE and estimate the parameters using ABC. In order to test the fit of our model, we divide the daily log adjusted closing prices into two parts, the training dataset and the testing dataset.
The first 4800 data points form the training dataset and the remaining 353 form the testing dataset. We tried a combination of normal and gamma priors. Given a tolerance level , the ABC algorithm accepts many numerical values for a single parameter. The table below summarizes the results for different levels.
107 Table 4.1: Table showing the number of simulations vs number of accepted parameters for different = 10, 000.
S.No. Number of simulations (n) Number of accepted parameters (R) 1) 100 50
108 Histogram of est_alpha1
14
12 10
8
6 Frequency
4 2
0 2 4 6 8 10 − − − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
16
8 14
12 6 10
8 4 Frequency Frequency 6
4 2
2
0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 2 4 6 8 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
14 12 12 10 10
8 8
6 6 Frequency Frequency
4 4
2 2
0 0 0.0 0.5 1.0 1.5 2.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − (g) Histogram of accepted values of ρ.
Figure 4.5: Histograms of accepted values of the parameters for = 10, 000 and 100 simulations. 109 Since the histograms of accepted values of the parameters appear to be skewed, we have used the median as a point estimate of the parameter. The estimated parameters of the model fit on SSE data from January 01, 1996 to April 08, 2016 are,
ˆ ˆ αˆ1 = 2.497, β1 = 0.066, σˆ1 = 0.211, αˆ2 = 1.812, β2 = 0.428, σˆ2 = 0.233, ρˆ = −0.358.
Using the estimated parameters given above, we simulate a synthetic dataset and compare the simulated dataset with the testing dataset. We use the same distance metric as was used in implementing the ABC algorithm to calculate the goodness of our fitted model. The smaller distance between the simulated dataset using estimated parameters and the testing dataset the better the model fits. Figure 4.6 shows the simulated dataset and the testing dataset. The distance between them was 67.63 units.
Comparision between simulated dataset and testing dataset from SSE
simulated dataset 8.5 testing dataset 8.4
8.3
8.2
8.1
8.0
7.9 Log Adjusted Closing Price Closing Adjusted Log
7.8
7.7 0 50 100 150 200 250 300 350 Value
Figure 4.6: Comparison between simulated dataset and testing dataset.
110 Parameter estimation using ABC for SSE during period 1
Using the method described above, we try to estimate the parameters using ABC.
In order to test the fit of our model, we divide the daily log adjusted closing prices into two parts, the training dataset and the testing dataset. The first 2800 data points form the training dataset and the remaining 292 form the testing dataset. We tried a combination of normal and gamma priors. The table below shows the number of accepted values of the parameters for different simulations.
Table 4.2: Table showing the number of simulations vs number of accepted parameters for different levels.
Number of simulations (n) Number of accepted parameters (R) 10,000 100 45 5,000 100 34 1,000 100 0
It can be seen from table 4.2 that the number of accepted values of the parameters increase with the increase in tolerance level (). For tolerance level () = 1, 000, the ABC algorithm does not accept any parameter values. Next, we look at the histograms of the accepted values of the parameters.
111 Histogram of est_alpha1
17.5 15.0
12.5
10.0
7.5 Frequency 5.0 2.5
0.0 60 80 100 120 140 160 180 200 − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
7 10 6
5 8
4 6
3 Frequency Frequency 4 2
2 1
0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
14 8
12 7
6 10 5 8 4 6 Frequency Frequency 3 4 2
2 1
0 0 0.0 0.5 1.0 1.5 2.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − (g) Histogram of accepted values of ρ. Figure 4.7: Histograms of accepted values of the parameters for = 10, 000 and 100 simulations.
112 Histogram of est_alpha1 Histogram of est_beta1
7 7
6 6
5 5
4 4
3 3 Frequency Frequency
2 2
1 1
0 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 Value Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
5 7
6 4 5
3 4
3 Frequency 2 Frequency
2 1 1
0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 2 4 6 8 10 12 14 16 18 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
8 10 7
8 6
5 6 4 Frequency Frequency 4 3
2 2 1
0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − (g) Histogram of accepted values of ρ. Figure 4.8: Histograms of accepted values of the parameters for = 5, 000 and 100 simulations.
113 We have used the median as a point estimate of the parameter as the histograms of the accepted values of the parameters appeared to be skewed. The estimated parameters of the model fit on SSE data during period 4 from January 01, 1996 to November 30,
2007 for different levels are,
Table 4.3: Table showing the estimated parameters for different levels (100 simula- tions).
ˆ ˆ αˆ1 β1 σˆ1 αˆ2 β2 σˆ2 ρˆ 10,000 19.283 1.349 0.173 5.774 0.414 0.222 -0.305 5,000 21.608 1.567 0.163 7.417 0.315 0.232 -0.286
Using the estimated parameters given above, we simulate a synthetic dataset and compare the simulated dataset with the testing dataset using the methods described above. For different levels in increasing order i.e. 5, 000 and 10, 000, Figures 4.9-4.10 show the simulated dataset and the testing dataset. The distance between them was
83.26 and 111.48 units, respectively.
114 Comparison between simulated dataset and testing dataset from SSE
simulated dataset 8.6 testing dataset
8.4
8.2
8.0
7.8 Log Adjusted Closing Price Closing Adjusted Log 7.6
7.4 0 50 100 150 200 250 300 Value
Figure 4.9: Comparison between simulated dataset and testing dataset for = 5, 000 for the first period.
Comparison between simulated dataset and testing dataset from SSE
simulated dataset 8.6 testing dataset
8.4
8.2
8.0
7.8 Log Adjusted Closing Price Closing Adjusted Log 7.6
7.4 0 50 100 150 200 250 300 Value
Figure 4.10: Comparison between simulated dataset and testing dataset for = 10, 000 for the first period.
115 Parameter estimation using ABC for SSE during period 2
Using the method described above, we try to estimate the parameters using ABC.
In order to test the fit of our model, we divide the daily log adjusted closing prices
into two parts, the training dataset and the testing dataset. The first 299 data points
form the training dataset and the remaining 100 form the testing dataset. We tried
a combination of normal and gamma priors.
Table 4.4: Table showing the number of simulations vs number of accepted parameters for different levels.
Number of simulations (n) Number of accepted parameters (R) 10,000 100 72 5,000 100 82 1,000 100 59
It can be seen from table 4.4 that the number of accepted values of the parameters
increase with the increase in tolerance level (). Next, we look at the histograms of the accepted values of the parameters.
116 Histogram of est_alpha1 Histogram of est_beta1
17.5 20.0
15.0 17.5
12.5 15.0
12.5 10.0 10.0 7.5 Frequency Frequency 7.5 5.0 5.0
2.5 2.5
0.0 0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 0 1 2 3 4 5 Value Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
40 17.5 35 15.0 30 12.5 25 10.0 20
Frequency Frequency 7.5 15
10 5.0
5 2.5
0 0.0 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 16 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
40 35 35 30 30 25 25 20 20
Frequency 15 Frequency 15 10 10
5 5
0 0 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
of est_rho
− − − − − − − − (g) Histogram of accepted values of ρ. Figure 4.11: Histograms of accepted values of the parameters for = 1, 000 and 100 simulations.
117 Histogram of est_alpha1 Histogram of est_beta1
14 14
12 12
10 10
8 8
Frequency 6 Frequency 6
4 4
2 2
0 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 1 2 3 4 5 Value Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
40 17.5
35 15.0
30 12.5
25 10.0 20 7.5 Frequency Frequency 15 5.0 10
5 2.5
0 0.0 0.0 0.2 0.4 0.6 0.8 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 50 30
40 25
20 30
15 20 Frequency Frequency 10
10 5
0 0 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
of est_rho
− − − − − − − − (g) Histogram of accepted values of ρ. Figure 4.12: Histograms of accepted values of the parameters for = 5, 000 and 100 simulations.
118 Histogram of est_alpha1 Histogram of est_beta1
20.0 12 17.5 10 15.0
12.5 8
10.0 6 Frequency Frequency 7.5 4 5.0 2 2.5
0.0 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Value Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
40 16
14
30 12
10
20 8 Frequency Frequency 6
10 4
2
0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
35 50
30 40 25
20 30
15 Frequency Frequency 20
10 10 5
0 0 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − (g) Histogram of accepted values of ρ. Figure 4.13: Histograms of accepted values of the parameters for = 10, 000 and 100 simulations.
119 We have used the median as a point estimate of the parameter. The estimated
parameters of the model fit on SSE data during period 2 from December 03, 2007 to
June 30, 2009 for different levels are given in table 4.5.
Table 4.5: Table showing the estimated parameters for different levels (100 simula- tions).
ˆ ˆ αˆ1 β1 σˆ1 αˆ2 β2 σˆ2 ρˆ 10,000 5.285 2.461 0.078 5.596 0.588 0.065 -0.045 5,000 5.877 2.803 0.055 5.545 0.387 0.074 -0.043 1,000 6.217 2.412 0.080 4.866 0.508 0.059 -0.031
Using the estimated parameters given above, we simulate a synthetic dataset and
compare the simulated dataset with the testing dataset using the methods described
above. For different levels in increasing order i.e. 1, 000, 5, 000 and 10, 000, Figures
4.14-4.16 show the simulated dataset and the testing dataset. The distance between them was 9.78, 9.67 and 10.62 units, respectively.
120 Comparision between the simulated dataset and testing dataset
8.0 Simulated dataset Testing dataset
7.9
7.8
7.7 Log Adjusted Closing Price Closing Adjusted Log
7.6
0 20 40 60 80 100 Value
Figure 4.14: Comparison between simulated dataset and testing dataset for = 1, 000 for the second period.
Comparision between the simulated dataset and testing dataset
8.0 Simulated dataset Testing dataset 7.9
7.8
7.7
7.6 Log Adjusted Closing Price Closing Adjusted Log 7.5
0 20 40 60 80 100 Value
Figure 4.15: Comparison between simulated dataset and testing dataset for = 5, 000 for the second period.
121 Comparision between the simulated dataset and testing dataset
8.2 Simulated dataset Testing dataset 8.1
8.0
7.9
7.8
7.7
Log Adjusted Closing Price Closing Adjusted Log 7.6
7.5
0 20 40 60 80 100 Value
Figure 4.16: Comparison between simulated dataset and testing dataset for = 10, 000 for the second period.
Parameter estimation using ABC for SSE during period 3
The first 1200 data points form the training dataset and the remaining 265 form the testing dataset. We tried a combination of normal and gamma priors.
Table 4.6: Table showing the number of simulations vs number of accepted parameters for different levels.
Number of simulations (n) Number of accepted parameters (R) 10,000 100 80 5,000 100 64 1,000 100 33
122 Histogram of est_alpha1
25
20 15 Frequency 10
5
0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 − − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
10 17.5
15.0 8 12.5
6 10.0
Frequency Frequency 7.5 4
5.0 2 2.5
0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0 1 2 3 4 5 6 7 8 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
25 12
20 10
15 8
6
Frequency 10 Frequency 4
5 2
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.00 0.05 0.10 0.15 0.20 0.25 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − − (g) Histogram of accepted values of ρ. Figure 4.17: Histograms of accepted values of the parameters for = 10, 000 and 100 simulations.
123 Histogram of est_alpha1
20
15
10
Frequency
5
0 0 1 2 3 4 5 − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
10 17.5
15.0 8
12.5 6 10.0
7.5 Frequency 4 Frequency
5.0 2 2.5
0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0 2 4 6 8 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 30
10 25
8 20
6 15 Frequency Frequency 10 4
5 2
0 0 0 1 2 3 4 0.05 0.10 0.15 0.20 0.25 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − − (g) Histogram of accepted values of ρ. Figure 4.18: Histograms of accepted values of the parameters for = 5, 000 and 100 simulations.
124 Histogram of est_alpha1 Histogram of est_beta1
8 14 7 12 6 10 5 8 4
Frequency 6 Frequency 3
4 2
2 1
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 1 2 3 4 5 Value Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
5
8 4
6 3
4
Frequency 2 Frequency
1 2
0 0 0.00 0.05 0.10 0.15 0.20 0.25 1 2 3 4 5 6 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
5 20.0
17.5 4 15.0
12.5 3
10.0
Frequency Frequency 2 7.5
5.0 1 2.5
0.0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.05 0.10 0.15 0.20 0.25 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
of est_rho
− − − − − (g) Histogram of accepted values of ρ. Figure 4.19: Histograms of accepted values of the parameters for = 1, 000 and 100 simulations.
125 The estimated parameters of the model fit on SSE data during period 3 from July
01, 2009 to May 29, 2015 for different levels are given in table 4.7.
Table 4.7: Table showing the estimated parameters for different levels (100 simula- tions).
ˆ ˆ αˆ1 β1 σˆ1 αˆ2 β2 σˆ2 ρˆ 10,000 0.644 1.101 0.132 2.523 0.777 0.139 -0.374 5,000 0.728 0.981 0.121 2.484 0.547 0.133 -0.356 1,000 0.567 0.838 0.122 2.016 0.176 0.142 -0.315
We have used the median as a point estimate of the parameter as the histograms
appear to be skewed. Using the estimated parameters given above, we simulate a
synthetic dataset and compare the simulated dataset with the testing dataset using
the methods described above. For different levels in increasing order i.e. 1, 000, 5, 000 and 10, 000, Figures 4.19- 4.21 show the simulated dataset and the testing dataset.
The distance between them was 54.76, 57.45 and 30.00 units.
126 Comparison between simulated dataset and testing dataset from SSE
simulated dataset 8.4 testing dataset
8.2
8.0
7.8 Log Adjusted Closing Price Closing Adjusted Log
7.6
0 50 100 150 200 250 Value
Figure 4.20: Comparison between simulated dataset and testing dataset for = 1, 000 for the third period.
Comparison between simulated dataset and testing dataset from SSE
simulated dataset 8.4 testing dataset
8.2
8.0
7.8
7.6
7.4 Log Adjusted Closing Price Closing Adjusted Log 7.2
0 50 100 150 200 250 Value
Figure 4.21: Comparison between simulated dataset and testing dataset for = 5, 000 for the third period.
127 Comparison between simulated dataset and testing dataset from SSE 8.8
8.6
8.4
8.2
8.0
7.8 Log Adjusted Closing Price Closing Adjusted Log 7.6 simulated dataset testing dataset 7.4 0 50 100 150 200 250 Value
Figure 4.22: Comparison between simulated dataset and testing dataset for = 10, 000 for the third period.
Parameter estimation using ABC for SSE during period 4
Using the method described above, we try to estimate the parameters using ABC.
In order to test the fit of our model, we divide the daily log adjusted closing prices into two parts, the training dataset and the testing dataset. The first 150 data points form the training dataset and the remaining 47 form the testing dataset. We tried a combination of normal and gamma priors.
Table 4.8: Table showing the number of simulations vs number of accepted parameters for different levels.
Number of simulations (n) Number of accepted parameters (R) 10,000 100 99 5,000 100 94 1,000 100 81
128 Histogram of est_alpha1
20.0
17.5
15.0
12.5
10.0
Frequency 7.5
5.0
2.5
0.0 0 2 4 6 8 10 12 14 16 − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2 25 60
20 50
40 15
30 10 Frequency Frequency 20
5 10
0 0 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
30
40 25
20 30
15 20 Frequency Frequency 10
10 5
0 0 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − (g) Histogram of accepted values of ρ. Figure 4.23: Histograms of accepted values of the parameters for = 10, 000 and 100 simulations.
129 Histogram of est_alpha1
20
15
10 Frequency
5
0 0 2 4 6 8 10 12 14 − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
35 50 30
40 25
30 20
15 Frequency Frequency 20 10
10 5
0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 4 5 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
30 40
25 30 20
15 20 Frequency Frequency
10 10 5
0 0 0 1 2 3 4 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − (g) Histogram of accepted values of ρ. Figure 4.24: Histograms of accepted values of the parameters for = 5, 000 and 100 simulations.
130 Histogram of est_alpha1
17.5
15.0
12.5
10.0
7.5 Frequency
5.0
2.5
0.0 0 2 4 6 8 10 12 14 16 − − − Value (a) Histogram of estimated values (b) Histogram of estimated values of α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
17.5 35
30 15.0
25 12.5
20 10.0
Frequency 15 7.5 10 Frequency
5 5.0
0 2.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Value 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (c) Histogram of estimated values Value of σ . 1 (d) Histogram of estimated values of α2.
Histogram of est_beta2 Histogram of est_sigma2
20 40
15 30
10 Frequency 20 Frequency 5
10 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Value 0 0.0 0.2 0.4 0.6 0.8 (e) Histogram of estimated values Value of β . 2 (f) Histogram of estimated values of σ2.
− − − − − − − (g) Histogram of estimated values of ρ. Figure 4.25: Histograms of estimated values of the parameters for = 1, 000 and 100 simulations. 131 As the histograms of the accepted values of the parameters appear to be skewed, we have used the median as a point estimate of the parameter. The estimated parameters of the model fit on SSE data during period 4 from January 01, 2015 to April 08, 2016 for different levels are,
Table 4.9: Table showing the estimated parameters for different levels.
ˆ ˆ αˆ1 β1 σˆ1 αˆ2 β2 σˆ2 ρˆ 10,000 3.756 1.063 0.006 0.823 0.602 0.044 -0.052 5,000 3.082 0.845 0.014 0.575 0.581 0.044 -0.079 1,000 4.109 1.157 0.020 0.731 0.544 0.039 -0.032
Using the estimated parameters given above, we simulate a synthetic dataset and compare the simulated dataset with the testing dataset using the methods described above. For different levels in increasing order i.e. 1, 000, 5, 000 and 10, 000, figures
4.25- 4.27 show the simulated dataset and the testing dataset. The distance between them was 2.51, 9.19 and 3.67 units.
132 Comparision between the simulated dataset and testing dataset
Simulated dataset 8.05 Testing dataset
8.00
7.95
7.90 Log Adjusted Closing Price Closing Adjusted Log
7.85
0 10 20 30 40 Value
Figure 4.26: Comparison between simulated dataset and testing dataset for = 1, 000 for the fourth period.
Comparision between the simulated dataset and testing dataset
8.1 Simulated dataset Testing dataset
8.0
7.9
7.8 Log Adjusted Closing Price Closing Adjusted Log 7.7
0 10 20 30 40 Number of data points
Figure 4.27: Comparison between simulated dataset and testing dataset for = 5, 000 for the fourth period.
133 Comparision between the simulated dataset and testing dataset 8.05
8.00
7.95
7.90
7.85
7.80 Log Adjusted Closing Price Closing Adjusted Log Simulated dataset 7.75 Testing dataset
0 10 20 30 40 Value
Figure 4.28: Comparison between simulated dataset and testing dataset for = 10, 000 for the fourth period.
4.3.2 Parameter estimation using ABC for NIKKEI 225
We fit the generalized Heston model to the data from NIKKEI 225 and estimate
the parameters using ABC. In order to test the fit of our model, we divide the daily
log adjusted closing prices into two parts, the training dataset and the testing dataset.
The first 700 data points form the training dataset and the remaining 227 form the
testing dataset. We tried a combination of normal and gamma priors. Given an , the ABC algorithm accepts many numerical values for a single parameter.
134 Table 4.10: Table showing the number of simulations vs number of accepted param- eters for different levels.
Number of simulations (n) Number of accepted parameters (R) 10,000 100 96 5,000 100 86 1,000 100 74
135 Histogram of est_alpha1
40 35
30 25
20 Frequency 15
10
5
0 0 1 2 3 4 5 − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
16 35
14 30
12 25 10 20 8 15 Frequency 6 Frequency 10 4
2 5
0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 1 2 3 4 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2 50 12
40 10
8 30
6 20 Frequency Frequency 4
10 2
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − − (g) Histogram of accepted values of ρ. Figure 4.29: Histograms of accepted values of the parameters for = 10, 000 and 100 simulations.
136 Histogram of est_alpha1
30 25
20
15
Frequency 10 5
0 0 1 2 3 4 5 − − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
12 40
35 10 30 8 25
6 20
Frequency Frequency 15 4 10 2 5
0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 1 2 3 4 5 6 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
12 25 10 20 8
15 6
Frequency 10 Frequency 4
5 2
0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − − (g) Histogram of accepted values of ρ. Figure 4.30: Histograms of accepted values of the parameters for = 5, 000 and 100 simulations.
137 Histogram of est_alpha1
25
20
15 Frequency 10
5
0 0 1 2 3 4 − − Value (a) Histogram of accepted values of (b) Histogram of accepted values α1. of β1.
Histogram of est_sigma1 Histogram of est_alpha2
12 17.5
10 15.0
12.5 8 10.0 6
Frequency Frequency 7.5 4 5.0
2 2.5
0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 1 2 3 4 Value Value (c) Histogram of accepted values of (d) Histogram of accepted values σ1. of α2.
Histogram of est_beta2 Histogram of est_sigma2
12 40 10
30 8
6 20 Frequency Frequency 4
10 2
0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Value Value (e) Histogram of accepted values of (f) Histogram of accepted values of β2. σ2.
− − − − − (g) Histogram of accepted values of ρ. Figure 4.31: Histograms of accepted values of the parameters for = 1, 000 and 100 simulations.
138 We have used the median as a point estimate of the parameter as all the histograms of accepted parameters appeared to be skewed. The estimated parameters of the model
fit on NIKKEI 225 data from January 05, 2015 to July 24, 2018 are given in 4.11.
Table 4.11: Table showing the estimated parameters for different levels.
ˆ ˆ αˆ1 β1 σˆ1 αˆ2 β2 σˆ2 ρˆ 10,000 0.642 1.021 0.152 0.691 0.328 0.142 -0.386 5,000 0.856 1.135 0.141 0.663 0.268 0.129 -0.340 1,000 0.702 0.716 0.157 1.081 0.236 0.146 -0.360
Using the estimated parameters given above, we simulate a synthetic dataset and compare the simulated dataset with the testing dataset. We use the same distance metric as was used in implementing the ABC algorithm to calculate the goodness of our fitted model. The smaller distance between the simulated dataset using es- timated parameters and the testing dataset the better the model fits. Using the estimated parameters given above, we simulate a synthetic dataset and compare the simulated dataset with the testing dataset using the methods described above. For different levels in increasing order i.e. 1, 000, 5, 000 and 10, 000, figures 4.32- 4.34 show the simulated dataset and the testing dataset. The distance between them was
33.88, 42.38 and 39.21 units.
139 Simulated dataset v/s testing dataset for NIKKEI 225
simulated dataset 10.4 testing dataset
10.3
10.2
10.1
10.0 Adjusted Closing Price Closing Adjusted
Log 9.9
9.8
0 50 100 150 200 Value
Figure 4.32: Comparison between simulated dataset and testing dataset for = 1, 000.
Simulated dataset v/s testing dataset for NIKKEI 225 10.6 simulated dataset 10.5 testing dataset
10.4
10.3
10.2
10.1
10.0 Log Adjusted Closing Price Closing Adjusted Log 9.9
9.8 0 50 100 150 200 Value
Figure 4.33: Comparison between simulated dataset and testing dataset for = 5, 000.
140 Simulated dataset v/s testing dataset for NIKKEI 225
10.2 simulated dataset testing dataset 10.1
10.0
9.9
9.8
9.7
9.6 Log Adjusted Closing Price Closing Adjusted Log
9.5
9.4 0 50 100 150 200 Value
Figure 4.34: Comparison between simulated dataset and testing dataset for = 10, 000.
141 Chapter 5: Contributions and Future Work
5.1 Results Overview
In chapter4, the generalized Heston model was fit to different indices of two
of the most important emerging markets of the world, namely the Shanghai Stock
Exchange (SSE) and NIKKEI 225. We used the ABC algorithm to estimate the
parameters of the model. As the histograms for almost all of the accepted values for
all the parameters appeared to be skewed, we used the median as a point estimate for
the parameters. If we look at the SSE, for a particular period, the number of accepted
parameters increased as the tolerance level () went up. The maximum number of accepted parameters was for tolerance level () = 10, 000. As mentioned in chapter
4, the data from SSE was divided into 4 separate periods. Among different periods
for the SSE data i.e., the number of accepted parameters was higher for periods
which were shorter i.e., period 2 and period 4. Overall, the generalized Heston model
was a good fit for the SSE data from 01 January, 1996 to 08 June, 2016. This is
evident from figure 4.6. Not only was the simulated dataset very close to the testing
dataset but it also is able to capture the variations in the testing dataset. All of the
estimated parameters fall within reasonable values and we believe these parameters
to be estimates for the true market parameters.
142 If we look at the SSE data for period 1, we observe that there are no accepted parameters for () = 1, 000. For tolerance levels () of 5, 000 and 10, 000, the distances between the synthetic dataset which was simulated using the estimated parameters of the generalized Heston model and the testing dataset were 83.26 and 111.48 units respectively. If we look at Figures 4.9-4.10 we would believe that the model has intermediate predictive power but this predictive power of the model reaches new heights during the second period. The second period was the period right before and right after the 2008 financial crisis. For tolerance levels () of 1, 000, 5, 000 and
10, 000, the distances between the synthetic dataset which was simulated using the estimated parameters of the generalized Heston model and the testing dataset were only 9.78, 9.69 and 10.62 units respectively. The model was beautifully able to capture the trend for this period which is evident from Figures 4.14-4.16. This was one example of a shorter time period for which the generalized Heston model had a high predicting power. For the SSE data for period 3, the distances between the synthetic dataset which was simulated using the estimated parameters of the generalized Heston model and the testing dataset for increasing tolerance levels () were 54.76, 57.45 and
30.00 units. As compared to period 2, these distances were relatively quite large.
From Figures 4.20-4.22 we observe that even though the model was accurately able to capture the general trend during this period, we feel that the predictive power of the model for period 3 was not on par with the predictive power of the model for period 2. The model performance for period 4 for SSE data is a different story from that of period 3 for the data from the same index. As can be seen from the
Figures 4.26-4.28 the generalized Heston model has very good predictive power. We should note that the SSE data for period 4 was from 01 June 2015 to 08 April, 2016
143 which coincides with Chinese Stock Market crisis period. For period 4, the distances between the synthetic dataset which was simulated using the estimated parameters of the generalized Heston model and the testing dataset for increasing tolerance levels () were 2.51, 9.19 and 3.67 units respectively. Out of the all the four periods in which the
SSE data was divided, we feel the model performed the best for period 2 and period 4, i.e., the period around the 2008 financial crisis and around the Chinese Stock Market crisis. We believe that the generalized Heston model has good predictive power for shorter time periods and during financial crunches or crisis.
We applied the model to the NIKKEI 225 data from January 05, 2015 to July 24,
2018. This dataset was really important as the interest rates in the Japanese economy were negative several times between this time period. The advantage of using the generalized Heston model over a Heston type model is that it allows for interest rates to be negative and can capture other local variations in the interest rates as well. The distances between the synthetic dataset which was simulated using the estimated parameters of the generalized Heston model and the testing dataset for increasing tolerance levels () were 33.88, 42.38 and 39.21 units. This model was a good fit and had good predictive power for the concerned data.
Even though we proposed this model in a financial realm, we would like to explore other applications of this model as well. This is explained in more details in the next section.
5.2 Future Work
In this project we propose a new model, the generalized Heston model, to predict the volatility of financial assets. The decision to build the generalized Heston model
144 was motivated by the fact that some economies and markets could have negative
interest rates as well. We use ABC to estimate the parameters of the generalized
Heston model and also test the validity of the model. Going forward, we would like
to approach this model from a more theoretical point of view. We would like to
calculate the moments of the generalized Heston model and estimate the parameters
using a likelihood based approach. Long term behavior is also of interest. Moreover,
we would like to explore other applications of the generalized Heston model. In
addition, we would like to test this model on a different market during a crisis period
and see how our model performs in comparison to the standard models.
5.2.1 Moments of generalized Heston model
In chapter3 we stopped at equation (5.1) for simulation purposes. Moving for-
ward, we could use the fact that µ(t) follows an OU process. For any two times t1, t2 such that t2 > t1, equation (3.12) translates to,
Z t2 Z t2 ν(s) p X X(t2) = X(t1) + µ(s) − ds + ν(s) dW (s). (5.1) t1 2 t1
This can be further simplified as,
Z t2 Z t2 Z t2 ν(s) p X X(t2) = X(t1) + µ(s)ds − ds + ν(s) dW (s). t1 t1 2 t1
Using (3.11) R t2 µ(s)ds is, t1 " # Z t2 Z t2 Z s Z s µ µ(s) ds = µ(t1) + α1(β1 − µ(p)) dp + σ1 dW (p) ds, t1 t1 t1 t1
Z t2 Z t2 Z s Z t2 Z s µ = µ(t1) ds + α1 (β1 − µ(p)) dp ds + σ1 dW (p) ds t1 t1 t1 t1 t1
145 Assuming that µ(t1) is known to us at time t2 and letting t2 − t1 = ∆t we get,
Z t2 Z t2 Z s Z t2 Z s µ µ(s) ds = µ(t1)∆t + α1 (β1 − µ(p)) dp ds + σ1 dW (p) ds (5.2) t1 t1 t1 t1 t1
After plugging the value of R t2 µ(s) ds from equation (5.2) into equation (5.1) we get, t1
Z t2 Z s Z t2 Z s µ X(t2) = X(t1) + µ(t1)∆t + α1 (β1 − µ(p)) dp ds + σ1 dW (p) ds t1 t1 t1 t1 Z t2 ν(s) Z t2 − ds + pν(s) dW X (s). (5.3) t1 2 t1
Using proposition6, we get,
Z t2 Z s Z t2 Z s µ X(t2) = X(t1) + µ(t1)∆t + α1 (β1 − µ(p)) dp ds + σ1 dW (p) ds t1 t1 t1 t1 Z t2 ν(s) Z t2 p Z t2 − ds + ρ pν(s) dW ν(s) + 1 − ρ2 pν(s) dW Z (s), (5.4) t1 2 t1 t1
where, dW Z and dW ν are independent of each other. [Using proposition6.] The
moments can be calculated using the properties of expectations and variance.
146 Bibliography
[1] https://www.bseindia.com.
[2] https://www.investopedia.com/articles/investing/102114/guide-japans-nikkei-
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