Introduction

Competition within financial markets is extremely high. With execution speeds paramount data processing that differs in fractions of seconds can result in extremely different investment returns. As competition has grown the requirements for developing suitable technology and pricing models have increased alongside incorporating online analysis with the potential to process data as new information becomes available. The growing maturity and invested interest of both individuals and investment companies in high-frequency Trading research alongside the improved power of computer processing presents an excellent opportunity to assess the effectiveness of modern statistical analysis within high-frequency traded capital markets through prevalent pricing models.

The intended readership of this project is a qualified statistician with a knowledge of Bayesian statistics with a basic understanding of financial trading. The primary focus has been given to statistical analysis with more complex financial terminology explained within key sections of the text to enhance reader comprehension and to relate statistical results to investment decisions.

The research question being considered is can the Extended Kalman filter when applied to the Variance-Gamma Stochastic Arrival model be used to track and forecast asset returns within a high-frequency, financial trading environment.

The projects key aim is to implement the Variance-Gamma Stochastic Arrival pricing model into a state-space framework which can be processed using the Extended Kalman filter in order to develop a forecasting model for asset returns supported by high-frequency BP stock market and simulated data Additionally, the following have been considered:

• An in-depth discussion regarding common pricing models used in practice
• Description of the key components underpinning the Variance-Gamma Stochastic Arrival model
• Provide information regarding state-space models and Bayesian statistics
• Completion of a simulation study to illustrate the model has the capacity to work theoretically
• Present a live data study of BP stock market data extracted from the New York Stock Exchange to illustrate the potential of the Extended Kalman filter and practically

The Variance-Gamma Stochastic Arrival model primarily overcomes limitations associated with legacy pricing models. The key problem with historic pricing models relate to the fundamental assumptions required for validation are not met in practice, causing model redundancy. For reasons discussed further in Chapter Two – Development and Characteristics of the Variance-Gamma Process the fundamental issue is that legacy models only consider the first two moments, mean and variance assuming normally distributed data. It is now well known that models need to accommodate for higher order moments of skewness and kurtosis with the additional capability of dealing with non-Normal, non-linear datasets. Accommodation of these inevitabilities has also created opportunities to develop new models more aligned with the attributes of financial data. Therefore, it has been vital as part of the model selection criteria to understand the probability distribution most suitable to the data and select a modelling framework permitting non-Gaussian characteristics. Implementing an appropriate importance density produces a solution to overcome this problem. The reasons for this are discussed in Chapter Three – Sequential Monte Carlo Methods. The format of the project consists of five chapters detailed below:

• Chapter One – Financial Markets and High Frequency Trading introduces financial markets, existing pricing model development, and the asset return calculation.
• Chapter Two – Development and Characteristics of the Variance-Gamma Process discusses the evolution of the Variance-Gamma process.
• Chapter Three – Sequential Monte Carlo Methods introduces key features of Bayesian statistics, Markov model requirement and the development of filtering algorithms, focusing specifically on the Extended Kalman filter as a non-parametric model.
• Chapter Four – Simulation Study analyses the theoretical performance of the Extended Kalman filter based on simulated Variance-Gamma Stochastic Arrival deviates.
• Chapter Five – Real-Data Analysis analysing the practical performance of the Extended Kalman filter based on live BP stock data from 25^th – 28^th June 2018

The live dataset for Chapter Five – Real-Data Analysis has been extracted from google finance through the establishment of an API connection within R Studio, (the selected analytic tool supporting the project analysis). This has allowed continuous pricing open-high-low-close recorded in American dollars, and discrete trading volume data to be extracted alongside time and date information. The data has been enhanced through the calculation of asset returns to differentiate from the standard practice of attempting to predict asset price solely. Additionally, focusing on returns has the added benefit of maintaining the range of data points within acceptable limits reducing the impact of outliers. The data collected consists of 389 rows accounting for active trading occurring between 9.30am – 4.30pm. Due to technical difficulties the failed API execution to collect data for Friday 29th June has been omitted. This has a slightly negative impact on data quality but does not affect the reader’s ability to understand the performance this model has in practical circumstances. The remainder of the data can be considered complete and there has been no reason to implement offset terms or additional correlation structures to reduce model bias as prices are shown to be independent meeting requirements for Markovian models.

Literature Review

Chapter One – Financial Markets and High-Frequency Trading

This initial introduction has been included to give the reader an overview of the main components of a financial market and demonstrates the approach to determine the stock price based on Brownian Motion.

Definition of Financial Markets

Portrayed as a fundamental law of economic theory, the price of a product or service is based on reflections of supply and demand. Financial markets implement this concept within pricing strategies in order to provide a platform through which investors can buy or sell assets expecting to exploit under- and over-priced opportunities.

Primarily established as a mechanism for trading, the activity, success and complexity of the financial industry have required market division into specific sectors. This project specifically analyses investment within the stock exchange sector of Capital Markets.

It is believed that the total capitalisation of the stock market globally stands at $76.3 trillion, of which the world’s largest exchange, The New York Stock Exchange (NYSE), manages $21 trillion (Agarwal, 2017). The highly liquid characteristic of stock markets provides a medium for easy and rapid transfer of equity between counterparties. Innovative modern technology has improved the efficiency of this process and, consequently most human interventions have been replaced by electronic trading systems. These technical advancements have created high-speed exchanges of real-time information and increased competition, generating a market dynamic. This dynamic controls asset pricing, facilitating the price discovery process. These advances have required the development of mandatory new procedures that can outperform legacy analytics. High-Frequency Trading is proficient to achieve this performance criterion.

Background to High-Frequency Trading

It is now estimated that high-frequency Trading accounts for 73% of daily trading volumes within US security markets (Branum, 2017). High-Frequency Trading software relates to the development and integration of computer algorithms to execute a large volume of market trades with extremely high-frequency. Speed of execution is paramount to business success. Studies are currently taking place that involves the implementation of microwave technology in order to further improve trade execution speeds (Anthony, 2016). These complex algorithms aim to capitalize on pricing inefficiencies and arbitrage. Their capacity to recognize patterns permits incorporation of machine learning and investment forecasting (Cross, 2015).

The strategy can also be referred to as Intraday Trading as all open trading positions will be closed prior to the end of the trading day. The advantage of this approach is that a very liquid market is created. Coupled with characteristics of lower volatility and risk, this attracts investors as returns can be turned into cash. These benefits are the primary reasons legacy trading activity through human interventions are greatly diminished, while technology trading more enhanced.

Stock Pricing and Asset Returns

Classical trading associated with the Efficient Market Hypothesis developed by Harry Max Markowitz in 1952, is based on the belief that when markets operate efficiently stock prices are unbiased and signify a random walk mean-variance model (Basu, 1997) (Mittal & Goel, 2012). Initial findings suggest that attempting to simulate a mean-variance model to limit risk will inaccurately mirror pricing movements. However, numerous studies contest this theory as random walks cannot include predictable trends (Bollen, Mao, & Zeng, 2011).

Fundamental laws of physics governed the earliest asset pricing model, leveraging the theory of Brownian Motion theoretically describe random particle movement and diffusion-based processes to model dependence. This modelling framework led to the development of stock pricing based on Brownian Motion, built by Robert C. Merton and Paul A. Samuelson, assuming continuous-time stochastic, that is to say, random processes in a transaction-free, market with perfectly divisible assets. Parameterized through systematic drift and random volatility components and assuming independent price changes, the asset pricing model is given in Equation 1.

Equation 1 Stock price formula based on Brownian Motion, parameterized through a drift and volatility component at discrete times

bt,θ,σ=θt+σWt

(Agustini, 2018, p. 4)

Advances in the understanding of attributes associated with financial data illustrated that classical modelling formats imposed narrow restrictions incapable of modelling real-world finance. Early developments attempted to simulate asset price through elliptical distributions, seen as viable representations for stochastic, multivariate processes (Zhou, 1993). Analytical tractability was fundamental to the popularity of these methods, but their failure related to the assumption that returns are elliptically distributed leading to practical restrictions when implementing the Capital Asset Pricing Model (Berk, 1996). The theoretic establishment of the determining dependence through Brownian Motion concludes that the modelling assumptions of a restrictive, symmetric dependence structure prove inadequate in representing heavy-tails and skewness attributed with pricing dynamics. Research concerning this diffusion process attributed with constant volatility makes log-price definitions using the Black-Scholes model unsatisfactory because the model does not have the capacity to explain the phenomenon of the volatility smile (Jasra, Stephens, Doucet, & Tsagaris, 2011). Consequently, its failure to take account of market jumps and poor performance accustomed with modelling extreme events, and the assumption of constant volatility, encouraged the development of three groups of sophisticated models more aligned to market reality;

1. Local Volatility Model (Derman, Kani, & Zou, 1996)
2. Stochastic Volatility Models: Markovian Stochastic processes with randomized volatility (Heston, 1993)
3. Jump Models: Account for market skewness through one-dimensional Markovian processes accommodating jump structures attributed to market shocks permitting increased flexibility and incorporation of both diffusion and drift components. (Seneta D. B., 1990) (Carr, Geman, Madan, & Yor, 2002)

Arnaiz, however, concludes that stochastic volatility models are infeasible to explain market reality (Arnaiz, 2002). Arnaiz demonstrates that as the period of sampling decreases, the kurtosis associated with returns increases. The opposite pattern is displayed for instances when volatility is driven by diffusion, suggesting an incorrect modelling application.

Lévy processes have become prevalent in modelling stock prices through work completed by Bachelier (Magdziarz, Orzeł, & Weron, 2011). Supplementary research by Mandelbrot et al. established the presence of long-memory components in asset returns, rendering auto-regressive moving average processes inadequate to capture correlations as they decay exponentially rather than hyperbolically as required for long-term persistence processes (Moody & Wu, 1997). It is believed that within both the equity and foreign Exchange markets Lévy models can adjust accordingly to explaining the volatility smile (skew) (Madan, Carr, & Chang, 1998). Consequently, this project determines the Lévy based Variance-Gamma Stochastic Arrival model as a plausible representation of stock market price and returns data as discussed in Chapter Two – Development and Characteristics of the Variance-Gamma Process.

The existence of financial bubbles emphasize the historic repetition of asset prices. Alternatively, the logarithm of stock price can be a neat way to highlight the proportional change between spot and future stock prices and to calculate asset returns. Spot price signifies the current price of the stock within the marketplace. The future price is deemed to be the price paid on the date of delivery, often seen as the expected value at a future point in time, and is commonly attractive as it permits hedging strategies and speculation to be incorporated within the stock market. Future stock prices are unknown at the present time, implying that stock price is a random variable through a process derived at

S=St,t≥0, where

Stdenoted the price. A basic data manipulation can be completed to transform asset price into returns (log-price increments), requiring only the current and previous stock price. This is advantageous as it meets the requirements for Markovian representation whilst also removing the obligation for large amounts of historic data normally essential to model with respective degrees of confidence.

Equation 2 Asset returns equation defined in Logarithm format (unless otherwise specified the quotation of log within this report implies the natural logarithm to an exponential base)

Asset Return=logSkSk-1

(Miskolczi, 2011, p. 129)

Conclusion

Academic research shows that the generalisation of this approach does not accurately support the characteristics of stock market returns, which commonly feature fat distribution tails, significant skewness and positive densities (Seneta E. , 2004). Therefore, pricing strategies developed via this method are arguably incorrect. Alternatively, more advanced models have been considered. Lévy processes have been nominated as most suitable due to the limitations posed by stochastic volatility frameworks. Additionally, vast amounts of research regarding stochastic volatility has already been completed, for example, Practical filtering for stochastic volatility models (Stroud, Polson, & M¨uller, 2004) and Bayesian Monte Carlo filtering for stochastic volatility models (Casarin, 2006). Consequently, the following sections attempt to improve the estimation of stock returns through the Variance-Gamma Stochastic Arrival model. The additional parameterization of this model compared with the simplistic model in Equation 1 is paramount to accurately capture the stochastic behaviour evident within high-frequency Traded financial markets.

Chapter Two – Development and Characteristics of the Variance-Gamma Process

Lévy Processes, Lévy-Khintchine Representation, and Jump-Diffusion Parameters

The distinguishing aspects of Lévy based models associates with independent and stationary incremental properties for stochastic processes supported by Ergodic Theorem. Sampling Lévy processes at definitive intervals with equal increments insinuates a random walk process in continuous time. Lévy processes possess the additional properties of a Markovian nature and with an infinitely divisible random variable

Xt. Stability of probability distributions relies on infinite divisibility. A formal definition is detailed in Theorem 1

Theorem 1 Infinite Divisibility

A random variable X is infinitely divisible if X can be written as a sum of n independent and identically distributed random variables for every positive integer

(Ducasse, 2017)

Accommodating for the infinitely divisible property permits the characteristic exponent of the marginal random variable

Xat time

tto be evaluated as per Equation 3. At unit time, the characteristic exponent of a Lévy process is shown as

ϕX1u. Advantageously, the property of infinite divisibility permits an assessment of

X1which will define the properties of the given distribution

Xtwithin finite time

t.

Equation 3 Characteristic exponent of a Lévy process

ϕXtu=EeiuXt=etψX1(u)

(Kyprianou, 2007, p. 5)

This result led to the Lévy-Khintchine formula, permitting all Lévy processes to be determined through an assessment of their characteristic functions.

Theorem 2 Lévy-Kintchine Representation

Let

Xtt≥0be a Lévy process on

R. The Lévy-Khintchine formula gives the representation for a characteristic exponent

ψX1(u)as follows:

ψX1u=bui-12σ2u2+∫R{0} 1-eiux+iux1x<1υdx

With

∫R{0} 1∧x2υdx<∞

(Kyprianou, 2007, p. 4)

This result deconstructs a Lévy process into three independent components:

• A trend determined by a drift component with rate

  θ ϵ R(negative skew if

  θ<0)

• A continuous path, stochastic diffusion component with volatility

  σ ϵ R+

• A jump process with a Lévy measure

  υdx

(Figueroa-Lopez, 2012)

Jumps are caused by sudden stochastic changes in market sentiment over a discrete time period. A Poisson Distribution is commonly used to model jump processes as jumps are considered counts of random shocks triggering disturbances within process flow (Klebaner, 1998). Through this area of research, it is now a widely accepted theory that asset prices contain jumps and thus, selection of a model accommodating this feature was paramount when deciding which financial models are deemed most suitable for the analysis (Jing, Kong, & Lui, 2012).

Jump components for Lévy processes are categorized based on the finite and infinite jump arrival rate.

Equation 4 Definition of finite and infinite activity associated with jump processes

Finite Activity=∫R{0} νdx<∞

Infinite Activity=∫R{0} νdx=∞

(Kyprianou, 2007, p. 5)

Pure-jump Lévy models are normally constructed based on infinite activity and variation logic.  The development of these classifications distinguishes between jump activity at the origin. By defining I as the total arrival rate and J as the total variation, a ‘pure-jump’ Lévy process follows:

1. Finite Activity Process

   I<∞, J<∞

2. Infinite Activity but finite variation process

   I=∞, J<∞

3. Infinite Activity and variation process

   I=∞, J=∞

The arrival rate of jumps with different sizes are adequately described through the Lévy measure and can be considered in the functional form

υdx=kxdx. The Lévy density is described by

k(x)with higher kurtosis and increased jump frequency evident with a larger Lévy measure value. Furthermore, the path properties of the parameterized Lévy process can be defined as the Lévy triplet

θ,σ,υ, influencing the basic structure for the Variance-Gamma process in Equation 7. A pure-jump process results from

σ2=0and reverts to a Standard Brownian Motion with continuous random paths when both

θ and υequal zero. In this instance the diffusion element of the Lévy process is eliminated due to jump kernels creating situations permitting the infinitely frequent arrival of small jumps to mimic continuous movements, where the arrival rate decreases monotonically with jump size.

In certain situations, however, it is important to be aware that pure-jump Lévy processes contract markets. Accordingly, the implied volatility curve for option products flattens under Lévy processes however, markets continue to highlight significant skewness. This feature is seen when the independent, identical incremental assumption holds. Acknowledgement of the central limit theorem states that the development of a Normal Distribution as the number of random variables increases initiates normally distributed properties. Therefore, Lévy processes lose their ability to accommodate skewness and kurtosis attributed to market perception. Additionally, constant variance assumed under Lévy models fail to adequately explain both leverage effects and volatility clustering portrayed by the realized volatility of historical data. An extension to the Variance-Gamma Stochastic Arrival model has been chosen to accommodate the volatility clustering limitation. Conversely, Carr et Al. overcome these shortcomings through the development of a model that incorporates an additional Markovian process to time-change the Lévy process accustomed with spot price dynamics (Carr, Geman, Madan, & Yor, 2002). The proposal combines the benefits of Lévy and stochastic models and removes the aforementioned deficiencies. Although Carr et al. invent a modelling improvement, further work remains in order to allow for Lévy processes to accurately represent real-world stock option market phenomena (Carr, Geman, Madan, & Yor, 2002).

Furthermore, the elimination of the diffusion component is a result of the dynamics associated with jumps being considered as capable to generate “non-trivial small-time behaviour” (Sonono & Mashele, 2015). The infinite activity feature permits model behaviour as evident in standard diffusion processes for small jumps (Sonono & Mashele, 2015). Sato completes further research in relation to this topic (Sato, 1999).

Incorporation of Lévy properties within Financial Modelling

As discussed, the fundamental reasoning for implementing Lévy based measures within financial modelling results from the comprehension that the log returns of stocks are not normally distributed. Moreover, their stationary increment and independence properties alongside their ability to accommodate various skew and kurtosis features permit a more realistic representation of real-life situations.

Determination of an appropriate stock price driven by Lévy processes assumes the market consists of two asset types, one risky and one riskless asset, defining price as

Bt=ert.

Equation 5 Representative model for a Risky Asset

St=S0eXt

(Yao, Yang, & Yang, 2009, p. 593)

Classification of returns as log returns develops a Lévy process, hence

logStS0=Xtwhere

S0=St-1in Markovian frameworks (discussed in Chapter Three – Sequential Monte Carlo Methods). Although Lévy processes generally fit the distribution of historical returns reasonably, well these models expose incomplete markets (Göncüac, Karahanbc, & Kuzubaş, 2016). The generation of incomplete markets develops from the infinite number of equivalent martingale measures (further detailed in Introduction to the Variance-Gamma Pricing Model). Several common martingale measures that exist and include; Esscher transform (Gerber & Shiu, 1999), minimal entropy (Sioutis, 2017), mean-correcting (Deville, 2008), or indifference pricing (Yoo, 2008). A popular choice in practice implements the mean-correcting martingale measure under the assumption in Equation 6.

Equation 6 Mean-Correcting Martingale measurement assumption of Risk-Neutral pricing with no dividends and a constant interest rate r

St=S0ereXtEeXt

reference

The mean correction martingale measure is recognized as being equal to

Xtremediated through

Xt+r-logϕ-iwhere it has been assumed that the interest is derived as

rand a dividend yield is not considered. This convexity correction measure modifies market incompletion and this inclusion of a martingale correction is implemented for continuous functions of calendar time. It is important to acknowledge that however complex, the clock always advances forward with increasing uncertainty although not at constant predetermined rates. In contrast to the model proposed by (Mercuri & Bellini, 2010) a selected alternative correction to the Esscher transform has been incorporated, as suggested in Calibration and Filtering of Exponential Lévy Option Pricing models (Sioutis, 2017). This modelling framework has been explored and proved successful on encouraging “convexity within the minimization functional and thus assuring the existence of a solution” regarding Sequential Monte Carlo methods to stock option pricing when defined by Stochastic volatility for Lévy processes for their Variance-Gamma model (Carr, Geman, Madan, & Yor, 2003).

Introduction and Development of the Variance-Gamma Process

The Variance-Gamma process is the result of the work completed by both Madan and Seneta in 1990. They argued that the “Bachelier” (classical) model, based on standard Brownian Motion and assuming Gaussian distributed data, illustrates characteristics of symmetry about the mean. The tails of stock price data however, are portrayed as too heavy to be represented accurately, if considered normally distributed (Seneta E. , 2007, pp. 5-6). This discovery prompted investigations for alternative representative probability distributions with heavier tails.

In 1986 the Variance-Gamma process was determined by Madan as a “pure-jump” process validated through the implementation of a Lévy process, where the characteristic function in log format,

Z(t)omits the inclusion of a Gaussian Component (Seneta E. , 2007, pp. 15-17). Attributed with the Variance-Gamma model exists an infinite arrival rate of activity and the model is considered symmetric when

θ=0. Arnaiz claims that the infinite activity of parameters creates instability (Carr & Madan, 1999). He therefore, encourages caution should an infinite variation process be implemented to model price dynamics, as inherently the parameters lack robustness of diffusions (Arnaiz, 2002). Yet, infinite activity assumptions have been shown to closely approximate highly liquid markets with a large amount of activity, a common feature linked with high-frequency Trading (Fiorani, 2004).

After the model’s initial introduction for defining asset returns, further developments were studied and implemented through fast Fourier transformations to successfully generalize model application to accommodate American-Style Option pricing (Kallsen & Kühn, 2004). Additionally, The Variance-Gamma Process and Option Pricing contrast Black-Scholes with the (symmetric) Variance-Gamma model through an F-test and conclude non-significance emphasizing no monetary bias, illustrating the success of the Variance-Gamma model on the evaluation of pricing bias, only requiring three parameter estimates rather than four (Madan, Carr, & Chang, 1998).

Introduction to the Variance-Gamma Pricing model

The belief of trading being continuous does not hold in reality as the stock price path is derived under tick-size movements that describe in discrete form how much the market moves. This result questions the inclusion of the Brownian component within parsimonious modelling. Some researchers have concluded that a continuous Brownian component is in fact insignificant provided movement can be reinforced by alternative structures (Ait-Sahalia & Jacod, 2010). Accordingly, under a pure-jump Lévy model framework Brownian Motion is excluded. The Variance-Gamma stock price model replaces Brownian Motion in the Black-Scholes model with a Variance-Gamma process (Daal & Madan, 2005). Beneficially, the asset price is described as an exponential Lévy process and Variance-Gamma offers analytic tractability. The Variance-Gamma model for a random variable can be seen as a drifted Brownian Motion, with time changed by an independent gamma process defined on a common probability space, featuring probabilistic behaviour controlled by three parameters.

Equation 7 The Variance-Gamma model

Xt=XtVG(θ,σ,υ)=θGt+σWGt

θ

– The drift in Brownian Motion

Gt

– Gamma process with unit mean rate and variance rate

νevaluated at time

t≥0

σ

–  The variance rate of the gamma time change, defined as a subordinator thus, a strictly increasing, monotonic Lévy process

W=Wt;t≥0

– Standard Brownian Motion with an independent subordinator

υ

– The volatility of Brownian Motion

(Daal & Madan, 2005, p. 2128)

The infinitely divisible increments of

Xt+s-Xsreflect independence and stationary attributes incurring

VGtθ,tσ,υtlaw. Additionally, the non-decreasing Gamma process of a random gamma variable

hwith parameters

α,βhas a density as defined in Equation 8.

Equation 8 Probability Density Function for a Gamma Distribution

h ~ Γα,β

fh=βαΓαhα-1e-βh

(Kyprianou, 2007, p. 9)

The usually seen leptokurtic properties of financial data can be viewed as an example of the Laplace Distribution with characteristics of tails that asymptotically converge towards zero more slowly than that seen under Gaussian. Subsequently, this produces more outliers that would be associated with the Normal Distribution. The Laplace transformations for Gamma are outlined in Equation 9 with the respective characteristic function in Equation 10.

Equation 9 Laplace Transformation for the Gamma Distribution

Ee-λGtv=1+λv-tv

(Sioutis, 2017, p. 10)

Equation 10 Characteristic function for the Variance-Gamma process

φVGu=EeiuXVGt=ϕVGu;θ,σ,ν=11-iuθν+σ2v/2u2-t/ν

(Arnaiz, 2002, p. 5)

Integration of the stock log return density for the Variance-Gamma model assuming that the density of

logStSt-1is known and is represented by modified Bessel function of the second kind and a hypergeometric function defined in Theorem 3 (Madan, Carr, & Chang, 1998). The returns density assesses how well the stock price model fits stock price data. The martingale correction, defined as

ω, is a convexity correction calculated by evaluation of the characteristic function at

-iand satisfying the discounted asset price,

Ee-rtSt=S0. The martingale correction must be attributed with the real number scale introducing a restriction “on the feasible parameters of the model” (Arnaiz, 2002, p. 6).

Equation 11 Definition of the convexity correction

ωof the characteristics function at -i

ω=-logϕ-i=1υlog1-θυ-12σ2υ

ERNST=erTS0=>ERNeXT=e-ωT=>ϕ-i=e-ωT

(Sioutis, 2017, p. 21)

r

– Instantaneous expected return of the stock evaluated at calendar time (Not interest rate as would be the case with future and option products) (Arnaiz, 2002)

T

– Time parameters indices

S0

– Stock price at time

0

ST

– Stock price at time

T

Xt

– Variance-Gamma random variable

ω

– A logarithm convexity correction by evaluating the characteristic function at

-iand must be a real number (Arnaiz, 2002). Also considered as a compensator term to ensure that the Variance-Gamma process is a martingale, preventing arbitrage opportunities (Sonono & Mashele, 2015)

ϕ-i

– Represents the function at

-i

-i

– Imaginary Number

-1

RN

– Represents the assumption of Risk neutrality

Lucas-type economic principles create a situation of an unstructured risk premium whereby the “equilibrium is general and not partial” (Arnaiz, 2002), permitting an alternative form of the Variance-Gamma process defined as the difference between two gamma processes. Acknowledgement of this trait permits the Variance-Gamma process to be epitomized through simulating the difference between two independent, monotonically increasing Gamma processes, thus accounting for increasing and decreasing movements of log-prices. The advantage of this representation permits a simple trading strategy to be incorporated whereby a monotonic increase relates to buy and sell signals.

Equation 12 Variance-Gamma process described as the difference between two Gamma processes

Xt=XtVG(θ,σ,υ)=XgammaC;1M-XgammaC;1G

C=1v>0

G=θ2v24+σ2v2-θv2-1>0

M=θ2v24+σ2v2+θv2-1>0

(Carr, Geman, Madan, & Yor, 2003, p. 350)

Equation 13 Characteristic function for the alternative representation of the Variance-Gamma process

ϕVGu=GMGM+M-Giu+u2C

(Carr, Geman, Madan, & Yor, 2003, p. 350)

The advantage of the Variance-Gamma model over the Heston (Heston, 1993) and Hull White (Hull & White, 2001) models pertains to the fact that these alternative models can only qualitatively capture the smile and in all cases fail to explain associated magnitudes (Arnaiz, 2002). The arrival rate of jumps represented by Equation 14 introduces a dependence on arrival frequency creating skewed distributions and volatility smile curves overall controlled by theta. A non-Uniform specification for the arrival of jumps resembles stock market tendencies whereby unpredictable news has irregular effects. The variance rate determines the likelihood of increased jump sizes. Increasing the variance rate leads to higher likelihoods of larger jumps increasing kurtosis and probabilities at the tails of the distributions with pronounced but symmetric smiles. Elegantly, this result permits the Lévy jump density to be expressed in Equation 14.

Equation 14 Lévy jump density for Variance-Gamma when defined as the difference between two Gamma processes

kVGxdx=CeGx|x|,x<0Ce-Mxx, x>0=eθσ2vxe-2v+θ2σ2σxdx

(Carr, Geman, Madan, & Yor, 2003, p. 6)

Deliberation on the time change attribute of the model has strong relations to economic perception. Understandably, the evolution of financial markets is unique on a daily basis. Accordingly, this creates variation pertaining to trading volumes and activity. Considering two clocks, one signifying standard calendar time change and another for a random business time a highly active trading day has a business clock running faster than the calendar clock. Therefore, trading evolution may be described by the business clock differentiating from the calendar clock. This logic can be used in order to describe a Variance-Gamma process by which Brownian Motion advances based upon a random Gamma business clock. In more turbulent markets the random time changes increase the speed of the calendar clock and vice versa. Such a process can capture seasonal trading (Jasra, Stephens, Doucet, & Tsagaris, 2011).

The Variance-Gamma Stochastic Arrival Pricing model

The Variance-Gamma Stochastic Arrival model is considered as an extension to the Variance-Gamma model capable of more precisely accommodating the clustering effect illustrated when a series of high (low) volatility contributes to another series of high (low) volatility. It is a mean-reversion time changed attribute of the square root process that permits modelling of volatility persistence.

Gt

defined in Equation 7 and determined by a time change respective of a Gamma Distribution

fυdt, τis replaced with

ytdtaccepting that

ytfollows a square root Cox-Ingersoll-Ross (CIR) mean reverting process with the arrival rate is considered stochastic.

Equation 15 Euler discretized Variance-Gamma Stochastic Arrival, a stochastic differential model

dyt=κη-ytdt+λytdWt

(Sioutis, 2017, p. 26)

κ

– Rate of mean reversion

η

– Rate of time change

yt

– Instantaneous rate of time change

dt

– Mean (time change) of the Gamma Distribution

λ

– Time changed volatility defining market risk

dWt

– Brownian Motion considered following a standard Normal Distribution

~ N0,1and independent from other processes within the model

The actual time change for all

tis achieved through integration of the Cox-Ingersoll-Ross process

ythat can be estimated through discretization and summation across all independent

ytwhen the sample size can be considered large. This permits an evaluation of stochastic time change through the development of an instantaneous stochastic clock. Inclusion of this amendment certifies log-price deviates to be simulated following Algorithm 1.

Equation 16 Integration of y in order to obtain the actual time change t of the Cox-Ingersoll-Ross process

Yt=∫0tyudu

Yt=∑k=0N-1yk∆t

(Sioutis, 2017, p. 26)

Equation 17 The equation representing a random variable simulated under the Variance-Gamma Stochastic Arrival process

ZVGSA=XVGYt;θ,σ,υ=θγYt;1,υ+σWγYt;1,υ

(Sioutis, 2017, p. 26)

Algorithm 1 Simulation of stock price deviates using the Variance-Gamma Stochastic Arrival model

1. Simulate the path of

   ykfor the Cox-Ingersoll-Ross time change

yk=yk-1+κη-yk-1∆t+λyk-1∆tdWt

2. Calculate

   YT

YT=∑k=0N-1yk∆t

3. Application of one-step simulations based on the cumulative density for Gamma distributed random variables with mean

   YTand variance simulated based on Uniform random deviates

   U0,1

T*=Fυ-1YT,U0,1

4. Calculate the log stock price and convert to log-returns based on Equation 2

logSt=logSt-1+μ+ωΔt+θT*+σT* dWt

Asset Return=logStSt-1

μ-

Real-world statistical drift of the stock log-returns illustrating the mean rate of return of the stock under the statistical probability measure (Madan, Carr, & Chang, 1998)

(Javaheri, 2004, p. 69)

Complex characteristic functions may also be implemented as an alternative to generate Variance-Gamma Stochastic Arrival deviates. A brief outline of this process has been included in Appendix 1.

Integrated Density

The problem associated with sequential probabilistic inference relates to defining a solution that permits estimation of hidden variables recursively as online observations are collected as discussed below. The integrated density permits the exact calculation of likelihood.

By marginalising the gamma variable, the unconditional density is obtained, and the Probability Density Function can be represented by a Bessel K function of the second kind and permits an exact estimation of the likelihood (Madan, Carr, & Chang, 1998). Additionally, the Bessel K function has the capability of dealing with complex numbers hence it is sufficient to support the requirements of the martingale correction function.

Theorem 3Density for the Variance-Gamma log return

Zt

For the log return

Zt=logStSt-1where the log is considered the natural logarithm to the base

e, the Probability Density Function is available in integrated form and can be developed through integration of Gaussian density function with respect to the random time density function. The result leads to a log-concave modified Bessel K function of the second kind (McDonald function)

fZ=pzk|z1:k-1=2eθxσ2vtv2πσΓtvx22σ2v+θ2t2v-14Ktv-122σ2v+θ2σ2x

t=tk-tk-1

x=z-μt-tυlog⁡1-θυ-σ2ν2

(Whittaker and Watson 1915, p. 367)

For

-∞<x<∞,

ν>0(all other parameters retaining previously defined properties) and

Kη(.)describing a modified Bessel function of the second kind, contrary to the ambiguity formed through the definition citing a Bessel function of a third kind (Jasra, Stephens, Doucet, & Tsagaris, 2011, p. 5)

As the Variance-Gamma Stochastic Arrival process is only partially observed, the density function is not available in closed form. Conditioning on a hidden parameter is required to determine the respective integrated density. Referring back to Equation 17, the Variance-Gamma Stochastic Arrival process is parameterized by Equation 18.

Equation 18 Variance-Gamma Stochastic Arrival parameterization

ZVGSA=XVGYt;θ,σ,υ=θγYt;1,υ+σWγYt;1,υ

(Sioutis, 2017, p. 26) 

Equation 19 Gamma Cumulative Distribution function

Fυh,x=1Γhυυhυ∫0xexp-tυthυ-1dt

h=tk-tk-1

(Sioutis, 2017, p. 40) 

 Equation 20 Gamma time change modelled by an integrated Cox-Ingersoll-Ross process

hdt=ytdt

dyt=κη-ytdt+λytdWt

(Javaheri, 2004, p. 69) 

Equation 21 Conditioning on the arrival rate as the hidden parameter the Variance-Gamma Stochastic Arrival conditional likelihood function with arrival rate dt* equals

fzk|h*=2eθxσ2vh*v2πσΓh*vx22σ2v+θ2h*2v-14Kh*v-122σ2v+θ2σ2x

dt*=ytdt

and

h*=yth

(Sioutis, 2017, p. 40) 

 Conclusion

The literature associated with the Variance-Gamma model and the model’s supplementary representations in many instances I deem incomplete: key features are omitted and contradictory information and non-standard practices for technical specification are portrayed, leading to confusion within this area of research. The intention of this section primarily has been to consolidate correctly and succinctly the key components of the Variance-Gamma Stochastic Arrival process including discussion on fundamental laws and numerical theorems which have led to the derivation of the log-returns under this pricing framework.

Chapter Three – Sequential Monte Carlo Methods

State-space modelling and Hidden Markov Chains

The nature of financial markets requires an appropriate representation. Consequently, the selected state-space should accommodate the markets’ non-Normal higher moments and the exhibited heteroscedasticity clustering of the series (Tech, 2010). These features also render the classical Kalman filter as an inappropriate tracking method for financial systems and render the Black-Scholes model redundant for high-frequency financial data (Jasra, Stephens, Doucet, & Tsagaris, 2011). The ability to perform parameter estimation is of significant importance within areas of finance. Implementation of state-space models assists with acquiring improved stability for parameter estimation. The Lévy based models permit improved performance in relation to modelling financial data over standard Gaussian approaches however, the increased sophistication of algorithms and the reduced computational efficiency should be accounted for, especially in high-frequency Trading where execution speed is vital to profitability.

Definition 1 State-space model

A state-space model represents “the dynamics of an Nth order system as a first order differential equation in an N vector called the state … the state summarizes the effects of past inputs on future output, commonly compared to system memory with the future output determined only on the current state and future input.”

(How & Frazzoli, 2010)

Markov Chains are simulated based on three key components: a state-space, a sequence of random variables and a specified transition probability matrix. The implantation of a Markovian procedure permits a relationship between adjacent observations were the probability of being in a defined state X’ at a given position

t+1depends only on the state at time

t(Doucet, Freitas, & Gordon, 2010). This feature is known as a Dynamic Bayesian Network. The framework requires the specification of two models; an observation model defining the probability of observing

ygiven the current state

x, and a transition model assuming time invariance that determines the likelihood of the next observation based on the previous,

pXt+1|Xt(Doucet, Freitas, & Gordon, 2010).

Hidden Markov models are a subset of Markov Chains developed by L.E. Baum in 1960, in which in contrast with a Markov Chain the states are considered unobservable hence “hidden”. This approach is reflected within many aspects of real-world systematics. By leveraging the available information from the observations appropriate state-space model estimates of the states can be completed at discrete time indexes.

Inference in Dynamic models

A dynamical system can be represented as a state-space. Dynamic state-space models incur a hidden system state

xtcoupled with an initial probability density

px0assumed to evolve through time conditionally

pxt|xt-1and established by determination of a partially observable first-order Markov Chain. Given a random process is stationary, to order one it can be assumed that the associated random variables share the same probability distribution thus, the expected value equates to the mean

μ,

EXt=μand the finite power of the process

EXt2<∞is common for all

t.Processes partially observed do not have an explicit density function therefore, a conditional density is derived through calculation of the hidden state. The state-space leverages information from observations

ytin order to filter noise from the system process. Observations are assumed conditionally independent from the state. The conditional probability density

pyt|xtlinks the transition and observation elements of the current state for successive states. Standard practice defines discrete time, state-space models as Equation 22.

Equation 22 General presentation for linear and non-linear state-space models

Transition equation

Xt=Gtxt-1,ut-1,ηt;w      ηt~N0,W(xt)

Measurement equation

yt=ftxt,ϵt;w                   ϵt~N0,V(xt)

xt

– State vector to be estimated at each discrete time index

t

yt

– Measurement vector describing how observations relate to the system

ut

– An assumed known, exogenous input of the system

ηt

–  White noise model residuals driving the dynamic system with an assumed known Probability Density Function

ϵt

– The disrupting white noise corrupting the observation of the state with an assumed known Probability Density Function and mutually independent of

ηt

Gtft

– Time variant linear or non-linear functions

w

– Corresponding parameters vector

(Stroud, Muller, & Polsen, 2011, p. 378)

This representation creates a state-space model with time-invariant parameters in addition to a vector of latent state variables driven by multi-factor Lévy processes. Asset price returns are linear functions of the current state variables with known model parameters. The assumption of allowing the model residuals to be normally distributed permits the estimation of maximum likelihood estimation during calibration.

Gtfully specifies the state transition density

pxt|xt-1with process noise distributed as

pηt. For the function

ftand corresponding noise distribution

pϵtthe observation likelihood can be determined as

pyt|xt.

Moody and Wu presented a state-space model for high-frequency FX data, based on the decomposition of price quotations into unobserved and underlying true price plus noise (Moody & Wu, 1997). However, the model exhibited poor performance, concluding that the correlation structures within high-frequency data fail to be appropriately represented by conventional random walk, efficient market theory. Unfortunately, within the report a reflection on model error statistics has been omitted, thus a quantitatively ‘poor’ is undefined.

The corresponding state-space for the Variance-Gamma Stochastic Arrival are discretized by leveraging Euler Methods, constructed via the auxiliary variable and detailed in Equation 23.

Equation 23 Proposed Variance-Gamma Stochastic Arrival state-space model

Measurement equation model

yk=yk-1+κη-yk-1Δt+λyk-1ΔtWk-1

State equation model

xk=Fυ-1yk∆t,U0,1

Observation model

zk=lnSk+1

zk=zk-1+μ+ωΔt+θxk+σxkBk

ω=1υlog1-θυ-σ2υ2

(Madan, Carr, & Chang, 1998, p. 87)

The Filtering Problem and Importance Sampling

Introduction of the filtering problem requires the definition of the state vector

xt ϵ Rnxwhere

ntrefers to the state variable dimension and time indexed at

t. The discrete-time stochastic model function can be used to describe the evolution of

xtgiven in Equation 24. Randomness is suitably defined by

υt-1and the propagation function

ft-1perturbed by this randomness is unforeseen disturbance during the state of motion. White noise is represented by

ϵtwith

ftbeing a known function that can be described as non-linear and

yt ϵ Rny.

Equation 24 Discrete-time state evolution and propagation functions

Xt=Gt-1xt-1,ηt-1

yt=ftxt,ϵt

(Stroud, Muller, & Polsen, 2011, pp. 378-379)

The fundamental problem is to extract knowledge regarding the states based on realized observations. The four-stage process, classified as initialization, prediction, update and, resampling is recursively completed through the establishment of an initial starting point

px0≜px0|Z0. Parameters at initialization are assumed known, that simulates independent, identically distributed random samples known as particles, according to the mean.

Particles are predicted for the next time step based on the Chapman-Kolmogorov equation according to Markovian properties such that at

xt-1incorporates all known information at

yt-1. The prediction phase passes each of the samples from the previous time point through the system model creating a set of prior samples.

Updating occurs through the establishment of normalized particle weights derived through the determination of the maximum likelihood according to the observation process. This step manages to effectively indicate regions of the state-space deemed suitable for representations of the observed measurement value. High likelihood values denote well-supported states by the measurement model and zero likelihood implies a non-existent state for the measurement model. Bayes theorem (Equation 25), is invoked in order to update the posterior distribution of

xtat each time point

t. The purpose of the Bayesian update is to provide a technique to combine a prior with the likelihood creating a density, conditional on a state incorporating all known information. The process to this point is known as importance sampling.

Equation 25 Bayes Theorem

pxk|z1:k=pzk|xkpxk|z1:k-1pzk|z1:k-1

(Sioutis, 2017, p. 40)

Comprehension of particle degeneracy due to sub-optimal prior selection, which causes an inadequate representation of the required Probability Density Function, has permitted the inclusion of an additional resampling step where highly likely trajectories are multiplied and those less likely are eliminated. This can create impoverishment which is the restriction imposed through sampling from a discrete rather than continuous distribution and inefficiency in high dimensions (Arulampalam, Maskell, Gordon, & Clapp, 2002). The problem is reduced through selection of an appropriate importance density or perturbation of particles post resampling to prevent significant decrease of system noise. Additionally, a dynamic check-point can be established by monitoring the effective sample size to control degeneracy by ascertaining an empirical threshold that signals a requirement to resample. Other methods such as Shannon Entropy and Coefficient of Variation also exist (Kong, Liu, & Wong, 1994). An approximation approach is instigated accordingly through the simulation of empirical filtering densities as a result of the predict-update procedure.

Monte Carlo simulation is commonly used in order to evaluate integrals that cannot be solved analytically by sampling random deviates from a selected probability distribution. Selection of an appropriate sampling probability distribution impacts efficiency. For example, assessment of rare-events using a brutal-force simulation can generate void deviates that realistically contribute minimally to the solution of the integral thus causing high variability and slow convergence.  Consequently, implementation of importance sampling techniques acts as variance reduction permitting samples to be obtained from an alternative distribution through which a high percentage of random draws make non-zero contributions to the overall evaluation of the integral. Moreover, the integral of interest can be transformed to permit more convenient simulations without modifying past simulated trajectories. The displacement portrayed is rectified through the multiplication of random deviates by respective weights determining which particles survive and which decay at each iteration. Resampling methods are the most common procedures used in order to rectify this issue. Much research has been done associated with this area, with the primary intention requiring the implementation of resampling strategies that do not highly impact computational efficiency.

Invoking the law of large numbers permits a Monte Carlo approximation in order to determine an expectation or an integral. The ability to calculate an a priori estimate of the state

x̂t|t-1based on knowledge from the observed data

y1:t-1. The process of recursive filtering is defined in Equation 26.

Equation 26 Definition of the process for recursive filtering

x̂t|t-1=Ext|y1:t-1=∫xkpxt|y1:t-1dxt

Reference

The posteriori estimate

x̂t|tcan be deduced acknowledging the fact that on the arrival of a new online observation

ytthe process equation can be considered first order Markovian.

Equation 27 Generic equation symbolising a first order Markovian process

x̂t|t=Ext|y1:t=Ext|yt

Reference

Consider the integral in Equation 28

Equation 28 Modification to establish a dependence between

f(x)and

p(x)

I=∫fxpxdx

Reference

A probability distribution is represented by

p(x). Sampling a large number of times from

p(x)permits an approximation to the integral as Equation 29.

Equation 29 Generic approximation to the integral of a probability distribution based on the law of large numbers

I’=∑fxi

Reference

Assuming independence between

pxand

fxan evaluation could result in

pxconcentrating within a specified region of

fxwhere values are insignificant. This creates inefficiencies due to wasted draws. Modification of the process through the inclusion of an Importance (proposal) Distribution

qxthe integral

Jmay be described as Equation 30.

Equation 30 Modification of an approximation of the Integral based on the additional inclusion of an Importance Distribution

J=∫fxpxqxqxdx

Thus,

J’=∑fxjp(xj)q(xj)

Reference

The advantage of this modelling platform is that the sampling space of

p(x)and

q(x)is shared. Conversely,

q(x)can be manipulated in a way that permits concentration within areas of non-zero values of

f(x)reducing variation and permitting an optimal solution to be derived

qtoptxt|x1:t-1=πtxt|x1:t-1. Equation 31 outlines the existence of an optimal solution when the variance of

ϵtx1:tis conditional on

x1:t-1=0and incorporates the corresponding incremental weight.

Equation 31 Optimal Solution for estimating the integral based on the implementation of a suitable proposal density

αtoptx1:t=ptx1:t-1pt-1×1:t-1=∫ptx1:tdxtpt-1×1:t-1

Reference

Equation 31 permits the expected value to be obtained through importance weights. The purpose of the returns density determines model fit and the importance density should be selected with the aim of permitting samples that can be easily be generated (Arnaiz, 2002). Within the Variance-Gamma model, the unconditional density can be devised through integrating out the latent time change component

Gt.Hence, due to the independent incremental properties of Lévy processes specification of Z and G increments are defined in Equation 32.

Equation 32 Independent incremental properties of random variables of the Variance-Gamma model

Z*u,t=Zt-Zu

G*u,t=Gt-Gu

(Madan, Carr, & Chang, 1998)

Madan, Carr & Chang prove the integral of

pZ*u,t|θis supported by the product of a gamma process and the modified Bessel K function of the second kind (Madan, Carr, & Chang, 1998).

Equation 33 Integral for the product of a Gamma process and the modified Bessel K function of the second kind

pZ*u,t|θ=∫pZ*u,tG*u,t,γ,σJpG*u,td(G*u,t)

(Szerszen, 2017)

The impact of this is that Bayesian Analysis of Stochastic Volatility models with Levy Jumps: Application to Value at Risk classifies Markov Chain Monte Carlo Methods infeasible in addition to the computational expense associated with Variance-Gamma models as expressions are represented as a power series (Szerszen, 2017). Furthermore, Jasra et Al 2007 show that Markov Chain Monte Carlo often fails in higher dimensions and additionally Particle Filters and Bayesian Inference in Financial Econometrics conclude that these schemes are arguably more suited for offline or batch sampling analysis (Lopes & Tsay, 2010).

The difference between importance sampling on random variable and finance is that finance involves specifically dealing with time-series of asset returns. The consequence of this is that each collection of random variables is indexed by time. Thus, a Radon-Nikodym derivative, which defines the weight ratio of the posterior distribution

px.to the Importance Distribution

qx., is required to properly perform the importance sampling. Exact details go beyond the scope of the project, with further details being described by Lévy processes and infinitely divisible distributions (Sato, 1999). The posterior distribution is considered as unknown, hence a plausible estimation may be made through normalized weights.

The Extended Kalman Filter

As the Variance-Gamma Stochastic Arrival model is classified a pure Lévy process the assumption of state variables following a Gaussian Distribution is inoperative, therefore a particle filtering alternative must be used in place of the Kalman filter (Mercuri & Bellini, 2010). The Kalman filter algorithm is a sequential Monte Carlo method, applicable when theoretical assumptions underpinning the measurement and propagation functions can be considered as linear and dynamics Gaussian. It is highly unlikely that financial data will follow these restrictive assumptions hence, alternative mechanisms are required.

The Extended Kalman filter is a variation of the Kalman filter in which the Markov Chain models the proposal density

pxk|zk-1and observation density

pxk|zkas non-linear but with perturbed Gaussian noise.  Assuming a non-linear transition equation represents the dynamic process

xksimilarly to Equation 24 the transition and measurement equations are respectively outlined in Equation 33.

Equation 34 Generic Transition and Measurement equation for the Extended Kalman filter

Transition equation

xk=Gxk-1,ηk

ηk~N0,Qk

Measurement equation

zk=fxk,ϵk

ϵk~N0,Rk

(Sioutis, 2017, p. 40)

The process and observation white noise terms

ηkand

ϵkare uncorrelated, mutually independent random variables assuming a Normal Distribution with the respective covariance matrices

Qkand

Rk. Considering all except the current observation an estimate based on the a priori system is defined as Equation 34 with the corresponding a posteriori estimate given in Equation 35.

Equation 35 A priori system estimate

x̂k|k-1=Exk|xk-1

(Sioutis, 2017, p. 37)

 Equation 36 A posteriori system estimate

x̂k|k=Exk|xk

(Sioutis, 2017, p. 37)

Equation 37 Covariance matrices linking a priori and a posteriori estimates

Pk|k-1=covxk-x̂k|k-1

Pk|k= covxk-x̂k|k

(Sioutis, 2017, p. 37)

Through Jacobian matrices, the non-linearity properties existing within the system can be transformed to linear. Equation 36 implements Jacobian matrices of the transition equation respective of the system and noise process parameters from

Aijand

Bij. Application is also required to the observation equation with respect to the system process and measurement noise

Cijand

Dijrespectively.

Equation 38 Definition of the Jacobian matrices required for the linearization of the transition and measurement equations

Aij=∂Gi∂xjx̂k|k-1,0

Bij=∂Gi∂ηjx̂k|k-1,0

Cij=∂fi∂xjx̂k|k,0

Dij=∂fi∂ηjx̂k|k,0

(Sioutis, 2017, p. 37)

Post calculation of the Jacobian matrices the time and measurement updates are calculated using Equation 37 and Equation 38.

Equation 39 Time update equation for the a priori state estimate and error covariance

x̂k|k-1=Gx̂k-1|k-1,0

Pk|k-1=AkPk-1ATk+BkQk-1BTk

(Sioutis, 2017, p. 38)

Equation 40 Measurement update equation for the respective a posteriori variables

x̂k|k=x̂k|k-1+Kkzk-fx̂k-1,0

Pk|k=I-KkCkPk|k-1

(Sioutis, 2017, p. 37)

The optimal Kalman gain is represented in matrix form as Equation 39 and is obtained through minimising the mean square error, or equivalently by calculating the trace

Pk|kin relation to all linear estimators.

Equation 41 Calculation of the Optimal Kalman Gain matrix based on the trace of

Pk|k

Kk=Pk|k-1CkTCkPk|k-1CkT+DkRkDkT-1

(Sioutis, 2017, p. 38)

Although the Extended Kalman filter overcomes the non-linear requirement of the Kalman filter, the calculation of Jacobian matrices are computationally expensive and the transformation into a linear format is not always feasible, especially in situations of highly non-linear and non-Gaussian states. The Unscented and Gaussian Quadrature Kalman filters are alternative proposals. All methods however, have posterior distributions based on Gaussian assumptions, that are regularly inoperative in real-world finance.  Particle filtering developed through research from (Gordon, Salmond, & Smith, 1993) relax the assumption of linearity and permit other posterior distributions to be considered, critical for the appropriate representation of Lévy based models.

Conclusion

The purpose of this section has been to inform the reader of the existence of state-space modelling and relate how the Variance-Gamma Stochastic Arrival model can be represented within this framework. A generalisation to the filtering process alongside the existence of some well-known filtering algorithms have been included with limitations clearly defined. The Extended Kalman filter is the focal algorithm used for the analysis in Chapter Four – Simulation Study and Chapter Five – Real-Data Analysis, hence it has been deemed appropriate to provide an in-depth discussion of how this modelling framework is developed and applied in practice.