Financial modelling

Financial modelling is the basic intellectual tool that supports any algorithmic implementation of financial operations. Unlike models in other areas of science, which require only a mathematical or formal representation to be suited for use in real-world environments, financial models often require to be expressed as contracts¹³⁵. In pricing securities, modelling is harder to describe. A lot of pricing algorithms have been proposed (a sample of which will be discussed here), but most of them aim to price specific instruments in specific situations and under specific circumstances. To get a feeling

133 Cuoco, Domenico and Kaniel, Ron. “Equilibrium prices in the presence of delegated portfolio management”. Journal of Financial Economics 101, pp. 264-296. Elsevier, 2011.

134 A performance related fee can be structured in one of two ways – asymmetric or symmetric.

Asymmetric fees have a base fee that is fixed as a proportion of assets under management as well as a performance fee where the managers earns a set portion of the upside performance. A symmetric fee, on the other hand, comprises a performance fee whereby the manager shares some proportion of both the up and downside of fund performance.

Source: Moore, Charlotte. ‘Making them pay’, Engaged Investor Magazine, January 6, 2015.

Available at: http://www.engagedinvestor.co.uk/making-them-pay/1474389.article . 135 Types of financial contracts:

Reproduced from:

Geiss, Christel. ‘Models in Financial Mathematics 1-discrete time-‘. Department of Mathematics and Statistics, University of Jyväskylä, July 6, 2010.

37 on how pricing algorithmic constructs are produced, it is indispensable to realize that pricing techniques are actually based on the settings of the financial environment they intend to serve. After all, pricing algorithms are computational techniques that get their required input from the financial environment hosting them, and their output has to be suitable for the conditions of that same environment as well. The key is to realize that particular settings of a financial environment are a consequence of two abstract conceptualisations:

‒ how the nature of the financial environment is assumed to be, and

‒ how systems under that same financial environment would be appropriately modelled.

Actually, these two “how’s” comprise two different dimensions of cognitive interpretation. The first one deals with the dimension of time. The financial system is a dynamical one, which means it evolves over time. Therefore, each financial variable and parameter get sense when observed for a period of time. The question here is if time is to be understood as a continuous or a discrete variable, because that distinction changes the financial setting drastically. Similarly, but with independence on how the dimension of time is conceptualized, also a decision has to be made whether the model we construct as a usable abstraction of the financial environment is continuous or discrete.

What are the practical differences implied by these choices and how these differences would translate into different pricing algorithms designs? Could there be a difference between prices that comes from a continuous-time model than from a discrete time one? After all, in both cases, we would be dealing with number only, let alone the fact that any conceptualization, when it lands into computational implementation, is discretized. Before trying to provide insights about the answers to these questions which really lie at the core of the computational understanding of finance, let’s take advantage of a simple matrix arrangement to depict the ideas being discussed:

Model implemented / State of the world

Continuous-time model Discrete time model

Table 2.1 Modelled vs. reality perspectives of design.

Discrete vs. continuous time modelling

Within dynamical systems theory, there is a dichotomy regarding how it is conceptualised and how it is modelled. The 2x2 array shown above intends to reflect the dichotomy of conceptualising-modelling perspectives. This constitute the more essential framework in which computational finance applications may be articulated¹³⁶:

— time: is both the independent variable and the main referential principle of the system. The evolution of a system can be described either as a continuous trajectory in the space of system states (what is called the “phase space”), or as a discrete sequence of successive states;

— real space (labelled as ‘state of the world’): the underlying d dimensional space might be seen either as a continuum, where positions are labelled by d-real-valued coordinates, or as a discrete arrangement modelled like a tiling of discrete cells or equivalently a lattice, where positions are labelled by d integers;

— phase space (labelled as ‘model implemented’): the representation of the system state may scan a continuum (a vector space or a manifold) or exist inside a discrete set (finite or countable) of configurations.

In its most general form, a discrete-time dynamical system consists of a set X (the “phase space”) and a map T: X → X (the “time-evolution law”). The set X is often assumed to have some additional

136 Lesne, Annick. “Discrete vs continuous controversy in physics”. Journal of Mathematical Structures in Computer Science, Volume 17 Issue 2, pp. 185-223, April 2007.

38 structure and the map T to be “compatible” with this structure. For example, in topological dynamics, X is a topological space and T is a continuous map on X.

39

Chapter III. Algorithmics and computational aspects of financial markets

“A person should always divide his money into three: one third in land, one third in commerce, and one third at hand.”

 Rebbe Yitzchak, The Talmud

At its core, finance is fundamentally the study of the trade-off between uncertainty and expected return. In the previous chapters I claimed that the topic of pricing algorithms hasn’t been studied neither properly nor systematically. But that doesn’t mean that computational finance techniques hasn’t been extensively studied and developed, and that includes computational finance algorithms as well. Computational finance is partially, but not totally devoted to price modelling; and the automatization of market operations is quite significant. So this chapter aims to provide a brief overview of the algorithmics and computational aspects of financial markets. It is worth noticing that any subfield of finance such as mathematical finance, computational finance or financial engineering, deals to some extent with the study of the variation of the markets’ prices, so the inclusion of this chapter within the thesis is relevant.

Financial markets are hybrid dynamical non-autonomous systems.¹³⁷

The theoretical assumptions for computational finance applications and by extension, for pricing algorithms, reviewed in the previous chapter may not hold at all times. When this happens, an anomaly is said to occur. A brief description of price anomalies is provided in Appendix 3.1.

3.1 The scope of computational finance

Modern computational finance is an interdisciplinary field which uses mathematical finance, stochastic and statistical methods, time series processing, numerical algorithms and computer simulations extensively. However, the way computational finance as a discipline is rooted in the mathematical theory of finance not explicit. Usually, the extensive application of computer based techniques and procedures in a field application, is preceded by a mathematical approach to the problems and issues within that area. Mathematics provide formal guidelines (theorems) for the practical implementation which come later. In this sense, the advent of mathematical techniques developed for financial analysis opened a natural passageway for computational finance and provided a bedrock for its growth.

In my opinion, Harry Markowitz did effectively outline the future of computational finance in its seminal paper about the efficient selection of investment portfolios on the basis of the tradeoff between and risk and return: <<The process of selecting a portfolio may be divided into two stages.

The first stage starts with observation and experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of portfolio>> (Markowitz, H., “Portfolio Selection”, The Journal of Finance, 1952). This classification regarding portfolio selection can serve as a blueprint

137 Financial markets are considered here as a hybrid dynamical system because its behaviour can be both stochastic and chaotic. A non-autonomous system is a set of differential equations which explicitly depends on an independent variable: time (i.e. x'(t) = f(x,t). Another characteristic is that nonautonomous systems cannot be act unforced without external input. Sources:

‘Is it correct to say that a nonautonomous system is the same as a time varying system?’.

ResearchGate. At:

https://www.researchgate.net/post/Is_it_correct_to_say_that_a_nonautonomous_system_is_the_sa me_as_a_time_varying_system

Sharma, Puneet and Raghav, Manish. “Dynamics of Nonautonomous Discrete Dynamical Systems”.

2016. Available at: https://arxiv.org/abs/1512.08868.

Koeden, Peter E. and Rasmussen, Martin. Nonautonomous Dynamical Systems. Mathematical Surveys and Monographs Volume 176. American Mathematical Society, 2011.

40 for understanding two of the main branches of the scope of computational finance. The first stage can be related to data mining, the modelling of the randomness in the market, and price prediction.

The second one actually relates to investment and trading, two key areas of computational finance applications.

Computational finance taxonomy

Being computational finance is a cross-disciplinary field that focuses on the financial services industry, aiming to make trading, hedging, and investment decisions, and also facilitating portfolio risk management, significant overlap exists with related areas, such as financial modelling, mathematical finance, and financial engineering.¹³⁸

The following table shows a simple computational finance taxonomy. A subdomain of computational science, computational finance consists of two distinct branches:

1) Data mining, which is a knowledge fetching and discovery technique that extracts hidden patterns from huge quantities of data, enabling formation of hypotheses.

2) Computer modelling, which provides simulation-based analysis that predicts system dynamics to test the validity of an underlying assumption.¹³⁹

From the table above, we can see that pricing algorithms are used in the implementation of the last two analytical methods: regression and simulation.

Domain of algorithmic procedures in computational finance

The main areas within the financial realm in which computational finance has had a major impact or play a significant role constitute the domain of algorithmic procedures in financial markets.¹⁴⁰

138 Financial modelling is the most general of the related terms, covering financial computations such as derivative pricing, with the central aim of modelling valuation under uncertainty.

Mathematical finance is the branch of applied mathematics concerned with financial markets.

Traditionally associated with stochastic calculus, in practice this discipline spans various areas of applied mathematics.

Financial engineering focuses on instrument design and creation, aiming to produce new securities.

139 Treleaven, Philip; Yingsaeree, Chaiyakorn and Nuti, Giuseppe. “Computational Finance”. © IEEE Computer Society, 2010.

140 The discussion on this section roughly follows the following article: Hilgers, Michael G.

“Computational finance models”. IEEE Potentials, 2001.

41 A) Prediction. Even though there is a line of thought that supports the view that it is theoretically possible to predict future price behaviour; and that in the imaginary of some there is the belief of the feasibility to predict financial markets’ movements through a skillful analysis of them or through smart algorithms; less effort than it could be expected is actually put in trying to predict the market’s prices. In any case, it is a quite difficult task, and most of those who claim the existence of such an strategy or algorithm are often orchestrating more or less elaborated scams; however, there is some serious work relating to the prediction of prices. For example, one approach relies on the view of financial markets as complex chaotic systems. In nonlinear dynamics theory¹⁴¹, chaotic systems behave around attractors which eventually can be viewed as a source of predictability.¹⁴² Another direction of forecasting in financial markets deals with attempting to forecast the volatility of time series, instead of the series themselves, like for example the prices. An approach to perform this suggested by Wang et. al. (2013) is by using iterative fractal models to forecast volatility while a support vector machine (SVM) handles the learning process and model the innovations. At least in the short-term span, the results have been good.¹⁴³.

B) Random model development. Investment/financial decision-making has always been considered a relevant topic for researching. Undoubtedly, the stock market would be quite attractive if its behavior could be predicted, but that has proven to be impossible for any practical purpose, especially in the long run. If we are going to believe that the EMH holds in the sense that changes in stock market prices are determined solely by new information, then because the new information is unpredictable, so it would be the stock market price. If we don’t believe in the validity of EMH, then the unpredictability problem turns even worse. As a result, many

141 A dynamical system can be characterized by (a) a set of parameters and the values of which define its state at a given point in time, and (b) a set of mathematical rules defining the change of state of the system in time. These rules are generally specified as differential equations, defining the rate of change of each of the parameters describing the system, as a function of the current state of the system. Also, a nonlinear system of equations is a set of simultaneous equations in which the unknown variables or functions (in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. Nonlinearity is ubiquitous in physical phenomena, but in a financial context it often manifest as Iteration, meaning repeated application of a function, which can be viewed as a discrete dynamical system, in which the continuous time variable has been “quantized” to assume integer values. Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The sequence of states the system passes through in time is called its orbit. If the system is dissipative, i.e. it loses energy in some way (and most systems do), the orbits converge to one of a small subset of all possible states called an attractor. An interesting property of attractor is that they perform a sort of principal component analysis of a chaotic system: it is a possibility that the chaotic behavior observed even in an infinite-dimensional system might be caused by a finite-dimensional attractor. Sources:

Wilkinson, M.H.F. “Nonlinear Dynamics, Chaos-theory, and the ‘Sciences of Complexity’ ”. Old Herborn University Seminar Monograph 10, pp. 111-130. 1997.

Farmer, J. Doyne. "Chaotic Attractors of an Infinite-Dimensional Dynamical System". Physica 4D, pp.

366-393. North-Holland Publishing Company, 1982.

Olver, Peter, J. “Nonlinear Systems”. Lecture Notes. Available at Peter Olver's Home Page (http://www.math.umn.edu/~olver/). Accessed online 11.03.2016.

Dawkins, Paul. “Non-Linear Systems”. Available at Paul's Online Math Notes (http://tutorial.math.lamar.edu/Classes/Alg/NonlinearSystems.aspx). Accessed online 11.03.2016.

142 For an example of a company using nonlinear dynamics theory to do stock market prediction visit http://www.predict.com/introduction.html.

143 Wang, Baohua; Huang, Hejiao, and Wang, Xiaolong. “A support vector machine based MSM model for financial short-term volatility forecasting”. Neural Computing and Applications Vol. 22, Issue 1, pp. 21–28, January 2013.

42 computational finance models then focus on modelling the randomness of a stock’s price. There are two ways to conceptualise the observed randomness in stock prices:¹⁴⁴

a) The first consists on focusing on the movements of the stock prices irrespective of their causes, also known as the random walk approach. Usually, this works as follows: at a fixed future time, a stock’s price is modeled as a random variable with a quasi-normal distribution centered about the current price adjusted with a simple growth multiplier. The standard deviation of this distribution depends on both the length of time into the future the model is asked to extend and the intrinsic volatility of the markets. The way the prices actually fluctuate is modelled after a concept borrowed from physics: the Brownian motion.¹⁴⁵ The basic assumption of the random walk model is that the changes in prices and returns are random and independent, so that past values are useless in predicting any future changes. The advantage of the random walk model is that it is a good first-approximation to real stock markets and provides a simple model on which other important problems in finance and economy, such as option pricing and volatility modeling can be addressed in a mathematically rigorous fashion while at the same time keeping them computationally tractable. The main disadvantage of the random walk model is that it does not provide a framework to study the microstructure of the price formation mechanism. Summarizing, the random model development draws conclusions from the fact that price behaviour is random, rather than attempt to predict it.

b) The second approach focuses on the “cause” of the price fluctuations, on the understanding that the primal cause are the agents that intervene in the markets, so these models are known as agent-based models. The agent-based models overcome the problem of the lack of a microstructure foundation of the modelling process by following a bottom-up approach to model directly the operations of different types of traders (different trader profiles). The amount of research that has been made on agent-based models for stock prices is huge, and so is the amount of literature available on the topic. The advantage of the agent-based computational approach is that different types of traders and various trading characteristics can be incorporated into the models and extensive simulations can be performed to study the resulting price dynamics. The problem with this approach is that is the number of parameters and degrees of freedom is so large that is difficult or impossible to identify the determinants of the observed price features. This complexity also makes it very hard to calibrate the models to real stock prices.

C) Derivative pricing.

The basic function of financial modelling and therefore the basic problem of computational finance is to determine the price of financial contracts, among of which derivatives securities occupy a most relevant place. Even though the variety of those instruments has expanded greatly in last decades, some elements remain constant. Most derivatives are contracts involving an underlying product that experiences random fluctuations in its price. And yet with all of this variety there are two dominant types of mathematical models used to determine the price of financial derivatives: partial differential equations (PDEs) and expected value integrals.

― Partial differential equation (PDE) models.

PDEs models arise when studying the distribution of information across individuals in an economy, e.g. beliefs about the value of a particular financial asset. These models are useful to understand the dynamics of asset prices and how these are affected when market participants

144 The discussion on the random walk and the agents approach roughly follows the article:

Wang Li-Xin. “Dynamical Models of Stock Prices Based on Technical Trading Rules. Part I: The Models”.

IEEE Transactions on Fuzzy Systems 23(4), pp. 1127-1141, 2015.

145 The relevant characteristic of the Brownian motion from the financial and computational finance perspective is that it can be building block for a wide spectrum of models as well as more complicated processes. Brownian motion lies in the intersection of several important classes of processes. Note that Brownian motion is a Markov process, and a Gaussian process. It has continuous paths, it is a process with stationary independent increments (a Lévy process), and it is a martingale.

43 do not share common beliefs about the ‘intrinsic’ value of a financial asset. The Black-Scholes equation is the prime example of PDE asset pricing model.

― Expected value integral models

An alternative model describing the European call option involves the use of an expected value integral. The generalisation of the one-period discrete model to a one period model with a

An alternative model describing the European call option involves the use of an expected value integral. The generalisation of the one-period discrete model to a one period model with a

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