Financial Modelling (eBook)

Theory, Implementation and Practice with MATLAB Source

Joerg Kienitz, Daniel Wetterau (Autoren)

eBook Download: EPUB

2012
John Wiley & Sons (Verlag)
978-1-118-41329-6 (ISBN)

Infos

Financial Modelling - Theory, Implementation and Practice is a unique combination of quantitative techniques, the application to financial problems and programming using Matlab. The book enables the reader to model, design and implement a wide range of financial models for derivatives pricing and asset allocation, providing practitioners with complete financial modelling workflow, from model choice, deriving  prices and Greeks using (semi-) analytic and simulation techniques, and calibration even for exotic options.

The book is split into three parts. The first part considers financial markets in general and looks at the complex models needed to handle observed structures, reviewing models based on diffusions including stochastic-local volatility models and (pure) jump processes. It shows the possible risk neutral densities, implied volatility surfaces, option pricing and typical paths for a variety of models including SABR, Heston, Bates, Bates-Hull-White, Displaced-Heston, or stochastic volatility versions of Variance Gamma, respectively Normal Inverse Gaussian models and finally, multi-dimensional models. The stochastic-local-volatility Libor market model with time-dependent parameters is considered and as an application how to price and risk-manage CMS spread products is demonstrated.

The second part of the book deals with numerical methods which enables the reader to use the models of the first part for pricing and risk management, covering methods based on direct integration and Fourier transforms, and detailing the implementation of the COS, CONV, Carr-Madan method or Fourier-Space-Time Stepping. This is applied to pricing of European, Bermudan and exotic options as well as the calculation of the Greeks. The Monte Carlo simulation technique is outlined and bridge sampling is discussed in a Gaussian setting and for Lévy processes. Computation of Greeks is covered using likelihood ratio methods and adjoint techniques. A chapter on state-of-the-art optimization algorithms rounds up the toolkit for applying advanced mathematical models to financial problems and the last chapter in this section of the book also serves as an introduction to model risk. 

The third part is devoted to the usage of Matlab, introducing the software package by describing the basic functions applied for financial engineering. The programming is approached from an object-oriented perspective with examples to propose a framework for calibration, hedging and the adjoint method for calculating Greeks in a Libor Market model.

Source code used for producing the results and analysing the models is provided on the author’s dedicated website, http://www.mathworks.de/matlabcentral/fileexchange/authors/246981

Jörg Kienitz is head of Quantitative Analytics at Deutsche Postbank AG. He is primarily involved in developing and implementing models for pricing complex derivatives structures and for asset allocation. He also lectures at university level on advanced financial modelling and implementation including the University of Oxford’s part-time Masters of Finance course. Jörg works as an independent consultant for model development and validation as well as giving seminars for finance professionals. He is a speaker at the major financial conferences including Global Derivatives, WBS Fixed Income or RISK. Jörg is the member of the editorial board of International Review of Applied Financial Issues and Economics and holds a Ph.D. in stochastic analysis from the University of Bielefeld.

Daniel Wetterau is senior specialist in the Quantitative Analytics team of Deutsche Postbank AG. He is responsible for the implementation of term structure models, advanced numerical methods, optimization algorithms and methods for advanced quantitative asset allocation. Further to his work he teaches finance courses for market professionals. Daniel received a Masters in financial mathematics from the University of Wuppertal and was awarded the Barmenia mathematics award for his thesis.

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Financial modelling Theory, Implementation and Practice with MATLAB Source J rg Kienitz and Daniel Wetterau Financial Modelling - Theory, Implementation and Practice with MATLAB Source is a unique combination of quantitative techniques, the application to financial problems and programming using Matlab. The book enables the reader to model, design and implement a wide range of financial models for derivatives pricing and asset allocation, providing practitioners with complete financial modelling workflow, from model choice, deriving prices and Greeks using (semi-) analytic and simulation techniques, and calibration even for exotic options. The book is split into three parts. The first part considers financial markets in general and looks at the complex models needed to handle observed structures, reviewing models based on diffusions including stochastic-local volatility models and (pure) jump processes. It shows the possible risk-neutral densities, implied volatility surfaces, option pricing and typical paths for a variety of models including SABR, Heston, Bates, Bates-Hull-White, Displaced-Heston, or stochastic volatility versions of Variance Gamma, respectively Normal Inverse Gaussian models and finally, multi-dimensional models. The stochastic-local-volatility Libor market model with time-dependent parameters is considered and as an application how to price and risk-manage CMS spread products is demonstrated. The second part of the book deals with numerical methods which enables the reader to use the models of the first part for pricing and risk management, covering methods based on direct integration and Fourier transforms, and detailing the implementation of the COS, CONV, Carr-Madan method or Fourier-Space-Time Stepping. This is applied to pricing of European, Bermudan and exotic options as well as the calculation of the Greeks. The Monte Carlo simulation technique is outlined and bridge sampling is discussed in a Gaussian setting and for L vy processes. Computation of Greeks is covered using likelihood ratio methods and adjoint techniques. A chapter on state-of-the-art optimization algorithms rounds up the toolkit for applying advanced mathematical models to financial problems and the last chapter in this section of the book also serves as an introduction to model risk. The third part is devoted to the usage of Matlab, introducing the software package by describing the basic functions applied for financial engineering. The programming is approached from an object-oriented perspective with examples to propose a framework for calibration, hedging and the adjoint method for calculating Greeks in a Libor market model. Source code used for producing the results and analysing the models is provided on the author's dedicated website, http://www.mathworks.de/matlabcentral/fileexchange/authors/246981.

Autor

About the authors JÖRG KIENITZ is the head of Quantitative Analytics at Deutsche Postbank AG. He is primarily involved in developing and implementing models for pricing complex derivatives structures and for asset allocation. He also lectures at university level on advanced financial modelling and implementation including the University of Oxford's part-time Masters of Finance course. Jörg works as an independent consultant for model development and validation as well as giving seminars for finance professionals. He is a speaker at the major financial conferences including Global Derivatives, WBS Fixed Income and RISK. Jörg is a member of the editorial board of International Review of Applied Financial Issues and Economics and holds a Ph.D. in stochastic analysis from the University of Bielefeld. DANIEL WETTERAU is a specialist in the Quantitative Analytics team of Deutsche Postbank AG. He is responsible for the implementation of term structure models, advanced numerical methods, optimization algorithms and methods for advanced quantitative asset allocation. Further to his work he teaches finance courses for market professionals. Daniel received a Masters in financial mathematics from the University of Wuppertal and was awarded the Barmenia mathematics award for his thesis.

Textauszug

1

Financial Markets – Data, Basics and Derivatives

1.1 INTRODUCTION AND OBJECTIVES

The first chapter is to introduce the models that appear in subsequent chapters and, in so doing, to highlight the necessity of applying advanced numerical techniques. Since we wish to apply mathematical models to financial problems, we first have to analyse the markets under consideration. We have to check the available data upon which we build our models. Then, we have to investigate which models are appropriate and, finally, we need to decide on numerical methods to solve the modelling problem.

We motivate using market data; we highlight the nature of risk and the problems which arise with inappropriate modelling. The final conclusion is that the observed market structures need sophisticated models, numerically challenging implementation and deeply involved special purpose algorithms. Furthermore, we provide answers and suggestions to the following questions:

• What kind of objects do we have to model?
• What kind of distributions are necessary? Do we need anything other than the Gaussian distribution?
• What kind of patterns do we observe and which model is capable of reproducing such patterns?
• How complex should a model be?
• Which mathematical methods do we need? PDE? SDE? Numerical Mathematics?

We do not rate the models, but we do give advice on the numerical methods which can be applied to implement the different models and on what kind of market observation is covered by a certain model. We work out several methods which can be applied. The reader can try the different solutions and – very important – check the implementation, the stability and the robustness. Furthermore, the code provided can be modified to fit the special modelling issues.

Since financial models have to be implemented as computer programs, or they have to be integrated into a pricing library, numerical methods are required. The most fundamental risk of a model is, of course, its inapplicability in a certain setting. To this end we have to analyse which risk factors can be modelled using a certain class of models and we have to be aware of the risk factors that have not been taken into account. But once we have decided to apply a particular model, and we think that we are applying it appropriately, we face the following challenges:

• Appropriate numerical techniques.
• Approximations used should be robust, efficient and accurate.
• Black box solutions should be avoided.
• The implementation should be stable and reliable.

1.2 FINANCIAL TIME-SERIES, STATISTICAL PROPERTIES OF MARKET DATA AND INVARIANTS

To use a mathematical model for gaining insights and applying it to financial market data we need to choose some quantities or risk factors which we model. To this end we consider the notion of a market invariant. Fix a starting point in time and an estimation interval τ. The interval τ could be one day or one month, for instance. Suppose from a market data provider we can get the data for an index , with

We regard X(t) as a random variable. A random variable X is called a market invariant for and estimation interval τ if the realizations

• are independent
• are identically distributed.

A simple but effective method to test if a random variable qualifies as an invariant is the following:

• Take a time series Xs, of the possible invariant.
• Split the time series into two parts

• Plot histograms corresponding to X1 and X2.
• Plot lagged time series against Xt.

Let us illustrate this test on time series for index and swap data. Before we actually start let us illustrate the dependence structure corresponding to independent, positively and negatively dependent random variables. To this end we take as an example the normal distribution with zero mean and a given covariance matrix, Σ. For our examples we choose three different covariance matrices, namely,

The dependence structure is displayed in Figure 1.1.

Figure 1.1 Time series generated from a normal distribution with covariance given by Σ0 (top left), Σ1 (top right) and Σ2 (bottom) reflecting independence, positive and negative dependence

We call a market invariant X time homogeneous if the distribution of X does not depend on the chosen time point tstart. In the sequel we consider Equity, Index, Interest Rate and Option markets. First, we consider index time series for the S&P 500, the Nikkei, the FTSE and the DAX. We argue that the prices of the indices do not obey the properties necessary to be an invariant.

The first observation regarding the data is that if we plot the lagged time series directly we get Figure 1.2. This clearly shows that the plain data are not independent and therefore not an invariant.

Figure 1.2 Lagged Time Series data of daily index closing prices for S&P 500 (top left), Nikkei (top right), FTSE (bottom left) and DAX (bottom right) calculated from daily index closing prices

Furthermore, when we plot histograms with respect to the observed data we cannot find a suitable distributional description. Figure 1.3 shows the corresponding histograms.

Figure 1.3 Histograms of daily index closing prices for S&P 500 (top left), Nikkei (top right), FTSE (bottom left) and DAX (bottom right)

Now, we consider the logarithmic returns computed from the time series. We see a very different picture. Figures 1.4 and 1.5 suggest that these quantities are invariants.

Figure 1.4 Lagged time series for the logarithmic returns for S&P 500 (top left), Nikkei (top right), FTSE (bottom left) and DAX (bottom right) calculated from time series of the daily closing prices

Figure 1.5 Histogram for logarithmic returns for S&P 500 (top left), Nikkei (top right), FTSE (bottom left) and DAX (bottom right) calculated from time series for daily closing prices. Furthermore, the figure shows a moment matched normal distribution

Thus, without further discussion we take as a suitable choice for a market invariant in the equity market the logarithmic returns, given by

(1.1)

In fact, let be a function, then g(H) is a market invariant.

Taking realized prices of an index it is difficult to assign probabilistic concepts. It is not clear how to obtain relevant statistical information using such prices. On the contrary, the logarithmic returns introduced in Equation (1.1) show that the observed time series are independent and some parametric probability distribution can be assigned.

Other market invariants can also be derived. For a general and formal treatment see Meucci, A. (2007). We consider the case of the interest rate market. The zero coupon bonds, DF(t, T) and the ratio might be considered as invariants. But since they tend to 1, respectively its redemption at expiry, zero coupon bonds are not time-homogeneous and therefore not market invariants. For further illustration let us take non-overlapping total returns, Rv, with maturity v. Thus,

and therefore g(Rv(t)) are market invariants since they are time homogeneous, independent and identically distributed. A convenient invariant is the change in yield to maturity (Y2m) and the changes in Y2m denoted by Y:

As in the equity market we observe that the time series generated by the changes of the yield to maturity shows the desired properties. The time series of the changes in the yield to maturity derived from different quoted swap rates are plotted in Figure 1.6.

Figure 1.6 Time Series for Changes in Yield to Maturity for 1Y swap rate (top left), 2Y swap rate (top right), 10Y swap rate (bottom left) and 20y swap rate (bottom right)

The corresponding historical distributions are plotted in Figure 1.7 and we again see that the assignment of some parametric probability distribution can be achieved.

Figure 1.7 Histograms for Changes in Yield to Maturity for 1Y swap rate (top left), 2Y swap rate (top right), 10Y swap rate (bottom left) and 20Y swap rate (bottom right)

Another method is to measure the logarithmic returns generated by investing a certain amount of currency using the current quoted rate for a pre-specified period.

Finally, let us consider the option market. For example, for a plain Vanilla Call option, the time value depends on the price of the underlying, on the yield and on the volatility. Therefore, we write the price of a European Call option, C, as

For reasons of liquidity we only consider ATM forward volatility. This means that the strike price is equal to the forward S(T). Since we have already identified the invariants for S(t) and r(t) we are left with the problem of determining the invariants...

Produkt-Details

• Erscheint lt. Verlag: 10.9.2012
• Reihe/Serie: The Wiley Finance Series
• Reihe/Serie: Wiley Finance Series
• Reihe/Serie: Wiley Finance Series
• Sprache: englisch
• Themenwelt: Recht / Steuern ► Wirtschaftsrecht
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Finanzierung
• Schlagworte: application • Book • combination • Design • Finance & Investments • Financial • Financial Engineering • financial problems • Finanztechnik • Finanz- u. Anlagewesen • implementation • MATLAB • Models • Practice • Quantitative • Range • Reader • source • techniques • theory • Unique
• ISBN-10: 1-118-41329-6 / 1118413296
• ISBN-13: 978-1-118-41329-6 / 9781118413296