Introduction: A Model Risk Primer

Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios

RISK, THE PRICING OF CAPITAL ASSETS, AND THE EVALUATION OF INVESTMENT PORTFOLIOS*

K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.

The dynamic of a financial asset's returns and prices can be expressed using a deterministic process if there is no uncertainty about its future behavior, or with a stochastic process in the more likely case when the value is uncertain. Stochastic processes in continuous time are the most used tool to explain the dynamics of a financial asset's returns and prices. They are the building blocks with which to construct financial models for portfolio optimization, derivatives pricing, and risk management. Continuous-time processes allow for more elegant theoretical modeling compared to discrete-time models and many results proven in probability theory can be applied to obtain a simple evaluation method. Keywords: Brownian motion; Poisson process; jump processes; Levy processes; stochastic process X =; gamma process; exponential process; variance gamma; process; stable process; standard CTS process; generalized tempered stable; process; modified tempered stable; process; standard MTS process; normal tempered stable; process; standard NTS process; normal inverse Gaussian; standard NIG process; Kim-Rachev tempered stable; standard KRTS process; rapidly decreasing tempered stable; standard RDTS process; Brownian motion; Independent increments; Stationary increments; time-changed Brownian motion; The Modified Tempered Stable Processes with Application to Finance; conditional expectation; change of measure; Esscher transform; Brownian Motion and Stochastic Calculus; Risk Assessment Decisions in Banking and Finance

SummaryAssuming duality and certain analytic conditions on the potential density u(x,y), we show that any harmonic function is a constant and the invariant measure, if one exists, is unique.We will assume that Xt is a standard process in duality with another standard process X t ′ relative to some fixed reference measure m on the state space E. The reader is refered to [l, VI, Sec 1] for the precise definition of duality and to either [1] or [2] for other usual definitions. We will also assume that both X and X′ are transient; for any compact K, TKo θt → ∞ as t → ∞ a.s.. However, only the transience of X is used in Proposition 1 and only that of X′ is used in the rest of the discussion.For simplicity, any subset A of E and any function f defined on E is automatically assumed to be measurable with respect to the Borel field of E.There is a common potential density u(x,y) which is excessive in x and co-excessive in y and satisfies:Uf(x) = ∫u(x,y)f(y)m(dy) and U′f(y) = ∫m(dx)f(x)u(x,y) for any f ≥ 0, where U and U′ are, repectively, the potential operators of X and X′.A function h ≥ 0 on E is said to be harmonic if for any compact subset K of E, PKch = h.For Brownian motion in Rn with n ≥ 3, any harmonic function is a constant. Our first proposition generalizes this classical result under a general setting, which includes as special cases, not only Brownian motion, but also transient symmetric stable processes and one sided stable processes. Our proof which uses the general theory of Markov processes is quite short, compared with the complicated analytic proof for symmetric stable processes given in [3].

In March 1996 the US introduced electricity futures contracts which provided for delivery at the California/Oregon border (COB) and Palo Verde switchyard. These were followed in July 1998 by contracts which provided for delivery of electrical energy into the Cinergy and Entergy systems, and in March 1999 by a contract for the Pennsylvania/New Jersey/Maryland (PJM) area. This paper employs the forecast error approach to study the informational efficiency of these markets (the model prediction and event studies approaches are not feasible). COB and Palo Verde spot and futures prices are stationary, while Cinergy spot and futures prices are I(1) and cointegrated (Entergy and PJM data were not utilised due to insufficient observations). All forecast errors, therefore, are stationary. Estimation is by instrumental variables in the presence of a lagged dependent variable and overlapping observations. For the period 1996 (04) to 1999 (12), with COB and Palo Verde data, the efficient markets hypothesis (EMH) is rejected. This outcome is confirmed by cointegration of the COB and Palo Verde futures prices. A significant M‐GARCH term in the forecast error relationship for both contracts supports the view that a time‐varying risk premium was present. A significant negative relationship between forecast errors and volume for the COB contract suggests that agents were still learning the true model driving this market, and that Stein’s ‘Bayesian error’ was non‐zero.For the period 1998 (07) to 1999 (12), with COB, Palo Verde and Cinergy data, the EMH cannot be rejected. Finally, an attempt is made to relate this study to the recent electricity crisis in California.

In the modern world, most gross product is created within Enterprise firms, project programs, state agencies, transnational corporations and their divisions, as well as various associations and compositions of the above entities. Enterprises, being, on the one hand, complex, and, on the other hand, widespread systems, are the subject matter of cybernetics, system theory, operations research, management sciences and many other fields of knowledge. However, the complexity of the system obstructs the development of mathematically rigorous foundations for Enterprise control. Moreover, methods of operations research and related sciences, which are widely used in practice, provide optimization of the constituents of an Enterprise, without modeling it as a whole system. But the optimization of parts does not lead to the optimality of the whole, and, also, the absence of top-down and holistic mathematical models of Enterprise contradicts the principle of holism and the system approach. The approach in this book looks first at Enterprise Systems and their essential aspects as complex sociotechnical systems composed of integrated sets of structural and process models (Chapters 1 and 2). A uniform description of all the heterogeneous fields of the modern Enterprise (marketing, sales, manufacturing, HR, finance, etc.) is then made, and the Enterprise Control Problem is posed as a top-down and holistic mathematical optimization problem (Chapter 3). Original models and methods of contract theory (Chapter 4), technology management (Chapter 5), human behavior and human capital (Chapter 6) and complex activity and resource planning (Chapter 7) are developed to solve the problem. Structural processes and mathematical models constitute an Optimal Enterprise Control Framework (Chapter 8) that provides a practical solution to the Enterprise Control Problem. This book is a resource for postgraduate and doctoral students, postdoctoral researchers and professors with research interests in the following fields of science: Fundamental Complex Systems study, Complex Systems Engineering, Enterprise Systems Engineering Applications of Operations Research, Optimization, Probability and Stochastic processes to Management Science, Economics and Business Theory of the Firm Business and Management – general, strategy/leadership, organization management, operations management and management information systems Theory of Business Processes, Business Processes Improvement and Reengineering

Geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion. It is used in mathematical finance to model stock prices in the Black–Scholes model (see http://en.wikipedia.org/wiki/ Geometric_Brownian_motion).

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

We model a stream of cash flows as an optional stochastic process, and value the cash flows by using a continuous and strictly positive linear functional. By applying a representation theorem from the general theory of stochastic processes we are able to study this valuation principle, as well as properties of the stochastic discount factor it implies. This approach to valuation is useful in the non-presence of a financial market, as is often the case when valuing cash flows arising from insurance contracts and in the application of real options.

Return Dynamics With Lévy Jumps: Evidence from Stock and Option Prices

A general set-up should include all basic failure time models, should take into account the time-dynamic development, and should allow for different information and observation levels. Thus, one is led in a natural way to the theory of stochastic processes in continuous time, including (semi-) martingale theory, in the spirit of Arjas (3; 4) and Koch (108). As was pointed out in Chap. 1, this theory is a powerful tool in reliability analysis. It should be stressed, however, that the purpose of this chapter is to present and introduce ideas rather than to give a far reaching excursion into the theory of stochastic processes. So the mathematical technicalities are kept to the minimum level necessary to develop the tools to be used. Also, a number of remarks and examples are included to illustrate the theory. Yet, to benefit from reading this chapter a solid basis in stochastics is required. Section 3.1 summarizes the mathematics needed. For a more comprehensive and in-depth presentation of the mathematical basis, we refer to Appendix A and to monographs such as by Bremaud (50), Dellacherie and Meyer (61; 62), Kallenberg (101), or Rogers and Williams (133).

The theoretical advancement in the field of modern nonlinear continuum mechanics is discussed. The paper includes elements of the mathematical apparatus, development of the foundations of a General tensor theory of mechanical processes and their representations, including generalization of concepts of objective derivatives and integrals, concepts of tensor measures of stresses and deformations, new approaches in theory of the resistance of solids to deformation.

We present an approach to model selection for a time series of data on a fine time scale. The underlying process generating the data is modelled as a continuous time stochastic process. The underlying continuous processes are assumed to be diffusions with time varying drift and diffusion coefficient. Several approaches to modelling the diffusion coefficient are described. To perform model selection, we propose an approximation to the Bayes factor that uses only the discrete data. We illustrate our approach for several well-known processes including: Brownian motion with drift, the Ornstein-Uhlenbeck process, a mean reversion process with drift, exponential Brownian motion, and a logistic growth model. Finally, we apply our technique to data from the Standard & Poor's 500 stock index by comparing a random walk to a mean reversion model.

A rational expectations competitive storage model was applied to the U.S. corn market, to assess the aptness of this framework in explaining monthly price behavior in an actual commodity market. Relative to previous models, extensive realism was added to the model, in terms of how production activities and storage costs are specified. By modeling convenience yield, “backwardation” in prices between crop years did not depend on the unrealistic assumption of zero ending stocks. Our model generated cash prices that were distributed with positive skewness and kurtosis, and mean and variance that increased over the storage season, comparable to the persistence and the occasional spikes observed in commodity prices. Futures prices were generated as conditional expectations of cash prices at contract maturity, and the variances of futures prices exhibited realistic time–to‐maturity and seasonal patterns. Model realizations of cash and futures prices over many “years” were used to demonstrate the wide variety of price behaviors that could be observed in an efficient market with a similar market structure, implying that economic and policy implications drawn from short, historical samples of prices could be misleading.