Narrow Results

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞.

In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the "right-copycat" local rule is equivalent globally to the classic "left-shift" Bernoulli map. Similarly, we prove the "left-copycat" local rule is equivalent globally to the "right-shift" inverse Bernoulli map.

Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b₁ ≠ 0, where b₁ is the "synaptic coefficient" associated with the cell differential equation developed in Part I.

Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened "Isles of Eden". They too have been identified and tabulated.

We apply compensated convex transforms to define a multiscale Hausdorff stable method to extract intersections between smooth compact manifolds represented by their characteristic functions or as point clouds embedded in ℝⁿ. We prove extraction results on intersections of smooth compact manifolds and for points of high curvature. As a result of the Hausdorff–Lipschitz continuity of our transforms, we show that our method is stable against dense sampling of smooth manifolds with noise. Examples of explicitly calculated prototype models for some simple cases are presented, which are also used in the proofs of our main results. Numerical experiments in two- and three-dimensional space, and applications to geometric objects are also shown.

This paper presents a new approach to perform a nearly unbiased simulation using inversion of the characteristic function. As an application we are able to give unbiased estimates of the price of forward starting options in the Heston model and of continuously monitored Parisian options in the Black-Scholes framework. This method of simulation can be applied to problems for which the characteristic functions are easily evaluated but the corresponding probability density functions are complicated.

We present new approximation formulas for local stochastic volatility models, possibly including Lévy jumps. Our main result is an expansion of the characteristic function, which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.

In this work, we consider a stock price process subjected to idiosyncratic Lévy jumps and global structural changes attributed to interventions due to a semi-Markov process. The semi-Markov process decomposes both the time and state domains of the price process into sub-intervals and price state sub-domains respectively, where a Lévy–Ito process operates. The Lévy jumps decompose the space domain of the currently operating Lévy process. We derive an infinitesimal generator for a stock price process and a closed form expression for the conditional characteristic function of a log price. The former result is used to derive a PIDE satisfied by option prices, while the latter could be used to retrieve risk neutral densities via Fourier transform and price European vanilla options. In the sequel, we derive the characteristic function of the residence time of a semi-Markov process. Incompleteness of the market is exhibited through a general change of measure. For pricing purpose, the minimum entropy martingale measure is defined as an Esscher transform.

The degree of relationship between financial products and financial institutions, e.g. must be considered for pricing and hedging. Usually, for financial products modeled with the specification of a system of stochastic differential equations, the relationship is represented by correlated Brownian motions (BMs). For example, the BM of the asset price and the BM of the stochastic volatility in the Heston model correlates with a deterministic constant. However, market observations clearly indicate that financial quantities are correlated in a strongly nonlinear way, correlation behaves even stochastically and unpredictably. In this work, we extend the Heston model by imposing a stochastic correlation given by the Ornstein–Uhlenbeck and the Jacobi processes. By approximating nonaffine terms, we find the characteristic function in a closed-form which can be used for pricing purposes. Our numerical results and experiment on calibration to market data validate that incorporating stochastic correlations improves the performance of the Heston model.

Correlation plays an important role in pricing multi-asset options. In this work we incorporate stochastic correlation into pricing quanto options which is one special and important type of multi-asset option. Motivated by the market observations that the correlations between financial quantities behave like a stochastic process, instead of using a constant correlation, we allow the asset price process and the exchange rate process to be stochastically correlated with a parameter which is driven either by an Ornstein–Uhlenbeck process or a bounded Jacobi process. We derive an exact quanto option pricing formula in the stochastic correlation model of the Ornstein–Uhlenbeck process and a highly accurate approximated pricing formula in the stochastic correlation model of the bounded Jacobi process, where correlation risk has been hedged. The comparison between prices using our pricing formula and the Monte-Carlo method are provided.

We propose an analytical pricing method for stop-loss reinsurance contracts and catastrophe insurance derivatives using a Cox process with jump diffusion Cox–Ingersoll–Ross (CIR) intensity. The expected payoff of these contracts is expressed by the Laplace transform of the integration of the jump diffusion CIR process and the first moment of the aggregate loss. To confirm that the proposed analytical formula provides stable and accurate insurance derivative prices, we simulate them using a full Monte Carlo method compared to those obtained from its theoretical expectation. It shows that it is much faster way to obtain them than the full Monte Carlo method. We also conduct sensitivity analysis by changing the relevant parameters in the loss intensity providing their figures.

Swing options are a type of exotic financial derivative which generalize American options to allow for multiple early-exercise actions during the contract period. These contracts are widely traded in commodity and energy markets, but are often difficult to value using standard techniques due to their complexity and strong path-dependency. There are numerous interesting varieties of swing options, which differ in terms of their intermediate cash flows, and the constraints (both local and global) which they impose on early-exercise (swing) decisions. We introduce an efficient and general purpose transform-based method for pricing discrete and continuously monitored swing options under exponential Lévy models, which applies to contracts with fixed rights clauses, as well as recovery time delays between exercise. The approach combines dynamic programming with an efficient method for calculating the continuation value between monitoring dates, and applies generally to multiple early-exercise contracts, providing a unified framework for pricing a large class of exotic derivatives. Efficiency and accuracy of the method are supported by a series of numerical experiments which further provide benchmark prices for future research.

In this paper, we offer a network model that derives the expected counterparty risk of an arbitrary market after netting in a closed-form expression. Graph theory is used to represent market participants and their relationship among each other. We apply the powerful theory of characteristic functions (c.f.) and Hilbert transforms to determine the expected counterparty risk. The latter concept is used to express the c.f. of the random variable (r.v.) $max(Y,0)$ in terms of the c.f. of the r.v. $Y$. This paper applies this concept for the first time in mathematical finance in order to generalize results of Duffie & Zhu (2011), in several ways. The introduced network model is applied to study the features of an over-the-counter and a centrally cleared market. We also give a more general answer to the question of whether it is more advantageous for the overall counterparty risk to clear via a central counterparty or classically bilateral between the two involved counterparties.

We generalize the notion of subisometric liftings of row contractions for liftings of completely contractive covariant representations of W*-correspondences. A theory of characteristic functions for such liftings of covariant representations is presented.

A technique is proposed to represent games in characteristic function form as games in normal form, enabling the former to exploit the concepts of solution of the latter. A new solution for games in characteristic function form is then introduced and some properties are found. A generalization of a result of von Neumann and Morgenstern is thus obtained.

Explicitly using the block structure of the unknown signal can achieve better reconstruction performance in compressive sensing. An unknown signal with block structure can be accurately recovered from under-determined linear measurements provided that it is sufficiently block sparse. However, in practice, the block sparsity level is typically unknown. In this paper, we propose a soft measure of block sparsity $k_{\alpha }(x)={(\parallel x{\parallel }_{2,\alpha }/\parallel x{\parallel }_{2,1})}^{\frac{\alpha }{1-\alpha }}$ with $\alpha \in [0,\infty ]$, and present a procedure to estimate it by using multivariate centered isotropic symmetric $\alpha$-stable random projections. The limiting distribution of the estimator is given. Simulations are conducted to illustrate our theoretical results.

This paper proves that: Let f be an entire function of finite order λ on Cⁿ. Then

(1) , where k(λ) is a nonnegative constant depending only on λ;

(2) If , then λ is a positive integer and equals the lower order of f.

In this paper, we investigate the uniqueness theory of algebroid functions and extend some uniqueness theorems of meromorphic functions (H.X. Yi [14], R. Nevanlinna [15,16]) to algebroid functions.

Our main focus in this and the next chapter is the application of Cooperative Game Theory (CGT) models to international water resource issues. In this chapter we will justify the use of CGT in water resource problems, and in particular, in international conflict-cooperation cases. The chapter reviews several important CGT concepts and demonstrates their use and calculation. After reading this chapter you will have a good grasp of basic CGT concepts and be able to apply them at both conceptual and empirical levels to simple cases.

In this chapter, we focus on basic Cooperative Game Theory solution concepts, explain their foundations (including assumptions), and demonstrate their empirical application (calculation and interpretation). You will understand the principles used to construct several commonly used CGT solution concepts, such as the Shapley Value and the Nash Equilibrium, and get some acquaintance with additional CGT and other solution concepts, although to a lesser extent. You will also be introduced to an important aspect of CGT, namely the power of the players in the game and the stability of the solution. These are very important concepts that have not been always an integral part of cooperative solutions applications. The chapter explains several of the power and stability concepts and applies them to the game that is being carried over from Chapter 5.

In this and next two chapters, we will use advanced calculus to formalize and extend the probability theory introduced in Chapter 2. The use of calculus enables us to investigate probability more deeply. A number of quantitative-oriented probability concepts will be introduced. In this chapter, we first introduce the concept of a random variable and characterize the probability distributions of a random variable and functions of a random variable by the cumulative distribution function, the probability mass function or probability density function, the moment generating function and the characteristic function, respectively. We also introduce a class of moments and discuss their relationships with a probability distribution. This chapter focuses on univariate distributions.

In the paper the novel approach for accurate representation and evaluation of uncertainty in intelligent measuring systems is suggested. According to the approach, available probabilistic knowledge of the quantity value is presented in the form of characteristic function. As an example, the on-line uncertainty calculation procedure for radiation thermometry is examined.