Showing papers on "Mathematical finance published in 1996"

Existence of strong solutions for Itô's stochastic equations via approximations

Abstract: Given strong uniqueness for an Ito's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on “any” probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ℝd and in domains in ℝd are considered.

Random Summation: Limit Theorems and Applications

Abstract: Examples Examples Related to Generalized Poisson Laws A Remarkable Formula of Queueing Theory Other Examples Doubling with Repair Mathematical Model A Limit Theorem for the Trouble-Free Performance Duration The Class of Limit Laws Some Properties of Limit Distributions Domains of Geometrical Attraction of the Laws from Class c Limit Theorems for "Growing" Random Sums A Transfer Theorem. Limit Laws Necessary and Sufficient Conditions for Convergence Convergence to Distributions from Identifiable Families Limit Theorems for Risk Processes Some Models of Financial Mathematics Rarefied Renewal Processes Limit Theorems for Random Sums in the Double Array Scheme Transfer Theorems. Limit Laws Converses of the Transfer Theorems Necessary and Sufficient Conditions for the Convergence of Random Sums of Independent Identically Distributed Random Variables More on Some Models of Financial Mathematics Limit Theorems for Supercritical Galton-Watson Processes Randomly Infinitely Divisible Distributions Mathematical Theory of Reliability Growth. A Bayesian Approach Bayesian Reliability Growth Models Conditionally Geometrical Models Conditionally Exponential Models Renewing Models Models with Independent Decrements of Volumes of Defective Sets Order-Statistics-Type (Mosaic) Reliability Growth Models Generalized Conditionally Exponential Models Statistical Prediction of Reliability by Renewing Models Statistical Prediction of Reliability by Order-Statistics-Type Models Appendix 1: Information Properties of Probability Distributions Mathematical Models of Information and Uncertainty Limit Theorems of Probability Theory and the Universal Principle of Non-Decrease of Uncertainty Appendix 2: Asymptotic Behavior of Generalized Doubly Stochastic Poisson Processes General Information on Doubly Stochastic Poisson Processes A General Limit Theorem for Superpositions of Random Processes Limit Theorem for Cox Processes Limit Theorems for Generalized Cox Processes Convergence Rate Estimates in Limit Theorems for Generalized Cox Processes Asymptotic Expansions for Generalized Cox Processes Estimates for the Concentration Functions of Generalized Cox Processes Bibliographical Commentary Index References

No-arbitrage, change of measure and conditional Esscher transforms

Abstract: This paper grew out of a seminar at the Department of Mathematics at the ETH, Zurich during the Summer Semester of 1995 on the subject of mathematical finance and insurance mathematics. It should be viewed as a contribution towards bridging the existing methodological gap between both fields, especially in the area of pricing derivative instruments. Both insurance and finance are interested in the fair pricing of financial products. For instance, in the case of car insurance, depending on the various characteristics of the driver, a so-called net premium is calculated which should cover the ecpected losses over the period of the contract. To this net premium, various loading factors (for costs, fluctuations, . . . ) are added. The resulting gross premium is also subject to market forces which imply that a market-conform premium is finally charged. The more an insurance market is liquid (many potential offers of insurance, deregulated markets), the more a “correct, fair” price may be expected to emerge. Very important in the process of determining the above premium is the attitude of both parties involved towards risk. Within the more economic literature this attitude towards risk can be described through the notion of utility. Utility theory enters as a tool to provide insight into decision making in the face of uncertainty. For a very readable introduction within the context of insurance, see Bowers et al. (1989). An alternative economic tool is equilibrium theory. Depending on the economic theory used, a multitude of possible premiums may result, one of which is the time-honoured Esscher principle. Rather than being based on the expected loss itself, the Esscher principle starts from the expectation of the loss under an exponentially transformed distribution, properly normalised. In Buhlmann (1980, 1983), the Esscher principle is discussed within the utility and equilibrium framework. Besides

Lectures on the Mathematics of Finance

Abstract: The model Part 1 Complete markets: Pricing Optimization Equilibrium Part 2 Incomplete markets: Hedging Optimization Pricing Transaction costs Appendix A Historical Notes Bibliography

Scenario Simulation: Theory and methodology

Abstract: This paper presents a new simulation methodology for quantitative risk analysis of large multi-currency portfolios The model discretizes the multivariate distribution of market variables into a limited number of scenarios This results in a high degree of computational efficiency when there are many sources of risk and numerical accuracy dictates a large Monte Carlo sample Both market and credit risk are incorporated The model has broad applications in financial risk management, including value at risk Numerical examples are provided to illustrate some of its practical applications

A hyperbolic diffusion model for stock prices

Abstract: In the present paper we consider a model for stock prices which is a generalization of the model behind the Black–Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black–Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.

Two renewal theorems for general random walks tending to infinity

Abstract: Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.

Mathematics for Economics and Finance: Mathematical models in economics

Abstract: Introduction In this book we use the language of mathematics to describe situations which occur in economics. The motivation for doing this is that mathematical arguments are logical and exact, and they enable us to work out in precise detail the consequences of economic hypotheses. For this reason, mathematical modelling has become an indispensable tool in economics, finance, business and management. It is not always simple to use mathematics, but its language and its techniques enable us to frame and solve problems that cannot be attacked effectively in other ways. Furthermore, mathematics leads not only to numerical (or quantitative ) results but, as we shall see, to qualitative results as well. A model of the market One of the simplest and most useful models is the description of supply and demand in the market for a single good. This model is concerned with the relationships between two things: the price per unit of the good (usually denoted by p ), and the quantity of it on the market (usually denoted by q ). The ‘mathematical model’ of the situation is based on the simple idea of representing a pair of numbers as a point in a diagram, by means of coordinates with respect to a pair of axes. In economics it is customary to take the horizontal axis as the q -axis, and the vertical axis as the p -axis.

Numerical and Statistical Approximation of Stochastic Differential Equations with Non-Gaussian Measures

Abstract: This monograph is based on methods and numerical tools from such fields as theory of stochastic differential equations (SDEs), stochastic modeling in computational physics, engineering and mathematical finance, statistical estimation methods, and Monte-Carlo type approximations.

Sample path large deviations for super-Brownian motion

Abstract: We derive two large deviation principles of Freidlin-Wentzell type for rescaled super-Brownian motion. For one of the appearing rate functions an integral representation is given and interpreted as ‘Kakutani-Hellinger energy’. As a tool we develop estimates for the Laplace functionals of (historical) super-Brownian motion and certain maximal inequalities. Also it is shown that the Holder norm of index α<1/2 of the processt↦〈f, X t 〉 possesses some finite exponential moments provided the functionf is smooth.

Dominating points and large deviations for random vectors

Abstract: We establish a representation formula useful for obtaining precise large deviation probabilities for convex open subsets of a Banach space. These estimates are based on the existence of dominating points in this setting.

Numerical and Statistical Approximation of Stochastic Differential Equations with Non-Gaussian Measures

Abstract: This monograph is based on methods and numerical tools from such fields as theory of stochastic differential equations (SDEs), stochastic modeling in computational physics, engineering and mathematical finance, statistical estimation methods, and Monte-Carlo type approximations.

Non-ideal Brownian motion, generalized Langevin Equation and its application to the security market

Abstract: Brownian motion has been extensively applied in the field of mathematical finance in modeling the stochastic processes of returns on securities. In this paper basic and generalized Langevin Equations with memory are used to augment Brownian motion to capture the well stylized facts of the financial market that frictions and imperfect information exist. The operator method of Fourier-Laplace transform with an appropriate kernel (influence function) is used to circumvent the difficulty associated with solving a time dependent nonlinear differential Equation, and a practical computational method is proposed.

Arbitrage-Based Pricing when Volatility is Stochastic

Abstract: Keywords: Mathematical Finance Reference SFI-PB-WORKING-1997-001 Record created on 2008-03-12, modified on 2017-05-12

Stability of semi-markov evolution systems and its application in financial mathematics

Abstract: We study the problem of stability of semi-Markov evolution systems and its application in financial mathematics.

Credibility for stationarity

Abstract: In this paper results for the linear credibility estimators are given for general stationary premium models The results are derived by applying recursions and formulas of the prediction theory of stationary stochastic processes The paper extends former results of 1982/83, given for ARMA-processes

A Hyperbolic Diffusion Model for Stock Prices

Abstract: In the present paper we consider a model for stock prices which is a generalization of the model behind the Black- Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black-Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.

Interest Rate Theory - CIME Lectures 1996

Abstract: This set of lecture notes constitutes a self contained overview of the martingale based theory of interest rates. The lectures were given by the author at the 1996 CIME Summer School on Mathematical Finance, in Bressanone, Italy. The topics covered include: Bond markets, interest rates, arbitrage, martingale measures, completeness, short rate models, affine term structures, forward rate models, change of numeraire, log-normal models, state price densities, point process models, risky bonds, minimization of arbitrage information.

A critical case for Brownian slow points

Abstract: LetXt be a Brownian motion and letS(c) be the set of realsr≧0 such that uXr+t−X r u≦c√t, 0≦t≦h, for someh=h(r)>0. It is known thatS(c) is empty ifc 1, a.s. In this paper we prove thatS(1) is empty a.s.

Martingale-Based Hedge Error Control

Abstract: Keywords: Mathematical Finance Reference SFI-PB-CHAPTER-1996-001 Record created on 2008-03-12, modified on 2017-05-12

A stochastic model for the financial market with discontinuous prices

Abstract: This paper models some situations occurring in the financial market. The asset prices evolve according to a stochastic integral equation driven by a Gaussian martingale. A portfolio process is constrained in such a way that the wealth process covers some obligation. A solution to a linear stochastic integral equation is obtained in a class of cadlag stochastic processes.