Bounds and Asymptotic Approximations for Utility Prices when Volatility is Random

Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework

Asymmetric risk and return: Evidence from the Australian Stock Exchange

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

We study the dependence of volatility on the stock price in the stochastic volatility framework on the example of the Heston model. To be more specific, we consider the conditional expectation of stochastic variance (square of volatility) given stock price log-return f as a function of f and time t. The behaviour of this function depends on the initial stock price log-return distribution density. In particular, we show that the graph of the conditional expectation of the stochastic variance is convex downwards near a special value of the stock price log-return . For the Gaussian distribution, this effect is strong, but it weakens and becomes negligible as the decay of distribution at infinity slows down.

This article derives semi-analytical pricing formulae for geometric average options (GAOs) within a stochastic volatility framework. Assuming a general mean reverting process for the underlying asset and a square-root process for the volatility, the cross-moment generating function is derived and the cumulative probabilities are recovered using the Gauss–Laguerre quadrature rule. Fixed and floating strikes as well as other exotic GAO on different assets such as stocks, currency exchange rates and interest rates are derived. The approach is found to be very accurate and efficient.

Statistical Tools for Finance and Insurance presents ready-to-use solutions, theoretical developments and method construction for many practical problems in quantitative finance and insurance. Written by practitioners and leading academics in the field this book offers a unique combination of topics from which every market analyst and risk manager will benefit. Features of the book: Offers insight into new methods and the applicability of the stochastic technology; Provides the tools, instruments and (online) algorithms for recent techniques in quantitative finance and modern treatments in insurance calculations; Covers topics such as heavy tailed distributions, implied trinomial trees, support vector machines, valuation of mortgage-backed securities, pricing of CAT bonds, simulation of risk processes, and ruin probability approximation; Presents extensive examples; The downloadable electronic edition of the book offers interactive tools.

Previous article Next article A Simple Proof of the Existence of a Solution of Itô’s Equation with Monotone CoefficientsN. V. KrylovN. V. Krylovhttps://doi.org/10.1137/1135082PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. V. Krylov and , B. L. Rozovskii, Stochastic evolution equationsCurrent problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, 71–147, 256 81m:60116 Google Scholar[2] I. Gyongy and , N. V. Krylov, On stochastic equations with respect to semimartingales. I, Stochastics, 4 (1980/81), 1–21 82j:60104 CrossrefGoogle Scholar[3] N. V. Krylov, Extremal properties of solutions of stochastic equations, Theory Probab. Appl., 29 (1984), 205–217 10.1137/1129033 0557.60042 LinkGoogle Scholar[4] L. A. Alyushina, Euler polygonal lines for Itô equations with monotone coefficients, Theory Probab. Appl., 32 (1987), 340–346 10.1137/1132046 0663.60049 LinkGoogle Scholar[5] Gisirō Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo (2), 4 (1955), 48–90 17,166f 0053.40901 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo modelApplied Numerical Mathematics | 1 Apr 2022 Cross Ref Approximation of BSDEs with super-linearly growing generators by Euler’s polygonal line method: A simple proof of the existenceSystems & Control Letters, Vol. 153 | 1 Jul 2021 Cross Ref Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic & 2D-Navier–Stokes Equations: The General CaseJournal of Mathematical Fluid Mechanics, Vol. 22, No. 4 | 4 September 2020 Cross Ref A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditionsInternational Journal of Computer Mathematics, Vol. 5 | 9 March 2020 Cross Ref Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility ModelsAndrei Cozma and Christoph ReisingerSIAM Journal on Numerical Analysis, Vol. 56, No. 6 | 11 December 2018AbstractPDF (1095 KB)Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded DomainsJournal of Mathematical Fluid Mechanics, Vol. 18, No. 1 | 22 October 2015 Cross Ref An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz CoefficientsJean-François Chassagneux, Antoine Jacquier, and Ivo MihaylovSIAM Journal on Financial Mathematics, Vol. 7, No. 1 | 14 December 2016AbstractPDF (602 KB)A New Type of Stability Theorem for Stochastic Systems With Application to Stochastic StabilizationIEEE Transactions on Automatic Control, Vol. 61, No. 1 | 1 Jan 2016 Cross Ref Loss of regularity for Kolmogorov equationsThe Annals of Probability, Vol. 43, No. 2 | 1 Mar 2015 Cross Ref Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay Under Local Lipschitz ConditionStochastic Analysis and Applications, Vol. 32, No. 2 | 28 February 2014 Cross Ref A Note on Euler Approximations for Stochastic Differential Equations with DelayApplied Mathematics & Optimization, Vol. 68, No. 3 | 24 August 2013 Cross Ref Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equationsThe Annals of Applied Probability, Vol. 23, No. 5 | 1 Oct 2013 Cross Ref Rate of convergence of Euler’s approximations for SDEs with non-Lipschitz coefficientsActa Mathematica Sinica, English Series, Vol. 29, No. 8 | 15 July 2013 Cross Ref Stochastic Itô inclusion with upper separated multifunctionsJournal of Mathematical Analysis and Applications, Vol. 400, No. 2 | 1 Apr 2013 Cross Ref Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficientsThe Annals of Applied Probability, Vol. 22, No. 4 | 1 Aug 2012 Cross Ref Stochastic 3D tamed Navier–Stokes equations: Existence, uniqueness and small time large deviation principlesJournal of Differential Equations, Vol. 252, No. 1 | 1 Jan 2012 Cross Ref On the Strong Solution for the 3D Stochastic Leray-Alpha ModelBoundary Value Problems, Vol. 2010, No. 1 | 1 Jan 2010 Cross Ref ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTSStochastics and Dynamics, Vol. 09, No. 04 | 21 November 2011 Cross Ref Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicityProbability Theory and Related Fields, Vol. 145, No. 1-2 | 19 July 2008 Cross Ref Large deviations in the Langevin dynamics of a random field Ising modelStochastic Processes and their Applications, Vol. 105, No. 2 | 1 Jun 2003 Cross Ref Viable solutions of set-valued stochastic equationOptimization, Vol. 48, No. 2 | 1 Jan 2000 Cross Ref Volume 35, Issue 3| 1991Theory of Probability & Its Applications411-623 History Submitted:05 September 1988Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135082Article page range:pp. 583-587ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Strategic capital investment decisions are being made every day in an increasingly uncertain world. While the traditional NPV approach does a reasonable job of valuing simple, passively managed projects, it does not capture the many ways in which a highly uncertain project might evolve, and the ways in which active managers will influence this evolution. In cases where managerial flexibility is a major source of strategic value, companies will want to use real options valuation methods.This article serves as a managerial tutorial on this newer, less understood approach. It uses simple examples to illustrate the essence of four basic categories of real options—timing, growth, production, and abandonment. The examples begin by taking a “binomial” approach to option valuation, in which the value of an investment initiative is allowed to take on two possible future values. Besides being used to illustrate the distinctive features of a real option, the binomial approach also serves to help the reader understand the alternative Black‐Scholes valuation approach (though without requiring the reader to master the complex mathematics underlying Black‐Scholes). Basic instructions for implementing both approaches are provided, along with a discussion of how to set appropriate discount rates and the important role of volatility assessment in the valuation process.

Solving equations—linear, algebraic, differential, difference, analytic, Diophantine …—is arguably the most central problem in mathematics. A case of this problem that can be efficiently tackled is that of linear systems of equations. What could be considered the level of difficulty immediately above that for linear systems, the case of quadratic, or more generally, polynomial equations, is substantially more complicated. Even for polynomials in one variable, classical results of Abel and Galois deprive us of any hope to actually compute their zeros. The best we can do is to approximate them (and a number of algorithms compute these approximations quite efficiently).For systems of multivariate polynomials we need to add complexity obstructions. The first that meets the eye is the possibly large number of solutions. A system of n quadratic equations in n variables has (generically) 2 n solutions in complex space \(\mathbb {C}^{n}\). But each polynomial in the system has \(\frac{1}{2}(n^{2}+3n+2)\) coefficients, and therefore the whole system is specified with Θ(n 3) coefficients. If we were to compute approximations for all its zeros, the size of the output would be exponential in the input size!A focal theme in this third part of the book is that of systems of polynomial equations and algorithms that approximate solutions of these systems. These algorithms have a “numeric” character, and it goes without saying that their analyses strongly rely on appropriate condition numbers. But the nature of these systems and their solutions suggests a view of their condition numbers within a more general framework than we encountered previously. The present chapter introduces this framework and provides some motivating (but also interesting per se) examples.

The knowledge state of an individual with respect to a particular body of information is conceptualized as the set of all the questions that this individual is capable of solving. The goal of an assessment procedure is to identify, by a sequence of appropriately chosen questions, the individual's state among all possible ones. A deterministic procedure is conceivable, but not realistic, in that it does not account for possible inconsistencies in the observed responses. Such inconsistencies may arise from careless errors or lucky guesses from the subject, but may also be of a more fundamental character. A stochastic framework is developed here, in which an individual ‘state’ is formalized as a distribution on the set of all possible knowledge states. On each trial, the assessor has a likelihood function on the set of knowledge states which provides the basis for selecting the question to be asked on that trial. The response is assumed to depend on the individual ‘state’. The question asked and the response observed are used to update the likelihood function. Several examples of questioning rules and of updating rules are discussed, which lead to Markov processes. A central problem is to describe conditions ensuring that the latent distribution corresponding to the subject's ‘state’ can be estimated. We show that, under general conditions, the ‘state’ of a subject is uncoverable if the latent distribution is concentrated on a particular knowledge state.

Quadratic constrained maximization appears as the central problem in many filter design methods and signal processing applications, whenever the processing or filtering entails the maximization of a signal to noise ratio. Furthermore, self reference systems for MSE filters, beamformers and equalizers which manage either time or frequency diversity of the desired signal, give rise to the same mathematical problem. This work reports an adaptive algorithm based on two coupled LMS-like algorithms. It is shown how the Lagrange multiplier, involved in constrained minimization problems, plays the role of gain control in forming the error signal for the filter updates. This gain is power controlled by the output signals keeping the proper equilibrium between the outputs of the two LMS loops.

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

In an analysis of the Lie point symmetries of the equation which allows for the inclusion of transaction costs into the Black–Scholes model (J Eng Math 82:67–75, 2013) one of the symmetries was omitted. We rectify that failing and demonstrate that the correctness of the subsequent analysis is not compromised.

In this paper, the stochastic optimal control problems, which frequently occur in economic and finance are investigated. First, using Bellman’s dynamic programming method the stochastic optimal control problems are converted to Hamilton-Jacobi-Bellman (HJB) equation. Then, obtained HJB equation is solved through the method of separation of variables by guessing a solution via its terminal condition. Also, the non-linear optimal feedback control law is constructed. Finally, the solution procedure is illustrated for solving some examples that two of them are financial models. In fact, to highlight the applications of stochastic optimal control problems in financial mathematics, some financial models are presented.

In this project, we study some properties of the Lévy process for a jump diffusion model and apply them to a problem in financial mathematics. We formulate the call option price in the situation where the underlying asset (share price) has higher than normal volatility and is considered on the basis of the Black-Scholes model. We also derive the formula of call option prices in stock prices with high valatility. Finally, the derived call option price formula is then used to predict and analyze future call option prices using real live data which then compared with the call option price formula that is derived from the results of the Black-Scholes model. The obtained results in this project can be employed in a decision-making for the investment in options derivatives.