1,721,117 research outputs found
Probability Theory II
No full textThis book offers a modern approach to the theory of continuous-time stochastic processes and stochastic calculus. The content is treated rigorously, comprehensively, and independently. In the first part, the theory of Markov processes and martingales is introduced, with a focus on Brownian motion and the Poisson process. Subsequently, the theory of stochastic integration for continuous semimartingales was developed. A substantial portion is dedicated to stochastic differential equations, the main results of solvability and uniqueness in weak and strong sense, linear stochastic equations, and their relation to deterministic partial differential equations. Each chapter is accompanied by numerous examples. This text stems from over twenty years of teaching experience in stochastic processes and calculus within master's degrees in mathematics, quantitative finance, and postgraduate courses in mathematics for applications and mathematical finance at the University of Bologna. The book provides material for at least two semester-long courses in scientific studies (Mathematics, Physics, Engineering, Statistics, Economics, etc.) and aims to provide a solid background for those interested in the development of stochastic calculus theory and its applications. This text completes the journey started with the first volume of Probability Theory I - Random Variables and Distributions, through a selection of advanced classic topics in stochastic analysis
Journal of Computational Finance
No full textThe Journal of Computational Finance is an international peer-reviewed journal dedicated to advancing knowledge in the area of financial mathematics. The journal is focussed on the measurement, management and analysis of financial risk, and provides detailed insight into numerical and computational techniques in the pricing, hedging and risk management of financial instruments
Valuation Adjustments for Improved Risk Management - ABC-EU-XVA
No full textThe EID aims to address significant challenges arising from the mathematical modelling, numerical computation and risk
management, in the form of valuation adjustments, of financial contracts. Valuation adjustments represent a major focus of
the on-going regulatory reform related to the recent global financial crisis. X-Value Adjustment (XVA) refers generally to
these different valuation adjustments. The purpose of XVA is two-fold: To hedge possible losses due to a counterparty
default event, and to determine the amount of capital required by the institution under the new regulations. The "X" in XVA
can be many letters, as institutions have to deal with CVA (credit value adjustment), FVA (funding value adjustment), KVA
(capital value adjustment), MVA (margin value adjustment), etc. This is reflected in the EID's title. As these adjustments
require deep understanding in terms of the mathematical modelling and efficient computation, we will work at the forefront
and consider huge financial portfolios and different market scenarios, inclusing extreme cases.
We thus wish to educate six ESRs in modern risk management and valuation adjustments, and we are in the unique setting
that four major European banks, one major European insurer plus a major consulting company agreed to join efforts with five
reputed academic beneficiaries, from Spain, Italy, Belgium and the Netherlands. The industry will host the ESRs for 18
months and will be active in the special organized Events
Sobolev embeddings for kinetic Fokker-Planck equations
Full text linkWe introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hormander condition. We prove continuous embeddings into Lorentz and intrinsic Holder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Holder spaces. The embedding at first order is proved by adapting a method by Luc Tartar which only exploits scaling properties of the intrinsic quasi-norm, while for higher orders we use uniform kernel estimates
On the Cauchy problem for a non linear Kolmogorov equation
No full textWe consider the Cauchy problem related to the partial differential equationLu ≡ Δ_x u + h(u)∂_y u − ∂_t u = f(·, u),where (x, y, t) ∈ R^N × R × ]0, T[, which arises in mathematical finance and in the theory of diffusion processes. We study the regularity of solutions regarding L as a perturbation of an operatorof Kolmogorov type. We prove the existence of local classical solutions and give some sufficient conditions for global existence
On the stochastic Magnus expansion and its application to SPDEs
Full text linkWe derive the stochastic version of the Magnus expansion for linear systems
of stochastic differential equations (SDEs). The main novelty with respect to
the related literature is that we consider SDEs in the It\^o sense, with
progressively measurable coefficients, for which an explicit It\^o-Stratonovich
conversion is not available. We prove convergence of the Magnus expansion up to
a stopping time {\tau} and provide a novel asymptotic estimate of the
cumulative distribution function of t. As an application, we propose a new
method for the numerical solution of stochastic partial differential equations
(SPDEs) based on spatial discretization and application of the stochastic
Magnus expansion. A notable feature of the method is that it is fully
parallelizable. We also present numerical tests in order to asses the accuracy
of the numerical schemes
Gaussian lower bounds for non-homogeneous Kolmogorov equations with measurable coefficients
Full text linkWe prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hörmander condition. The bound is independent of the smoothness of the coefficients and generalizes classical results for uniformly parabolic equations
LEVERAGED ETF IMPLIED VOLATILITIES FROM ETF DYNAMICS
Full text linkThe growth of the exchange-traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local-stochastic volatility models. A closed-form approximation for prices is derived for European-style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed-form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well-known CEV and SABR models
Explicit implied volatilities for multifactor local-stochastic volatility models
Full text linkWe consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three-halves stochastic volatility, and SABR local-stochastic volatility
Pricing approximations and error estimates for local Lévy-type models with default
Full text linkWe find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar Lévy-type stochastic processes. We derive rigorous error bounds for the approximate solutions. We also provide numerical examples illustrating the usefulness and versatility of our methods in a variety of financial settings
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