Introduction to Stochastic Calculus Applied to Finance

Lamberton, Damien; Lapeyre, Bernard

doi:10.1201/9781420009941

Cited by 260 publications

(206 citation statements)

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“…In [9] Bensoussan and Lions characterized price of a put option in terms of a solution of a system of partial-integro differential inequalities (see also [21]). In [39] and [38] Wang et al investigated a penalty method for solving a linear complementarity problem using a power penalty term for the case without jumps in the underlying asset dynamics.…”

Section: Existence Results For Nonlinear Pide Option Pricing Modelsmentioning

confidence: 99%

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On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models

Cruz

Ševčovič

2020

Japan J. Indust. Appl. Math.

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In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel.As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black-Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Lévy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Lévy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Lévy stochastic processes. We also present numerical results and comparison of option prices for various Lévy stochastic processes modelling underlying asset dynamics.

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Section: Existence Results For Nonlinear Pide Option Pricing Modelsmentioning

confidence: 99%

“…Theorem 5.1. Let V (t, S) be the price of an American style put option on underlying asset S following a geometric Lévy process with an admissible activity Lévy measure ν satisfying the structural inequality (21). Then V is a solution to the linear complementarity problem:…”

Section: Existence Results For Nonlinear Pide Option Pricing Modelsmentioning

confidence: 99%

On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models

Cruz

Ševčovič

2020

Japan J. Indust. Appl. Math.

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“…The first term of equation (8) is related to the local curvature of M. The second term relates to the position-specific alignment of the BM by transforming the standard BM B.t/ in R d on the basis of the metric tensor g. For simulating BM sample paths, the discrete form of equation (8) is first derived in equation (9). Specifically, the Euler-Maruyama method is used (Kloeden and Platen, 1992;Lamberton and Lapeyre, 2007), which yields…”

Section: Simulating Brownian Motion On Manifoldsmentioning

confidence: 99%

Intrinsic Gaussian Processes on Complex Constrained Domains

Niu

Cheung

Lin

et al. 2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of double-struckR, double-struckR2, double-struckR3 and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples.

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“…Using Itō calculus (e.g., Lamberton and Lapeyre, 2007), given L y 0 the expectation and variance of the current location L y t are…”

Section: Multivariate Ornstein Uhlenbeck Processmentioning

confidence: 99%

Modeling Interdependent Animal Movement in Continuous Time

2016

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Summary. This article presents a new approach to modeling group animal movement in continuous time. The movement of a group of animals is modeled as a multivariate Ornstein Uhlenbeck diffusion process in a high-dimensional space. Each individual of the group is attracted to a leading point which is generally unobserved, and the movement of the leading point is also an Ornstein Uhlenbeck process attracted to an unknown attractor. The Ornstein Uhlenbeck bridge is applied to reconstruct the location of the leading point. All movement parameters are estimated using Markov chain Monte Carlo sampling, specifically a Metropolis Hastings algorithm. We apply the method to a small group of simultaneously tracked reindeer, Rangifer tarandus tarandus, showing that the method detects dependency in movement between individuals.

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Introduction to Stochastic Calculus Applied to Finance

Cited by 260 publications

References 0 publications

On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models

On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models

Intrinsic Gaussian Processes on Complex Constrained Domains

Modeling Interdependent Animal Movement in Continuous Time

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