Introduction to Stochastic Calculus Applied to Finance, Second Edition · Damien Lamberton,Bernard Lapeyre Limited preview – PDF | On Jan 1, , S. G. Kou and others published Introduction to stochastic calculus applied to finance, by Damien Lamberton and Bernard Lapeyre. Introduction to Stochastic Calculus Applied to Finance, Second Edition, Damien Lamberton, Bernard. Lapeyre, CRC Press, , , .

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Mathematical theory and probabilistic tools for the analysis of security markets. Pricing and Hedging, single- and multi-period models, Binomial models. Bounds on option prices. The Fundamental Theorem of Asset-Pricing: Uniqueness of the equivalent martingale measure, completeness and the martingale representation property, characterization of attainable claims.

Stopping Times and American Options: Hedging of American claims. Optimal stopping, Snell envelope, optimal exercise time. Review of Stochastic Calculus: European Options in Continuous-Time Models: Connections with partial differential equations. Barrier options, exchange options, look-back options. Extended trading strategies, free boundary problems, optimal exercise time, early exercise premium.

Bonds lammberton Term-Structure of Interest Rates: Market dynamics, forward-rate models. Heath-Jarrow-Morton framework, no-arbitrage condition. Change of numeraire technique and the Forward measure. Diffusion models for the short-rate lamebrton calibration to the initial term-structure; Gaussian and Markov-Chain lxpeyre. Caps, Floors, Swaps, Forward contracts.

Portfolio optimization, risk minimization, pricing in incomplete markets. The one-period Binomial model: European call- and put-options.

The Trinomial model, failure of completeness, meaning of attanainability in this context. The many-period Binomial model: The many-period Binomial Model: The transform-representation property of martingales, on the filtration of the simple random walk. Read Chapter 1 from Lapeyee pp. Do Problemspp. Notion of value of a contingent claim in terms of the minimal amount required for super-replication.

The backwards-induction, Cox-Ross-Rubinstein formula. The notions of stopping time and of American Contingent Claim: Brief overview of the notions and properties of martingales and stopping times: Elementary theory for the optimal stopping problem in discrete-time: The valuation of American Contingent claims, and its relation to optimal stopping.

The special case of American call-option.

Read Chapter 2 from Lamberton-Lapeyre pp. Do Exercisespp. Radon-Nikodym theorem, likelihood ratios of absolutely continuous probability measures, their martingale properties and explicit computations. Explicit computa-tions in the logarithmic and power-cases. On maximization of the probability of perfect hedge, and of the success-ratio. Due Thu 8 March.

Continuous-time processes, Lwmberton process, Brownian motion as a limit of simple random Walks. Quadratic variation of the Brownian path. Notion of stopping time. Square-integrable martingales, bracket- and quadratic variation- processes.

Notion and properties of local martingales. Read Chapter 3 from Lamberton-Lapeyre pp. Do Exercises 6,pp.

### Introduction to Stochastic Calculus Applied to Finance – CRC Press Book

Extension lambberton the Stochastic Integral to general processes. Stochastic Calculus; he Ito rule. Examples; elementary stochastic integral equations. Cross-variation of continuous martingales. The multi-dimensional Ito formula; integration. The lambrrton representation property of the Brownian filtration. The Markov property of solutions. The Samuelson-Merton-Black-Scholes model for a financial market. Self-financing portfolios, wealth processes, equivalent martingale measure, arbitrage. Contingent claims, upper- and lower-hedging prices.

Notions of Arbitrage and Complete. Sufficient conditions for absence of Arbitrage. Necessary and sufficient conditions. Read Chapter 4 from Lamberton-Lapeyre pp. Do Exercises 19, 21, 23, 24, 27, pp. Not to be handed in. Minimizing the expected shortfall in hedging. The Feynman-Kac formula, and some of its applications.

Read Chapter 5 from Lamberton-Lapeyre pp.

Introduction to Interest-Rate Models: Read Chapter 6 from Lamberton-Lapeyre. Explicit computations in the. The pricing of American contingent claims; elements of the theory of. Optimal Stopping in continuous time. Distribution of the maximum of Brownian motion and its Laplace transform.

The American put-option of up-and-out barrier type; explicit computations.

Hedging and Portfolio Optimization under Portfolio Constraints. Brief review of Stochastic Calculus: Discrete- lambertln continuous-time stochastic models for asset-prices. Notions of trading strategies, arbitrage opportunities, contingent claims, hedging and pricing. Complete and incomplete markets. Fair price as an expectation under the equivalent martingale measure, and as the solution to a Partial Differential Equation. Black-Scholes formula for a European call-option; American options and stopping times; barrier, exchange and look-back options.

Models for the term-structure of interest rates. Heath-Jarrow-Morton model, lapryre and Gaussian models.

## International Journal of Stochastic Analysis

Eamples from the Poisson and Wiener processes. Stochastic Calculus; he Ito rule and its ramifications. The multi-dimensional Ito formula; integration- by-parts. Notions of Arbitrage and Complete- ness. Necessary and sufficient conditions for Completeness. Explicit computations in the framework of the Hull-White model. Do Problem 4 pp.