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Computer experiments & Stochastic calculus in Finance

Presentation

Program (detailed content) :

 

Computer Experiments

  • Introduction: computer experiments and metamodelling. Examples of applications
  • Two famous metamodels : chaos polynomials and Gaussian process regression (Kriging)
  • Simulation of unconditional / conditional Gaussian processes
  • Accounting for external knowledge and covariance kernel customization
  • Metamodel-based optimisation (Bayesian optimisation)
  • Design of computer experiments: focus on space-filling design
  • Global sensitivity analysis: focus on ANOVA decomposition (Sobol decomposition)
  • Industrial application: uncertainty quantification

 

Stochastic calculus in Finance

  • Review on Stieltjes integral.
  • Elements of continuous martingales in continuous time.
  • Notion of quadratic variations.
  • A brief introduction to the Brownian motion.
  • Construction of the Itô integral.
  • Main elements of the Itô calculus.
  • Introduction to Stochastic Differential Equations.
  • The Black-Scholes model.
  • Some methods of Pricing and Hedging in the Black-Scholes model

 

Organization :

 

Computer Experiments

  • Course, exercises, computer lab with R software.

 

Stochastic calculus in Finance

  • Project

Objectives

At the end of this module, the student will have understood and be able to explain (main concepts) :

 

Computer Experiments

  • Metamodelling for optimization / uncertainty quantification of a computer code
  • At least the two main families of metamodels : chaos polynomials and Gaussian processes
  • Kernel customization to account for external knowledge
  • Design of computer experiments
  • Global sensivity analysis

 

Stochastic calculus in Finance

  • The theoretical background of stochastic calculus associated to a continuous martingale.
  • The modeling of a time-continuous financial market.
  • The pricing and hedging techniques in the Black-Scholes model.

 

The student should be able :

 

Computer Experiments

  • At a theoretical level, to do computations for:
    • covariance kernels and Gaussian process
    • ANOVA decomposition, Sobol indices
  • At a practical level, to perform the complete methodology for analyzing a computer code
    • design of experiments
    • metamodel construction / evaluation
    • application to optimization / uncertainty quantification of a computer code

Stochastic calculus in Finance

  • to understand the main mathematical concepts allowing one to define a stochastic integral and to perform the associated calculus.
  • to perform the main computations associated to a continuous martingale in continuous time.
  • to understand the main concepts for solving and manipulating Stochastic Differential Equations.
  • to use pricing and hedging methods in the Black-Scholes model.

Needed prerequisite

  • Gaussian vectors.
  • Discrete-time Martingales

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...

Bibliography

  • Shreve E. Steven, “Stochastic calculus for Finance II : Continuous-time models”, Springer, 2013,
  • Lamberton Damien et Lapeyre Bernard, « Introduction to Stochastic Calculus Applied to Finance, Second Edition » (dipsonible également en Français), Chapman and Hall/CRC, 2007,
  • Le Gall Jean-François, «Mouvement Brownien, Martingales Et Calcul Stochastique », Sringer, 2016,