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,

• Gaussian process for machine learning, C. E. Rasmussen and C. K. I. Williams, The MIT Press, 2006.
  http://www.gaussianprocess.org/gpml/

• B. Iooss. Revue sur l'analyse de sensibilité globale de modèles numériques. Journal de la Société Française de Statistique, 152:1-23, 2011.
• http://journal-sfds.fr/article/view/53