# Stochastic calculus

## Summary

This course gives an introduction to probability theory and stochastic calculus in discrete and continuous time. We study fundamental notions and techniques necessary for applications in finance such as option pricing and hedging.

## Content

Topics include :

- Random variables, characteristic functions, limit theorems
- Markov processes
- Kalman filter
- Ito calculus
- Stochastic differential equations
- Martingale representation
- Girsanov theorem
- Optimal stochastic control
- Jump processes
- Numerical simulation

## Keywords

probability, Ito calculus, diffusion, martingale representation, change of measure, Brownian motion, Poisson process

## Learning Prerequisites

## Important concepts to start the course

calculus

## Learning Outcomes

By the end of the course, the student must be able to:

- Explain the stochastic integral with respect to a Brownian motion
- Explain the notion of an Ito processes with finite activity jumps and its quadratic variation
- Apply Ito's formula to multivariate Ito processes with finite activity jumps
- Compute the stochastic exponential of an Ito process with finite activity jumps
- Explain the notion of a stochastic differential equation, the existence, uniqueness, and Markov property of its solution
- Apply the Feynman-Kac theorem on the stochastic representation of solutions to partial differential equations
- Solve a stochastic differential equation formally, for the linear case, and numerically, for the general case
- Derive the HJB equation for some basic stochastic optimal control problems
- Compute the optimal strategy for some basic optimal portfolio choice and consumption problems, via the HJB equation and the martingale method
- Explain the three pillars of stochastic calculus: Ito's formula, Girsanov'€™s theorem, and the martingale representation theorem
- Work out / Determine moment generating functions, conditional moment generating functions, conditional and unconditional moments for multi-dimensional random vectors
- Apply the Law of Large Numbers and the Central Limit Theorem
- Analyze multi-dimensional Gaussian distributions and derive the corresponding conditional expectations and conditional variances
- Apply Kalman filter to a general linear model and derive the filter and the optimal Kalman gain

## Transversal skills

- Use a work methodology appropriate to the task.

## Teaching methods

Lectures, exercises, homework

## Expected student activities

attendance at lectures, completing exercises

## Assessment methods

- 20% homework
- 30% midterm exam
- 50% final exam

## Resources

## Bibliography

Björk, T. (2004), "Arbitrage Theory in Continuous Time", Oxford University Press

Glasserman, P. (2004), "Monte Carlo Methods in Financial Engineering", SpringerVerlag

Lamberton, D. and Lapeyre, B. (2000), "Introduction to Stochastic Calculus Applied to Finance", Chapman&Hall/CRC

Oksendal, B. (2007), "Stochastic Differential Equations. An Introduction with Applications", Springer Verlag

Shreve, S. (2004), "Stochastic Calculus for Finance I. The Binomial Asset Pricing Model", Springer Verlag

Shreve, S. (2004), "Stochastic Calculus for Finance II. Continuous-Time Models", Springer Verlag

## Ressources en bibliothèque

- Stochastic Differential Equations. An Introduction with Application / Oksendal
- Stochastic Calculus for Finance II. Continuous-Time Models / Shreve
- Monte Carlo Methods in Financial Engineering / Glasserman
- Arbitrage Theory in Continuous Time / Bjork
- Stochastic Calculus for Finance I. The Binomial Asset Pricing Model / Shreve
- Introduction to Stochastic Calculus Applied to Finance / Lamberton

## Prerequisite for

- Derivatives
- Advanced derivatives
- Interest rate and credit risk models
- Real options and financial structuring

## In the programs

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Stochastic calculus**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Stochastic calculus**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Exam form:**Written (winter session)**Subject examined:**Stochastic calculus**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Stochastic calculus**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

## Reference week

Mo | Tu | We | Th | Fr | |

8-9 | |||||

9-10 | BS260 | ||||

10-11 | |||||

11-12 | |||||

12-13 | |||||

13-14 | EXTRANEF126 | ||||

14-15 | |||||

15-16 | |||||

16-17 | |||||

17-18 | |||||

18-19 | |||||

19-20 | |||||

20-21 | |||||

21-22 |