The financial crisis of 2008-2009 has once again made it clear that the extreme behavior of financial instruments cannot be described by using the traditional models based on Gaussian processes. The Gaussian processes based models are not capable to capture events such as market crashes, changes in credit ratings, defaults, etc. Such event-driven uncertainties are more appropriately modelled by jump processes. After the financial crisis, jump processes are becoming increasingly significant for modelling real world situations. This has led to increased interest in jump processes amongst academicians and practitioners alike.

The goal of this course is to show that jump processes, and in particular Lévy processes, provide an easy-to-use toolbox for evaluating and hedging financial risks. After a short introduction to the theory of Lévy processes, we will study their applications to risk management and explore the financial problems where the use of jump processes is particularly important. The lecture session on the Lévy processes and their applications in risk management and finance will be accompanied by hands-on training and implementation of several numerical simulations using Python in additional tutorial sessions.

The course will consist of 10 lectures of 1 hour each and 10 exercise/tutorial sessions of 1 hour each.

The primary objectives of the course are as follows:

i. To understand why Lévy processes are more realistic models in risk management and finance.
ii. To learn the intuitive and mathematical descriptions of the Lévy processes and their applications in risk management and finance.
iii.To implement numerical simulations of several problems arising in risk management and finance by using Python computing tools.
iv. To enhance skills of the participants to tackle problems arising in financial industry.
v. To motivate participants for research.


Module 1: Introduction. Compound Poisson processes and jump-diffusions. Characteristic functions. Numerical simulation of compound Poisson processes.
Module 2: Examples of Lévy processes with finite jump intensity: Kou’s model, Merton’s model. Poisson random measures.
Module 3: Trajectories of Lévy processes and Lévy-Khintchine formula. Further examples of Lévy processes used in financial modeling: variance gamma model.
Module 4: Further examples of Lévy processes used in financial modeling: normal inverse Gaussian model. Numerical simulation of Lévy processes.
Module 5: Basic stochastic calculus for Lévy processes. Exponential Lévy models.
Module 6: Stochastic exponential of a Lévy process. Application: evaluating the risks of the CPPI strategy (constant proportion portfolio insurance).
Module 7: Option pricing in Lévy models. Absence of arbitrage and market incompleteness.
Module 8: Market incompleteness. Fourier transform methods for option pricing.
Module 9: Hedging options in Lévy models. Quadratic hedging strategy.
Module 10: Utility-based hedging and its approximations.

Number of participants for the course will be limited to fifty.

You Should Attend If…

• You are quantitative researcher, trader or risk manager in a financial institution, interested in becoming proficient with models based on jump processes.
• You are a research scientist / professor interested in learning the theory and financial applications of Lévy processes.
• You are a student at master or PhD level studying probability theory, stochastic processes, or mathematical finance and interested in pursuing a research career in mathematical finance or a professional career in quantitative analysis / risk management.


Dr Chaman Kumar
Assistant Professor
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667