Sumários
Esscher transform, mean correcting martingale measure and Arbitrage free Lévy models
23 Novembro 2020, 10:00 • João Guerra
The equivalent martingale measures in n incomplete Lévy market model: the Esscher transform martingale measure and the mean-correcting martingale measure.
Arbitrage free exponential Lévy models. Theorem. Examples.
Lévy market models and complete/incomplete markets
18 Novembro 2020, 11:30 • João Guerra
Existence and uniqueness of equivalent martingale measures in Lévy market models.
The Girsanov theorem - brief review.
The equivalent martingale measure existence and uniqueness general conditon. Examples: Brownian motion, Poisson process, sums of Poisson processes and of Brownian motion with Poisson processes.
Exponential martingales. Complete and incomplete markets
16 Novembro 2020, 10:00 • João Guerra
Exponential martingales of Lévy-type stochastic integrals. Example: exponential martingale of an Itô process.
Basic terminology pf option pricing. Replicating portfolios and self-financing portfolios. Arbotrage opportunity. Complete market.
The two fundamental theorems of asset pricing. The concept of equivalent martingale measure or risk neutral measure. The risk-neutral valuation formula.
The meta-theorem for complete/incomplete markets and arbitrage free markets. Examples of application.
Lévy processes simulation and option pricing
11 Novembro 2020, 11:30 • João Guerra
Simulation of a Gamma process and of the variance Gamma process.
The equivalent martingale measure or the risk neutral measure Q. Option pricing under the equivalent martingale measure and risk-neutral valuation formula. The mean-correcting martingale measure and the change of drift in a Lévy model for option pricing. The m_{new} parameter. How to price an option using an exponential Lévy model and the Monte-Carlo method.
Lévy processes simulation
9 Novembro 2020, 10:00 • João Guerra
Lévy processes simulation.
Simulation of standard Brownian motion.
Simulation of standard Poisson processes: two different methods.
General simulation of Lévy processes by approximating the very small jumps by a Brownian motion and the other jumps by a sum of d independent Poisson processes (compound Poisson process).
General method can be applied to CGMY, Meixner and NIG models.
Example with epsilon=0.001 and with 14 Poisson processes.