# Finansielle modeller med Lévy-processer

Mikael Oskar Engelund

Studenteropgave: Kandidatafhandlinger

## Abstrakt

In this paper I will cover the exponential Lévy model. The paper starts off by introducing the Lévy process and some of its most important properties and how it can be characterized. The focus will be on the difference between Lévy processes with finite versus infinite activity, as infinite activity gives rise to numerical problems, and hard characteristics. Thereafter I will introduce the exponential Lévy model, were I will derive ways to price call options with numerical methods. The first way to calculate prices will be with Fourier transforms where I show that prices can be calculated as an integral. The second way is where I show how to derive a partial integro differential equation for the pricing function, and then explaining how to solve it numerically. I then compare it to the Black Scholes model from which the exponential Lévy model is an expansion off. I look at which is most realistic and which is better to use as a financial institution as both precision and time is a valuable factor. Then I look at the implied volatility when the true prices are generated from the exponential Lévy model, to see if the model could fit real world data. Then I make a hedging experiment to see how bad it would be if the Black Scholes delta hedging method was used when the true price process is as described in the exponential Lévy model. Here I derive a stochastic variable for the profit and loss and simulate the experiment to see how bad the distribution looks when the Lévy process is of infinite activity. It seems like it does not go as bad as you could fear, so you can find empirical densities to calculate risk measures as Value at Risk and expected shortfall. Lastly, I discuss what you could use this model for in practice, when selling call options, and I relate to another way off expanding the Black Scholes model called hestons Stochastic volatility model.

Uddannelser Other, (Kandidatuddannelse) Afsluttende afhandling Dansk 2019 97 Anders Rønn-Nielsen