Fractional Brownian motion and weather derivatives

Abstract

The fractional Brownian motion (fBm) has recently drawn a lot of attention and has been studied in several directions, such as stochastic integration, stochastic differential equations, financial applications, and solutions for many other theoretical problems. This Master thesis focuses on investigating the financial applications which is built on the fBm platform, and it studies weather deriva- tives as a classical example. In the first part of this thesis, Wick Itô Skorohod (WIS) integrals are introduced as the stochastic integrals of the financial model based on fBm. To establish a parallel fractional financial model to the well-known Black-scholes model, which is driven by the classical Brownian motion, a fractional version of Itô formula and the Girsanov theorem are presented. The solution of the fractional Ornstein-Uhlenbeck equation is also given in this part. In the second part of this thesis, the weather market is studied in two aspects: on one side, the stochastic model for temperature-based derivatives and its analytical solutions for pricing; and on the other side, data analysis from five Norwegian districts and the Monte Carlo pricing. This thesis tries to give an overall understanding of fBm from the theoretical interest to financial model and real-world significance.