Valuing Volatility and Variance Swaps for a Non‐Gaussian Ornstein–Uhlenbeck Stochastic Volatility Model

Abstract

helping me out when I most needed it. His continuous support and encouragement has been invaluable, and this thesis would not have been realised without his profound understanding of mathematics, finance and supervision.Many thanks go out to my co-authors, Paul C. Kettler, Rodwell Kufakunesu, Dr. Carl Lindberg and Olli Wallin, who have shared their time and knowledge with me. Without their help I would not have accomplished this. I am obliged to Docent Roger Pettersson at Växjö University for his enthusiastic efforts in the first years. The encouragement from Docent Magnus Wiktorsson, Lund University, the opponent at my Licentiate defense, was much appreciated.The friendly and inspiring group of colleagues at the Centre of Mathematics for Applications is responsible for making work a pleasure. All facilitated by the excellent Administrative director Helge Galdal. I am grateful to all the participants at the Fourth Scandinavian Ph.D. workshop in Mathematical Finance for accepting my invitation. I also want to thank the Ph.D. committee at the Department of Mathematics for their unconditional support.On a personal level I want to thank all my friends for their unceasing pursuits to make my days include more than research. Proofreading is an unavoidable but unrewarding task and I am in debt to Camilla Malm for her kindness to carefully and courteously correct my mistakes. I owe everything to my parents for their endless devotion, and my siblings with families for always caring for me. Finally, Christina for all her support and love during the final year.

On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps

“…Portfolio optimization has been studied in [30][31][32]. Paper [33] considers Variance Swaps. Multivariate extensions and the related option pricing results are presented in [34].…”

Exact discrete sampling of finite variation tempered stable Ornstein–Uhlenbeck processes

“…By the arguments in Section 17 of Sato [21], there exists an Ornstein-Uhlenbeck process ¹Y t W t 0º whose marginal has the infinitely divisible distribution with the tempered stable Lévy density (2.5), if the initial state Y 0 is chosen to have the same distribution to the stationary infinitely divisible distribution. In particular, the Ornstein-Uhlenbeck process with inverse Gaussian stationary marginal (˛D 1=2) is often abbreviated to IG-OU and is applied in Benth [5] to stochastic volatility modeling of [3] for volatility and variance swap valuations.…”

“…(For details, see Section 17 of Sato [21], Masuda [18] and references therein.) On the other hand, in practice, non-Gaussian Ornstein-Uhlenbeck processes have been used in mathematical physics under the name of exponentially correlated colored noise, and more recently in financial economics and mathematical finance (for example, Barndorff-Nielsen and Shephard [3,4] and Benth et al [5]). Due to the growing practical interest, many authors have proposed statistical inference methods for non-Gaussian Ornstein-Uhlenbeck processes.…”

Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

“…(We refer the reader to Masuda [10] and the references therein for details about Lévy-driven OU processes.) In particular, OU process with inverse Gaussian invariant law (α = 1/2) was applied in Benth [3] to stochastic volatility modeling of [2] for volatility and variance swap valuations. Let w(z) be the Lévy density of the marginal Z 1 of the background driving Lévy process.…”

“…Also, it was shown in Zhang and Zhang [18] that the transition law is selfdecomposable when the stability index is no less than 1/2. In practice, due to its distributional flexibility and the positivity of sample paths, they have been used in financial economics and mathematical finance (for example, Barndorff-Nielsen and Shephard [2] and Benth et al [3]). …”