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.
We propose a quasi-Monte Carlo (qMC) algorithm to simulate variates from the normal inverse Gaussian (NIG) distribution. The algorithm is based on a Monte Carlo technique found in Rydberg [13], and is based on sampling three independent uniform variables. We apply the algorithm to three problems appearing in finance. First, we consider the valuation of plain vanilla call options and Asian options. The next application considers the problem of deriving implied parameters for the underlying asset dynamics based on observed option prices. We employ our proposed algorithm together with the Newton Method, and show how we can find the scale parameter of the NIG-distribution of the logreturns in case of a call or an Asian option. We also provide an extensive error analysis for this method. Finally we study the calculation of Value-at-Risk for a portfolio of nonlinear products where the returns are modeled by NIG random variables.
Abstract. We develop and apply a numerical scheme for pricing options for the stochastic volatility model proposed by Barndorff-Nielsen and Shephard. This non-Gaussian OrnsteinUhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To price numerically options with respect to this risk neutral measure, one needs to consider a Black & Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black & Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.
We derive derivative-free formulas for the Delta and other Greeks of options written on an asset modelled by a geometric Brownian motion with stochastic volatility of Barndorff-Nielsen and Shephard type. The method applies the Malliavin calculus in Wiener space which moves differentiation of the payoff function of the option to a random weight function. Our method paves the way for simple Monte Carlo approaches, illustrated by several numerical examples.
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