Option pricing with a natural equivalent martingale measure for log-symmetric levy price processes

This thesis examines the pricing of options when the stock price follows a log-symmetric Levy process. Models in continuous time and discrete time are considered. We identify situations when there is an equivalent change of measure that preserves the Levy property, symmetry and the family of symmetric distributions of the returns (log-returns if discrete time), and makes the discounted price process into a martingale. We call such measures natural equivalent martingale measures. In continuous time, when a natural equivalent martingale measure exists it is unique. It can be obtained by changing only the location or the scale parameter of the symmetric distribution if the Brownian component in present or absent, respectively, in the Levy process. The analogous natural equivalent martingale measure in discrete time always exists but not unique. It can be obtained by changing the location and the scale parameters of the symmetric distribution. Option pricing with natural equivalent martingale measure is arbitrage-free. We apply this approach to obtain new and elegant option pricing formulae for log-symmetric variance gamma and log-symmetric normal inverse Gaussian models.