In recent decades, stochastic processes with non-Gaussian noise are widely utilized in financial models. The a-stable Levy motion, one type of... Show moreIn recent decades, stochastic processes with non-Gaussian noise are widely utilized in financial models. The a-stable Levy motion, one type of non-Gaussian noise processes, provides robust data ts and events simulations in financial world. Due to "heavy" tails and path jumps property, the a-stable Levy motion modeling becomes extremely popular among financial decision makers and risk hedgers. The a-stable Levy motion, however, usually has neither closed form of probability density function nor the higher moments, which raises implement obstacles. We exhibited distributions of a-stable random variables by different values. In contrast to the Gaussian distribution, the a-stable distribution illustrated the "heavy" tails and shape skews with various parameters. We analyzed jump behaviors along with calculating tails probabilities. We exploited scenario simulation method to solve stochastic differential equations with a-stable Levy motions. Except Euler scheme, we derived two strong convergence 1.0 order numerical schemes via the Wagner-Platen expansion. After we executed the schemes on the Merton Jump-Diffusion model, we roughly proved the convergence order of the schemes. We successfully applied the derived schemes to simulate a sophisticated stochastic volatility model with skewed a-stable Levy motions. With the approximated underlying asset process, we priced an european call option value and visualized implied volatility curve. As the result, we concluded the logarithm of underlying asset follows a skewed distribution rather than a symmetric one. M.S. in Applied Mathematics, May 2015 Show less
