Application Research On Moore-Penrose Inverse In Option Pricing

Since the creative study of Black and Scholes[1] in 1973, theory of contingent pricing has been developed rapidly, especially in arbitrage-free pricing theory. This theory shows that the arbitrage-free market hypothesis is equivalent to the existence of martingale measures. If the market is complete, the arbitrage-free market hypothesis is equivalent to the existence of unique martingale measure. Thereby, any contingent of complete market must have a unique arbitrage-free price. However, if the market is incomplete, there may exist many equivalent martingale measures, and many different arbitrage price systems. Therefore, the valuation and hedging issues are not definite in this situation. Many empirical results show that actual market is incomplete, so studying on option pricing in incomplete is much more practical significance.In this paper, we solve a kind of linear stochastic equations by M-P inverse of stochastic processes, and study equivalent martingale measures in incomplete market. The research contents and results as follow.In introduction, we introduce the history of mathematical finance, especially the main research contents, results and hot topics of option pricing theory.In Chapter 2, firstly, we extend Esscher transform to two-parameters by decom-posing Levy processes as the sum of two independent processes, and we get a cluster of (two parameters) probability measures. What's more, a necessary and sufficient con-dition for two-parameters Esscher transform measures to be equivalent martingale mea-sures is given. Secondly, a probability measure Qmo is constructed by mean correcting transform for geometric Levy process model. This paper proves Qmo is an equiva-lent martingale measure if and only if Le'vy processes has Brownian part, and gives Radon-Nikodym formula. Although Qmo can not be equivalent to physical measure for a pure jump Levy process, we show that a European call option price under Qmo is still arbitrage free.In Chapter 3, we extend the conclusion of Dzhaparidze and Spreij[2] which says the M-P inverse of any Rd value, predictable, locally square integrable martingale is predictable, and we prove that the M-P inverse of any Rd×n value, predictable stochas-tic process is predictable. Further, the property and construction of predictable solution of linear stochastic equation are discussed. In Chapter 4, we discuss the diffusion model under the condition which dispersion matrix is not always full rank once again. All equivalent martingale measures in the diffusion model are depicted and the concrete formula of some equivalent martingale measures are given by using of the M-P inverse of dispersion matrix.In Chapter 5, by using of M-P inversewe, we derive general formula of the mini-mal martingale measures in semimatingale model, especially in diffusion model, jump diffusion model and geometry Levy model.