Option Pricing In Geometric Lévy Processes Model

In the theory of mathematical finance,option pricing is the most fundamental,important and challenging problem.The traditional Black-Scholes model has been very popular.American option,Asian option and various financial innovations in Black-Scholes framework have been studied deeply and extended.However,the asset prices are described by geometric Brownian motion and there are many shortcomings.In order to describe the underlyings' movements better,people are trying to promote the Black-Scholes model to the models based on Lévy processes.The promotion brings many new issues to be resolved,such as:there are many martingale measures since market is incomplete;the complex of model leads to the increase of calculation difficulty and so on.To solve these problems we must use the following mathematical tools deepgoingly: stochastic processes,statistics,time series,stochastic simulation and numerical calculation. In this paper,we discuss the option pricing when the underlyings are described by geometric Lévy processes.Firstly,we discuss the option pricing in subordinated market.If the market is a fair market and there is no arbitrage opportunity,there is a measure under the non-arbitrage constrain such that the summation of the price process and the cumulated dividends is a martingale under this measure,which is called equivalent martingale measure.If the market is a complete market,that is to say any contingent claim is replicable by the portfolio of the underlying asset and the bond,there is a unique equivalent martingale measure such that the price of the contingent claim can be expressed by the mathematical expectation under the equivalent martingale measure.However,in Lévy system,the market is incomplete.In incomplete market,there are many equivalent martingale measures. We can only get a scope of the option's value if we value option by no arbitrage approach alone.Therefore,the first thing we must solve is how to choose and set the martingale measure.We choose the minimum entropy as the target.We give the representations of the minimal entropy martingale measure(MEMM) of subordinated market and the pricing formula of European option under this measure.Research shows that the MEMM of subordinated market can be determined by the MEMM of one-dimensional market.Exchange option has two underlyings.Margrabe derived the exact analytic price formula for the European exchange option in the Black-Scholes framework.Margrabe suggested the valuation problem can be reduced to that of a one-asset option by treating asset two as numeraire.Then price the exchange option under the martingale measure of one-asset market.In this paper,firstly,we investigate the minimal entropy martingale measure and give the density processes for 2-dimensional geometric Lévy processes when the continuous parts of driving processes have correlation coefficient and the jump parts of diving processes have function relation.Secondly we validate whether Margrabe's conclusion is proper or not in our model.Research shows that Margrabe' conclusion is not estabilished when the model has jumps.Many foreign currency options fall into the class of multi-state models.We discuss the foreign currency option pricing under the assumption as exchange option that the continuous martingale parts of driving processes have correlation coefficientρ;the jump parts of driving processes have mapping relations.Three kinds of foreign currency options are studied.We give the integro-differential equations and FFT approximate calculate method for these option prices.Finally,a special example shows the influence of the relevance of the underlyings to option prices.The convertible bond carries two main components:the component as potential bond and the component as call option.We find this call option can be regarded as an American exchange option and this American exchange option can be converted to American put by the method of measure transformation.Under Lévy setting no explicite analytical expression is available.But there are many studies about the approximation of American put when the underlyings are particular Lévy processes.We can make use of these results to CB call and obtain the approximation.As an example,we give the approximation representations of CB call in double exponential jump diffusion model.Finally,We compare the Asian options with three kinds of averaging procedures by Monte Carlo simulation and numerican illustration.We demonstrate that European Asian options with harmonic averaging behave better than Asian options with arithmetic and geometric averaging procedures for the stock with remarkable fluculation. Furthermore,approximation method fbr the valuation of harmonic average options and numerical illustrations are also given.