European Option Pricing With Transaction Costs Under The Environment Of Lévy Jump

In recent years,derivatives pricing models with transaction costs are attracting more and more attentions.It is also of importance to select asset pricing models.This article focuses on European option pricing with transaction costs,selecting three different asset pricing models to make it closer to the real situation of the financial markets.First,we select Lévy jump-diffusion model with constant parameters.Using Ito lemma on Possion jump process and combining with arbitrage pricing theory and dividend discount principal,we obtain asset price's formula.Under the no-arbitrage principle condition,using the ?—hedging strategy and the discretization of time,we obtain European option price's explicit formula,considering transaction costs in the each little period of time.Second,considering Markov-modulated Lévy jump-diffusion model,we define the process of economic states' changes in the economic market as a Markov chain.With the changes of economic states,asset pricing models can modulate in different,model-s.Because financial market is incomplete,we can use Esscher transform to find an equivalent martingale measure that we can prove it is minimal in some sense.Using Ito lemma on Markov-modulated Lévy jump-diffusion process,martingale theory and dividend discount principal,we get asset price's formula.Under the no-arbitrage prin-ciple condition,we make full use of Leland(1985)thought about option pricing with transaction costs to the Markov-modulated Lévy jump-diffusion model.At last,we obtain European option price's explicit formula with transaction costs.Finally,we focus on Lévy jump-diffusion model with stochastic interest rate and stochastic volatility.Apparently,we increase the number of parameters in the model.Although the model become complicated.it is closer to the real financial market.Com-bined with Ito lemma on Lévy jump-diffusion process and the ?-hedging strategy,we can get European option price's implicit formula with transaction costs.Namely,we have a partial differential equation about European option price with transaction costs.Under the same parameters condition,using the initial conditions and boundary conditions,we perform simulations about European option price with no transaction cost,and with transaction cost,respectively.We use iterative method and the finite different method for the numerical analysis.