Option Pricing Of Renewal Jump Diffusion Model With Multiple Sources Jumps

In modern finance,the research on option pricing is always a core problem.The option pricing theory focus on two aspects: the one is how to construct new options to meet the needs of the changing market investment;the other is how to determine the value of these increasingly complex options to carry out effective management of risk.The martingale method and the insurance actuarial approach are the main methods of determining the value of options.The martingale method of option pricing is based on the assumption that the financial market is arbitrage-free,equilibrium and complete.When the market is incomplete,the insurance actuarial approach which can overcome the difficulties in finding the equivalent martingale measure is applicable as well.Therefore,we choose insurance actuarial approach in this paper.In this paper,we consider that the fluctuation of real stock price is not continuous in the financial market.The stock prices may be affected by several kinds of major information or occasional events so that it would be caused more than one kind of jump.Therefore,we establish a jump diffusion model influenced by multiple sources of jumps and the jump process is a special kind of Renewal process which is more general than Poisson process.In this paper,we first establish the renewal jump diffusion model with multiple sources of jumps on the stock price process.When the interest rate is a function,the pricing formula of European option is derived by using actuarial approach,which generalize the conclusion of European option pricing with a single Poisson jump diffusion model.The pricing formula of reset option is also obtained,which generalize the pricing formula of reset option based on jump diffusion model with multiple sources of jumps when the expected rate,volatility and risk-less rate are constant.Secondly,we generalize the interest rate to the Vasicek interest rate model and derive the pricing formulas of the European option and Chooser option using the insurance actuarial approach.Finally,under the two conditions of constant interest rate and Vasicek interest rate,the price of the European call option is simulated based on three different modelswhich the stock price without jump,with the single jump and with two sources of jumps.We find that jump process of stock price has a great influence on the value of the option by comparing the numerical results of the three models and analyzing the real financial market.The value of the option is underestimated if we ignore the jump of stock price or consider a single jump of it.At the same time,it also shows that the model is reasonable and the price is reliable.