Pricing Barrier Options In The Two-factor Stochastic Volatility Jump-diffusion Model

With the development of the financial industry,a large number of new financial products and financial means are emerging,which makes the international financial market more vigorous.Option,as one of the important financial instruments for risk management,has the function of controlling investors risk to a certain extent,thus creating its indispensable position in the financial society.With the development of option market,Barrier options are popular among new options.Compared with conventional options,they are cheaper and more flexible,and their risks are better controlled.Therefore,the research on the pricing of the option is of practical significance.After the price of the underlying asset reaches the prescribed value,barrier options are more popular.The state of an option can be divided into knock-out option and knock-in option.When the underlying asset reaches the specified value,the option loses its meaning,which is called knock-out option.When the underlying asset reaches the specified value,the option becomes effective,and this option is called knock-in option.The classical Black-Scholes option pricing model is the earliest model in option pricing,and it has many idealized fake.For example,short-term interest rate is a constant without risk and no change,and the price fluctuation of underlying assets follows a fixed law.However,the financial environment is complex and unstable.If the assumption is too ideal,it will certainly affect the actual value of the model.Nowadays,more and more empirical studies show that there are many factors that can affect the underlying asset return.They will more or less lead to the instability and irregularity of the underlying asset returns,so volatility can not be stable as a constant.In order to make the research results closer to the real economic life,this paper uses a more comprehensive model to price barrier options.In the complex and changeable financial market environment,considering the volatility of the underlying assets caused by emergencies or major events in the market,this paper uses jump-diffusion model to add an independent jump term to the underlying assets,and then takes into account the impact of macroeconomic and financial trends and other short-term risks on the return of the underlying assets,and proposes to adopt long-term volatility.The jump-diffusion model of two-factor stochastic volatility is designed by replacing single constant volatility with short-term volatility(affected by short-term risk).Based on this model,the relevant lemmas needed in the paper are introduced and proved.The multi-dimensional characteristic function is deduced by mathematical induction,and then the measure transformation and mathematical induction are used.The pricing formulas of European barrier options with multiple discrete time points are obtained.Then,the influence of fluctuation rate on the price of barrier options is explored by numerical examples,and relevant conclusions are drawn.Compared with Black-Scholes model,this model is more suitable to show the stochastic characteristics of volatility,better to describe the economic law of the market and closer to the actual economic life.After that,we can see the impact of long-term and short-term volatility and other factors on the underlying assets under several different parameters of the model.We can conclude that the volatility parameters of both have different degrees of impact on the price of barrier options,and the price volatility of barrier options is greater under short-term volatility.Therefore,in the actual financial market.,investors should try their best to consider all-round factors.The more specific the factors affecting option pricing in the model,the more applicable the option pricing formula derived under the model is to the real economic market.Using the option pricing model designed in this paper to study the pricing problem of barrier options,it can more adapt to the needs of investors in all aspects and control the premise of their investment risk.The next step is to increase its earnings as much as possible,which brings a new direction for financial investors.