论文标题
延迟的跳跃模型 - 选项定价和对数Euler-Maruyama计划
Jump Models with delay -- option pricing and logarithmic Euler-Maruyama scheme
论文作者
论文摘要
在本文中，我们获得了延迟随机微分方程的解决方案的存在，独特性和积极性。然后，将该方程式用于建模金融市场中风险资产的价格变动，并与套头衫投资组合一起获得了欧洲期权价格的黑色chcholes公式。期权价格通过使用傅立叶变换技术在最后一个延迟期进行分析评估。但是通常，期权价格没有分析表达。为了以数值评估价格，我们然后使用蒙特卡洛方法。为此，我们需要使用跳跃模拟延迟的随机微分方程。我们提出了一个对数Euler-Maruyama计划，以近似方程式，并证明所有近似值保持正面，该计划的收敛速度被证明为$ 0.5 $。
In this paper, we obtain the existence, uniqueness and positivity of the solution to delayed stochastic differential equations with jumps. This equation is then applied to model the price movement of the risky asset in a financial market and the Black-Scholes formula for the price of European options is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. But in general, there is no analytical expression for the option price. To evaluate the price numerically we then use the Monte-Carlo method. To this end, we need to simulate the delayed stochastic differential equations with jumps. We propose a logarithmic Euler-Maruyama scheme to approximate the equation and prove that all the approximations remain positive and the rate of convergence of the scheme is proved to be $0.5$.
