Development of the algorithm for stochastic modeling and option pricing of risky bond

在三年的时间里，我们开发了风险债券的随机建模和期权定价算法。在第一年，考虑使用适当的带有边界噪声的随机抛物系统进行期限结构建模。在找到无套利机会的充分条件后，我们解决了不完全市场中均值-方差最优控制问题。利用非线性估计理论研究了赫斯顿随机波动模型的滤波问题。为了解决随机波动过程的估计问题，我们采用了随机时间变化方法。推导出滤波的基本方程为Zakai方程，并借助分裂法提出了其数值实现算法。数值模拟研究表明了该方法的优越性。在第二年，我们考虑构建最优投资组合，使最终的电力效用最大化。为了管理投资组合，我们控制银行账户的金额和几种不同期限的债券。通过带边界噪声的抛物线型无限维因子模型，给出了债券价格的动态变化。利用动态规划方法，得到了不完全市场下的最优投资组合。去年，我们利用美国国债数据考虑了抛物线型因子模型的参数辨识问题。首先对产量数据进行插值，估计出系统噪声的协方差核。利用该估计，得到了双曲型和抛物型模型的未知参数的修正极大似然估计。最后，通过对所得结果的比较，可以看出抛物线因子模型的有效性。我们提出了一个关于去年成果发展的新项目。

During three years, we developed the algorithm for the stochastic modeling and option pricing for risky bonds.In the first year, consider the term structure modeling by using an appropriate stochastic parabolic systems with boundary noises. After finding a sufficient condition for the no arbitrage opportunity, we solve the mean-variance optimal control problem in the incomplete market. We also study the filtering problem for the stochastic volatility model of Heston by using the nonlinear estimation theory. To solve the estimation problem for the stochastic volatility process, we use the random time change method. The derived basic equation for the filtering is the so-called Zakai equation and its numerically realized algorithm is proposed with the aid of the splitting-up method. Some numerical simulation studies are demonstrated to show the advantage of the proposed method.In the second year, we consider the construction of optimal portfolio for maximizing a power-utility at the final time. For managing the portfolio, we control the amounts of the bank account and several bonds with different maturities. The dynamics of bond price is given through the parabolic type infinite-dimensional factor model with boundary noises. By using the dynamic programming approach, we obtain the optimal portfolio in the incomplete market.In the last year, we consider the parameter identification problem for the Parabolic type factor model by using the US treasury bond data. First interpolating the yield data, we can estimate the covariance kernel of the system noise. With the aid of this estimate, the modified maximum likelihood estimates of the unknown parameters are obtained for the hyperbolic and parabolic models. Finally, comparing the obtained results, we can show that the parabolic factor model works well. We proposed a new project concerning for the development of the last year results.