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

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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.