Ahighly Nonlinear Model Of Spot Interest Rate And The Euler-Maruyama Approximations

This thesis gives a framework on the spot interest rate，including the basic theory ofspot interest rate，the importance of research and the development process. Manystochastic differential equations have been used to describe the spot interest rates process.Some of them hold "good" properties, for example, the linear growth condition and (local)Lipschitz condition, which guarantee the existence and uniquenese of the solution toequations. Though some equations don’t satisfy the linear growth condition or (locally)Lipschitz continuous, as CIR model, the existence and uniquenese of the solution andother analytical properties have been proved. The properties of the solution ofmean-reverting-process, including existence and uniquenese, boundedness, convergenceof EM numerical solution, have showed by Wu. Moreover, there are also some nonlinearmodels, for example, Ait-Sahali’s model, whose drift and diffusion coefficients do notobey the classical linear growth condition. The analytical properties of these models havealso been examined widely.Based on CIR model and mean-reverting-process, this thesis will propose thefollowing highly nonlinear stochastic differential equation model on spot interest rates,Obviously，this model include CIR model and mean-reverting-process. We mainly dothe following two work，Firstly, we choose the data of short-term state debt yield from1Jan.2002to17Feb.2012published by The Federal Reserve System to estimate the parameters of the equation.The result show thatα﹥1，β﹥1Secondly, with the result of the parameters estimate, this equation holds a unique global nonnegative solution, and examines the stochastic boundedness of the solution. Inparticular, we show that the Euler-Maruyama approximations converge to the true solutionin probability for sufficiently small stepsize. The convergence result justifies clearly thatthe Monte Carlo simulations based on the Euler-Maruyama scheme can be used tocompute the expected payoff of financial products e.g. options.