Parameter Estimation/Financial Applications For Several Classes Of Mean-Reverting Processes

In recent decades,due to the periodicity of the mean-reverting process,it has been widely used to characterize the seasonal and periodic phenomena in finance,economy,physics and other fields,such as the classical OU(Ornstein-Uhlenbeck)process,the earliest it is a model used in physics to calculate the velocity of massive Brownian particles under the influence of friction.Because of its characteristics of stationarity,Gaussianity,Markovianity,and mean reversion,it is widely treated in operations research management,finance and stochastic analysis theory.When the OU processes describe the evolutions of interest rates,commodity futures,and inventory,they need to be ensured non-negativity,and sometimes need to be taken into account the phenomenon of random coefficients and time-dependent coefficients.Therefore,the studies of the reflected OU process,the threshold OU process,and the time-dependent OU process are interesting and useful.Compared with the classical OU process,the reflected OU process and the threshold OU process maintain the property of mean reversion,in addition,they also have the characteristics of being limited to an interval(reflected OU)and random coefficients(threshold OU).There are five chapters in this dissertation.Chapter 1 explains the research background and the motivation of this dissertation,as well as the structure of this dissertation.In Chapter 2 and Chapter 3,this dissertation mainly aims to estimate the parameters of the reflected OU process and the threshold OU process based on the discrete observations obtained in practice.For the reflected OU process,using the spectral representation of the transition density and the ergodicity of the reflected OU process,this dissertation estimates the drift and diffusion coefficients simultaneously,which solves the problem in [1] that the volatility cannot be estimated simultaneously.For the threshold OU process,using the ergodic theorem,this dissertation solves the existence and uniqueness of the solution of the parameter estimators.The estimators are strongly consistent and asymptotically normal.For the mean-reverting process with the timedependent drift,notice a special kind of process – the bridge process.The well-known Brownian bridge process is the conditional Brownian motion from the initial point a to the end point b in the finite time interval [0,T].Such a Brownian bridge can characterize the phenomenon that the stock price is fixed at a value on the expiration date of an option,as well as the Kyle-Back model of insider trading.For the existing theoretical research,they are all based in a finite time interval.Chapter 4 aims to construct a general bridge process in an infinite time horizon through the stochastic differential equation,which is called the infinite time bridge,by giving the growth conditions that the drift term needs to be satisfied.From stochastic differential equations to stochastic partial differential equations,notice that the solution of the linear stochastic partial differential equation can be represented by the Green function,when the Green function does not depend on time,if the appropriate function is specified,the solution is a stationary stochastic process that depends on two parameters,which is the so-called OU sheet.The two-parameter random field is different from the simple multi-factor model with the uncertainty caused only by the Brownian motion factor in which the random term associated with each parameter is a different random factor,so it can enrich the uncertainty of the underlying process.Chapter 5 mainly studies the application of such two-parameter random fields for interest rate spread options.Chapter 6 concludes the dissertation and discusses the future research.Therefore,this dissertation mainly studies some problems in the continuous-time mean-reverting process,including the parameter estimation of the reflected OU process and the threshold OU process,and the property of the mean-reverting process with time-dependent drift.As an extension,this dissertation also investigates the pricing problem of interest rate spread options under the random field LIBOR market interest rate.Among them,a special case of random fields is the OU sheet.The main contributions of this dissertation are as follows:In Chapter 2,it is assumed that a reflected OU process is observed at discrete times,and the time step can take any values.Based on the ergodic theorem and the spectral representation of the transition density,the moment estimation of its drift and diffusion terms is proposed,which solves the problem of [1] that the volatility parameter can not be estimated through their ergodic estimation.Compared with the existing method to estimate the volatility based on continuous observations,the proposed estimators are more accurate.Chapter 3 investigates the strong consistency and asymptotic normality estimation of the threshold OU process based on discrete observations.Numerical simulations show the accuracy of the results based on both high-frequency observations and lowfrequency observations.The advantage of the method in this dissertation is that the proposed estimators can be easily calculated,and the existence and uniqueness of the solution can be solved.Chapter 4 studies the construction of the bridge in an infinite time horizon.The infinite-time bridge is a special example of the mean-reverting process with the timedependent drift.From a theoretical point of view,it is interesting to find that when the time-dependent drift term of the mean-reverting process satisfies some growth conditions,it becomes a bridge process in an infinite time interval.It also develops the existing theory about the finite-time bridge process.Regarding the theoretical properties of the infinite-time bridge,Chapter 4 studies its local time approximation and asymptotic behaviour.Chapter 5 derives the option pricing formula of interest rate spread options under the random field framework.For spread options with a zero strike price,an analytical pricing formula is given,which can be seen as an extension of the Margrabe formula in [2] under the term structure of random field interest rates.For spread options with non-zero strike prices,an approximation of the pricing formula is given.From the numerical results of specific examples of OU sheet and Brownian sheet,it can be seen that the pricing formula is accurate and stable.