Inference for Stochastic Processes and Applications

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Volatility is a measure of the amount by which an asset price is expected to fluctuate over a given period,*and the greater the volatility, the higher the risk. Filtering and recursive parameter estimation for stochastic*volatility (SV) models have many applications in financial decision making. SV models are commonly used *in financial applications as their dynamics are flexible enough to model observed asset and derivative prices. *Many applied decision making problems such as portfolio selection and option pricing are recursive in nature.**Inference for the volatility plays an important role in option pricing applications. Constant volatility has been *assumed in the basic Black-Scholes-Merton approach to option pricing. Significant correlation among the *squared values of the log returns points at a need to model beyond this constant volatility. As a consequence, *the world of nonlinear generalized autoregressive conditional heterocedastic (GARCH) modeling together *with nonlinear stochastic volatility models has emerged.**Practitioners use random coefficient volatility (RCV) models in finance and economics. Nonlinear GARCH *models have been very popular and effective for modeling volatility dynamics in many asset markets. *We have developed a data driven method for option pricing and demonstrated the superiority of GARCH/SV *option pricing models using real data.**(a) In this proposed research, we study inference problems for stochastic processes such as GARCH models,*recently proposed ACP ( autoregressive conditionally Poisson)/RCV models, duration models, integer *valued models, nonlinear stochastic volatility models, circular time series models and semimartingale models.*The unified method of estimating function theory for continuous time as well as for discrete time models will*be used to obtain joint maximum informative recursive estimates/filtered estimates and will be applied to *inferences from option prices and to inference based on censored data.**(b) There has been a growing interest in stochastic processes with infinite variance, for example Fama *(Nobel Price Winner for econometric modelling this year) studied estimation and prediction for infinite regression*models. This is due to the inherent challenge and theoretical interest provided by the non-normal stable laws as *well as the possibility that the processes constructed from these laws may be appropriate models for many *diverse phenomena. In practice, any time series which exhibits sharp spikes or occasional bursts of outlying *observations suggests the possible use of a model with stable errors having infinite variance. For time series *models with infinite variance stable errors, for which closed form expressions for the density are not available *and hence the maximum likelihood estimate cannot be obtained.We have used combined sine and cosine *estimating functions to study estimation. Recently I developed a maximum informative recursive method *and applied to financial data. In this proposal, I will also study maximum informative filtering/joint recursive *estimation for infinite variance processes using transformation based estimating functions.**(c) One of the problems with the implementations of stochastic interest rate models was that the theoretical *model prices did not fit the existing observed market prices of bonds. The reason is that at any time there is a*vector of current bond prices, and a model with a few parameters simply cannot fit the entire set of bond prices.*It is useful to have an interest rate model (with time varying parameters) which can fit the observed prices more*accurately. We use the nonparametric estimation method to study bond prices based on recently proposed *interest rate models with time varying parameters.