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.