Abstract: An option is defined as a financial contract that provides the holder the right but not the obligation to buy or sell a specified quantity of an underlying asset in the future at a fixed price (called a strike price) on or before the expiration date of the option. This paper examined two approaches for derivation of Partial Differential Equation (PDE) options price valuation formula for the Heston stochastic volatility model. We obtained various PDE option price valuation formulas using the riskless portfolio method and the application of Feynman-Kac theorem respectively. From the results obtained, we see that the two derived PDEs for Heston model are distinct and non-unique. This establishes the fact of incompleteness in the model for option price valuation.
Abstract: This work is devoted to the study of modeling
geophysical time series. A stochastic technique with time-varying
parameters is used to forecast the volatility of data arising in
geophysics. In this study, the volatility is defined as a logarithmic
first-order autoregressive process. We observe that the inclusion of
log-volatility into the time-varying parameter estimation significantly
improves forecasting which is facilitated via maximum likelihood
estimation. This allows us to conclude that the estimation algorithm
for the corresponding one-step-ahead suggested volatility (with ±2
standard prediction errors) is very feasible since it possesses good
convergence properties.