Problem statement: Modeling of the Dow Jones Industrial Average is frequently attempted in order to determine trading strategies with maximum payoff. Changes in the DJIA are important since movements may affect both individuals and corporations profoundly. Previous work showed that modeling a market as a random walk was valid and that a market may be viewed as having the Markov property. Approach: The aim of this research was to determine the relationship between a diverse portfolio of stocks and the market as a whole. To that end, the DJIA was analyzed using a discrete time stochastic model, namely a Markov Chain. Two models were highlighted, where the DJIA was considered as being in a state of (1) gain or loss and (2) small, moderate, or large gain or loss. A portfolio of five stocks was then considered and two models of the portfolio much the same as those for the DJIA. These models were used to obtain transitional probabilities and steady state probabilities. Results: Our results indicated that the portfolio behaved similarly to the entire DJIA, both in the simple model and the partitioned model. Conclusion: When treated as a Markov process, the entire market was useful in gauging how a diverse portfolio of stocks might behave. Future work may include different classifications of states to refine the transition matrices.

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In this paper, we present the white noise methods for solving linear stochastic differential equations of anticipating type. Such equations may be solved using the S-transform, an important tool within the white noise theory. This approach provides a useful remedy to the fact that the Itô theory of stochastic integration is inapplicable to such equations. The technique is presented with several examples, including an application to finance.

Weather derivatives are becoming prominent features in multiasset class portfolios of alternative risk. The pricing of these securities is nonetheless challenging since it requires an incomplete market framework. We discuss pricing formulas for temperature-based weather derivative options, constructing mean reverting stochastic models for describing the dynamics of daily temperature with a constant speed of mean reversion for three cities. Truncated Fourier series are used to model the volatility, and assuming a constant market price of risk, we introduce a novel approach for estimating this constant, using Monte Carlo simulations.

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