Stochastic Modelling and Techniques in Financial Management

Aim

It is actuaries are increasingly using the term ‘stochastic modeling’. It means the most common interpretations of a stochastic model tends to be something that allows for uncertainty of future outcome, a very complicated calculation engine with lots of technical math’s involved something that must be run several hundred or thousand times.

Individuals and corporate are facing the problem in day-to-day life related to maximize the profits. They can either spend it immediately, or save it, or partially spend and partially save. In either of the possibilities, they must decide how to do that process. In the case of saving, they prorogue their immediate consume in favor of investment and also most of the individuals now-a-days participated in some type of retirement plan, either through their previous employer or in a self-business plan. The reason is that they want to get best financial security for them after retirement of their service. For this reason, most of the senior citizens and companies are invested huge money in the stock market. Changes of share prices on every day make it more volatile and difficult to predict the future price. When purchasing a stock, it does not guarantee anything in return. Thus, it makes stocks risky in investment, but investors can get high profit in return. When investors are taking wrong decision in choosing the counters, it may end up in capital loss. The behavior of stock market returns has been deeply discussed over some years. Stock market prediction is the art of trying to evaluate the future value of a company stock or other financial instrument traded on a financial exchange. There is no single method can accurately predict changes in the stock market every day. So, there are many stochastic models introduced many papers in predicting share prices and they are birth and death processes, random walk models, Markov chain, hidden Markov model (HMM) and Brownian motion model. In this paper, we discussed about the role of application perspective discussions on stochastic models in the field of finance.

Methods

Markov Chain

Consider a stochastic process {Xn, n=0, 1, 2, …} that takes on a finite or countable numbers of possible  states with some known probabilities Pij, If the chain is currently in state si, then it moves to state sj at the step with a probability denoted by pij, where Pij is the probability of moving from state i to j. If Xn =i, then the process is said to be in state i at time n. Suppose we say that whenever the process is in state i,  there is a fixed probability Pi j that it will next be in state j. 

Suppose say that,

P{Xn+1 = j/Xn = i, Xn-1=  in-1, …, X1 = i1, X0 = i0} = Pij   For all states i0, i1,…,in-1, i, j and all n ≥ 0. Such type of stochastic process is known as a Markov chain.

Markov Chain is a time-indexed random process with the Markov property. Having the Markov property that means it gives the present state, the future states are independent of the past state, i.e. that for all t, the process [X(t + s)-X(t)|s ≥ 0] has the same distribution as the process [X(s) | 0 ≤ s ≤ t]

Hidden Markov Model

An HMM is a doubly stochastic process in which an underlying stochastic process is unobservable, which means that the state is hidden. This can only be observed through another stochastic process that produces a sequence of observations. Thus, if S = {Sn, n=1, 2,...} is a Markov process and F = {Fk , k=1, 2, …} is a Function of S, then S is a hidden Markov process or hidden Markov mode that is observed through F.

Modeling to Forecast the Future

Decisions are made based upon judgments that are influenced by analysis and instincts. Key to analysis is the use of historical and current performance characteristics as a basis for forecasting future performance, understanding that the future is difficult to predict. Consequently, the future might best be viewed in relation to statistically informed probabilities.

Deterministic Modeling: Contracting for Professional Services

It is important in financial models that they are a critical element for decision making and are routinely created in the process of budgeting as well as new project planning throughout health care. Models can be deterministic or stochastic. Deterministic models assume perfect predictability, whereas stochastic models accommodate randomness and probabilities to reflect uncertainties of future inputs.

Datasets

Source: International Journal of Scientific and Innovative Mathematical Research

From the above set of values, we observed that the probability of decrease to decrease, decrease to increase, increase to decrease and increase to increase values having same probability in 7th and subsequence days. From this empirical evidence, Markov chain is more useful and powerful to predict next day values; it is not reliable for subsequent days. In my observation all the above mentioned three types of Stochastic models were partially supported to prediction of the stock market, even some of the authors were strongly agree with the stock market prediction is not reliable to the investors. Some of the authors said that, their prediction method was more reliable for short term investors only. The above empirical result also shows that the Markov chain analysis is powerful tool for predict next day value only, which is not useful for lateral days.

Stochastic model for financial datasets

Analysis

Use of the Arbitrage Pricing Theory, (APT) also known as Arbitrage Pricing Model, APM, serves as a generalization of the single factor CAPM to a multifactor model. The idea behind the APT is that the returns vary from their expected values due to unanticipated changes in production, inflation, term structure, and other economic factors. In the multifactor model it is supposed that the return on an asset is explained in terms of a linear combination of more factors or indexes. Note that in CAPM, the expected return on an asset is a linear function of the expected market return only. The development of APT is based on the assumptions of an efficient market, see I.9. A technical realization of APT uses two popular statistical methods, regression analysis and factor analysis.

Techniques

The techniques based on the Markov structure of data. It takes into account a given suitably labelled tree structure already in the course of simulation. Moreover, the sequential procedure can be adopted for an iterative refinement of the discrete representation of the underlying continuous data process. The main advantage of the aforementioned common optimization algorithms and software tools is that they are easy to use and well-tested for mid-size problems.

However, the main disadvantage is that they are inefficient for large-scale stochastic programs as they do not utilize their special structure.

Interpretation

This system of financial markets is to facilitate mutually beneficial inter temporal exchanges. Financial markets promote these exchanges by organizing trading in a variety of financial securities. The financial market influences personal corporate financial lives and the economic health of a country. Finance is the essential, which helps in the formation of new businesses and to grow the current business. Financial trends also define the state of the economy on a global level, so central banks can plan appropriate monetary policies.

Conclusion

The behaviour of stock market returns is a common issue to the theory and practice of asset pricing, asset allocation, and risk management. There are various techniques implemented for the prediction of stock market. In this study we see that a complete theoretical survey of all the stock market prediction techniques are used for the different types of stochastic models that are given. Cognitive Biases in decision Making even with robust historical data, logical financial projections, and well-considered strategy, decisions can be shortcut by cognitive biases and heuristics, influencing the perception of risk.

A comprehensive discussion of how cognitive biases affect decision-making is beyond the scope of this article, but because an awareness of their existence is needed to recognize and mitigate them, three common cognitive biases are highlighted in brief. When a preconceived valuation for an activity or project is influenced by experiences and perceptions, anchor bias can colour an individual’s or an organization’s financial assumptions. Financial forecasts supporting an investment in interventional radiology facilities may be rejected based upon experiences during an era with less sophisticated technology, consideration for operational efficiency, or prioritized patient comfort. Anchoring to past valuations may lead a hospital administrator to reject a more favourable forecast for a new radiology procedure suite in favour of a less financially favourable surgery suite, simply because surgery has generated larger profits in the past.

References

1. N. Keyfitz and H. Caswell, Applied Mathematical Demography, 3rd ed., Springer (2005).
2. H. R. Lerche, Boundary crossing of Brownian motion, Berlin, Springer-Verlag (1986).
3. K. Levenberg, A Method for the Solution of Certain Non-Linear Problems in Least Squares, The Quarterly of Applied Mathematics 2: 164–168 (1944).
4. D. Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters, SIAM Journal on Applied Mathematics 11 (2): 431–441 (1963).
5. C. H. Skiadas, Innovation Diffusion Models Expressing Asymmetry and/or Positively or Negatively Influencing Forces. Technological Forecasting and Social Change 30, 313- 330 (1986).
6. C. H. Skiadas, Life Expectancy at Birth, Estimates and Forecasts in the Netherlands (Females), arXiv:1112.0796v1 [q-bio.PE] (4 Dec 2011)
7. C. H. Skiadas and C. Skiadas, A modeling approach to life table data, in Recent Advances in Stochastic Modeling and Data Analysis, C. H. Skiadas, Ed. (World Scientific, Singapore), 350–359 (2007).
8. C. H. Skiadas and C. Skiadas, Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Forms, CRC / Taylor & Francis (2008).
9. C. H. Skiadas, C. Skiadas, Comparing the Gompertz Type Models with a First Passage Time Density Model, in Advances in Data Analysis, C. H. Skiadas Ed. (Springer/Birkhauser, Boston), 203-209 (2010).
10. C. Skiadas and C. H. Skiadas, Development, Simulation and Application of First Exit Time Densities to Life Table Data, Communications in Statistics 39, 444-451 (2010).
11. C. H. Skiadas and C. Skiadas, Exploring life expectancy limits: First exit time modelling, parameter analysis and forecasts, in Chaos Theory: Modeling, Simulation and Applications, C. H. Skiadas, I. Dimotikalis and C. Skiadas, Eds. (World Scientific, Singapore), 357–368 (2011).