applications of various stochastic models in financial ...

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Special Issue 3, July 2015, PP 852-857, ISSN 2347-307X (Print) www.arcjournals.org

APPLICATIONS OF VARIOUS STOCHASTIC MODELS IN FINANCIAL PREDICTION R. Sasikumar

A. Sheik Abdullah

Assistant Professor Department of Statistics Manonmaniam Sundaranar University Tirunelveli,Tamil Nadu, India.

Research Scholar Department of Statistics Manonmaniam Sundaranar University Tirunelveli,Tamil Nadu, India.

Abstract: The purpose of a 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. Stochastic models like birth-death process, random walk model, Markov chain, hidden Markov model and Brownian motion have a vital role on prediction, finding steady state solutions and forecasting. This paper attempts to provide a comprehensive review of stochastic models especially theoretical as well as application perspective in finance, and also it provides an overview of a number of currently proposed methods in stochastic models namely birth-death process, random walk model, Markov chain and hidden Markov model to finance. Keywords: birth-death process, random walk, Markov chain, hidden Markov model, finance, stock market.

1. INTRODUCTION 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 mentioned possibilities, they must decide how to do that process. In the case of saving, they prorogue their immediate consume in favour 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 their 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 any guarantees are anything in return. Thus, it makes stocks risky in investment, but investors can get high profit return. When investors are taking wrong decision in choosing the counters, it may end up in capital loss. The behaviour 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.

2. STOCHASTIC MODELS IN FINANCE In real life situation, there are so many techniques have been used to predict financial market, such as data mining, artificial neural network technique, machine learning techniques and various types of stochastic models and so on. This paper exclusively discusses different types of stochastic models in finance especially focusing on stock market. ©ARC

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Applications of Various Stochastic Models in Financial Prediction

2.1 Birth-Death process The birth-death model generalizes the random walk model in the sense that the probability of no transition in a given time interval is not null. The birth-death process is a continuous time Markov chain with states 0, 1, 2, …, in which transitions from state i can only go to either i+1 or state i-1. That is, a transition either causes an increase in state by one or decreases in state by one. For a birth-death process, we define the following transition rates from state i:

λi = vi pi (i + 1)

µ i = vi pi (i − 1)

Thus, λi is the rate at which a birth occurs when the process is in state i, and µi is the rate at which a death occurs when the process is in state i. The sum of these two rates is λi + μi = νi, which is the rate of transition out of state i.

2.2 Random Walk The random walk theory is the occurrence of an event determined by a series of random movements. Let {Xn, n = 0, 1, 2, 3, …} be a sequence of independent random variables taking the integer values only and Sn = X1 + X2 + … +Xn (n = 0, 1, 2, …). Then the sequence {Sn} is a Markov chain [Basu, 2003], whose transition probabilities are given by, (m) Pij = P ( S m +1 = j | S m = i ) = P ( X m +1 = j − i ), i, j = ..., − 2, − 1, 0, 1, 2, ... The chain represents a random walk of a particle along a straight line, the magnitude of ‘jump’ at time n being given by the random variable Xn. If X0 denotes the initial position of a particle then its position after n jumps (at time n) is given by Sn. when Xn’s are also identically distributed, Pij(n) = Pj-i where Pj = P (Xn = j). Then we have a homogenous random walk. Such random walks occur in fluctuation theory (sums of discrete or continuous random variables). The sequence {Sn, n ε N} with S0 = 0 and Sn = X1 + X2 + . . . + Xn is called a random walk. In classical random walk is, P(Xn = +1) = p, P(Xn = -1) = q = 1-p.

2.3 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 next 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. We suppose say that whenever the process is in state i, there is a fixed probability Pij that it will next be in state j. ie., we 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 means that gives the present state, the future states are independent of the past state, ie) 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]. Thus, when the state of the process is known at time t the probability law of the future change of state of the process will be determined as if the process started at time t, independently of the history of the process unto time t. Generally, Markov Chain is described by vector p(nxn) which gives the unconditional probability distributions of states and transition probability matrix p which gives the conditional probabilities Pij = P(Xn+1 = Sj / Xn = Si), i,j = 1, 2, …, k where Pij may depend on n. In many real-world situations, Markov chain is used in several ways such as values of stocks over a period of time, weather patterns from day to day, Predict daily income of the business or profit of the company, results of some particular election party over a period of elections, social science, economic status, finance, computer science, computer-generated music, and many other fields. Analysts would like to be able to predict the future, both in the short term and in the long term. Here, how the probability and the matrix theory can be combined to analyze the short and long term behavior of certain kinds of phenomena which can be modeled as “Markov Chains”.

2.4 Hidden Markov Model A 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, and International Journal of Scientific and Innovative Mathematical Research (IJSIMR)

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R. Sasikumar & A. Sheik Abdullah we can regard S as the state process that is hidden and F as the observation process that can be observed. We call the observed event a “symbol” and the invisible factor underlying the observation a “state”. A HMM usually defined as a 5-tuple (S, F, P, ψ, π), where S = {s1, s2, …,sn} is a finite set of n states. F = {o1, o2, …, om} is a finite set of m possible symbols. P = {pij} is the set of state-transition probabilities, where pij is the probability that the system goes from state si to state sj. ψ = {ψi (ok)} are the observation probabilities, where ψi (ok) is the probability that the symbol ok is emitted when the system is in state si. π = {πi} are the initial state probabilities; that is the probability that the system starts in state si.

3. DISCUSSIONS In this section we have discussed stock market predict techniques using various types of stochastic models. Fama (1995) proposed two techniques to predict the stock prices; they were (i) technical or chartist theories (ii) intrinsic value analysis or theory of fundamental. The techniques of the chartist had always been surrounded by a certain degree of mysticism, however and as a result most of the market professionals had been found them suspect. So, probably we said that the pure chartist is relatively rare among the stock market analysts. Abraham et al., (2002) have examined the random walk properties and weak form efficiency of three growing up gulf stock markets. They used variance ratio test and nonparametric run test to evaluate the weak form of efficiency of the stock markets. From the run test, they determined that the successive price changed that were independent of each other and also they examined the direction of change of any position. Changes may be the following three possibilities (+) positive, (-) negative and (0) no change. Stock market prediction technique in Markov chain models are discussed here. Svoboda and Lukas (2005) used Markov chain analysis to predict the stock index trend of Prague stock exchange PX. Discrete state spaces are defined for Markov chain models then appropriate transition probability matrix and conditional probability matrices are calculated. They reveal that the prediction methods are only for short term investors and not for long term investors. Hassan and BaikunthNath (2005) used HMM to predict next day closing price for some of the airlines. They considered four input attributes for a stock, and they were the opening price, highest price, lowest price and closing price. These four attributes of previous day were used to predict next day closing price. They predicted only on the next day closing price. They did not predict both other three attributes of next day as well as for the subsequent days. The investors might feel that, that was the major drawback of that analysis. Hassan (2009) introduced the new combination of HMM and Fuzzy model to forecast the stock market data. He classified the data set as daily opening, high, low and closing prices to predict the next day’s closing price. He revealed that the prediction method was only for the short term investors and not for the long term investors. Also he compared with other forecasting models; HMM-fuzzy is more reliable and profitable than the other model. Deju and Xiaomin (2009) used Markov chain prediction method which is the probability forecasting methods. For predicting the ith day closing price of a stock, they used the state interval vector formula which is n(i) = n(i-1)p, where p is a state transition probability. An efficient stock market, whose price randomly fluctuates, reflects the homogenous distribution of market information. But we can predict possible future trend of the stock market through analysis of past information. Agwuegbo et al., (2010) had examined the random walk approach that was applied on the daily stock returns of 60 quoted companies in the Nigeria stock market. They studied the behavior of the market for two successive days. They suggested that the investors were not allowed at any time to invest in the market. It may lead to get failure of their investment. The investor is not allowed at any time to invest in the market in a way which might involved on his losing more than his investment. So this research work was very helpful to know the correct time of investing for investors. ZHAO Peng-ju (2010) proposed birth-death process for stock market evolution and studied the velocity of both rational traders and irrational traders entering and quitting the stock market. From the analysis, this is widely accepted by scholars that the market selection can eliminate irrational traders in competitive market. Borges (2011) had used random walk test for the Lisbon stock market and he mainly concentrated to add international evidence on the random walk theory of stock market prices by testing the Portuguese benchmark index (PSI – 20). Also, he used serial correlation test, a run test, an Augmented Dickey-Fuller (ADF) test, Unit root test and the multiple variance ratio test for testing the Portuguese benchmark index. ADF test, which is effective and clearly favourable to the null hypothesis. It may be accepted, in other International Journal of Scientific and Innovative Mathematical Research (IJSIMR)

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Applications of Various Stochastic Models in Financial Prediction words stock market index follows a random walk. Soloviev et al., (2011) applied higher order Markov chain or complex Markov chain for predicts financial time series. The procedure of prediction and splicing is iterative and conducted starting from smaller increments, adding a prediction with the bigger time increment on every step. They generate variants of the time series continuation according to the relations between the sequences of absolute and relative changes discovered with the help of complex Markov chains. From their research they suggested prediction is more reliable to find out the help of complex Markov chains. Doubleday and Esunge (2011) used Dow Jones Industrial Average Stock Market is analyzed by Markov chain by the use of discrete time stochastic model namely simple model and partitioned model. From the above two model, there is no significant difference among the models, both behaved similar to the entire analysis. Gupta and Dhingra (2012) used four different stocks for identifying the fluctuations. They predicted only for the next day values not for the subsequent days. One particular stock is independent of the other stocks. But in real life situation the fluctuation of the stocks are highly correlated with one another. Tuyen (2013) used HMM to estimate the parameter of the Markov Black-Sholes model to predict the option prices in the stock market. The historical data of daily VN-Index (Vietnam Stock Market) were taken from 2009 to 2011 for finding the four hidden states corresponding to the Normal distribution N ( µ i , σ i ) for i = 1,2,3,4 with the help of HMM. This model helps to analyze and predict option price in stock market. Angelis and Pass (2013) analyzed weekly changes in the US stock market index over a period of twenty years using HMM and also analyzed the stock market dynamic pattern detecting the different regimes. Stock market can be analyzed by referring to a simple and flexible model. They used Markov-switching volatility approaches, which was useful for defining periods characterized by a high conditional variance value. But investor cannot investigate fluctuations within these periods; this is the drawback of Markov-switching volatility approach. Kavitha et al., (2013) had applied HMMs for one day close value of a particular period for stock market trend analysis using Baum-Welch algorithm. They used MATLAB to get hidden states sequence. Therefore, the investors can observe easily the behavioral pattern of the stock market. Qing-xin Zhou (2014) studied the stock price of China Sport Industry. The weight values of every state were calculated with the method of weighted Markov chain theory and prediction intervals of the industry’s future stock prices were obtained. The prediction result through the weighted Markov chain is closer to the actual value of the closing price. The error of Markov chain prediction was relatively larger than the weighted Markov chain. And also the prediction result through the weighted Markov chain is closer to the actual value of the stock price of China Sport Industry. Kai Cui (2014) explained that, the variation of financial time sequence for Shangai composite index was predicted through introducing a dual state HMM. Also, he justified that the HMM was the best tool to predict the variation of financial time sequence. It was compared with GARCH Model.

4. EMPIRICAL RESULT We are interested in predicting the stock market. Stock market can take on one of the three values such as increase, decrease or no change. The closing price on a given day is dependent only on the closing price of the previous day. ie., P ( X t | X t −1 ,. . . , X 1 ) = P ( X t | X t −1 )

4.1. Dataset The complete set of data for the proposed review has been taken BSE Oilgas Stock data from 1st October 2014 to 13th November 2014. The table 1 given below shows the closing values of a stock market. Table 1: The closing value of a stock market S. No 1 2 3 4

Close Value 10517.71 10646.5 10705.28 10634.22

1 day close value Difference 128.79 58.78 -71.06

Observing Symbol

I I D

S. No 14 15 16 17

Close Value 11088.64 11081 11053.34 10975.65

1 day close value Difference 56.71 -7.64 -27.66 -77.69

International Journal of Scientific and Innovative Mathematical Research (IJSIMR)

Observing Symbol I D D D Page 855

R. Sasikumar & A. Sheik Abdullah 5 6 7 8 9 10 11 12 13

10683.32 10521.71 10508.71 10684.41 10702.54 10854.86 11141.78 11179.85 11031.93

49.1 -161.61 -13 175.7 18.13 152.32 286.92 38.07 -147.92

I D D I I I I I D

18 19 20 21 22 23 24 25

11067.88 11184.9 11364.76 11482.69 11484.65 11423.62 11456.14 11661.18

92.23 117.02 179.86 117.93 1.96 -61.03 32.52 205.04

I I I I I D I I

The observed series is simply considered as a series of two states are Increase(I) and Decrease(D) with a first-order Markov chain which is explaining the dependence between Increase and decrease stock price on successive days. From the Table 1, we found that the transition frequency matrix is D I

F=

D I

3 5  5 10  

D and transition probability matrix is P =

I

D 3 / 8 5 / 8 0.375 0.625 =     I 1 / 3 2 / 3 0.333 0.667 

We can find P2, P3, . . ., Pn values using the Chapman-Kolmogorov equation.

0.348750 P2 =  0.346986 0.347599 P5 =  0.347599

0.651250 0.653014 0.652400 0.652400

0.3476475 P3 =  0.3475734 0.3475992 P6 =  0.3475992

0.6523525 0.6524266

0.3476012 P4 =   0.3475981

0.6524008 0.6524008

 0.347599 P7 =  0.347599

0.3475992 ... P n =  0.3475992

0.6523988 0.6524019

0.652400 0.652400

0.6524008 0.6524008

From the P2 to Pn 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.

5. 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 paper, a complete theoretical survey of all the stock market prediction techniques are used for the different types of stochastic models that are given. Most of the papers were predicted only for next day stock prices. Some other people were predicted lateral days also.

ACKNOWLEDGEMENTS The authors acknowledge University Grant Commission for providing the infrastructure facility under the scheme of SAP (DRS-I). The second author acknowledges UGC for financial support to carry out this research work under Basic Science Research Fellowship.

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K. Basu, Introduction to Stochastic Process, Narosa Publishing House, 2003. Oliver C. Ibe, Markov Processes for Stochastic Modeling, Elsevier Academic Press, 2009. Fama, E. F., Random walks in stock market prices, Financial Analysts Journal, 51(1), (1995).

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