Adenegan, K. E. and Adewoye, R. A. 12 Abstract— Stochastic volatility models are models which assume that the volatility of the stock price process is not constant but stochastic itself. Volatility is a measurement of the change in price process over a given period of time. In this paper, a survey of various stochastic volatility models were x-rayed and benchmarked against the constant volatility. Analytical comparisons were made on time series model, Nadaraya-Watson Model, constant volatility and integrated variance. The models governing equations were showcased for valid financial markets. Keywords— Financial Market, Model, Stochastic Volatility, Stock Price, Volatility. I. INTRODUCTION olatility is a change in the price process. Stochastic volatility models show that the price process is not constant. In practice, correlations are usually estimated on the basis of past historical observations. This is an important consideration in the construction and analysis of a portfolio, as the associated risks will depend to an extent, on the correlation between its constituents. It should be apparent that from a portfolio perspective, a positive correlation increases risk. If the returns on two or more instruments in a portfolio are positively correlated, strong movements in either direction are likely to occur at the same time. The overall distribution of returns will be wider and fatter, as there will be higher joint probabilities associated with extreme values (both gains and losses). A negative correlation indicates that the assets are likely to move in opposite directions, thus reducing risk. Bengtsson and Olsbo (2003) reported that it has been argued that in extreme situations, such as market crashes or large- scale market corrections, correlations cease to have any relevance, because all assets will be moving in the same direction. However, under most market scenarios using correlations to reduce the risk of a portfolio is considered satisfactory practice, and the Value-at-Risk number for diversified portfolio will be lower than that for an undiversified portfolio. K. E. Adenegan is with the Department of Mathematics, Adeyemi College of Education, Ondo, Ondo State, Nigeria. (phone: +2348036576466; e-mail: adeneganke@aceondo.edu.ng , adenegankehinde@gmail.com ). R. A. Adewoye is with the Department of Mathematics/Statistics, Rufus Giwa Polytechnic Owo, Ondo State, Nigeria. II. LEVY PROCESS Definition 1: A collection { } ()/ 0 Xt t ≥ of random variable is called a stochastic process. For each point ω ∈Ω , the mapping (, ) t Xtw → is the corresponding sample path. Simply, a stochastic process indexed by a set T R ⊂ is a , ( , ) d mapX T L R → Ω . This means that () Xt sometimes written as t X -valued random variable for each t T ∈ . Definition 2: A R valued − stochastic process { } : 0 X t ≥ is a Levy process if the following conditions are satisfied  For any choice of 0 1 2 1 0 ,,, , n and t t t t ≥ ≤ ≤ ≤ ≤ ≤ , random variables 2 0 2 1 1 , , , .... i n n n t t t t t t t X X X X X X X − − − − are independent.  0 0 X = a.s. (i.e. almost surely)  The distribution of s t s X X + − does not depend on S i.e. the process has stationary increments.  For all, it holds that lim ( ) 0 t s s t PX X + − > = (i.e stochastic continuity).  t X Is right continuous for 0 t ≥ and has left limit for 0 t > a.s. This is also known as cadlag (continu a droit are limites a gauche) Remark: The most important levy process is Brownian motions where ( ) V , st is the autocovariance. As discussed earlier, we work under the assumption that the price of an asset can be modeled by an exponential Levy process. A special case is the Bachelier-Samuelson model in which the price process t S is taken to be exponential Brownian motion 0 ,0 t t t t S Se t T µ σβ + = ≤ ≤ (1.0) Where t µ are t σ respectively the mean and the (constant) volatility? Remark: The Bachelier-Samuelson model makes the log- returns log log t t t t X S S −∆ = − (2.0) Gaussian distributed, where ∆ typical is one day of say a stock price at the time t. a model that seems to better fit the price process is the exponential Levy motion A SURVEY OF SOME SELECTED STOCHASTIC VOLATILITY MODELS Adenegan, Kehinde Emmanuel and Adewoye, Raphael A. V