Modern pricing models - Essay Example

MODERN PRICING MODELS (GEOMETRIC BROWNIAN MOTION) By The Geometric Brownian motion (GBM)Definition and the History of the GBM In every financial modeling for pricing options, the main purpose of applying the financial model is through the determination of correct prices after evaluating a number of generated predictions. Similarly, the same case applies to the geometric Brownian motion model, which is a pricing model, used through simulations to determine the future prices of stock, especially in the binary options and stock prices for investors.

As such, investors get to learn how best to place their investments judging from the future expected price changes of a given stock price or of a binary option. A geometric Brownian motion (GBM), or an exponential Brownian motion, refers to a stochastic process that runs continuously over time in which case the logarithm of the randomly varying quantity follows a Wiener process, or the Brownian motion with a drift. According to Vose (2008), the Brownian motion is a significant example of the stochastic processes that satisfy a stochastic differential equation (SDE).

Most applications of Brownian motion incorporate, in particular, mathematical finance, especially in consideration of the model stock prices, such as is the case with Black-Scholes model. As such, the geometric Brownian motion is a core building block of modern finance. This is particularly in the case of the Black Scholes model whereby the underlying stock price is assumedly in line with the principles and expectations of the GBM dynamics (Vose 2008, p.37). From a technical perspective, the definition of SDE is the stochastic process (St), said to follow a GBM if it satisfies the following stochastic differential equation (SDE)Wt = a Brownian motion or a wiener process = the percentage drift (constant) used to model deterministic trends = the percentage volatility (constant) used to model a set of unpredictable events occurring during this motion.

When an investor wants to make an investment in the binary options, the most important element that he or she should account for is the fluctuation in the price that a particular commodity or good is likely to experience over a specific period of the trading process. As such, once an investor is able to track the volatility of the price changes with some degree of accuracy, they get to a better elevated position to determine the right the price of an option at the point when it expires, thereby increasing significantly his or her chances of being in the money at the right time.

Therefore, such an investor will be in a better position to collect the highest level of return on his or her investment (Vose 2008, p.115).The Stochastic Process of the Brownian motion According to Benth (2004), it is quite fortunate that top economists conducted a great deal of research in the field of Wiener motion, such as Fischer Black and Myron Scholes who came up with the infamous Black-Scholes formula instrumental in predicting the market volatility of a stock. The geometric Brownian motion forms an important element of this formula, and as such, serves as an approximate model for commodity price fluctuations when traced on a graph.

These economists argue that the Wiener motion follows a stochastic process in the manner in which it responds to particular changes in prices of commodities (Benth 2004, p.44).The Brownian motion model, named after Robert Brown, associates closely with the Wiener process in such a way that both have a characteristic of a series of random variables that form a consistent time chart of motion. This type of process, known by analysts as the stochastic process, is a very useful model for tracking the stock prices over time because it incorporates a certain degree of randomness that is consistent with the fluctuation of commodity price changes in the market.

By taking the Wiener process into account in its formula, and harnessing the stochastic process followed by commodity prices over a particular time-period, the Black-Scholes formula accounts for a level of randomness in the market volatility thereby making the formulae more reliable. While it is impossible to predict accurately all the changes that affect a commodity on the market, particularly over a very short period of time, the reliance on Brownian motion and the Wiener process makes it easier to make an informed assessment of the binary options and the chances of being in the money.

This is important as it enables investors to determine the right prices or the relative changes expected in a given stock price or binary optionApplying the Stochastic Process to the Real World Economists have used numerous mathematical formulas, even those that chart real events in the world, to solve theoretical problems. As such, nowadays they continually use the Wiener process and the Brownian motion model widely in approaching theoretical problems consigned to the economy. For instance, a wide usage of the Black-Scholes formulae incorporates the stochastic process in the world of economic.

Glasserman (2004) argues that this formula is a significant contributor to the rise in the binary options trading. Considering the advent application of internet sites that make it easy for prospective investors to invest in binary options without having to pay fees to specialized brokers, the Black-Scholes formula has gained even more prominence as more and more people look at the binary options as an attractive form of investment. The Brownian motion model remains the most reliable model available today in determining price fluctuations in the binary options despite the numerous warnings given continuously by economists saying that the model is not the perfect option for monitoring stock price volatility (Glasserman 2004, p.441).Analysis of the GBM Equation According to Benth (2004), an analysis of the GBM equation requires one to solve the SDE equation, which in turn results to the variety of options available for users to take advantage of while making financial decisions.

In solving the SDE, one considers So which is an arbitrary initial value of the equation to be solves. As such, the analytic solution of the SDE, given under the Itōs interpretation is:In order to arrive at this formula, one has to undertake a number of mathematical calculations and simulation, such as dividing the SDE by the St, and later write it in the Itō integral form:A close analysis of the formula provides that appears somewhat related to the other derivative. As such, analysts need to use the Itō calculus because is an Itō process.

An application of the Itōs formulae provides thatAnalysts obtain the following equation when they plug back the equation derived from the SDE. It is evident from the formulae that exponentiating gives rise to the solution claimed above (Benth 2004, p.44).Monte Carlo Simulation with GBM According to Glasserman (2004), the Monte Carlo simulation makes great use of the analysis from GBM by enabling the financiers and economic analysts to determine the level or amount of risk that might accrue from a given investment option.

As such, the Monte Carlo simulation makes use of the GBM equation analysis to estimate risks and expected changes in the financial markets. The Monte Carlo simulation is among the most common methods applied by financial analysts to estimate risks accruing from a financial investment or a binary option. For instance, in order to calculate the Value of Rick (VaR) accruing from a given portfolio, investors have the option of running the Monte Carlo Simulation as this simulation makes effort to predict what could be the worst likely loss that the portfolio might suffer considering a given confidence interval, as well as spanning across a specified horizon of time.

Notably, it is imperative for investors to specify all the time the two crucial conditions that affect the value or risk, which are horizon and confidence. In order to understand the effect of the Monte Carlo simulation (MCS) on the determination of the value of risk, it is best to review the basics of the MCS as applied to a given price of stock. The geometric Brownian motion comes in at this stage of the simulation since this also requires a model to specify the behavior of a price stock, as GBM is one of the most common models used in financial analysis and simulation.

As the simulation process begins, it is imperative for investors to note that the Monte Carlo simulation is just an attempt by the financial analysts to predict the future of a stock price many times over. From the simulation, analysts get thousands to millions of random trials that produce a distribution of outcomes usable for analysis and prediction of risk in associated cases. The basic steps to adhere to include the specification of the financial analysis model to apply, and in this case, the Geometric Brownian motion is the best option.

The second step is to generate random trials, whilst the third and last step is the process the output (Glasserman 2004, p.64).Specify a model, such as the GBMThe best model to use in this simulation is the geometric Brownian motion model, technically a Markov process because it enables financiers to estimate the value of risk, as well as the level of exposure to certain risks in the financial market and stock trading. As such, it is evident from the simulations that the stock price follows a random walk, and as such, remains consistent to even the weakest form of the efficient market hypothesis (EMH).

In such a case, the simulation incorporates beforehand the past price information, whereby the next price movement remains conditionally independent of the past price movements. This then follows the application of the GBM equation whereby “S” reflects the price of stock, “m” reflects the expected return, “s” reflects the standard deviation on the returns under consideration, “t” reflects the time horizon over which the investment process takes place, while “e” reflects the random variable in the application (Glasserman 2004, p.19).A rearrangement of the formula in order to solve for the change experienced in the price of stock provides the stock price “S” as a product of two terms as outlined in the parenthesis below The first term in the equation represents a “drift” while the second term in the equation represents a “shock”.

For every time-period under consideration, the GBM model assumes that the stock price will undergo a “drift” upwards in line with the expected value of return. However, in the event of a shock within the drift, such as an addition or a subtraction, the stock price is said to have suffered a random shock. As such, the derivation of the random shock incorporates the product of the standard deviation “s” by a random number “e”, as a simple way of scaling the standard deviation. This is the key essence of applying the GBM model in this simulation.

The stock price follows a given series of steps in which every step it undergoes either a drift plus or a drift minus a random shock, which is itself a function of the standard deviation of the stockFigure 1 The figure below explores a drift and a shock that occurs to an option price. A shock and a drift refer to either a subtraction or an addition in the stock priceFigure 1: [online] Available at http://studentblogs.warwick.ac.uk/morse/tag/statistics/Generate random trialsAfter deriving a model specification, the next step is to run random trials.

For instance, the illustration below on Microsoft excel shows a run of 40 trials. However, it is imperative to note that this is an unrealistically small sample because most simulations or “Sims” runs at the least, in several thousands of trials. In this case, the assumption is that the stock begins on day one at the price of $ 10. The following illustration is a chart of the outcome of the simulation whereby each interval or time step is twenty-four hours, and the series or time-period for simulation is ten consecutive days on the trading period, a summary of forty trials that have daily steps over a period of over ten days.

The result of this run is forty simulates stock prices as at the end of the time-period of ten days. From the results, it is evident that none of them happened to fall below $ 9 but one is above $11. Figure 2Figure 2: [online] Available at https://www.google.com/search?tbs=simg:CAESoAEanQELEKjU2AQaAggKDAsQsIynCBo8CjoIAhIU-gjuC_1EL5giBCfMLjwyODNUL8AsaIBzE75rvClAYDaQ8wt5FOLue5yjoVoDZgnIsIiOPt6udDAsQjq7-CBoKCggIARIEsra0pQwLEJ3twQkaMQoGCgRwbG90Cg4KDGJvZHkgamV3ZWxyeQoGCgRsaW5lCgYKBGFyZWEKBwoFc2hhcGUM&tbm=isch&sa=X&ei=P7ljVLe5KqndsASDuoLgAw&ved=0CCwQsw4&biw=1366&bih=625Process the outputFrom the simulation, the result is a distribution of future outcomes that are hypothetical.

As such, analysts can make use of this output through several ways. For instance, if the analysts want to estimate the VaR with a confidence range of 95%, the analysts will have to locate the 38th ranked outcome, which is also the third worst outcome of the simulation. This is because 2 out of 40 results in 5%, while the two worst outcomes are in the lowest 5%. Consequently, a stack of the illustrated outcomes into bins, whereby each bin holds one third of $1, provides three bins covering the interval from $9 to $10 and produces the following histogramGraph 1Graph 1: [online] Available at http://images.frompo.com/048525db92891adea94899ec4e2c678aIt is important for the analysts to bear in mind that one of the assumptions of the GBM model is normality, whereby it assumes that price returns are on a normal distribution with the expected returns, or mean “m”, as well as the standard deviation “s” of the product under consideration.

However, an interesting factor about the above histogram is that it does not look normal. On the contrary, if it undergoes more trials, it will not tent towards normality as outlined in the assumptions of GBM. It will however tend towards a lognormal distribution, characterized by a sharp drop off to the left of the mean, as well as a highly skewed long tail towards the right of the mean. This to some extend may lead to a certain degree of confusing dynamics on first timers, especially students whereby the price returns remain normally distributed whilst price levels remain log-normally distributed.

A simple example to approach this dynamic is to analyze a stock that can return either up or down 10% or 5% of its fair value. However, with the passage of a given time period, the price of the stock cannot be a negative. Furthermore, the increments of price on the upside have a significant compounding effect. Similarly, a decrease in the prices on the downside reduces the base, such as if one loses 10% of the value price, he or she is left with a lesser value to lose next time he or she undertakes trade.

The following chart illustrates the lognormal distribution superimposed on the given assumptions of $ 10 starting price (Linghao 2010, p.47).Graph 2Graph 2: [online] Available at http://lr2234.simple-files.info/j5GIeEzkjBZF9Z9zRZWCCkPpmQZGiZYFO4KPCl+foAFb6pEkH+HdahPs320e6MhFXamXDlbdyFsG18ldHJhgAUqCYAUdmW1GddJ5DSued+oto2H+ImUkuXBnceY5YxyheHhB5D14RugSKVn3D3lV0AB5QYkRT1SNAkBO8BlHNN0KXjyKyXVvyeNVIeDpVDig5WM7qK9xf7/yNwKm6nVQ46tiVOGQflvviGVa+d85LovPLx3ahFBJioBcRdaIWuPHiV21wvtD7J6lTL2b+07zZqtFp2X09fg2kMK7Y721xH3478Rj372YPZbmwnuf+8V8h/iIFseGy0qYwIAZypv1FYqRog== In summary, the Monte Carlo simulation must apply a selected model.

This model is the one that specifies the behavior of the Geometric Brownian motion. This model applies to a large set of random trials with the key goal of producing a plausible set of possible future outcomes. As such, the geometric Brownian motion is the best and most commonly used model in simulating prices of stocks. Furthermore, the GBM always assumes that a constant drift comes along with certain random shocks. On the other hand, it is imperative to note that period returns under GBM are normally distributed, whilst the price levels for a consequent multi-period, such as the 10 trading period used in the above simulation, are log normally distributed.

As such, this information comes in handy for investors who want to trade in stocks and binary options so that they can make wise investments, and as such, invest as experts rather than investing as ignorant fools.Properties of GBM According to Chambers (2008, p.221), the properties of the GBM model enable it to apply effectively to various applications of financial analysis. For instance, the properties of the GBM model come through a further analysis of the SDE equation. From the solution provided for the SDE simulation, which stands for any value of t, is a random variable that is log-normally distributed.

This formulae comes with an expected value and variance derived through the following formulae This is the probability density function of a St, also derived through the following formulae:In the process of deriving further properties of the GBM, it is possible to make use of the SDE, which has the GBM as its solution, or makes use of the explicit solution as given above. For instance, in consideration of the stochastic process, the log (St) is a process that is very interesting considering that fact that the Black-Scholes model relates closely to the Log return of the stock price.

In addition, through the usage of the Itōs lemma with f(S) = log(S) givesIt further follows that the,The derivation of this result can be through the application of the logarithm to the explicit solution of GBM such as in the following equationIn addition, in taking the expectation yields from the same result as the ones above provides that Another possible property of the GBM equation is the multivariate version whereby the GBM can extend to a case, which contains multiple correlated price paths.

As such, each price oath follows an underlying process as provided in the following equation.,In this case, the Wiener processes are correlated such that where .For the multivariate case, this implies that.Uses in Finance of GBM The geometric Brownian motion is an effective model for financial analysts to use in determining the various changes that a stock price might undergo through the application of the Black-Scholes model. Furthermore, the GBM is among the most widely used model of analyzing the stock price behavior.

According to Linghao (2010), the largest usage of the GBM model is in undertaking an analysis of the stock price, such as in the Monte Carlo simulation, the binary options trading, as well as the changes in the stock prices of the commodities and services within the market. Consequently, a number of arguments are fronted by these analysts to support their usage of the GBM model to analyze and evaluate changes in stock prices, as well as estimating the risks associated in the said changes that investors might have to consider before making an investment in a given stock.

Some of the arguments given by these analysts in support of them using the GBM model in the analysis of financial risks associated with changes in stock prices are that the expected returns of GBM are always independent of the stock price, which is the value of the processes. This is a concept that is in agreement with what appears as the expected performance in a real market scenario. Another argument fronted by these analysts is that the GBM process only assumes positive values, just as is the case with real stock prices.

In addition, the GBM processes exhibit the same kind of roughness in its path. This situation is also similar as observed in real stock prices. Another argument placed in favor of using the GBM model in these analyses is that the calculations relating to the usage of GBM process are relatively easy to apply and undertake as opposed to the application of other financial models in the same spirit of financial risk analysis (Linghao 2010, p.3).On the other hand, it is imperative to note that the GBM model is not entirely a realistic model while making these financial analyses.

As such, it is noteworthy to remember at all times when applying the GBM on financial analysis of stock prices that the model may fall short of reality in some points. One point is the fact that GBM assumes volatility to be a constant factor, a concept that is very different in real stock prices whereby the volatility of the stock prices changes over a given period, possible in a stochastic manner. Another point that makes the GBM model irrelevant in such cases is that the returns in real stock prices are usually not under a normal distribution platform since real stock returns have a higher level of kurtosis, or “fatter tails”.

This means that there exists a higher chance for occurrence of large changes in these prices. Furthermore, it is notable that these returns have a negative skewness. In extensions of the financial applications of the GBM model, the GBM model attempts to be a model that is more realistic when it comes to the evaluations and analysis of the stock prices. As such, when making such considerations, an analyst can drop the assumption adopted by the GBM equation, which considers the volatility () function as a constant.

Consequently, in the assumption that the volatility of a value is a deterministic function of the stock, as well as the time of trading, the derived model in the end is the local volatility model. On the other hand, if the assumption is that volatility creates a certain degree of randomness of its own, a concept, which in most cases is described by a different equation driven by a different version of the Brownian motion, the model created as a result, is known as the stochastic volatility model.

Financial application of the GBM model According to Linghao (2010, p.27), it so appears, and as mentioned earlier, that the time series generated by the geometric Brownian motion is among the simplest, as well as the most commonly used model in undertaking financial analyses and estimating risks associated with changes in stock prices. Take for instance the following formula, which provides that the value of the variables change in one unit of time by a given amount that has normal distribution with a variance of 2 and a mean of Such a Normal distribution as recorded in the above formula provides a suitable first choice for a number of variables because it is possible for analysts to consider that the model outlines that numerous but independent random variables affect additively the x variable.

As such, analysts can proceed to iterate the equation in order to deduce a relationship that exists between the following functions xt and xt+2:The next step is to generalise to any time interval of T:This equation is rather convenient because of a number of parameters, such as because it continues to use normal distributions, as well as it enables users to make possible predictions between and time intervals that they may select while undertaking the simulation. As such, the above equation focuses on discrete units.

However, it is still possible to write these in a continuous time form whereby the analysts have to consider a small interval of time, and the change incurred within this minor interval provided as tThis provides the Stochastic Differential Equation (SDE) which is equivalent to                In this case, dz refers to a generalised Wiener process, also referred to in some cases as the “innovation”, the “perturbation”, or the “error, whilst the refers to a Normal (0, 1) distribution.

The notation might appear like an unnecessary complication. However, with its use in the SDE, it provides one of the most succinct descriptions of a stochastic time series. A generalised version of the above equation is as follows                In a case whereby g and f are two functions, which is really just shorthand for writing:                GBM with mean reversion According to Linghao (2010, p.37), another aspect of the GBM that draws particular interest to the financial analysts is the long-run properties of the time series and its effect on the equity prices of the stock.

This is just one among the other numerous variables that are of key interest to the financial analysts when they make use of the GBM model in evaluating changes in stock prices. As such, this concept attracts a strong interest of these financial analysts in the process of determining whether the characteristic of the stock price can be a random walk process, or a mean reverting process. This is especially the case considering the important effect this characteristic has on the value of the asset.

In normal cases, a stock price will follow a mean reverting process if it has a tendency to go back to or to return to some value that is of average over a given period in the trading process. This means that the investors might forecast future returns in a manner that is much better and almost accurate through application of the past returns of the given stocks in order to determine the reversion level to the long-term trend path. Investors applying the GBM model in determining changes in stock prices must note that a random walk has no particular memory.

As such, this means that a given large move in the prices of stock that come after a random walk process is permanent. Therefore, there exists no tendency for the level of prices to return to a trend path over a given trading period. On the other hand, the random walk property further implies that a stock price’s volatility has the ability to grow without any form of bounds especially in the end period of trading. This is because the increased volatility reduces the value of the stock, and as such, a reduction in the volatility owing to the mean reversion would lead to an increase in the value of the stock.

For instance, taking stock of a variable x that follows a random walk under the Brownian motion model, the analysts encounter an SDE similar to the one encountered before, denoted as followsHowever, when considering the mean reversion, it is possible to reverse the equation in the following platform as denoted belowIn this case,  > 0 represents the speed of reversion. In addition, the effect of the dt coefficient produces an expectation of a downward movement in the event x currently stays above  and vice versa.

 The production of the mean reversion models is in terms of S or r, denoted as followsThe process denoted in the equation above is the Ornstein-Uhlenbeck process. This was among the very first models used to describe interest rates on a short-term level during the initial stages when the model was still known as the Vasicek model. However, this model leads to a problem in the equation because it can lead to negative prices in the stock prices, especially when modelling in terms of rIt is also imperative to keep the stock price positive.

As such, integrating this last equation over time provides the following equationThis equation is very simple and easy to simulate. For instance, the following graph denotes a simulation of the GBM simulation, the plots on the right show some typical behaviour for rt. Typical values of a would be in the range of 0.1 to 0.3.Graph 3Plots of sample GBM series with mean reversion for different values of alpha ( = 0,  = 0.001) Graph 3: [online] Available at https://www.google.com/search?tbs=simg:CAESngEamwELEKjU2AQaAggKDAsQsIynCBo8CjoIAhIU_1QjmCPwI8giGCPoI-wjVC-oI8wgaIFEKlKU8kjT8uE1DLBPCTZaejsQTLdANK9l4cqhFbJ2gDAsQjq7-CBoKCggIARIEyz2tzwwLEJ3twQkaLwoJCgdkcmF3aW5nCgcKBXNoYXBlCgkKB2FydHdvcmsKBgoEYXJlYQoGCgRsaW5lDA&tbm=isch&sa=X&ei=fr1jVPvuL4flsATq-IDICQ&ved=0CDQQsw4&biw=1366&bih=625 A slight modification of the above simulation is the Cox-Ingersoll-Ross, or the CIR model, usually used in some cases for analysing short-term interest rates.

As such, it has the useful property of not allowing negative values because it has a volatility that goes to zero as S approaches zero. This is the best model to use in analysing variable S, denoted as follows:                  Integration of this equation over time provideswhere    As such, Y becomes a non-central Chi-squared distribution with degrees of non-centrality parameter and freedom denoted at . This makes it little difficult to simulate since one must have an uncommon non-central Chi-squared distribution in his or her simulation software.

GBM with jump diffusion According to Linghao (2010, p.37), a Jump diffusion represents a sudden shock that occurs randomly to the variable in a given period. The idea behind this concept is to give recognition to these changes that goes beyond the ordinary randomness experienced in the backgrounds of these time series variables. This is because there is usually an event or two that has a significantly higher or more extensive impact on the given variable, such as a CEO of the company resigning or stepping down from his or her post at the organization, the occurrence of a terrorist attack, or a given drug getting approval from the FDA.

Modelling of the frequency of these jumps follows the Poisson distribution model with intensity  such that, in the event of a given period of time increment, the T results into Poisson (T) jumps. As such, the modelling for the jump size for r is as Normal (J, J) which also provides mathematical convenience, as well as provides ease in making estimates of the parameters in question. By adding jump diffusion to the discrete time provided in the GBM equation for a given single working period, the following equation is the resultIf we define k = Poisson (), this reduces to:On the other hand, for T periods we have:           These formulas are easy to model in relation to the Monte Carlo simulation discussed above.

In addition, it is also easy to estimate the parameters for through matching of moments. Analysts warn that one must be very careful and alert when undertaking these processes in order to ensure that the  estimate does not turn out as an arbitrary huge figure, such as > 0.2. This is because such Poisson jumps are usually rare events in the financial markets, and as such, they do not form part of the volatility experienced or recorded during each trading period. For instance, the following plot is an illustration of a typical jump diffusion model that gives rise to both r and Graph 4 The graph below represents jump diffusion as experienced in the Geometric Brownian Motion model.

In this graph, the points noted with circles refer to the areas that experienced a jump diffusionSample of a GBM with Jump Diffusion with parameters  = 0,  = 0.01, J = 0.04, J = 0.2, and  = 0.02.Graph 4: [online] Available at http://marcoagd.usuarios.rdc.puc-rio.br/stoch-a.html#gen-stochasticGBM with jump diffusion and mean reversion If under an imaginable market scenario, the return r receives a large shock from the trading practises, a correctional transaction will take place over time thereby leading to a normalcy of the expected returns on the  of the series.

As such, a combination of the jump diffusion with a mean reversion allows analysts to model the characteristics involved in a much easier platform considering the few parameters involved. However, in this case, the GBM with a jump diffusion model leads to the following equation since both variance and mean no longer apply. This is the case particularly in consideration of the case when the reversion speed large, and as such, an analyst has to model at the time, within which the period of the jump occurred.

For instance, if the case occurred at the beginning of the period, then the jump might have had a strong revertion before on can observe the value at the end of the period.As such, the best procedure is to employ the Euler’s method, which is the most practical solution in this case, and requires that the analysts split up the time period into numerous small increments. The quantity of these increments might be sufficient if the model produces outputs that are similar as for the case of decision purposed in consideration to any given great number of increments.

Benefits of GBM The geometric Brownian motion is beneficial because it is very effective when applied to different cases of market analysis as discussed above, such as in evaluating the changes in the stock market, especially the way the prices change from time to time. As such, this enables investors to predict future trends or come up with a given pattern that certain selected stock follows. The GBM is also beneficial to financial analysts because it provides optimal dynamic trading that has advantage constraints (Vose 2008, p.21). In addition, the model also allows investors to undertake an appropriate analysis that allows for investments, uncertainty, as well as creation of price stabilization schemes.

Consequently, the results from the analysis of the Brownian motion equation and application of the model gives investors the right information to make important informed decisions concerning their investments, especially those for the stock market.Reference ListBenth, F. (2004) Option Theory with Stochastic Analysis: An Introduction to Mathematical Finance. New York: Springer Science & Business Media, pp.33-37 [Online] Available at http://books.google.co.ke/books?id=1vB0Gfh1omsC&printsec=frontcover&dq=Option+Theory+with+Stochastic+Analysis:+An+Introduction+to+Mathematical+Finance&hl=en&sa=X&ei=sHNcVILmNaKe7gaI34HYCQ&redir_esc=y#v=onepage&q=Option%20Theory%20with%20Stochastic%20Analysis%3A%20An%20Introduction%20to%20Mathematical%20Finance&f=falseChambers, J. (2008). Software for Data Analysis: Programming With R.

New York: Springer Science & Business Media, pp.67-71 [Online] Available at http://books.google.co.ke/books?id=UXneuOIvhEAC&printsec=frontcover&dq=Software+for+Data+Analysis:+Programming+With+R.&hl=en&sa=X&ei=mXNcVKXdKsGP7AatmoHoBw&redir_esc=y#v=onepage&q=Software%20for%20Data%20Analysis%3A%20Programming%20With%20R.&f=falseFigure 1: [online] Available at http://lss799.filedatabase.biz/j5GIWGzErGRF1b9TZeecJnrUuiNhn5AlSYK0IXumgjZBnIExBKSaZhfg3God7sFQF+nTFEeCkA8L3cRWCdEnA1+ecRknglYSIsk+WT6KcOUolXbYJnlp7ms1O+w2YUL4Yz0D4wZjdvMJZ1miHnBUwgpvW8IGDlrEGwJJywl3ItwLQjHF+w0C/qBGKN7mZzKv518+mPwvZPuwMDm8xSER7q91UvmTZ1D1l2Vf8ZxqV5bPAx2GwQhE3YZUHNmCULjBjA6xwPNAvML1Eeic8kj1Mvge8GLrsvsto+WgSZyuzXTj6csxgfHdJtOy3W2F/MRAmsPcQ8+Sn1mXytkSy53wBsyKsQpvFigure 2: [online] Available at https://www.google.com/search?

tbs=simg:CAESoAEanQELEKjU2AQaAggKDAsQsIynCBo8CjoIAhIU-gjuC_1EL5giBCfMLjwyODNUL8AsaIBzE75rvClAYDaQ8wt5FOLue5yjoVoDZgnIsIiOPt6udDAsQjq7-CBoKCggIARIEsra0pQwLEJ3twQkaMQoGCgRwbG90Cg4KDGJvZHkgamV3ZWxyeQoGCgRsaW5lCgYKBGFyZWEKBwoFc2hhcGUM&tbm=isch&sa=X&ei=P7ljVLe5KqndsASDuoLgAw&ved=0CCwQsw4&biw=1366&bih=625Glasserman, P. (2004) Monte Carlo Methods in Financial Engineering. New York: Springer Publishers. [Online] Available at http://books.google.co.ke/books?id=e9GWUsQkPNMC&printsec=frontcover&dq=Glasserman+P.

+,+Monte+Carlo+Methods+in+Financial+Engineering,+2004&hl=en&sa=X&ei=dkddVOiiGJKw7AbywoDIDw&redir_esc=y#v=onepage&q=Glasserman%20P.%20%2C%20Monte%20Carlo%20Methods%20in%20Financial%20Engineering%2C%202004&f=falseGraph 1: [online] Available at http://lss799.filedatabase.biz/j5GIWGzErGRF1b9TZeecJnrUuiNhn5AlSYK0IXumgjZBnIExBKSaZhfg3God7sFQF+nTFEeCkA8L3cRWCdEnA1+ecRknglYSIsk+WT6KcOUolXbYJnlp7ms1O+w2YUL4Yz0D4wZjdvMJZ1miHnBUwgpvW8IGDlrEGwJJywl3ItwLQjHF+w0C/qBGKN7mZzKv518+mPwvZPuwMDm8xSER7q91UvmTZ1D1l2Vf8ZxqV5bPAx2GwQhE3YZUHNmCULjBjA6xwPNAvML1Eeic8kj1Mvge8GLrsvsto+WgSZyuzXTj6csxgfHdJtOy3W2F/MRAmsPcQ8+Sn1mXytkSy53wBsyKsQpvGraph 2: [online] Available at http://lr2234.

simple-files.info/j5GIeEzkjBZF9Z9zRZWCCkPpmQZGiZYFO4KPCl+foAFb6pEkH+HdahPs320e6MhFXamXDlbdyFsG18ldHJhgAUqCYAUdmW1GddJ5DSued+oto2H+ImUkuXBnceY5YxyheHhB5D14RugSKVn3D3lV0AB5QYkRT1SNAkBO8BlHNN0KXjyKyXVvyeNVIeDpVDig5WM7qK9xf7/yNwKm6nVQ46tiVOGQflvviGVa+d85LovPLx3ahFBJioBcRdaIWuPHiV21wvtD7J6lTL2b+07zZqtFp2X09fg2kMK7Y721xH3478Rj372YPZbmwnuf+8V8h/iIFseGy0qYwIAZypv1FYqRog==Graph 3: [online] Available at https://www.google.com/search?tbs=simg:CAESngEamwELEKjU2AQaAggKDAsQsIynCBo8CjoIAhIU_1QjmCPwI8giGCPoI-wjVC-oI8wgaIFEKlKU8kjT8uE1DLBPCTZaejsQTLdANK9l4cqhFbJ2gDAsQjq7-CBoKCggIARIEyz2tzwwLEJ3twQkaLwoJCgdkcmF3aW5nCgcKBXNoYXBlCgkKB2FydHdvcmsKBgoEYXJlYQoGCgRsaW5lDA&tbm=isch&sa=X&ei=fr1jVPvuL4flsATq-IDICQ&ved=0CDQQsw4&biw=1366&bih=625Graph 4: [online] Available at https://www.google.com/search?

tbs=simg:CAESngEamwELEKjU2AQaAggKDAsQsIynCBo8CjoIAhIU_1QjmCPwI-gjyCIYIjAyACYEJ8wgaIJ42Wf69DrwG57ZRqxPCYpk5c7yhXRj63gbE0XXVnJVPDAsQjq7-CBoKCggIARIE3C4_1NwwLEJ3twQkaLwoJCgdkcmF3aW5nCgYKBHBsb3QKCQoHYXJ0d29yawoGCgRhcmVhCgcKBXNoYXBlDA&tbm=isch&sa=X&ei=wL1jVI3NFe3CsATr_oCYAQ&ved=0CCUQsw4&biw=1366&bih=625Graph 5: [online] Available at http://en.wikipedia.org/wiki/Geometric_Brownian_motionLinghao, Yi 1964 (2010) The Pricing Of Options With Jump Diffusion And Stochastic Volatility. Skemman, [Online] Available At http://skemman.

is/stream/get/1946/6224/17806/1/Linghao_Yi.pdfVose, D. (2008) Risk Analysis: A Quantitative Guide. Hoboken, New Jersey: John Wiley & Sons, pp.25-58 [Online] Available at http://books.google.co.ke/books?id=9CaoAqaRcVwC&printsec=frontcover&dq=Risk+Analysis:+A+Quantitative+Guide.&hl=en&sa=X&ei=eHNcVM2yDayO7AakuIHIDw&redir_esc=y#v=onepage&q=Risk%20Analysis%3A%20A%20Quantitative%20Guide.&f=falseAppendicesGraph 5Two sample paths of Geometric Brownian motion, with different parameters. The blue line has larger drift; the green line has larger variance.

Graph 5: [online] Available at http://en.wikipedia.org/wiki/Geometric_Brownian_motion