# The Monte Carlo Simulation Is A Method For Evaluating A Deterministic Model Accounting Essay

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Introduction

The Monte Carlo Option Model method was actually established by Stanislaw Ulam and John von Neumann while working on the Manhattan Project. The Manhattan project is the project that develops the first nuclear weapon during World War II. The Monte Carlo method helps to solve a problem by stimulating directly to the physical process, and is not necessary to write down the differential equations that describe the behaviour of the system.B. Ricky Rambharat_and Anthony Brockwell (2006) mentioned that the Monte Carlo method is a very general process and is a valid approach in scientific areas such as physics, chemistry and computer science. In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features.

The Monte Carlo Option Model can be used to calculate some type of option. For example, the option that relate to various sources of uncertainty and calculating their values with other model that is difficult. In addition, Claudia Ribeiro and Nick Webber (2003) indicated that the Monte Carlo Option Model also can be used to calculate the option when that exist in the market but have very complicated features. Besides, Michael B. Giles (2009) highlighted that other option is when the arbitrage-free valuation of a definite derivative that consists of a large number of dimensions.

## BACKGROUND OF MONTE CARLO OPTION MODEL

In year 1930, an Italian physict who name is Enrico Fermi; he is the first founder of the Monte Carlo Option Model. Boyle (1977) stated that in year 1946, an American mathematician who name is Stanislaw Ulam had also found the Monte Carlo Option Model. A Greek American Physicist, Nicholas Metropolis had given the name of the model as "Monte Carlo". Herath and Park (2001) explained that the model was intended to compute the value of particular option invented the technique and solving certain types of differential equation using probabilistic methods by using the Monte Carlo Methods. Phelim Boyle is the first applicant of using this option pricing which was applies to calculate the value of European option in year 1977. M. Broadie and P. Glasserman show how to process and price the Asian securities by using the Monte Carlo Option Model. In year 2001, E.S.Schwartz and F.A.Longstaff discovered the process and the model were applied for determining the values of American option.

Jamison (1999) indicated that the Monte Carlo simulation is a method for evaluating a deterministic model using sets of random number as inputs. This method is used when the model is complex, nonlinear or involves more than a couple of uncertain parameters. In the past, Schwartz (1977) suggested that the simulation only practically is using super computers. Mark S. Joshi (2006) indicated that the Monte Carlo simulation methods are especially useful in simulating various types of uncertainty in inputs and with complicated features which would make them difficult to value. Chitro Majumdar (2005) discovered that specific areas of application include physical sciences, design and visuals, finance and business, telecommunication and games. The model also used on the Manhattan Project, towards the development of nuclear weapons in the past.

In theory, Monte Carlo valuation rely on risk neutral valuation where the price of the option in its discounted value. In addition, M R Samis, G A Davis and D G Laughton (2007) mentioned that in order to calculate the associated exercise value of the option for example the payoff, it had to apply the model to generate several possible prices via simulation. The payoff are then averaged and discounted. The result is the value of the option. Besides, some of the theory allows the increasing of complexity. For example, Barraquand and Martineau (1995) developed the option of equity is one of the approaches that may model with one source of uncertainty.

Wim Schoutens and Stijn Symens (2002) highlighted that in mathematical finance field, the Monte Carlo Option Model uses the Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty, to value and analyze instruments, portfolio and investment by simulating various types of uncertainty or with complicated features through the straightforward Black-Scholes process.

According to Aniela Karina Iancu, M.S. (2004) The Monte Carlo Option Method is used to calculate the options that relate to various source of uncertainty and calculating their values with other models is difficult like the Monte Carlo simulation which can be used to value options where the payoff depends on the value of multiple underlying assets such as the Basket option or the Rainbow option. Options that exist in the market but have very complicated features and arbitrage-free valuation of a definite derivative that consists of huge number of dimensions.

Sergey A. Maidanov (2000) indicated that the Brownian motion is one of the forms of

Markov processes used as a model for stock price movements as far back as in 1900 by L. Bachelier. The Brownian motion model states that the value S of a security follows the Stochastic process

dS = Î¼Sdt +ÏƒSdW

Î¼ = the drift rate

Ïƒ = the volatility

W (t) =variable is a standard Brownian motion

dt = time increment

At t = 0, the value S (0) is S0

## Literature Review

Marius Holtan (2002) highlighted that the advance of technology had made the Monte Carlo simulation fast becoming the choice for evaluating and analyzing assets, be it pure financial derivatives or investments in real assets. Sibel Kaplan (2008) mentioned that the Monte Carlo method is a technique that involves using random numbers and probability to solve problems. According to the John R.Birge (1995), Monte Carlo can be used to represent many random phenomena that can influence the option price. Nowadays, reasonable random numbers generators can be found in every programming language. Aniela Karina Iancu (2004) stated that the Monte-Carlo simulation is also an approach that can accommodate complex payoffs, stochastic volatility and variable interest rates. John R.Birge (1995) highlighted that the Monte Carlo method has its basic in statistical theory, in particular, the central limit theorem that allows the calculation of confidence intervals about the mean value of a random variable based on a sequence of independent random observations. Other than that, Victor Podlozhnyuk & Mark Harris (2007), indicated that the Monte Carlo option pricing is "embarrassingly parallel", because the pricing of each option is independent of all others. Barry R. Cobb & John M. Charnes (2007) studies describes the use of Monte Carlo simulation and stochastic optimizations for the valuation of real options arise from the abilities of managers to influence the cash flows of the projects under their control. According to Chitro Majumdar(2005),the Monte Carlo simulation in forecasting model is based on selecting a random value for each of the task , based on the range of estimates.

## Limitations of Monte Carlo Simulation

Susana Alonso Bonis(2006),indicated that the limitation of the Monte Carlo approach is that it can be used only for European style derivatives securities and Monte Carlo simulation cannot handle early exercise since there is no way of knowing whether early exercise is optimal when a particular price is reached at a particular time. Susana Alonso Bonis (2006) also stated that the Monte Carlo simulation is a forward induction procedure, which generates future values of the variable from its previous value and therefore is not suitable for valuing assets generating cash flows contingent on future events, such as is the case of American-type options. However, Aniela Karina Iancu, M.S.(2004) mentioned that even we being given high speed computers, the method is time consuming, as both n and N have to be very large to yield good estimates for the option price . In addition, John R.Birge (1995) mentioned that the Monte Carlo simulation's validity of a pseudo-random sequence for statistical randomness is somewhat questionable.

## Advantages of Monte Carlo Simulation

Marius Holtan (2002) highlighted that the two of the main virtues of simulation are flexibility and simplicity. Marius Holtan (2002) the simulation is also easy to implement and models can easily be constructed in spreadsheet packages. Michael J. Sanislo P.E (2003) indicated that the Monte Carlo simulation become proficient at recognizing opportunities with high market risk and controllable unique risk. Michael J. Sanislo, P.E. (2003) mentioned that the Monte Carlo simulation can help to separate the unique risk and market risk. Besides, John R.Birge (1995) mentioned that even we assuming the randomness, the error from central limit arguments decrease slowly, inversely proportional to the square root of the number of observations. Hongbin Zhang (2009) indicated that the disadvantage of using Monte Carlo methods for path dependent options is the large number of calculations that are necessary to update the path dependent variables throughout the simulation. N Bolia and S Juneja (2005) stated that even the Monte Carlo techniques can be quite slow as the problem-size increases, motivating research in variance reduction techniques to increase the efficiency of the simulations. Xiang Tian, Khaled Benkrid, and Xiaochen Gu(2008) highlighted that the computational time of the Monte Carlo simulations increases approximately linearly with number of variables, whereas in most other methods, computational time increases exponentially with the number of variables. In addition, the multiple independent paths are computed by using the parallelism which is one of the most important characteristics of the Monte Carlo simulation.

## Usages of Monte Carlo Simulation

The fields that can use the Monte Carlo simulation are expanding production capacity, building new plants, real investment problem or investment in IT. Yongzheng Lai & Jerome Spanier(2003) indicated that Monte Carlo simulation is also a method used to understand the impact of risk and the uncertainty in financial, project management, cost, and other forecasting models. Xiang Tian, Khaled Benkrid, and Xiaochen Gu (2008) indicated that Monte Carlo simulation also used to forecast a wide range of events and scenarios, such as the weather, product sales and consumer demand .Claudia Ribeiro(2003) studies shown that Monte Carlo option can used in determine uncertainty forecasting model which the forecasting model is the model that plan ahead for the future .There are some assumptions that can determine using Monte Carlo simulation in the forecasting model. For example the investments return on a portfolio, the cost of a construction project or the length of period that the project will be completed.

## Examples of Monte Carlo Simulation

For example of the case: To estimate the total time completion of the period of the project. In this case, it's a construction project, with three parts. The parts have to be done one after the other, so the total time for the project will be the sum of the three parts. All the times are in months.

Time Estimate

Job 1

5 Months

Job 2

4 Months

Job 3

5 Months

Total

14 Months

Table1: Basic Forecasting Model

Minimum

Most Likely

Maximum

Job 1

4 Months

5 Months

7 Months

Job 2

3 Months

4 Months

6 Months

Job 3

4 Months

5 Months

6 Months

Total

11 Months

14 Months

19 Months

Table 2: Forecasting Model Using Range Estimates

Time

Number of Times (Out of 500)

Percent of Total (Rounded)

12 Months

1

0%

13 Months

31

6%

14 Months

171

34%

15 Months

394

79%

16 Months

482

96%

17 Months

499

100%

18 Months

500

100%

Table 3: Results of a Monte Carlo Simulation

The original estimate for the "most likely", or expected case, was 14 months. From Monte Carlo simulation, however, we can see that out of 500 trials using random values, the total time was 14 months or less in only 34% of the cases.

Conclusion: In the simulation there is only a 34% chance, it about 1 out of 3 which that any individual trial will result in a total time of 14 months or less. On the other hand, there is a 79% chance that the project will be completed during 15 months. Further, the model demonstrates that it is extremely unlikely, in the simulation, which we will ever fall at the absolute minimum or maximum total values. This demonstrates the risk in the model. Based on this information, we might make different choices when planning the project.

## Related Theory

Black Box Testing Model

In the Monte Carlo option, the related theory that related to the model is the Black Box Testing Model. According to Umar Saeed and Ansur Mahmood Amjad (2009), it is the computer program which user enters the information and the system utilized the logic to form an output to the users. Umar Saeed and Ansur Mahmood Amjad (2009) mentioned when the logic is form, the portion of the system creates formulas and calculations for the user to use the system. Regard to Umar Saeed and Ansur Mahmood Amjad (2009) the system often used to determine optimal trading and practices that generate many different types of data including buy and sell signal.

According to Umar Saeed and Ansur Mahmood Amjad (2009) the black box model is recommended by International Software Testing Qualification Board (ISTQB) and it is important and uses to determine the defects finding, sophistication, defect guessing, effort and quality of test cases. Regard to Umar Saeed and Ansur Mahmood Amjad (2009), the black box testing strategies and techniques are important in equivalence partitioning, boundary value analysis (BVA), decision table based testing, classification tree method, cases based testing and state diagram based testing. Umar Saeed and Ansur Mahmood Amjad (2009) said that there are all essential and frequently used black box testing strategies (BBTS) in software organizing nowadays. Umar Saeed and Ansur Mahmood Amjad (2009) mentioned the boundary value analysis is a black box testing techniques to identify test cases. It is use to correct the critical error in the input or output of boundaries by mechanics calculation or manipulated the data with an objective. Umar Saeed and Ansur Mahmood Amjad (2009) stated the system developed that cause bugs and the bugs catch the error in the system, the two variables x1, x2 of function input and their range are as follow:.

E.g. F(x1, x2) = 2x1+x22

For the test case selection, Umar Saeed and Ansur Mahmood Amjad (2009) stated that there are single versus multiple range checking fault which to determined the number of cases, for the single valuable, the boundary-adjacent values are helpful to exercise the program checking logic which, may use "<", ">" or "<=" on most keyboard layout. Umar Saeed and Ansur Mahmood Amjad (2009) mentioned the set of test cases selected at the minimum would be the value with the baseline where x+ is value greater than x and y- is value less than y and the z is nominal value which lies between x+ and y-.

E.g. [x- , x, x+, z, y-, y, y+]

Umar Saeed and Ansur Mahmood Amjad (2009) highlighted that there is also having the multiple sub-ranges for single variable which will be the union of two set of test cases as given by the following:

Lbaseline = {a, a+, e, b-, b} U {b, b+, f, d-, d} = {a, a+, e, b-, b, b+, f, d-, d}

For multiple variables, Umar Saeed and Ansur Mahmood Amjad (2009)stated that there will be a single fault model, which has two inputs X and Y, is failure due to the low probability of faults occurring simultaneously and the baseline single variable test case are as follow:

Xbaseline = {a, a+, b, c-, c}

And

Ybaseline = {d, d+, e, f-, f}

Umar Saeed and Ansur Mahmood Amjad (2009) stated the advantages of black box which using several methods as follow:

It is easy to learn, simple and the most defective technique.

Some of the methods have quality of sophistication and simplicity which generates to cover discovers errors of the boundaries.

It is versatile in controlling the changes by controlled on the base of single and multiple fault assumptions.

It can also facilitate the test case to generate and help identifying any problem before the coding start moreover it also helps full for testing purpose.

It is a reliability theory by using baseline or robust to determines the cases and the BVA in the black box model is a useful tools in determining the quality and completeness of the program which make the test easy to apply.

Umar Saeed and Ansur Mahmood Amjad (2009) mentioned the disadvantages of the black box as below:

The quality of the test depend on the investment of time and effort and any confusion understate in the program is easy to neglect and decrease the chances of discovering the errors.

The black box model does not contain any input criteria hence it depend on the selection of equivalence class types, although it depend on the selection but it does not ensure best test cases.

Some of the black box method is reluctant and the test suite will have different result as it used by different people so it will affect the accurately of the result.

Besides that exhausting in using the model will result in impractical in addition the test cases may be infeasible and eventually it will be removed from the program.

Black Scholes Model

The Black Scholes Model is another related theory that can be connected to the Monte Carlo Model. Simon Benninga and Zvi Wiener (1997) said that the model is introduced by Robert Merton and Myron Scholes in year 1973 which used in binomial pricing option formula. The Black Scholes model involves two underlying assets which are riskless asset Cash Bond and risky asset Stock. According to Peter Denteneer (2009), the share price St of the risky asset Stock at time t is assumed to follow a stochastic differential equation of the form; it is also the Brownian motion:

dSt = Î¼tSt dt + ÏƒSt dWt

where {Wt}t â‰¥ 0, Î¼t is a nonrandom function of t and Ïƒ > 0 is a constant volatility of the stock. Besides that, Simon Benninga and Zvi Wiener (1997) shows that the value of the Call at time T is (ST - K). According to the Fundamental Theorem of Arbitrage Pricing, the price of the asset Call at time t = 0 must be the discounted expectation under the risk-neutral measure which is:

C = SN (d1) - Xe-rT N ( d2 )â€š where d1 = lnS/X)+(r + Ïƒ2/2)T and d2=d1-Ïƒ(T)1/2

Ïƒ (T)1/2

According to Peter Denteneer (2009) there are several assumptions for the Merton-Black-Scholes model to idealized financial market which are:

Trading takes place continuously and the standard form of the capital market model holds at each instant.

The selling of the assets is possible at any time.

There are no transaction costs and short selling is allowed, i.e. an investor can sell a security that he does not own.

All market participants with using this model can lend and borrow money at a constant interest rate.

Assume there are no dividend payment between t=0 and t=T.

According to Georges Dionne, Geneviève Gauthier, Nadia Ouertani and Nabil Tahani (2006), the basket pricing option is useful in exposed to the currency risk as every market in the country are very active and liquid, hence it is more important to adopt a portfolio approach as it allows the firm to account for the correlations between these difference financial market simultaneously with different financial risk. Georges Dionne, Geneviève Gauthier, Nadia Ouertani and Nabil Tahani (2006) describe that basket option is a type of exotic option whose payoff is depend on the value of the basket and is usually cheaper than a portfolio of the standard option. In addition, the basket options are traded and designed to meet the needs of the buyer.

Georges Dionne, Geneviève Gauthier, Nadia Ouertani and Nabil Tahani (2006) studies describe that as the pricing of the basket options have no explicit analytical solution hence it is more challenging than the standard option. There are several approaches for the price basket option: the numerical methods, upper and lower boundary and analytical approximations. The contributions of the option are:

To design a framework that are stochastic

To compute the heterogeneous market under the T-forward measure.

To show that some of the existing analytical approximation may used in general setting.

To qualify the approximation errors.

According to Georges Dionne, Geneviève Gauthier, Nadia Ouertani and Nabil Tahani (2006), St denotes the time value of commodity, Î´t is its continuously-compounded convenience yield at time t, Ct is the value at time t of one unit of foreign currency expressed in domestic currency, and hence the equation is given by:

dSt = st[(Î±s - Î´t)dt + ÏƒsdWt] ,

dÎ´t = Îº(Î¸ - Î´t)dt + ÏƒÎ´dWt ,

dCt = Ct[Î±cdt + ÏƒcdWt],

GARCH Option Pricing Model

Steven L. Heston and Saikat Nandi (1997) said that the GARCH option pricing model is a very popular and effective tools in modeling the volatility of the dynamics market which is efficient use than the Black Scholes model in determining the volatility of the market. As the Black Scholes model is updated every period to determine the market option pricing while the GARCH model is held constant from the historical price. Steven L. Heston and Saikat Nandi (1997) stated that the advantages of this model are:

Its analytical solution that is available in pricing.

Perform well even when the parameter is not re-estimated for long period.

More easy to implement as the volatility is identifiable.

The volatility can be observed from the historical price.

Able to generate in illiquid market where the information is not exist.

Ability to capture the correlation of volatility with return.

Steven L. Heston and Saikat Nandi (1997) shows that there are two assumptions made for the model, the log-spot price follow by a particular GARCH process and the value of call option prior to expiration. The log-spot price formula is as follow:

Log(S(t)) = log (S(t-âˆ†)) + r + Æ›h(t) + âˆšh(t)z(t),

Where r is the continuously compounded interest rate for the time interval âˆ† and z(t) is a standard normal disturbance, h(t) is the conditional variance of the log return between t - âˆ† and t. Steven L. Heston and Saikat Nandi (1997) shows that the formula of second assumption is as follow:

Log (S(t)) = log (S(t - âˆ†)) + r - 1/2 h(t) + âˆšh(t)z*(t),

Steven L. Heston and Saikat Nandi (1997) derive that the prices written as functions of the spot asset price and since the function of S(t) and h(t + âˆ†) can be written as function of S(t). Since the second assumption is concerns with the pricing option hence the variance of the model is not stochastic and we assume the Black Scholes formula hold.

Heath-Jarrow-Morton Model (HJM)

Carl Hiarella and Oh Kang Kwon (1999) indicate that the Heath Jarrow Morton model provides a very general interest framework and capable to incorporate most of the market features to allows the measureable of the errors from the estimation of data. Carl Hiarella and Oh Kang Kwon (1999) mentioned that it is useful to provide an expression of discount bond price in term of appropriate variable and allow us to price most of the European type term structure. Carl Hiarella and Oh Kang Kwon (1999) shows that the heath Jarrow Model is often time consuming than the Monte Carlo simulation. In order to overcome this problem, many authors have considered many ways to transform the HJM model to Markovian system.

According to Andrew Jeffrey, Oliver Linton, Thong Nguyen and Peter (2001), there are two factor models in HJM, the first factor is according to the short term interest rate meanwhile the second factor are including the long rate, short rate, inflation, central tendency and volatility. For our information, Andrew Jeffrey, Oliver Linton, Thong Nguyen and Peter (2001) shows that the study determines the issues that analyze the HJM model factor is in a rigorous fashion.

Andrew Jeffrey, Oliver Linton, Thong Nguyen and Peter (2001) stated the one factor nonparametric HJM model determines the term structure in a forward rate which is sensitive to the method adopted. Besides that, Andrew Jeffrey, Oliver Linton, Thong Nguyen and Peter (2001) highlighted that the model's estimation of yields is less sensitive as the yield is the average rate. Below is the equation that introduces by the framework that using for the uncertainty of the bond market.

dy (t, T) = Î±y(É¯,t,T)dt + Î³(É¯,t,T)dW(t),

Where W (t) is a Brownian motion, É¯ indicates the possible dependence structure up to time t. specifically É¯ÏµÓºt, where Óºt indicate the information before time generated by information.

In conclusion, the above model are all can be use by the public and company to determine and calculate the error, risk, volatility and interest rate of a company bond, stock and many other transaction that related to the company with using the function formula. The most appropriate model that uses to determine the risk of the market is the Black Scholes Model. The GARCH option pricing model is most useful in determine the volatility of the market compare to others model as its ability to updated the information every period meanwhile it is easy to implement. For measuring and correcting the error, the Black Box model is the most useful model with compare to others as its ability to transform the information into the logic output. Among the several related theory that can be instead of the Monte Carlo Option, the most useful and efficient model that most of the company use is the Black Scholes Model. It is being use by the company as it almost same with the Black Box model which is easy and simple to learn by the users. It is able to generate and identify the problem faster. Besides that, the assumption of the model had also brought many benefits either in term of cost or payment of a transaction in the company.

## Conclusion

In this paper, we have showed that the example of Monte Carlo simulation in the predicting the completion period of the project. From that, we conclude that the Monte Carlo methods can provide useful solutions to many problems arising in finance. Besides, we also agree with the theory that suggested by Marius Holtan (2002) which he highlighted that the two of the main virtues of Monte Carlo simulation are flexibility and simplicity. However, Monte Carlo simulation also has their limitation which Susana Alonso Bonis (2006) indicated that Monte Carlo approach is that it can be used only for European style derivatives securities. Besides that, we also conclude that there is several related theory that can be instead of the Monte Carlo Simulation. Among several theory we found that the Black-Box Model is the most suitable model to replace the Monte Carlo model as it is easy and simple to learn by the users and it able to generate and identify the problem faster. But if possible it is better to use the Monte Carlo model as its ability to determine and calculate the various type of problem that occurs accurately.

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Wan, H.(2002, March). Pricing American-Style Basket Options By Implied Binomial Tree. Applied Finance Project of University of California.

Xiang Tian., Benkrid, K., Xiaochen, G.(2008,August 20).High Performance Monte-Carlo Based Option Pricing on FPGAs. The University of Edinburgh, School of Electronics and Engineering: Engineering Letters.

Available :<http://www3.iam.metu.edu.tr/iam/images/d/d6/Sibelkaplanterm.pdf.html>

Yongzeng, Lai., Spanier, J.(2003). Application of Monte Carlo/ Quasi-Monte Carlo Methods in Finance: Option Pricing. Department of Mathematics: Claremont Graduate University.

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## Reference of Related Theory

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Denteneer, P.(2009, October 22). The Standard Model of Finance: Merton-Black-Scholes model for option pricing. Introductie Econofysica.

Available :< http://www.ilorentz.org/~pjhdent/introefcollegeII.pdf.html>

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Heston, S, L., Nandi, S.(1997, November). A Closed-Form GARCH Option Pricing Model. Federal Reserve Bank of Atlanta Working Paper 97-9.

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Hiarella, C., Kang Kwon, O.(1999, November 1). A Class Of Heath-Jarrow-Morton Term Structure Models With Stochastic Volatility. School of Finance and Economics.