The Subject Area of Monte Carlo Methods in Financial Mathematics - Term Paper Example

The paper mainly discusses about the application of Monte Carlo methods in recent developments in financial mathematics. In modern finance, the prices of security derivatives which are complex are mainly signified as high dimensional integrals. The Carlo Monte method has proved to be important in integral evaluation. Instead of applying random points in evaluating the integrals as in standard Monte Carlo an individual is able to use deterministic sequence which has appropriate properties. These sequences are referred to as low discrepancy sequence and the method is named quasi-Monte Carlo. The paper will discuss mainly this approach and its application in dealing with financial problems. Introduction In financial mathematics, numerical methods have been of great importance in the current years. There are numerous reasons as to why these methods are useful. First, the principal model that defines the progression of prices of the significant state variables and basic securities has become more refined and sophisticated. Second, the securities types and their associated offspring of security derivatives are complex. For one to compute the risks sensitivities and the prices of these instruments one needs to assess a high dimensional integral. Third, advancements in management technology of risks and practice nowadays mandate a wide range and complex evaluation at the portfolio level. For instance, because of regulation requirements numerous financial institutions nowadays allocate enough resources which will be used in determining or in computation of value at risk (VaR). The Value at Risk computations comprises numerous different variables. Credit risks computations also encompass an extensive numerical work. Of course there is much advancement that has been made in calculation of power thus making calculations more feasible. There is a comprehensive numerical approaches used in computation. These methods are discussed as follow; first there those problems which have analytical solution or are said to have closed form though we still need numerical work. Even if, the analytical expression is existing one should raise numerical algorithm to perfect the computation. In addition, to get numerical values one will evaluate two functions which are normally distributed but cumulatively. Though there are many derivative conventions where there is no existence of analytical solutions. Many issues of interest are governed by the price of the derivative in order of partial differential equation (PDE). In modern world the modern methods are the ones which are employed to derive the efficient pricing of different types of exotic options. The advantage of this method is that it deals with small number of variables. In cases where the problem is evaluating a large number of variables for instance in the case of path-dependent securities and mortgage –backed securities. In cases of these type of securities will be written as a multi-dimensional integral because the price of the derivative security payoff is a function of the state variables. However, for problems like this Monte Carlo (MC) method will be employed because it is a flexible and a powerful tool. Monte Carlo method currently is applied in wide variety of problems which are complex. Even there are some problems which cannot be solved by use of Monte Carlo approach though the numerous approaches that have been recommended to speed up merging. Quasi Monte Carlo is another method which has been developed. In evaluation, the method applies the use of deterministic sequence instead of random sequences. There is a property in deterministic sequence which makes the case to be well dispersed in the domain of integration. Sequences which have these characteristics are referred to as low discrepancy sequences. Generally, low discrepancy sequences outdoes the standard Monte Carlo in cases where one is dealing with high dimensional problems though Monte Carlo seems to be more perfect when it comes in dealing with high dimensional sequence except in cases where one uses a large numbers. The importance of low discrepancy sequence approach in application to financial mathematics is that it appears to hinder the smoothness of integrands use and due to the fact that application of dimensions in finance which are effective may lower nominal dimensions at times. On the other hand, it is not the same situation for quasi Monte Carlo outdoes the standard Monte Carlo in application in finance. In Quasi Monte Carlos its efficiency and effectiveness highly lies on the characteristics of Low discrepancy sequence and nature of the problem. The main known limitation of applying low discrepancy sequence is the lack of workable error bound though in quasi Monte Carlo there is deterministic error of upper bound for integration but it is of no value when it comes to financial application. Overview of Monte Carlo methods These financial methods of Monte Carlo were introduced by Metropolis, Fermi and Neumann in 1940`s in solving a project that involved integral-differential equations. However, though the Monte Carlo methods were practically in use as today they were still in existence because people had basic ideas. The Monte Carlo approach covers any type of method which is used in statistical sampling to achieve a solution to problems which are quantitative. Basically the MC methods answers a resolves a problem by simulating the essential process directly and then computing the average outcomes of the process. Monte Carlo is a general technique which is mainly applied in areas like computer science, engineering, chemistry, accounting and physics. The metropolis method is mostly used in cases which are dealing with high dimensional problems. There have been many implementations for this method which have widely employed a proposal distribution which has a short length as compared to the relative length, L. The main reason for choosing a length which is small is that in cases which are dealing with high dimensional problems are most likely to give results which have low probability. The low probability is as a result of large random steps which are not likely to be accepted. In computing problems which have a large length scale there will be a movement in the state space which occurs only when there is a shift to the state due to low probability and it is accepted or when there are chances that a large random step will land on another state which is in risk. Hence, the progress rate will be slow except when small steps are used (Boyle, 1977). The major limitation which is associated with small steps is that metropolis approaches have explored distribution probability by a random walk. Random walks are time consuming to reach anywhere. For instance in considering random walks which have one-dimension, the steps applied to the state will either move right or left having uniform probability. The rule of thumb which is applied in metropolis method states that, `` lower bound on number of iterations of a metropolis method”. This means that in case where the scale of the space has the largest length the probability of the state been L, and having a step size which is originating from random walk of metropolis method with a proposal distribution will result in T~(L/)2 which are obtained by iterations in an independent sample. This rule provides only a lower bound and these may lead to worsening of the situation. For example, in case where the probability is composed of numerous islands which have high probability and they are mainly parted by areas consisting of low probability. In financial mathematics, the method is applied to simulate the different sources of uncertainty which have an effect on portfolio, the investment in question, instrument and then one is usually supposed to compute a value which is needed to provide possible values of the inputs. Moreover, there are numerous high-dimensional integrals of variables which arise in finance. These integrals should be calculated numerically in a threshold. The computational complexity will be in this order if the integral dimension is said to be in the worst case and one is assured of an error of at most. This means that the problem is suffering from dimensionality curse. The Monte Carlo method formula is defined by, , is an evaluation points which are randomly chose, expected error is in order of and the cost of the process which is having an error is computed in order of thus breaking the dimensionality curse. The Monte Carlo approach gives appropriate solutions to numerous mathematical problems by performing some statistical experiments in computers. Monte Carlo method has been in use for many centuries. Currently the method has gained it status and it recognized as a numerical method which is in a position to address many complex applications. This method is applied to solve both problems which have probability structures and those financial problems that do not have probability content (Boyle, 1977). In the practice of Monte Carlo method in computation the necessary random vectors and random numbers are generated in a deterministic subroutine by a computer. These vectors are referred to as pseudorandom vectors and pseudo random numbers (Eckhard & Peter, 1992). These methods have several advantages. Firstly, the application of Monte Carlo method is widely used in solving complex financial mathematics. It is also easy to implement and apply. In implementing the technique one needs minimal requirements to make the approach applicable in solving complex integration problems. The evaluation of the method needs only the capability to sample random points in terms of X and be able to evaluate f (X) of the points. The second advantage of this method is that it provides a better way of dealing with numerical integration which has high dimensionality. Another usefulness of Monte Carlo method is that the standard error is not affected by curse dimensionality because it does not depend on integral dimensionality like other techniques which are used to solve financial problems likes of Simpson’s method and trapezoidal rule which are highly affected by curse dimensionality. The final advantage of Monte Carlo method is that simulation is in a position to mimic phenomena’s which are too complex by use of random numbers which are generated by computer so that they can simulate fluctuations in nature. Though the method has several strengths it has also its weakness. The method has many deficiencies which limits the usefulness of the method. The disadvantages of the method are; in order to use the method one is supposed to decide prior the number of points which to choose and how fine it is supposed to be. Immediately the random numbers is chosen all the points been sampled starts to complete. It is not convenient to use grids to sample the points because it is difficulty to generate random samples using Mento Carlo method. Moreover, the method converges slowly. Also the results are numerical in nature. This means that the estimates obtained are wrong hence leading to probability error. The error which is bound does not replicate any rules of integrand. Application of Monte Carlo methods The application and uses of the methods by Monte-Carlo involves a wide range of areas. These include simulation, discrepancies, integration, pricing, portfolio analysis and in other financial application ( Guias, 1997). Option pricing Provided the high discrepancy of dimensional problems in finance that have existed for a long period of time, it is not surprising that Monte Carlo methods have raised attraction to research staff and practitioners in presentation fields. Several studies have been conducted over the past to integrate low discrepancy sequences in most finance applications. It was noted that low discrepancy sequences provide accurate measures and results when applied in contracts relating to pricing options. The pricing of these financial derivatives is essential in usage in finance. The reasons for this is attributed to the fact that financial institutions need to be aware of prices when coming up with new products in finance or when planning to market some non-liquid assets. Besides this, pricing of financial derivative is essential because a benchmark for comparing the budgeted price and the actual price can be conducted. Should the variance be big, it means that either a profit opportunity or a chance of misprizing. The first breakthrough in the concept of option pricing was advanced by Black and Scholes who advanced a mathematical theory based on prices of stock for option pricing. They found closed form solution that was used to price vanilla European options. Since then however, more another complex derivatives have been advanced on the basis of stochastic equations like multifactor models in mathematics. Analytically, these complex problems in pricing make them hard to solve using these evolved methods. In order to get any feasible solutions, we resort to analytical techniques which fall into three categories relating to finite difference methods, lattice methods or the Monte Carlo methods (Eckhard & Peter, 1992). The application of the MC method of simulation was first accomplished by Boyle in finance. In the option pricing method, a fair price or a set of these fair prices is needed in time, t which equals zero. That is for the prices which give risk free gains against the other party for both the writer and the holder of contract. In order to get these fair prices a model for evolution of underlying stock needs to be assumed. The most common model assumed is that the stochastic process follows a differential equation in stochastic of the form S = (St) t∈[0,T] This equation describes the stock behavior of prices and follows a Brownian motion where St follows a differential equation in stochastic form. dSt = St(μ · dt + σ · dWt) The μ and σ, the vitality of S and the trend are constants that have been previously determined. The solution of the above stochastic equation in differentiations is presented in the form St = S0 · exp(μ −σ2/2)t + σpt · z) where S0 refers to the stock price at a time 0 and z is a normal random variable. The security in charge of options need not necessarily be represented by stock in this equation; rather it can be represented by interest rates or even currencies. Current developments show that, MC methods do not only apply to European derivative styles, but to also to the American options by simulation. There are numerous important derivatives which are either path dependent or high dimension. This means that the price of these derivatives mainly lies on the past history of the security value. Such important examples are mortgage-backed security, Asian options, and American options which mainly applies Monte Carlo simulation because it is effective in computation method so that they will be evaluated. These derivatives cannot apply path dependent securities because they are not valued by use of partial differentiation equation (Akesson & Lehoczy, 2000). Mortgage backed securities are securities which are financed by a collection of mortgage loans. These securities are divided into collateralized mortgage requirements, mortgage securities which are stripped and mortgage securities which are pass through. Mortgage rate is the interest rate which is charged on a loan. For one to price a mortgage security he or she needs to project his cash flows. Cash flows are mainly projected based on the assumption of the rate of prepayment over the span life from the mortgage pool (Mcleish, 2005). Path dependent interest rates are evaluated based on the periodic cash flows which are associated with the mortgage-back security. A path dependent cash flow is defined as cash flows of one period which relies on interest path of the current level and interest rate of current level. Monte Carlo method of simulation is the most effective and flexible method which can be used to value rate of interest of path-dependent income securities which are fixed. The MC method encompasses generating randomly many future rates of interest path scenarios. These rates of interest are generated based on the assumption of volatility of the interest rates. This Monte approach in cases of mortgage securities involves generation of a set of cash flows founded on simulated mortgage refinancing in the future. Random path of rate of interest is created from arbitrage-free approach of the future term structure of rate of interest rates. An arbitrage free model can be defined as a model that duplicates present structure of terms of the rate of interests (Peter, 2002). In order to get reliable estimate of mortgaged backed security price there is need to adjust the paths of interest rate so that one avoids rates of interest reaching a level which are unreasonable like interest rate of zero, high or below the zero. Application of Monte Carlo methods in solving a more complex options like path dependent options involves a partitioning the life’s options into time periods. When binomial model is applied the life option should be longer so that the dimension of the problem is higher. Also more calculation time is needed in order to price the option (Akesson & Lehoczy, 2000). The Mento Carlo method is not well applied in case of American options because of its dissimilarity to the equation of partial differentials; the approach of Monte only assesses the value of the option with assumption of a specified starting time and point. The prices of American options are easily obtained by the use of black Scholes PDE technique because they apply simulation running backwards from the date of expiry. The major factors to consider in the simulation are the intermediate time which is found in between the expiry date or the start time of the option. It is not possible to obtain this information by use of Monte Carlo methods. Derivative pricing with MC. In this case, discrete version of the model by Black-Scholes is considered Log St − log St−1 = μ · dt + σ · dW; t = 1, where μ= r, r is a return which is risk-free, σ is a fixed volatility, dt=one day and dW is a normal random variable whose mean is zero and dt variance. In a given present stock price S0, the price ST at the time when T = τ, the derivative can be obtained by ST (z1, . . . , z_ ) = S0 · e(r−_22 )_dt+_√dtP_i=1 zi When constructing sample N paths, it follows that a single simulation experiment is conducted. 1. Simulation methods in risk management Over the past years, financial institutions have been greatly involved in the quantification of risk exposure as much as the pricing of derivatives. Despite the fact that the models of markets are similar, the approaches and requirements are very varied. In the pricing of derivatives, the main issue is to find out the fair price of a particular contract as accurately as possible. In this case, several risks have to be contented with even for portfolio with several contracts. The measurement of risk in this case is necessary and the management of this risk is essentials and it involves statistical work where a physical measure is used. An important measure of risk which is the value at risk is presented using the Monte Carlo technique (Dessislava & Frank, 2010). When focusing on market risk we suppose that a portfolio consisting of many contracts such as bonds, shares and derivatives is held. At a time, 0, the worth of the portfolio should be well known or can be potentially estimated and the worth of the portfolio at a future time T is unknown and uncertain. The future worth can be expressed by as random variable X to represent this uncertainty. The discounted profit or even loss for the portfolio produced within a range of period [0, T], can be represented by a random variable X/ (1 + r) − a, where r is the riskless return in this range of period. In practice however, t represents one to ten days mostly and the generalization becomes straight forward in non-zero interest rates. The X – a distribution in this case is referred to as the profit/loss distribution schedule (Mcleish, 2005). In determining the risk and management of the risks involved in finance, the most commonly used method is the Value at Risk, VAR. For instance, the VAR at a target horizon [0, T ] and confidence level, c, means that with a probability of c, the loss over the targeted period is less than or equal to VAR. In mathematical analysis, this basically means that VAR is the negative of (1-c) which is a quartile of the profit/loss distribution. Therefore VAR is defined as the number VaR_(X):= −inf{x: P[X − a _ x] > α} where α = 1 − c. The value at risk is not coherent because it does not have any sub-addition properties. This property states that the risk of the portfolio should be less and smaller than the total sum of the risks of the sub- portfolios. In a related example, if we assume that the market forces influencing the portfolio have a normal distribution and that the portfolio depends on each linear factor, then the profit/ loss distribution would be normal. The next step would be to estimate parameters such as mean, variance and covariance of the distribution (Dessislava & Frank, 2010). However, this assumption cannot hold because stock returns do not follow a normal distribution and extreme events are likely to occur in practice. It is therefore the extreme situations and events that the managers in finance want as they lead to losses which necessitate the need for risk management. Besides this, most contracts are not follow a linear distribution when plotted against the underlying market factors. Derivatives form a big generator of losses or profits as a result of the leverage effect. The simulation method aspect is therefore necessary because we cannot compute the profit/loss distribution without using the linear and normal assumptions. The estimation of the profit/loss distribution can be achieved by following four steps relating to: 1. Mark the whole portfolio to the market and identify the current worth, a 2. Generate any possible N scenarios on market evolution in the future 3. Generate and evaluate the net worth of the portfolio at time, T for each scenario and subtract te net worth a. T 4. Lastly , develop an empirical function on distribution his gives N values of the profit and loss that have been simulated F(x) =k/N where k refers to the number of xi that are less than x. This formula is called the standard Quasi-Monte Carlo scheme of integration where the second and third steps are so delicate that they may require appropriate simulation in their own capacity, like for instance, where the portfolio has exotic options. Generation of scenarios A scenario refers to one realization of the market factors in any likely state. Market factors can relate to implied volatilities, interest rates, prices of stock among other. There are two ways in which the scenarios can be generated relating to historical simulation and parametric simulation (Dagpunar, 2007). Historical simulation entails taking the scenarios from past evolution of the market. For instance, if we have a portfolio containing M shares and need to determine the profit/ loss distribution over a horizon of a day and having recorded the daily price movements of all the shares over the previous N days; then we take the daily joint price movements as the N possible scenarios in the future. The advantage of historical simulation is that it is not a must to make any assumptions about the distribution of the factors affecting the markets. As long as the distribution of the market factor changes does not change over time, we can then expect very accurate results where N is made large The more back we relate in history, the larger the value of N becomes and the more likely it becomes for the assumption of constancy to be corrupted. This is the biggest disadvantage of this method. An example of this would be that, taking N=1000 scenarios of daily movements correspond to records relating to four years which present a long time in the current financial markets where a number of financial instruments cannot exist for so long a period. Another disadvantage of historical simulation is that a person cannot test to the parameters of the market, the sensitivity of VAR (Dessislava & Frank, 2010). Parametric simulation fits certain probability distributions such as normal, parametric and hyperbolic distributions to past data. For instance, deterministic simulation has been found to be superior to the classical Monte Carlo approach to simulation. In this approach, two examples were considered. First is a 34-dimentional problem which consisted of 34 at-the-money equity and European call currency options. The second consisted mortgage obligations that had been collateralized from a pool of 30years mortgages with monthly pays, giving a problem with 360-dimention. It was found out that, the estimation error of the classical formulae were ten times higher than that of low discrepancy method after testing the results. In case of the need to allow more generalized distributions, the issue becomes more subtle. The main problem relates to creation of sample points whose correlation structure is predefined. The method that is used to solve this problem is called the Stein’s Algorithm. The method generates a sample having a correlation structure which is prescribed, transforms it into a unit cube without altering the correlation structure and applies the inverse of cumulative function of distribution to all coordinates. In this case, a stratified sample that lacks any random accumulation points is obtained. The need to allow for more complex portfolios such as the dynamic portfolios or those that have derivatives that are path-dependent, takes into account all intermediate values so that specific scenes relate to entire paths (Dagpunar, 2007). Evaluation of net worth There are two ways of conducting an evaluation of a portfolio in a special scenario relating to full valuation and parametric valuation. Full valuation entails marking for each scenario to the market, the whole portfolio. In most cases, the portfolio may contain several contracts which may require separate simulation so as to value them. The cost of computing in such phenomena can be quite infeasible (Mcleish, 2005). Contrary to this, parametric valuation entails replacing by a suitable approximation the profit/loss distribution of the portfolio which can be achieved by interpolation or through Taylor approximation. This approximation may not be the best choice because we are interested in big relative changes which may not be possible because, the method is very accurate for small movements and not accurate for large ones. The important part is to compute profit/loss approximation function for the sub-portfolio that has been chosen from the portfolios. This is achieved by conducting full valuations for a few relative scenarios and interpolating them n between by using a regular grid. However, the presence of several market factors leads to a great number of scenarios. The global or total profit/loss function of approximation can be achieved by summing up the profit/loss approximation functions of the sub-portfolios (Eckhard & Peter, 1992). One big defect of the Monte Carlo methods is that they involve computer experiments that derive estimators of quantities of interests. The estimators are determined by the sampling distributions that are relevant in obtaining the samples. Monte Carlo methods allow one to implement most probability distribution in a powerful manner and they answer most questions relating to P(x). In problems that are of high dimension, the most satisfactory methods that are used are the Markov chain Monte Carlo. Even though simple algorithms are mainly applied, they perform poorly because they explore space in slow walks that re are random in nature and they make use of proposal densities that provide fast movements through space. Simulation optimization is used in providing feasible solutions to problems that are too complex. Recent advancements in this area explore the power of simulation optimization in management of risk. In most cases, this approach is the best in provision of quality solutions and it is more practical. In cases of uncertainty, there are risks which can be defined as the probability of an event occurring which would negatively influence a goal. The portion of the probability that brings forward potential harm is the one that is identified in risk management. Risk management involves identifying, selecting and implementing procedures so as to control risk in a specific issue to an acceptable level. This level of acceptance depends on the attitude of the decision makers, marginal rewards that are expected from having an additional method of mitigating risk and the available situation (Dagpunar, 2007). The optimization method contains a natural appeal because it incorporates mathematical formulations which help arrive at the correct answers which are not subjective. In these methods, there are various approaches designed to cope with uncertainty and the parameters may not be known with certainty but may vary depending on the nature of the values that they represent. The traditional approaches which are scenario- based relate to scenario and robust optimization. These offer feasible solutions and minimize the deviation of the total solution from the optimal solution of all scenarios. The methods however, only use small subsets of potential scenarios and have limited size and complexity in models they can relate to. Monte Carlo methods therefore form a big and useful tool in addressing problems in mathematical applications and thus improve the access of feasible solutions in application. These tools are relevant in eradication of risks and help managers realize their roles in risk assessment and management. The methods which are widely applied in various financial management issues help the managers in realizing their objectives and goals in the management process ( Guias, 1997). Reference list Akesson, F., & Lehoczy, p. 2000. Path generation for quasi Monte Carlo simulation of mortgage-backed securities. Management science. Boyle,P., 1977, `options: A Monte Carlo approach’, journal of Financial Economics, Vol.4, pp.323-338. Dagpunar, J. 2007. Simulation and Monte Carlo: with applications in finance and MCMC. Chichester,England: John Wiley. Dessislava, P. & Frank, J. 2010. Simulation and optimization in finance. John wiley and sons. Eckhard, P. & Peter,E. 1992.numerical solution of Stochastic Differential equations. Springer-Verlag. Guias,F., 1997,` A monte Carlo approach to smoluchowski equations’, Monte Carlo Methods and applications, vol. 3, no.4. Mcleish, D. 2005. Monte Carlo simulation and finance.Hoboken,N.J: wiley. Niederreiter, H. & Talay,D., 2006. Monte Carlo and Quasi-Monte Carlo methods 2004. Berlin: Springer. Peter, J., 2002. Monte Carlo methods in finance. John Wiley and sons. Wang, H. 2012. Monte Carlo simulation with application to finance. Boca Raton: CRC Press. Read More