A Practical Application of Monte Carlo Simulation for Options Pricing

Abstract

In this paper, an attempt has been made to describe a practical application of the Brownian-walk Monte Carlo simulation in option pricing. This simple Monte Carlo routine is useful in option pricing and forecasting productivity, installation rates, labour trends, etc. While Monte Carlo simulation is very useful and relevant to calculate the “P50 value” for contingency planning, the true strength of Monte Carlo simulation is in data extrapolation or forecasting. This paper throws light on some basic elements of Monte Carlo simulation approach for its application. The model can assist corporates to develop unique and accurate nearterm market insights and trends in order to compete in the marketplace on analytics. Hence, in this paper, in particular, an attempt has been made to first study an options pricing (OP) model that produces an analytical solution, and then analyze two numerical options pricing models in terms of accuracy.

Introduction

Monte Carlo simulation (also known as the Monte Carlo Method) lets us see all the possible outcomes of our decisions and assess the impact of risk, allowing for better decision making under uncertainty. Monte Carlo simulation is a computerized mathematical technique that allows people to account for risk in quantitative analysis and decision making. Monte Carlo simulation furnishes the decision-maker with a range of possible outcomes and the probabilities that will occur for any choice of action. It shows the extreme possibilities—the outcomes of going for broke and for the most conservative decision—along with all possible consequences for middle-of-the-road decisions. The technique was first used by scientists working on the atom bomb; it was named after Monte Carlo, the Monaco resort town renowned for its casinos. Since its introduction in World War II, Monte Carlo simulation has been used to model a variety of physical and conceptual systems. Probability is a way to bracket the volatility of short-term forecasts (seemingly random data). Monte Carlo simulation is a specialized probability application that is no more than an equation where the variables have been replaced with a random number generator. In other words, Monte Carlo is another computer approximation routine or numerical method that replaces geometry, calculus, etc. A Monte Carlo simulation is a method for iteratively evaluating a deterministic model using sets of random numbers as inputs. This method is often used when the model is complex, non-linear, or involves more than just a couple uncertain parameters.

Monte Carlo Simulation Function – Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—a probability distribution—for any factor that has inherent uncertainty. It then calculates results over and over, each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete. Monte Carlo simulation produces distributions of possible outcome values. By using probability distributions, variables can have different probabilities of different outcomes occurring. Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis. Common probability distributions include:

a. Normal –The user simply defines the mean or expected value and a standard deviation to describe the variation about the mean. Values in the middle near the mean are most likely to occur. It is symmetric and describes many natural phenomena such as people’s heights. Examples of variables described by normal distributions include inflation rates and energy prices.

b. Lognormal – Values are positively skewed, not symmetric like a normal distribution. It is used to represent values that don’t go below zero but have unlimited positive potential. Examples of variables described by lognormal distributions include real estate property values, stock prices, and oil reserves.

c. Uniform – All values have an equal chance of occurring, and the user simply defines the minimum and maximum. Examples of variables that could be uniformly distributed include manufacturing costs or future sales revenues for a new product.

d. Triangular – The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur. Variables that could be described by a triangular distribution include past sales history per unit of time and inventory levels.

e. PERT– The user defines the minimum, most likely, and maximum values, just like the triangular distribution. Values around the most likely are more likely to occur. However, values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized. An example of the use of a PERT distribution is to describe the duration of a task in a project management model.

f. Discrete – The user defines specific values that may occur and the likelihood of each. An example might be the results of a lawsuit: 20% chance of positive verdict, 30% change of negative verdict, 40% chance of settlement, and 10% chance of mistrial.

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