# Option Pricing Theory

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### Option Pricing Theory

### Background Information

### INTRODUCTION

The portfolios of investor's today incorporate extensive investments in funds, stocksand bonds in which option is an additional form of security that exuberates abundant opportunities to participating investors.The growth in option trading has been accompanied by vast interest among academics and practitioners in the valuing of option contracts. [1]

The efficacy of options is established by its compliance and ability to integrate the business position and adjust it to the situation of the market accordingly. Options can be as hypothetical or as conservative depending on the present circumstances of a business position, in which appropriate propositions can be taken to fully adjust and safeguard the business from the likelihood of a decline and future predicaments. Hence to also use it as a tool for profitable future outlooks but in spite of this options can be a collateral measure and too risky and not always suitable for all businesses within a market.

Therefore bearing in mind the current position it is always beneficial to be fully aware of the ‘risks' of options before making a final decision, which haphazardly does not mean to infinitely straying from options that may signal a gold opportunity to bloom an operation. Therefore it is imperative to fully understand the options available and suitable to be invested in to prevent the transition of the current position to anything insignificant or even procuring a mere loss.

### WHAT ARE OPTIONS

An option is a contract that entitles the buyer the right to buy or sell an element of an asset at or within a certain period of time at a particular price, which solely enables the buyer the right and not the obligation to buy or sell the underlying asset. This option is considered as a safeguard contract with stringent terms and conditions. For example, an investor is engrossed to buying a store that may have significant potential in a specific market and begin to negotiate with its merchandiser in order to purchase it. Unfortunately the investor does not have the financial capital to make the payment for 2 months. After a negotiation with the merchandiser it is agreed to convey an option to secure the purchase in 2 months for £150,000 with £5000 liability. The option acquisition of £5,000 liability may lead to the likelihood of two situations arising theoretically: A) It is discovered that the store is worth more than initially asserted and is worth double the price; therefore you have £150,000 in profit (£150.000 x 2) – £150,000 = £150,000+. Or b) After purchase it is discovered that on the apartment above the store a mother and a child were brutally murdered and individuals are no longer comfortable visiting that area hence low sales, therefore the store is now worthless. Constructively since an option was purchased you are under no obligation to proceed with the sale and make a loss of £5000 for that option and under no obligation to make the final purchase.

Fortunately, buying such an option and the regard of not being under any obligation to purchase meant a risky investment was void in the second scenario. And you were under no obligation to make the final purchase. Or if the expiration date had gone by, the option would have become worthless together with the 100% loss of investment. Therefore an option is a mere contract that accords with an underlying asset which generates a financial value in relation to having rights without any obligation. Hence options are known as derivates, meaning that an option derives its value from something else. In this example the store is the underlying asset in which the underlying asset is a stock or an index.

### Pay off Diagram for Equity as a Call Option

Acall optionauthorises the holder the right to buy an asset at a particular price within a specific period of time, also known as the ‘strike' or the ‘exercise' price. Figure 1 illustrates an investor that buys a European call option with a strike price trading at $50 expiring in a month's time. Distinctly it exhibits the pattern of price for an option originally purchased for $4. Suppose that the current stock price is $46 with an initial investment of $400. The investor can only exercise on the expiration date since the option is European. For share prices above $50 on the expiration date, it would be secure to exercise the call and gain by the difference between the share price and exercise price. Let's assume that the stock price is $60. By exercising the option, the investor is able to buy 100 shares for $50 per share. If the shares are sold immediately, the investor secures $10 per share.

(100 x $10=$1000). Thus, taking into account the initial cost of the option, the net profit to the transaction is $1000-$400=$600. For stock prices less than $54, the investor makes a loss considering that the pay off from exercising the option is less than the cost of the call. In such cases, the investor may not be inclined to exercise the option as it may lead to an overall loss of $400 which is substantial than what could be gained by exercising the option.

### Pay off Diagram for Equity as a Put Option

Unlike the call option, the holder of the option would be waiting for the stock prices to rise; whereas the holder of the put option would be hoping for the stock prices to decline. Let's examine figure 2 which illustrates an investor buying a European put option with a strike price trading at $50 expiring in a month's time. Figure 2 exhibits the pattern for the price of an option originally sold for $4. Suppose that the current stock price is $46 with an initial investment of $400. The investor can only exercise on the expiration date since the option is European. For share prices above $50 on the expiration date, it would pay to exercise the call and gain by the difference between the share price and exercise price. Let's assume that the stock price is $41 at the expiration of the option. By exercising the option, the investor is able to buy 100 shares for $50 per share. If the shares are sold immediately, the investor secures a gain of $9 per share (100*$9=$900). Thus, net gain is $900-$400=$500 after taking into account the initial cost of the option. For stock prices more than $46, the investor makes a loss since the amount gained from exercising the option is less than the cost of the call. The investor gains profit if the price of stock strikes below $46, since the amount gained from exercising the option is more than the cost of the call.

### Option Style

Options appear in various contract limitations. The two most commonly exercised are European and American. European options are only exercised at a fixed pre-determined expiration date whereas American options are exercised at any time before and up to the agreed contract length.

### Underlying Asset

The underlying asset represents the stock that the option can be used to purchase. Options trade on four main underlying assets such as stock option, currency options, index options and futures options.

### Specification of Stock Options

v Strike Price - The price at which underlying assets can be purchased or sold at when the option is exercised. The strike price intervals are spaced $2.5, $5, or $10 apart. The spacing increases as the stock price escalates and vice-versa in which the spacing of $2.5 is used when the stock price is between $5 and $25; $5 is used when the stock price is between $25 and $200, and $10 for stock prices over $200. [2]

v Expiration Date - Option contracts expire after a period of time. When option contracts are expired, the right to exercise does not exist any longer, therefore the stock option becomes worthless. The expiration date for each option contract depends on the type of option. For instance, for U.S. exchange-listedstockoption contracts expire is the Saturday following the third Friday of theexpirationmonth unless that Friday is a market holiday, in which case the expiration is on the Friday.

### Binomial Option Pricing Model

Binomial option pricing model is a simple and popular technique but yet powerful technique used to figure out many complex option pricing problems. Binomial option pricing model is mathematically simple and it is based on the assumption of no arbitrage. The model begins with a binomial tree of discrete future possible underlying stock prices and then constructs a riskless portfolio of an option and stock using a simple formula to calculate the option price at the end of each node. The binomial tree illustrates different paths that might be tracked by the stock price over the life the option.

### One Step Binomial Tree

Consider a very simple situation where a share of stock is currently selling at $50 per share and at the end of next month it will either increase $75 or decrease to $25. It is not possible to have any other outcomes over the next month for this stock's price. If investor is interested in buying the stock for $65 and the stock turns out to be $75, the value of option will be $15; if the stock price turns out to be $25, the value of the option will be zero. The only assumption need in this model is that arbitrage opportunities do not exist.

An option can be priced when stock price movements are provided, with the only assumption to be no arbitrage opportunities. Thus,

Equation 1

Equation 2

### Two Step Binomial Trees

Suppose we were to extend the one-step to two-step binomial tree, and instead of having only a single price change during the month, let's assume that the price change occurs once every 2 weeks. Thus, by dividing the month into 2 periods, 3 possible outcomes are concluded at the end of the month. In this case, the aim is to calculate the option price at the initial node of the tree by repeatedly applying the equations established earlier. The option prices at the final nodes of the tree are calculated as the pay-offs from the option.

### Estimating the Binomial Stock price processes

Subsequently, knowing the value of an option at period one leads to calculation of the value at period zero by playing as if there is one period to go. In this iterative manner the Equation 3 can be derived and the value of the call with n periods to go would be Equation 3

### References:

Elton E.J. et al (2007) Modern portfolio Theory & Investment Analysis, 7th edition, USA: Wiley & Son. pp. 575-601.

Hull, J.C. (2008) Fundamentals of Futures and Options Markets, 6th edition. New Jersey: Pearson. pp.247-261.

Hull, J.C. (2009) Options, Future and Other Derivatives, 6th edition, New Jersey: Pearson. pp,179-186, 237-251.

### Journal Resources:

Cox J.C et al (2001) ‘Option Pricing: A Simplified Approach', Journal of Financial Economics, vol.3, September, pp 12-27.

Pollard M. (2005) ‘Monte Carlo and Binomial Tree Methods in Pricing of Options', Journal of finance, vol.1, July, pp 12-18.

### Online Resources:

[http://www2.imperial.ac.uk/~mdavis/course_material/MOP/ROBUST.PDF]

[http://www.aims.ac.za/resources/archive/2005/femi.pdf]

[http://www.cs.rpi.edu/~magdon/ps/conference/martingaleNIPS05.pdf]

[http://pages.stern.nyu.edu/~adamodar/pdfiles/option.pdf]

[http://www.finint.ase.ro/Materiale/Manuale/Investment%20Valuation_Damodaran/ch5.pdf]

[http://pages.stern.nyu.edu/~adamodar/New_Home_Page/lectures/opt.html]

### Draft Plan

### Aim:

To create an application in VBA that is capable of decomposing the FTSE 100 stock price data through the use simulation techniques to forecast the value of an option. This price of the option can then be obtained by discounting the average of the simulated pay-offs. It is intriguing to make this comparison by using a progressively larger number of simulations.

The encoded VBA program is expected to deliver the following tasks:

ü Calculate the corresponding past returns and the volatility

ü Model the stock price after the specified time using simulation via Geometric Brownian motion

ü Model the option's pay-off (based on the type of the option, whether is a call or a put)

ü Calculate the price of the option using either Binomial tree approximation or Black-Scholes option pricing model.

### Historical Data:

Download historical prices (preferably observed monthly) of a stock for the past 100 months from the FTSE100, these should therefore be frequently traded. The data can be accessed and downloaded via data stream.

### Software Design and Implementation

Microsoft Excel 2007, particularly VBA will be utilised for inputting the opted modular system into an excel spreadsheet. VBA application represents systematic design, implement, and test medium-sized program. This application is used for the purpose of using pseudo code to input data to produce accurate theoretical values and graphs. It is also used for its ease of use and time constraints.

### Conclusion

The development of several models for pricing possible claims is a fairly recent and important contribution. [3] The theory behind these models and their use in pricing options has been examined. Binomial option pricing model develops estimates of the price process of the underlying asset over the maturity of the option and then overlay the option payoffs given the values of the underlying asset. Binomial model is more flexible compared to the other approaches resulting in widespread use of it to price a variety of options.

### Project Structure:

### Abstract

### Contents

### Chapter 1: Introduction

This chapter focuses on modern Option Pricing techniques and provides the reader with a good general understanding of the reasons behind carrying this project and what I am trying to achieve. This chapter gives the readers a brief description of the main analysis techniques and evaluation methods and financial background on the data. A Gantt chart will also be included to visualize the estimated time scale, project deadlines.

### Chapter 1.1: Data Collection

The aim is to collect historic Data from Data Stream and to be used for testing purposes and also to describe how it was collected, stored and executed.

### Chapter 1.2: Methodology

This carries out software implementation of the project using VBA and how to build the model application using pseudo code. Variables, functions and macros are also elaborated in details.

### Chapter 2: Option Pricing Theory Background

The background will give more in-depth information about different aspects that corresponds to option pricing, what it is composed of.

### Chapter 2.1: Basics of Option Pricing

This chapter takes it in depth to understand the variety aspects of option pricing, and their contributions to value options.

### Chapter 2.2: Binomial Option Pricing Model

This chapter expands on the Binomial tree technique and valuation of an option manually via this model.

### Chapter 3: Testing results and Evaluation of software

This section discusses the accuracy of the findings obtained from the application in relation to the original literature reviewed. What further features shall be added to the software and what unnecessary features shall be axed to improve the software!

### Chapter 4: Data Comparisons

This chapter will describe the outputs of the program. Present some graphs of processed and raw data, describe them. Describe the numeric outputs of the program, what they mean.

### Chapter 5: Conclusion

This chapter will discuss the strengths and weaknesses of the software application and what has been achieved throughout this project.

### Chapter 6: Recommendations

This chapter looks closely at possibility of any improvements could be made to the application either by adding new features or extending the models used to achieve a better accuracy of results.

### Bibliography:

This will containlist of all the sources and references used in order to gather relevant information to complete this project.

### Appendix:

This includes any extra information to support this project.

### User Manual:

This manual is used for novice users that explain how to use or operate around the host application. It will also include sub-function procedures for amendment purposes in the future.