### Option Pricing Theory

### Background Information

### Introduction

The portfolios of investor ‘s today incorporate extended investings in financess, stocksand bonds in which option is an extra signifier of security that exuberates abundant chances to take parting investors.The growing in option trading has been accompanied by huge involvement among faculty members and practicians in the valuing of option contracts. [ 1 ]

The efficaciousness of options is established by its conformity and ability to incorporate the concern place and adjust it to the state of affairs of the market consequently. Options can be every bit conjectural or as conservative depending on the present fortunes of a concern place, in which appropriate propositions can be taken to to the full set and safeguard the concern from the likeliness of a diminution and future quandaries. Hence to besides utilize it as a tool for profitable future mentalities but in malice of this options can be a indirect step and excessively hazardous and non ever suited for all concerns within a market.

Therefore bearing in head the current place it is ever good to be to the full cognizant of the ‘risks ‘ of options before doing a concluding determination, which randomly does non intend to boundlessly rolling from options that may signal a gold chance to blossom an operation. Therefore it is imperative to to the full understand the options available and suited to be invested in to forestall the passage of the current place to anything undistinguished or even securing a mere loss.

### WHAT ARE OPTIONS

An option is a contract that entitles the purchaser the right to purchase or sell an component of an plus at or within a certain period of clip at a peculiar monetary value, which entirely enables the purchaser the right and non the duty to purchase or sell the implicit in plus. This option is considered as a precaution contract with stringent footings and conditions. For illustration, an investor is engrossed to purchasing a shop that may hold important potency in a specific market and get down to negociate with its merchant in order to buy it. Unfortunately the investor does non hold the fiscal capital to do the payment for 2 months. After a dialogue with the merchant it is agreed to convey an option to procure the purchase in 2 months for ?150,000 with ?5000 liability. The option acquisition of ?5,000 liability may take to the likeliness of two state of affairss originating theoretically: A ) It is discovered that the shop is worth more than ab initio asserted and is deserving double the monetary value ; therefore you have ?150,000 in net income ( ?150.000 x 2 ) – ?150,000 = ?150,000+ . Or B ) After purchase it is discovered that on the flat above the shop a female parent and a kid were viciously murdered and persons are no longer comfy sing that country therefore low gross revenues, hence the shop is now worthless. Constructively since an option was purchased you are under no duty to continue with the sale and do a loss of ?5000 for that option and under no duty to do the concluding purchase.

Fortunately, purchasing such an option and the respect of non being under any duty to buy intend a hazardous investing was void in the 2nd scenario. And you were under no duty to do the concluding purchase. Or if the termination day of the month had gone by, the option would hold become worthless together with the 100 % loss of investing. Therefore an option is a mere contract that agreements with an implicit in plus which generates a fiscal value in relation to holding rights without any duty. Hence options are known as derivates, intending that an option derives its value from something else. In this illustration the shop is the implicit in plus in which the implicit in plus is a stock or an index.

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

Acall optionauthorises the holder the right to purchase an plus at a peculiar monetary value within a specific period of clip, besides known as the ‘strike ‘ or the ‘exercise ‘ monetary value. Figure 1 illustrates an investor that buys a European call option with a work stoppage monetary value trading at $ 50 run outing in a month ‘s clip. Distinctly it exhibits the form of monetary value for an option originally purchased for $ 4. Suppose that the current stock monetary value is $ 46 with an initial investing of $ 400. The investor can merely exert on the termination day of the month since the option is European. For portion monetary values above $ 50 on the termination day of the month, it would be unafraid to exert the call and addition by the difference between the portion monetary value and exercising monetary value. Let ‘s presume that the stock monetary value is $ 60. By exerting the option, the investor is able to purchase 100 portions for $ 50 per portion. If the portions are sold instantly, the investor secures $ 10 per portion.

( 100 x $ 10= $ 1000 ) . Therefore, taking into history the initial cost of the option, the net net income to the dealing is $ 1000- $ 400= $ 600. For stock monetary values less than $ 54, the investor makes a loss sing that the wage off from exerting the option is less than the cost of the call. In such instances, the investor may non be inclined to exert the option as it may take to an overall loss of $ 400 which is significant than what could be gained by exerting 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 monetary values to lift ; whereas the holder of the put option would be trusting for the stock monetary values to worsen. Let ‘s examine figure 2 which illustrates an investor purchasing a European put option with a work stoppage monetary value trading at $ 50 run outing in a month ‘s clip. Figure 2 exhibits the form for the monetary value of an option originally sold for $ 4. Suppose that the current stock monetary value is $ 46 with an initial investing of $ 400. The investor can merely exert on the termination day of the month since the option is European. For portion monetary values above $ 50 on the termination day of the month, it would pay to exert the call and addition by the difference between the portion monetary value and exercising monetary value. Let ‘s presume that the stock monetary value is $ 41 at the termination of the option. By exerting the option, the investor is able to purchase 100 portions for $ 50 per portion. If the portions are sold instantly, the investor secures a addition of $ 9 per portion ( 100* $ 9= $ 900 ) . Therefore, net addition is $ 900- $ 400= $ 500 after taking into history the initial cost of the option. For stock monetary values more than $ 46, the investor makes a loss since the sum gained from exerting the option is less than the cost of the call. The investor additions net income if the monetary value of stock work stoppages below $ 46, since the sum gained from exerting the option is more than the cost of the call.

### Option Style

Options appear in assorted contract restrictions. The two most normally exercised are European and American. European options are merely exercised at a fixed pre-determined termination day of the month whereas American options are exercised at any clip before and up to the agreed contract length.

### Underliing Asset

The implicit in plus represents the stock that the option can be used to buy. Options trade on four chief underlying assets such as stock option, currency options, index options and hereafters options.

### Specification of Stock Options

V Strike Price – The monetary value at which underlying assets can be purchased or sold at when the option is exercised. The work stoppage monetary value intervals are separated $ 2.5, $ 5, or $ 10 apart. The spacing increases as the stock monetary value escalates and vice-versa in which the spacing of $ 2.5 is used when the stock monetary value is between $ 5 and $ 25 ; $ 5 is used when the stock monetary value is between $ 25 and $ 200, and $ 10 for stock monetary values over $ 200. [ 2 ]

5 Expiration Date – Option contracts expire after a period of clip. When option contracts are expired, the right to exert does non be any longer, therefore the stock option becomes worthless. The termination day of the month for each option contract depends on the type of option. For case, for U.S. exchange-listedstockoption contracts expire is the Saturday following the 3rd Friday of theexpirationmonth unless that Friday is a market vacation, in which instance the termination is on the Friday.

### Binomial Option Pricing Model

Binomial option pricing theoretical account is a simple and popular technique but yet powerful technique used to calculate out many complex option pricing jobs. Binomial option pricing theoretical account is mathematically simple and it is based on the premise of no arbitrage. The theoretical account begins with a binomial tree of distinct hereafter possible implicit in stock monetary values and so constructs a risk-free portfolio of an option and stock utilizing a simple expression to cipher the option monetary value at the terminal of each node. The binomial tree illustrates different waies that might be tracked by the stock monetary value over the life the option.

### One Step Binomial Tree

See a really simple state of affairs where a portion of stock is presently selling at $ 50 per portion and at the terminal of following month it will either increase $ 75 or diminish to $ 25. It is non possible to hold any other results over the following month for this stock ‘s monetary value. If investor is interested in purchasing the stock for $ 65 and the stock turns out to be $ 75, the value of option will be $ 15 ; if the stock monetary value turns out to be $ 25, the value of the option will be zero. The lone premise demand in this theoretical account is that arbitrage chances do non be.

An option can be priced when stock monetary value motions are provided, with the lone premise to be no arbitrage opportunities. Therefore,

Equation 1

Equation 2

### Two Step Binomial Trees

Suppose we were to widen the one-step to two-step binomial tree, and alternatively of holding merely a individual monetary value alteration during the month, allow ‘s presume that the monetary value alteration occurs one time every 2 hebdomads. Therefore, by spliting the month into 2 periods, 3 possible results are concluded at the terminal of the month. In this instance, the purpose is to cipher the option monetary value at the initial node of the tree by repeatedly using the equations established before. The option monetary values at the concluding nodes of the tree are calculated as the pay-offs from the option.

### Estimating the Binomial Stock monetary value procedures

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

### Mentions:

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

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

Hull, J.C. ( 2009 ) Options, Future and Other Derived functions, 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.

### On-line Resources:

[ hypertext transfer protocol: //www2.imperial.ac.uk/~mdavis/course_material/MOP/ROBUST.PDF ]

[ hypertext transfer protocol: //www.aims.ac.za/resources/archive/2005/femi.pdf ]

[ hypertext transfer protocol: //www.cs.rpi.edu/~magdon/ps/conference/martingaleNIPS05.pdf ]

[ hypertext transfer protocol: //pages.stern.nyu.edu/~adamodar/pdfiles/option.pdf ]

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

[ hypertext transfer protocol: //pages.stern.nyu.edu/~adamodar/New_Home_Page/lectures/opt.html ]

### Draft Plan

### Purpose:

To make an application in VBA that is capable of break uping the FTSE 100 stock monetary value informations through the usage simulation techniques to calculate the value of an option. This monetary value of the option can so be obtained by dismissing the norm of the fake pay-offs. It is fascinating to do this comparing by utilizing a increasingly larger figure of simulations.

The encoded VBA plan is expected to present the undermentioned undertakings:

u Calculate the corresponding yesteryear returns and the volatility

u Model the stock monetary value after the specified clip utilizing simulation via Geometric Brownian gesture

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

u Calculate the monetary value of the option utilizing either Binomial tree estimate or Black-Scholes option pricing theoretical account.

### Historical Datas:

Download historical monetary values ( sooner observed monthly ) of a stock for the past 100 months from the FTSE100, these should hence be often traded. The informations can be accessed and downloaded via informations watercourse.

### Software Design and Implementation

Microsoft Excel 2007, peculiarly VBA will be utilised for inputting the opted modular system into an excel spreadsheet. VBA application represents systematic design, implement, and test moderate-sized plan. This application is used for the intent of utilizing pseudo codification to input informations to bring forth accurate theoretical values and graphs. It is besides used for its easiness of usage and clip restraints.

### Decision

The development of several theoretical accounts for pricing possible claims is a reasonably recent and of import part. [ 3 ] The theory behind these theoretical accounts and their usage in pricing options has been examined. Binomial option pricing theoretical account develops estimations of the monetary value procedure of the implicit in plus over the adulthood of the option and so cover the option final payments given the values of the implicit in plus. Binomial theoretical account is more flexible compared to the other attacks ensuing in widespread usage of it to monetary value a assortment of options.

### Undertaking Structure:

### Abstraction

### Contentss

### Chapter 1: Introduction

This chapter focuses on modern Option Pricing techniques and provides the reader with a good general apprehension of the grounds behind transporting this undertaking and what I am seeking to accomplish. This chapter gives the readers a brief description of the chief analysis techniques and rating methods and fiscal background on the information. A Gantt chart will besides be included to visualise the estimated clip graduated table, undertaking deadlines.

### Chapter 1.1: Datas Collection

The purpose is to roll up historic Data from Data Stream and to be used for proving intents and besides to depict how it was collected, stored and executed.

### Chapter 1.2: Methodology

This carries out package execution of the undertaking utilizing VBA and how to construct the theoretical account application utilizing pseudo codification. Variables, maps and macros are besides elaborated in inside informations.

### Chapter 2: Option Pricing Theory Background

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

### Chapter 2.1: Basicss of Option Pricing

This chapter takes it in deepness to understand the assortment facets of option pricing, and their parts to value options.

### Chapter 2.2: Binomial Option Pricing Model

This chapter expands on the Binomial tree technique and rating of an option manually via this theoretical account.

### Chapter 3: Testing consequences and Evaluation of package

This subdivision discusses the truth of the findings obtained from the application in relation to the original literature reviewed. What farther characteristics shall be added to the package and what unneeded characteristics shall be axed to better the package!

### Chapter 4: Datas Comparisons

This chapter will depict the end products of the plan. Show some graphs of processed and natural informations, depict them. Describe the numeral end products of the plan, what they mean.

### Chapter 5: Decision

This chapter will discourse the strengths and failings of the package application and what has been achieved throughout this undertaking.

### Chapter 6: Recommendations

This chapter looks closely at possibility of any betterments could be made to the application either by adding new characteristics or widening the theoretical accounts used to accomplish a better truth of consequences.

### Bibliography:

This will containlist of all the beginnings and mentions used in order to garner relevant information to finish this undertaking.

### Appendix:

This includes any excess information to back up this undertaking.

### User Manual:

This manual is used for novitiate users that explain how to utilize or run around the host application. It will besides include sub-function processs for amendment intents in the hereafter.