This study chiefly covers the different pricing theoretical accounts that are used chiefly to monetary value the European options. The study briefly explains the basic option constructs in the first portion covering the option rudimentss, different places possible with options for the option holders and the factors that affect the monetary value of the option. The study so describes both Binomial pricing theoretical account and Black-Schole option pricing theoretical account. The ulterior portion of the study covers some algorithms and simulations done to find the monetary value of the options utilizing excel and VBA.

1. Introduction

## 1.1 Background

Fiscal derived function is a fiscal instrument whose value is determined by an implicit in plus or variable. The implicit in plus is largely a tradable plus but can be a non tradable plus besides. These derived functions are largely traded on the exchanges all over the universe, but the trade can besides be Over the Counter ( OTC ) . The derived functions are by and large used for guess or to fudge the market hazard against any security or plus.

The Assets that are used as underlying for the derived functions, largely determine their monetary values. For illustration, a stock option derived function will find its value from the value of the underlying stock. Option is one of the most popular derivative instruments that are used by the investors all over the universe. An Option is a contract between two parties where the purchaser pays a amount of money, the option monetary value or premium to another party and the marketer receives the right and non the duty to purchase or sell an implicit in plus at a fixed monetary value during the specified clip period[ 1 ].

There are different types of the options available in the market. The most popular 1s are European and American options. These options are different from each other merely because the European option can be exercised by the option holder merely when the option matures nevertheless the American option can be exercised by the option holder any clip during the life clip of the option.

## 1.2 Report Structure

This study is divided into four different subdivisions. Section one covers a brief debut of this study and provides a brief background of derived functions and options. The 2nd subdivision covers some basic option constructs, defines assorted places that are possible for the option holders and the different factors that affect the option pricing. This subdivision besides covers the two chief option pricing theoretical accounts ( Binomial Option pricing theoretical account and Black-Schole Model ) that are widely used for option pricing. Section three covers the simulation, algorithm and VBA codes that I used for option pricing. The last subdivision discusses the simulation consequences obtained from the VBA codifications and concludes the thesis.2. Basic Option Concepts

This chapter discusses the basic types of Options that are available in the fiscal markets, their chief features and belongingss.

## 2.1 Option Basicss

There are two cardinal types of Options in the fiscal market. A Call option and a Put option. A Call Option gives the holder of the option the right to purchase an plus by a certain day of the month for a certain monetary value. A Put Option gives the holder the right to sell an plus by a certain day of the month for a certain monetary value. The day of the month specified in the Option contract is called Expiration day of the month or Maturity day of the month and the monetary value specified in the Option contract is called Exercise monetary value or Strike monetary value. Options can either be American Options or European Options. American Option can be exercised by the holder at any clip up to the termination day of the month ; nevertheless European option can be exercised merely on the termination day of the month. Most of the times the, options that get traded on the exchange are American options.

Options at times are referred as ‘in the money ‘ , ‘at the money ‘ , or ‘out of money ‘ . An ‘in the money ‘ option would give a positive hard currency flow to the option holder, if the option is exercised instantly. Similarly, ‘at the money ‘ option would give a nothing hard currency flow and ‘out of money ‘ option would give a negative hard currency flow, to the option holder if exercised instantly. If S is the stock monetary value and K is the work stoppage monetary value so whenever[ 2 ],

There are two sides for every option contract. One side is for the investor who has taken the long place i.e. has bought the option, and other side is for the investor who has taken a short place i.e. has written or sold the option. The individual who has written the option receives the hard currency up front, but may hold possible liabilities subsequently. The author ‘s net income or loss is precisely the antonym of that of the buyer of the option. There are four types of option place that are possible.

Some stock exchanges like Chicago Board Options Exchange species a place bound and exercising bound on the option contracts. These bounds are chiefly to forestall the market from being influenced by the activities of a individual investor or a group of investors. With the place bound, the maximal figure of contracts an investor can keep on the one side of the market is defined whereas for the exercising bound defines the maximal figure of option contracts an investor or a group of investors can put to death in a period of five back-to-back concern yearss.

## Current Stock Monetary value

For a call option, the final payment is ever the difference between stock monetary value and work stoppage monetary value. Hence as the stock monetary value of the implicit in increases the monetary value of the call option additions and as the stock monetary value decreases the monetary value of the call option lessenings. The monetary value of the option is straight relative to the monetary value of the stock. For a put option its merely frailty versa of a call option i.e. the monetary value of the put option is reciprocally relative to the monetary value of the stock.

## Strike Monetary value

As the work stoppage monetary value of a call option increases the monetary value of the option lessenings and as the work stoppage monetary value of a call option lessenings, the option monetary value additions. i.e. the relationship is reverse. For the put option as the work stoppage monetary value of a put option increases the option monetary value additions and as the work stoppage monetary value of the put option decreases the option monetary value lessenings i.e. the relationship is direct.

## Time to Expiration

This will non hold any consequence to the European options as the option holder can merely put to death the option at termination, but for the American options, As the option comes closer to the termination day of the month ( both Put and Call ) , the value of the option additions, or at-least remain the same. This is chiefly because the proprietor of the longer continuance option ever has the right and chance to put to death it even at a shorter clip continuance. Hence there should non be any difference in the monetary value, nevertheless sometimes when a dividend is expected on an implicit in stock, this may take to diminish in the monetary value of the option to set the entire hard currency flow with the longer term option.

## Volatility

The volatility of the stock is the step of uncertainness about the future stock monetary value motions. As the volatility increases the chance of a stock making really good or really bad additions. Since the call option will bask more returns when the stock monetary value additions and will non be affected by the same degree when the stock monetary value lessenings. Besides for a put option the option holder will bask the returns when the stock monetary value lessenings but will non be affected by the same grade if the stock monetary value additions. Hence for both call and put options, the monetary value of the option increases as the volatility in the stock monetary value additions, and decreases as the volatility of the stock monetary value lessenings.

## Risk Free Interest Rate

As the hazard free rate increases the needed return by an investor on a stock besides increases and the present value of any future hard currency flow of a holder of a option decreases. Hence the value of a call option give that the involvement rates are increasing and all other factors are changeless additions. However, in the instance of a put option the value of the option decreases with increasing involvement rates.

## Future Dividends

Dividends by and large cut down the monetary value of the stock on the ex-dividend day of the month. Since it is impacting the stock monetary value the behaviour of the stock option is same as that of increase/decrease of the stock monetary value of the stock.

## 2.2 Option Pricing Models

This subdivision describes some option pricing theoretical accounts

## 2.2.1 Binomial Pricing Model

The binomial pricing theoretical account besides known as Cox-Ross-Rubinsten pricing theoretical account was foremost used by Sharpe in 1978 as an intuitive manner to explicate Option pricing. The theoretical account was further enhanced by Cox et Al, Rendleman and Bartter in 1979, who explained the execution of this theoretical account and demonstrated the nexus between the Binomial theoretical account and the Black-Scholes theoretical account and proved that this theoretical account provides a traceable manner to monetary value options for which early exercising may be optimum. The basic premise here for this theoretical account is that the monetary value of the implicit in plus follows a binomial distribution i.e. the monetary value can either travel up or down by certain value. This method assumes that in every clip measure, the stock has a certain chance of increasing by a certain sum and have certain chance of diminishing by a certain sum. For illustration, the stock monetary value will travel up or down from the current monetary value of Second: up to Su and down to Sd with chance P and ( 1 – P ) severally.

## 2.2.1.1 Derivation of the Binomial Model

“ Hazard less portfolios, must in the absence of arbitrage chances, earns the hazard free rate of return. ” ( From John C. Hull – Options, hereafters and other derived functions, page 238 ) . To farther generalise the no arbitrage statement let us see a stock whose monetary value is S0. Besides let ‘s see an option whose monetary value is f and the clip period of option is T and during the life span of the option the monetary value of the stock can either travel up S0u or down S0d. Besides lets assume that the stock monetary value either moves up by a praportional sum U or moves down by a praportional sum d. Hence the per centum addition in the stock monetary value during the up motion is ( u-1 ) and down motion the per centum lessening is ( 1-d ) . When the monetary value moves up the final payment for the option is fu and when the monetary value moves down the final payment is fd. Let us see a portfolio consisting of a long place in I” portions and a short place in one option. The end is to happen the value of I” that will do the portfolio hazard less.

As we can see for the above expression, the monetary value of the option is non affected by the chance of the stock monetary value traveling up or down. This is chiefly because we are non valuing the option in the absolute footings, instead we are ciphering its value based of the value of the underlying stock. As a consequence the chance of the stock monetary value traveling up or down is already incorporated in the monetary value of the stock and it is non required to include it once more for the option pricing.

## 2.2.2 Black-Scholes Model

In 1970, Fischer Black, Myron Scholes and Robert Merton came up with a new option pricing theoretical account that redefined the methodological analysiss used for option pricing. In the twelvemonth 1997 the Nobel Prize for economic science was awarded to All these three gentlemen for the find of this new option pricing methodological analysis[ 7 ]. This theoretical account has been used since last 3 decennaries for valuing the European call and put options. The theoretical account explains how the volatility can either be estimated from historical informations or implied from option monetary values. We can utilize this theoretical account to cover with the European call and put options on dividend paying stocks.

This subdivision discusses and compares two different option pricing algorithms. To transport out the simulation for both the theoretical accounts I created some VBA codifications that are discussed below. Apart from this I found an excel sheet provided as a supplement stuff along with the text book “ An Introduction to Derived functions and Risk Management ” written by Don M. Chance and Robert Brooks. This text book has this sheet with a better macro written in VBA covering more scenarios and besides ciphering the option Greeks every bit good. I have used this sheet for the consequence analysis for my thesis[ 9 ]. The instructions to utilize this excel sheet are given in the appendix for mention.

## 3.1 Black-Scholes Model Simulation

This subdivision has some sample VBA codification, I wrote for the Black-Schole Model simulation. For simulation this theoretical account uses the Put-Call para belongings of options to find the option monetary value. Besides I have assumed that there are no dividends paid for the stock.

On this sheet we can input the parametric quantities as shown on the right manus side and the value of both Put option and Call option is automatically calculated utilizing the Black-Schole pricing theoretical account. To run the Binomial theoretical account, we enter the value of the figure of stairss we want the theoretical account to run and press the button “ Run Binomial Model ” and the monetary values for both European and American options are automatically calculated.

The theoretical account besides allow the user to stipulate the dividend output of the plus, and see the output is it is uninterrupted or distinct. In instance of distinct output, we can come in the output in the tabular array on the left underside of the sheet and can cipher the monetary value of both Put and Call option utilizing the Black-Schole theoretical account.

Besides we can detect that as the figure of stairss for the Binomial theoretical account additions to a big value the option monetary value for both call and set option converges to the value calculated by the Black-Schole theoretical account.

Insert values in highlighted cells. Risk-free rate, standard divergence and output can be entered as denary or per centum ( e.g. , .052 or 5.2 for 5.2 % ) . Choice signifier ( distinct or uninterrupted ) for riskless rate. Black-Scholes values automatically recalculate. Click on “ Run Binomial Option Pricing Model ” button to recalculate binomial values. Input cells have dual boundary lines. Output cells have individual boundary lines. Up to 1,000 clip stairss can be used in the binomial theoretical account.

Input a uninterrupted dividend output or up to 50 distinct dividends. Do non come in both or the distinct dividends will be ignored.

This spreadsheet can be used to cipher options on forwards or hereafters utilizing the Black fluctuation of the Black-Scholes theoretical account. Input the forward or hereafters monetary value alternatively of the plus monetary value and input the riskless rate as both the riskless rate and the dividend output. Do non come in distinct dividends.

To monetary value foreign currency options, input the topographic point rate as the plus monetary value, the domestic involvement rate as the riskless rate and the foreign involvement rate as the dividend output. Do non come in distinct dividends.