Norman Adu Bamfo

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ACTS 467 Page 1

MATHEMATICAL MODELS IN FINANCE ACTS 467

Mini-Thesis: ACTUARIAL SCIENCE (IV)

TOPIC:

OPTION PRICE SENSITIVITY ANALYSIS (a Black- Scholes model approach)

October, 2012


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This mini thesis (report) is submitted in fulfillment of our assignment in the Mathematical

Model in Finance (ACTS 467) Class (lesson 2), based on Option price and its sensitivity

to some factors.

PRESENTED TO

Dr. LORD MENSAH (Lecturer, Mathematical Model in Finance)

PRESENTED BY

Elaine Ablorh Quarcoo (3961809)

Norman Adu Bamfo (3962109)

Benneth Kweku Koufie (3967009)

Francis Kuditcher ( 3967209)

DECLARATION

We humbly declare this report is a true reflection of data forCALBANKs monthly stock

prices (2007 to 2010) loaded (from actuarial class e mail) on due provision by

Lecturer: Dr. Lord Mensah.


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ABSTRACT

This work is designed to actualize the factors that affect the pricing of options (call and

put). An assessment of this nature is done by a data set obtained on monthly CALBANK

stock prices from 2007 to 2010.

Applying the Black Scholes Pricing Model (Formular), highlights how variations in

some factors such as Volatility ( standard deviation), strike price, stock price, time to

expiration and even dividend pay outs affect the valuation or the pricing of options.


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TABLE OF CONTENT

I. Declaration

II. Abstract..

III. Introduction

IV. Methodology..

V. Application of Method..

VI. Results ..

VII. Conclusion .


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INTRODUCTION

OPTIONS (CALL AND PUT)

An option is a typical example of a derivative (securities that get their value from the

price of other securities). Options are traded both on exchanges and in the over-the-

counter market. There are two basic types of options. A call optiongives the holder the

right to buy the underlying asset by a certain date for a certain price. A put optiongives

the holder the right to sell the underlying asset by a certain date for a certain price. The

price in the contract is known as the exercise priceor strike price;the date in the

contract is known as the expiration dateor maturity. American optionscan be

exercised at any time up to the expiration date. European optionscan be exercised only

on the expiration date itself.


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OPTION PRICING/ VALUATION

In many ways, options are just like any other investment in that you need to understand

what determines their price in order to use them to take advantage of moves the

market.

A stock investor who is interested in using options to capture a potential move in a stock

must understand how options are priced. Besides the underlying price of the stock, the

key determinates of the price of an option are its intrinsic value - the amount by which

the strike price of an option is in-the-money - and its time value.

DETERMINANTS OF OPTION PRICING

Five main factors may influence the price of an option, also called the premium:

The strike price of the option

The market price of the underlying asset

Volatility, the price uncertainty of the underlying asset

Remaining life (the time length until the expiry date)

The interest of a loan with a term similar to the options remaining life

If there are payments attached to the underlying asset during the life of the contract,

e.g. share dividend, the expected size and time of payment will also have an impact on

pricing.


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The strike price of an option

The price of an option is naturally related to the strike price. A lower strike price implies

that the buyer of a call option is willing to pay more to acquire the option. Similarly, a

higher strike price will cause the price of a put option to rise since the buyer of the right

to sell the underlying shares may sell at a higher price.

The price of the underlying asset

The option price depends on the market price of the underlying asset. If the price of the

underlying asset increases, the premium of a call option will rise and the premium of a

put option will fall. If the price of the underlying asset drops, the premium of a put option

will rise and the premium of a call option will fall.

Volatility

Volatility expresses the expectations to fluctuations in the price of the underlying asset.

The volatility has an influence on the value of an option because it is one of the factors

that determine the probability to what extent the option will end in the money, and thus

the size of payoff at expiry; The higher the volatility, the higher the value of the option

price. The option price will therefore rise if the volatility of the underlying assets market

price increases.


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Interest rates

Buying a call option may be considered as an alternative to buying the underlying asset.

The purchase of a call option postpones the investment until the options expiry date,

and the excess liquidity can be placed on the money market. For that reason the seller

of a call option will naturally demand payment for having to finance the underlying asset

during the life of the option. A higher interest rate will therefore imply a higher price on

call options. The same applies to put options, the price of which will accordingly drop as

interest rates go up. The interest rate level of both call and put options is, however,

quite low.


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METHODOLOGY

There are several options pricing models (methods) that use these parameters (the

determinants) to determine the fair market value of the option. The Binomial Option

Pricing, Black - Scholes Pricing Model etc. Of these, the Black-Scholes model is the

most widely used.

In finance, the Binomial Options Pricing Model (BOPM) provides a generalizable

numerical method for the valuation ofoptions. The binomial model was first proposed by

Cox, Ross and Rubinstein (1979). Essentially, the model uses a discrete-time (lattice

based) model of the varying price over time of the underlying financial instrument. In

general, binomial options pricing models do not have closed-form solutions.

Valuation is performed iteratively, starting at each of the final nodes (those that may be

reached at the time of expiration), and then working backwards through the tree towards

the first node (valuation date). The value computed at each stage is the value of the

option at that point in time.

Option valuation using this method is, as described, a three-step process:

1. price tree generation,

2. calculation of option value at each final node,

3. sequential calculation of the option value at each preceding node
http://www.investopedia.com/terms/f/fairmarketvalue.asphttp://www.investopedia.com/terms/b/blackscholes.asphttp://en.wikipedia.org/wiki/Financehttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Option_(finance)http://en.wikipedia.org/wiki/John_C._Coxhttp://en.wikipedia.org/wiki/Stephen_Ross_(economist)http://en.wikipedia.org/wiki/Mark_Rubinsteinhttp://en.wikipedia.org/wiki/Lattice_model_(finance)http://en.wikipedia.org/wiki/Lattice_model_(finance)http://en.wikipedia.org/wiki/Lattice_model_(finance)http://en.wikipedia.org/wiki/Underlyinghttp://en.wikipedia.org/wiki/Underlyinghttp://en.wikipedia.org/wiki/Lattice_model_(finance)http://en.wikipedia.org/wiki/Lattice_model_(finance)http://en.wikipedia.org/wiki/Mark_Rubinsteinhttp://en.wikipedia.org/wiki/Stephen_Ross_(economist)http://en.wikipedia.org/wiki/John_C._Coxhttp://en.wikipedia.org/wiki/Option_(finance)http://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Financehttp://www.investopedia.com/terms/b/blackscholes.asphttp://www.investopedia.com/terms/f/fairmarketvalue.asp


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The tree of prices is produced by working forward from valuation date to expiration.At

each step, it is assumed that the underlying instrument will move up or down by a

specific factor (u or d) per step of the tree (where, by definition, u 1 and 0 < d 1).

So, ifSis the current price, then in the next period the price will either;

Sup = S .u or Sdown =S.d

THE BLACK SCHOLES PRICING MODEL

The Black Scholes Model is one of the most important concepts in modern financial

theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes

and is still widely used today, and regarded as one of the best ways of determining fair

prices of options.The model assumes that the price of heavily traded assets follow

a geometric Brownian motion with constant drift and volatility. When applied to a

stock option, the model incorporates the constant price variation of the stock, the time

value of money, the option's strike price and the time to the options expiry. This is

known as the Black-Scholes-Merton Model.
http://en.wikipedia.org/wiki/Underlying_instrumenthttp://en.wikipedia.org/wiki/Underlying_instrument


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ASSUMPTIONS

Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the

market or an individual stock. The market operates continuously with share prices

following a continuous It process. An It process is simply a Markov process in

continuous time.

No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even

floor traders pay some kind of fee, but it is usually very small. The fees that Individual

investor's pay is more substantial and can often distort the output of the model

Returns are log normally distributed

This assumption suggests, returns on the underlying stock are normally distributed,

which is reasonable for most assets that offer options.


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LIMITATIONS

The stock pays no dividends during the option's life

Most companies pay dividends to their shareholders, so this might seem a serious

limitation to the model considering the observation that higher dividend yields elicit

lower call premiums. A common way of adjusting the model for this situation is to

subtract the discounted value of a future dividend from the stock price.

Only European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration

date. American exercise term allow the option to be exercised at any time during the life

of the option, making American options more valuable due to their greater flexibility.

However, this limitation is not a major concern because very few calls are ever

exercised before the last few days of their life. This is true because when you exercise a

call early, you forfeit the remaining time value on the call and collect the intrinsic value.

Towards the end of the life of a call, the remaining time value is very small, but the

intrinsic value is the same.


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FORMULAE

ON CALL OPTION

Co = So N(d1) - XN(d2)

d1 = [ln (So/X) + (r+ /2) T] / ()

d2 = d1 - ()

ON PUT OPTION

P = X N(d2)) - So ( 1 - N(d1))

d1 = [ln (So/X) + (r + /2) T] / ()

d2 = d1 - ()

Alternative:

Using put call parity: P = Co + X

OPTION (CALL OR PUT) MODELLED ON DIVIDEND

Replace the stock price with a dividend adjusted stock price.

Sowith So PV (DIVIDEND)

Or Sowith So(,where d = dividend


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Where:

Co = Current call option value

So = Current stock price

N(d) = probability that a random draw from a normal distributionwill be less d.X = Exercise price

e = 2.71828, the base of the natural log

r = Risk-free interest rate (annualized, continuously compounded with the same maturity

as the option)

T = time to maturity of the option in years

LN = Natural log function

= Standard deviation of the stock


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THE BLACK SCHOLES MODEL COMPUTED IN EXCEL ON CALBANK STOCKS

We considered the Black Scholes Formular as the ideal approach for the assessment of

how the price of the option ( call and put ) on CALBANKs stock ( 2007 to 2010 ) is

sensitive to factors such as time to maturity, the volatility of the option, the risk free rate,

stock price, exercise or strike price and even dividend pay-out.

A spreadsheet (Microsoft Office Excel application) was used to diagnose the model to

seek out the effects of the determinant factors aforementioned.

Table 1.0 below shows the distribution of stock prices on CALBANK stocks and

the corresponding Treasury bill on due dates.

DATE CALBANK STOCK T-bills

monthly

return(m) LN (1+m)

2007 Jan 0.2205 0.007599202

Feb 0.2119 0.007501389 -0.039002268 -0.03978323

Mar 0.2331 0.0074964 0.100047192 0.095353081

Apr 0.2500 0.007520577 0.072501073 0.069993372

May 0.2620 0.007518658 0.048 0.046883586

Jun 0.272 0.007553186 0.038167939 0.037457563

Jul 0.3203 0.007627951 0.177573529 0.16345599

Aug 0.34 0.007654774 0.061504839 0.059687561

Sep 0.3425 0.007971086 0.007352941 0.00732604


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Oct 0.3515 0.008201035 0.026277372 0.025938054

Nov 0.4152 0.008228834 0.181223329 0.166550621

Dec 0.442 0.008305713 0.064547206 0.06254955

2008 Jan 0.506 0.008349452 0.14479638 0.135226787

Feb 0.700 0.00842661 0.383399209 0.324543666

Mar 0.700 0.008825426 0 0

Apr 0.620 0.010061599 -0.114285714 -0.121360857

May 0.660 0.011757428 0.064516129 0.062520357

Jun 0.700 0.013502762 0.060606061 0.0588405

Jul 0.700 0.016472541 0 0

Aug 0.700 0.017533092 0 0

Sep 0.690 0.017528274 -0.014285714 -0.014388737

Oct 0.670 0.017574725 -0.028985507 -0.029413885

Nov 0.600 0.017563373 -0.104477612 -0.110348057

Dec0.600 0.017573005 0 0

2009 Jan 0.600 0.017581605 0 0

Feb 0.450 0.01794516 -0.25 -0.287682072

Mar 0.340 0.018171359 -0.244444444 -0.280301965

Apr 0.300 0.018202115 -0.117647059 -0.125163143

May 0.260 0.018255743 -0.133333333 -0.143100844

Jun 0.200 0.018277253 -0.230769231 -0.262364264

Jul 0.220 0.018303537 0.1 0.09531018

Aug 0.250 0.018301148 0.136363636 0.127833372

Sep 0.240 0.018282374 -0.04 -0.040821995

Oct 0.220 0.018051997 -0.083333333 -0.087011377Nov 0.200 0.016969025 -0.090909091 -0.09531018

Dec 0.200 0.01472062 0 0

2010 Jan 0.200 0.013184043 0 0

Feb 0.170 0.011667985 -0.15 -0.162518929

Mar 0.190 0.010430344 0.117647059 0.111225635

Apr 0.230 0.01002651 0.210526316 0.191055237

May 0.280 0.0098561 0.217391304 0.196710294

Jun 0.270 0.009811202 -0.035714286 -0.036367644

Jul 0.260 0.009749807 -0.037037037 -0.037740328

Aug 0.300 0.009623893 0.153846154 0.143100844

Sep 0.290 0.009513201 -0.033333333 -0.033901552

Oct 0.280 0.009463252 -0.034482759 -0.03509132

Nov 0.300 NaN 0.071428571 0.068992871

Dec 0.310 NaN 0.033333333 0.032789823


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The generation of monthly returns and the LN (1+M) column

Monthly Return

This is measured by accounting for the rate of change in the stock prices over time. In

Table 1.0 We found the difference between the current and the previous stock price as

a rate on the current price.

Monthly return = (CT - CT- 1)/ CT- 1

The LN (1+M)

This is the natural log of an increased value of +1 on the monthly return.

LN (1+M) = LN (1+ (CT - CT- 1)/ CT- 1)

Evaluation for input (Monthly and Annual Volatility) ()

This is calculated by using the STDEV function in excel on the LN (1+M); Such as: =

STDEV (LN (1+M)), where LN (1+M) are the range of values in table 1.0.

Monthly Volatility = 0.122811668

Aftermath, we estimated annual volatility by multiplying the monthly volatility with

.Such as: = SQRT (STDEV (LN (1+M))

Annual Volatility = 0.425432096


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Evaluation for input (Risk Free Rate) ()

This also calculated by finding the average on the T-bill for the period (2007 to 2010).

Such as: = AVERAGE (RANGE OF VALUES ON T.BILL) in excel

Risk Free Rate = 0.012494247

SUMMARY OF ALL INPUTS FOR THE MODEL

Volatility () = 0.425432096

Strike Price (X) = 0.35

Stock Price (S0) = 0.37

Time or duration of the option contract (T) = 3 years

Risk Free Rate = 0.012494247

OUTPUT OF THE MODEL: THE CALL PRICE AND THE PUT PRICE

ON CALL OPTION

d1 = [ln (0.37/0.35) + (0.01249 + 0.425432/2) T] / (0.42542 *) = 0.494715882

d2 = 0.49471588 - (0.4252*) = - 0.242154123

Co = 0.37* N (0.494715882) (0.35) N(-0.242154123) = 0.118842455


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ON PUT OPTION

P = (0.35) N(-0.242154123) - 0.37* N (0.494715882) = 0.08596632

SUMMARY OF ALL OUTPUTS OF THE MODEL

Call value = 0.118842455

Put value = 0.08596632


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STOCK PRICE CALL PRICE PUT PRICE D1 D2 N(D1) N(D2)

0.37 0.118842455 0.08596632 0.494715882 -0.2421541 0.6896 0.40433

0.38 0.125802223 0.082926087 0.530907133 -0.2059629 0.702258 0.41841

0.39 0.132885756 0.080009621 0.566158244 -0.1707118 0.714357 0.432225

0.4 0.140087573 0.077211438 0.600516825 -0.1363532 0.725919 0.445771

0.41 0.147402428 0.074526293 0.634026959 -0.102843 0.736968 0.459044

INTERPRETATION OF RESULTS


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