View
235
Download
0
Category
Preview:
Citation preview
7/31/2019 Norman Adu Bamfo
1/21
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
7/31/2019 Norman Adu Bamfo
2/21
ACTS 467 Page 2
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.
7/31/2019 Norman Adu Bamfo
3/21
ACTS 467 Page 3
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.
7/31/2019 Norman Adu Bamfo
4/21
ACTS 467 Page 4
TABLE OF CONTENT
I. Declaration
II. Abstract..
III. Introduction
IV. Methodology..
V. Application of Method..
VI. Results ..
VII. Conclusion .
7/31/2019 Norman Adu Bamfo
5/21
ACTS 467 Page 5
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.
7/31/2019 Norman Adu Bamfo
6/21
ACTS 467 Page 6
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.
7/31/2019 Norman Adu Bamfo
7/21
ACTS 467 Page 7
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.
7/31/2019 Norman Adu Bamfo
8/21
ACTS 467 Page 8
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.
7/31/2019 Norman Adu Bamfo
9/21
ACTS 467 Page 9
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.asp7/31/2019 Norman Adu Bamfo
10/21
ACTS 467 Page 10
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_instrument7/31/2019 Norman Adu Bamfo
11/21
ACTS 467 Page 11
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.
7/31/2019 Norman Adu Bamfo
12/21
ACTS 467 Page 12
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.
7/31/2019 Norman Adu Bamfo
13/21
ACTS 467 Page 13
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
7/31/2019 Norman Adu Bamfo
14/21
ACTS 467 Page 14
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
7/31/2019 Norman Adu Bamfo
15/21
ACTS 467 Page 15
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
7/31/2019 Norman Adu Bamfo
16/21
ACTS 467 Page 16
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
7/31/2019 Norman Adu Bamfo
17/21
ACTS 467 Page 17
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
7/31/2019 Norman Adu Bamfo
18/21
ACTS 467 Page 18
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
7/31/2019 Norman Adu Bamfo
19/21
ACTS 467 Page 19
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
7/31/2019 Norman Adu Bamfo
20/21
ACTS 467 Page 20
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
7/31/2019 Norman Adu Bamfo
21/21
Recommended