Finance

1. Options Pricing: Introduction
Simple Mathematical Models, MTMM.00.292
Urmas Aunin, 2014

2. Options Pricing: Introduction
• By Jean Folger , (Investopedia)
• Options are derivative contracts that give the holder the right, but not the obligation, to buy or
sell the underlying instrument at a specified price on or before a specified future date. Although
the holder (also called the buyer) of the option is not obligated to exercise the option, the option
writer (known as the seller) has an obligation to buy or sell the underlying instrument if the
option is exercised.
• Depending on the strategy, option trading can provide a variety of benefits including the security
of limited risk and the advantage of leverage.
(Note: In finance, leverage is any technique to multiply gains and losses).
• Options can protect or enhance an investor's portfolio in rising, falling and neutral markets.
• Regardless of the reasons for trading options or the strategy employed, it is important to
understand the factors that determine the value of an option. This tutorial will explore the factors
that influence option pricing, as well as several popular option pricing models that are used to
determine the theoretical value of options.

3. Options contract specifications: example
RAPESEED OIL OPTIONS CONTRACT SPECIFICATIONS
• Contract Rapeseed oil option contract (OSO).
• Option type American option.
• Underlying Rapeseed oil futures contract.
• Trading Unit One rapeseed oil futures contract.
• Strike prices Spread between two strike prices: 2.5 euros.
• Price quotation Premium in Euro per metric tonne.
• Tick size 0,10 Euro per metric tonne, (2 euros per contract).
• Expiry months September, December, March and June such that 10 delivery months are open simultaneously
• Last trading day Close of business on the 15 calendar day of the month immediately preceding the expiry month
(if not a business day then the first business day immediately preceding).
• First trading day The first trading day of the underlying futures contract.
• Exercise Assignment of one Rapeseed Oil futures contract for the expiry month at the strike price.
Automatic exercise of in the money options.
• Trading hours 10.45 am to18.30 pm (Paris time).
https://derivatives.euronext.com/en/search/node/option

4. Options Pricing: Modeling
Option traders utilize various option price models to attempt to set a current
theoretical value. Models use certain fixed knowns in the present – factors
such as underlying price, strike and days till expiration – along with forecasts
(or assumptions) for factors like implied volatility, to compute the theoretical
value for a specific option at a certain point in time.
Variables will fluctuate over the life of the option, and the option position's
theoretical value will adapt to reflect these changes.
Most professional traders and investors who trade significant option
positions rely on theoretical value updates to monitor the changing risk and
value of option positions and to assist with trading decisions.
Many options trading platforms provide up-to-the-minute option price
modeling values, and option pricing calculators can be found online at
various Web sites, including the Options Industry Council

5. Options Pricing: Modeling

6. Options Pricing: Black-Scholes Model
The Black-Scholes model for calculating the premium of an option was introduced in 1973 in a
paper entitled, "The Pricing of Options and Corporate Liabilities" published in the Journal of Political
Economy.
The formula, developed by three economists – Fischer Black, Myron Scholes and Robert Merton –
is perhaps the world's most well-known options pricing model. Black passed away two years
before Scholes and Merton were awarded the 1997 Nobel Prize in Economics for their work in
finding a new method to determine the value of derivatives (the Nobel Prize is not given
posthumously; however, the Nobel committee acknowledged Black's role in the Black-Scholes
model).
The Black-Scholes model is used to calculate the theoretical price of “European“-style put and call
options, ignoring any dividends paid during the option's lifetime.
“European“-style option can only be exercised on the expiration date, while an “American“-style
option can be exercised at any point prior to the expiration date.
While the original Black-Scholes model did not take into consideration the effects of dividends paid
during the life of the option, the model can be adapted to account for dividends by determining
the ex-dividend date value of the underlying stock.
The model makes certain assumptions, including:

7. Options Pricing: Black-Scholes Model
• The options are „European“-style and can only be exercised at expiration
• No dividends are paid out during the life of the option
• Efficient markets (i.e., market movements cannot be predicted)
• No commissions
• The risk-free rate and volatility of the underlying are known and constant
• Follows a lognormal distribution; that is, returns on the underlying are normally distributed.
The formula, (shown in Figure), takes the following variables into consideration:
• Current underlying price
• Options strike price
• Time until expiration (The time remaining until a financial contract expires. Also called time to maturity)
• Implied volatility (Price volatility is a measure of price variation from one period to the next)
• Risk-free interest rates

8. Options Pricing: Black-Scholes Model Formula

9. Options Pricing: Black-Scholes Model
• The model is essentially divided into two parts: the first
part, SN(d1), multiplies the price by the change in the call
premium in relation to a change in the underlying price. This
part of the formula shows the expected benefit of purchasing
the underlying outright.
• The second part, N(d2)Ke^(-rt), provides the current value of
paying the exercise price upon expiration (remember, the Black-
Scholes model applies to European options that are exercisable
only on expiration day).
• The value of the option is calculated by taking the difference
between the two parts, as shown in the equation.

10. Options Pricing: Black-Scholes Model
The mathematics involved in the formula is complicated and can
be intimidating.
However, traders and investors do not need to know or even
understand the math to apply Black-Scholes modeling in their
own strategies.
As mentioned previously, options traders have access to a variety
of online options calculators and many of today's trading
platforms boast robust options analysis tools, including indicators
and spreadsheets that perform the calculations and output the
options pricing values.

11. Options Pricing: Black-Scholes Model

12. Options Pricing: put/call parity
Options traders use put/call parity as a simple test for their „European“-style options pricing
models.
There are several formulas to express put/call parity for „European“-style options.

13. Options Pricing: put/call parity
c= S + p – Xe – e*r(T-t)
p = c – S + Xe – e*r(T-t)
Where
c = call value
S = current stock price
p = put price
X = exercise price
e = Euler's constant (exponential function on a financial calculator equal to approximately 2.71828
r = continuously compounded risk free rate of interest
T = Expiration date
t = Current value date
The following formula provides an example of a formula that can be used for non-dividend paying
securities. The tree of prices is produced by working forward from valuation date to expiration.
The up and down factors are calculated using the underlying volatility and the time duration of a
step, measured in years. From the condition that the variance of the log of the price is, we have:

14. Definition of 'Stock'
A type of security that signifies ownership in a corporation and represents a claim on part of the corporation's assets and
earnings.
There are two main types of stock: common and preferred.
Common stock usually entitles the owner to vote at shareholders' meetings and to receive dividends.
Preferred stock generally does not have voting rights, but has a higher claim on assets and earnings than the common shares.
For example, owners of preferred stock receive dividends before common shareholders and have priority in the event that a
company goes bankrupt and is liquidated.
Also known as "shares" or "equity.„
A holder of stock (a shareholder) has a claim to a part of the corporation's assets and earnings.
In other words, a shareholder is an owner of a company. Ownership is determined by the number of shares a person owns
relative to the number of outstanding shares. For example, if a company has 1,000 shares of stock outstanding and one
person owns 100 shares, that person would own and have claim to 10% of the company's assets.
Stocks are the foundation of nearly every portfolio. Historically, they have outperformed most other investments over the
long run.

15. Definition of 'Stock Market'
• The market in which shares of publicly held companies are issued and traded either through
exchanges or over-the-counter markets. Also known as the equity market, the stock market is
one of the most vital components of a free-market economy, as it provides companies with
access to capital in exchange for giving investors a slice of ownership in the company.
• 1.22 million traders have learned to trade since 1982.
• The stock market lets investors participate in the financial achievements of the companies whose
shares they hold. When companies are profitable, stock market investors make money through
the dividends the companies pay out and by selling appreciated stocks at a profit called a capital
gain. The downside is that investors can lose money if the companies whose stocks they hold
lose money, the stocks' prices goes down and the investor sells the stocks at a loss.
• Stocks are traded through exchanges. The two biggest stock exchanges in the United States are
the New York Stock Exchange, founded in 1792, and the Nasdaq, founded in 1971. Today, most
stock market trades are executed electronically, and even the stocks themselves are almost
always held in electronic form, not as physical certificates.
• If you want to know how the stock market is performing, you can consult an index of stocks for
the whole market or for a segment of the market. Examples include the Dow Jones Industrial
Average, Nasdaq index, Russell 2000, Standard and Poor’s 500, and Morgan Stanley Europe,
Australasia and Far East index.

16. Stock Market in Estonia
The Estonian stock market works by the principle – fewer in number, but more good companies.
* Although the Tallinn stock market lists the smallest number of companies – 15 – all (except for one) have been listed on the
Main List since 2008.
* As in the other Baltic States, the stock market was an important instrument in the process of privatization of the state-
owned and municipal companies. In 1995, 10 banks, 9 brokerage companies and state institutions (Bank of Estonia, Ministry
of Finance and Compensation Fund) established the Stock Exchange of Tallinn, which started trading in shares of 11
companies in 1996.
* In 2005 the exchange in Tallinn has reached the historically largest turnover – EUR 1,9 billion, which was furthered by
Hansapank leaving the regulated market. The following year the turnover decreased by 63%.
* Over 16 years, substantial changes have taken place with respect to the companies listed on the exchange. However, the
stock market of Estonia was still aiming for qualitative indicators – shares in very good companies were quoted on the lists.
* Over a period of seven years until 2005, some companies (mainly banks) left the Tallinn stock market, however, this process
was stopped when the privatized municipal water management company Tallinn Vesi, was listed on the stock market. Since
then, various new companies, which used the exchange as a platform for raising capital, including the media company
Express Grupp, the real estate company Arco Vara, the shipping company Tallink Grupp, the gambling, casino operator
Olympic Entertainment Group, food company Premia Foods, have also entered the securities market. In 2010, the company
Norma was delisted from the Stock Exchange, the departure of Eesti Telecom was also a heavy blow.
* In early 2012 only one company Tallinna Kaubamaja, which started trading in 1996 – was still trading on the exchange.
* Strategic and financial investors hold shares in the companies quoted on the exchange. At the same time, there are a large
number of minority shareholders (holding less than 10% shares), which ensures a higher free float or number of shares in
free circulation, making a company attractive for active trading in its shares.
* The further development of NASDAQ OMX Tallinn will also be determined by the alternative securities list First North,
which is intended mostly for small and medium companies, and which was formed on the stock markets of the Baltic
States several years ago. Furthermore, NASDAQ OMX Tallinn has entered into the cooperation memorandum withKredEx
to develop the local capital market and ensure additional financing opportunities for the companies intending to start
trading in shares on the First North.

17. Stock Market in Estonia
Facts for 2011
* Total capitalization of the 15 companies quoted on the Tallinn exchange has reached EUR 1,24 billion by the
end of the year, 26,33% less than at the end of 2010, and, compared with the pre-crisis period – as of the end
of 2007, down by 69,8%.
The substantial decline in capitalization resulted from the crisis on the global financial markets, because due to
high liquidity, investors were able to easily divest their holdings in companies on the Tallinn exchange.
* In 2011, prices of nearly all shares decline, with the largest drop – by 74% – experienced by shares in Baltika.
The largest rise in prices was only 12.6%, by shares in Viisnurk.
* A total of 83 700 trades were made on the exchange in 2011, with 145,5 million shares changing hands for
EUR 187,4 million
* The most trades – 20 500 in 2011 – were made with shares of Tallink Grupp and turnover resulting from
these transactions was the highest on the exchange in 2011 – EUR 50,3 million. A year earlier this honor went
to Olympic Entertainment Group, with 24 000 trades in its shares amounting to turnover of EUR 63 million.
* Compared to the Lithuanian exchange, the capitalization companies on the Tallinn was significantly smaller
(2,5 times, but compared to the Riga stock exchange, 1,5 times larger), turnover on the exchange was
somewhat larger than in Lithuania (EUR 176 million) and several time larger than that of the Riga exchange.
Useful links
Stock Exchange – www.nasdaqomxbaltic.com
Financial and Capital Market Commission – www.fktk.lv
The Securities Commission of the Republic of Lithuania – www.vpk.lt
Financial Supervision Authority – www.fi.ee
Securities Register – www.e-register.ee
http://www.nasdaqomxbaltic.com/market/?lang=en

18. Most active Nasdaq-traded stocks
• American Realty Capital Properties Inc. fell 8.5 percent to $8.23 with 33,770,100
shares traded.
• Apple Inc. fell 1.4 percent to $108.23 with 66,344,000 shares traded.
• Cisco Systems Inc. fell .7 percent to $26.68 with 27,283,200 shares traded.
• Facebook Inc. fell 1.1 percent to $76.99 with 28,959,200 shares traded.
• Intel Corp. fell .8 percent to $35.92 with 30,446,500 shares traded.
• Micron Technology Inc. fell 1.9 percent to $33.36 with 26,386,000 shares traded.
• Microsoft Corp. fell .6 percent to $46.67 with 29,044,500 shares traded.
• Riverbed Technology Inc. rose 8.4 percent to $20.31 with 19,270,000 shares
traded.
• Sirius XM Radio Inc. fell 2.1 percent to $3.34 with 60,683,700 shares traded.
• Staples Inc. rose 2.3 percent to $16.86 with 23,292,100 shares traded.

19. Stock Price Quote of Microsoft Corp (MSFT)
Chart Snapshot
Stock Price Quote of Microsoft Corp (MSFT) 46.67-0.28 (-0.60%) 4:15P EST (NASDAQ)
http://www.barchart.com/quotes/stocks/MSFT
.
Stock Price Quote as of Tue, Dec 16th, 2014
High 46.34 Low 45.13
50.0452Wk High
(-9.75%)Since 11/14/14
34.6352Wk Low
(+30.41%)Since 01/14/14
Open 45.90 Prev Close 46.67
Volume 47,765,699 Avg Volume 29,330,865
EPS 2.58 Div & Yield $1.24 (2.64%)
P/E Ratio 18.20 Market Cap $372.25 B
Weighted
Alpha
+19.53 Standard Dev -2.89
Sectors Computer - Software, All MSFT Sectors

20. MSFT Option Chain | Yahoo! Inc. Stock -
Yahoo! Finance
Calls
Strike
∵ Filter
Contract Name
Last
Bid Ask Change %Change Volume Open Interest Implied Volatility
60.00 MSFT141220C00
060000
0.01 0.00 0.01 0.00 0.00% 1 36 84.38%
56.50 MSFT141220C00
056500
0.01 0.00 0.02 0.00 0.00% 0 113 73.44%
56.00 MSFT141220C00
056000
0.01 0.00 0.02 0.00 0.00% 0 84 70.31%
55.50 MSFT141220C00
055500
0.01 0.00 0.02 0.00 0.00% 11 11 67.19%
54.50 MSFT141220C00
054500
0.01 0.00 0.02 0.00 0.00% 194 282 62.50%
57.00 MSFT141220C00
057000
0.01 0.00 0.02 0.00 0.00% 0 222 75.00%
57.50 MSFT141220C00
057500
0.01 0.00 0.01 0.00 0.00% 264 574 71.88%
51.50 MSFT141220C00
051500
0.01 0.00 0.04 0.00 0.00% 32 381 54.69%
55.00 MSFT141220C00
055000
0.02 0.00 0.02 0.00 0.00% 20 8126 64.06%
49.00 MSFT141220C00
049000
0.02 0.01 0.02 0.00 0.00% 19 21806 30.86%
49.50 MSFT141220C00
049500
0.02 0.00 0.04 0.00 0.00% 2 3045 39.06%
54.00 MSFT141220C00
054000
0.02 0.00 0.02 0.00 0.00% 80 212 59.38%
51.00 MSFT141220C00
051000
0.02 0.00 0.02 0.00 0.00% 12 1081 45.31%
Microsoft Corporation (MSFT)
Watchlist
45.82 -0.85(-1.82%) NasdaqGS - As of 10:40am EST
December 20, 2014In The Money

21. Definition of 'Portfolio Investment'
A portfolio investment is made with the expectation of earning a return on it.
This expected return is directly correlated with the investment's expected risk.
Portfolio investment is distinct from direct investment, which involves taking a sizeable stake in a target
company and possibly being involved with its day-to-day management.
• Portfolio investments can span a wide range of asset classes – stocks, government bonds, corporate bonds,
Treasury bills, real estate investment trusts, exchange-traded funds, mutual funds, certificates of deposit and
so on. Portfolio investments can also include options, warrants and other derivatives such as futures, and
physical investments like commodities, real estate, land and timber.
• The composition of investments in a portfolio depends on a number of factors, among the most important
being the investor’s risk tolerance, investment horizon and amount invested.
• For a young investor with limited funds, mutual funds or exchange-traded funds may be appropriate
portfolio investments. For a high net worth (HNW) individual, portfolio investments may include stocks,
bonds, commodities and rental properties.
• Portfolio investments for the largest institutional investors such as pension funds and sovereign funds
include a significant proportion of infrastructure assets like bridges and toll roads. This is because their
portfolio investments need to have very long lives, so the duration of their assets and liabilities match.

22. References
• http://www.investopedia.com/
• Investopedia is a premiere resource for investing education, personal
finance, market analysis and free trading simulators.
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Articles - Tutorials - The Biggest IPO Flops - Terms - ...

23. Stock Price Quote of Microsoft Corp (MSFT)
Stock Price Quote as of Mon, Dec 15th, 2014
High 47.67 Low 46.55
50.0452Wk High
(-6.73%)Since 11/14/14
34.6352Wk Low
(+34.77%)Since 01/14/14
Open 47.20 Prev Close 46.95
Volume 29,266,466 Avg Volume 28,458,510
EPS 2.58 Div & Yield $1.24 (2.63%)
P/E Ratio 18.28 Market Cap $384.69 B
Weighted
Alpha
+25.16 Standard Dev -0.63
Sectors Computer - Software, All MSFT Sectors