Options Pricing: Introduction

Simple Mathematical Models, MTMM.00.292Urmas Aunin, 2014

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.

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

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

Options Pricing: Modeling

Options Pricing: Black-Scholes ModelThe 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:

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

Options Pricing: Black-Scholes Model Formula

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.

Options Pricing: Black-Scholes ModelThe 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.

Options Pricing: Black-Scholes Model

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.

Options Pricing: put/call parity

c= S + p – Xe – e*r(T-t)p = c – S + Xe – e*r(T-t)

Wherec = call value

S = current stock pricep = put price

X = exercise pricee = Euler\'s constant (exponential function on a financial calculator equal to approximately 2.71828

r = continuously compounded risk free rate of interestT = 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:

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.

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.

Stock Market in EstoniaThe 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 with KredEx to develop the local capital market and ensure additional financing opportunities for the companies intending to start trading in shares on the First North.

Stock Market in EstoniaFacts 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.comFinancial and Capital Market Commission – www.fktk.lvThe Securities Commission of the Republic of Lithuania – www.vpk.ltFinancial Supervision Authority – www.fi.eeSecurities Register – www.e-register.eehttp://www.nasdaqomxbaltic.com/market/?lang=en

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.

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, 2014High 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

MSFT Option Chain | Yahoo! Inc. Stock - Yahoo! Finance

Calls

Strike∵ Filter

Contract NameLast

Bid Ask Change %Change Volume Open InterestImplied Volatility

60.00 MSFT141220C00060000

0.01 0.00 0.01 0.00 0.00% 1 36 84.38%

56.50 MSFT141220C00056500

0.01 0.00 0.02 0.00 0.00% 0 113 73.44%

56.00 MSFT141220C00056000

0.01 0.00 0.02 0.00 0.00% 0 84 70.31%

55.50 MSFT141220C00055500

0.01 0.00 0.02 0.00 0.00% 11 11 67.19%

54.50 MSFT141220C00054500

0.01 0.00 0.02 0.00 0.00% 194 282 62.50%

57.00 MSFT141220C00057000

0.01 0.00 0.02 0.00 0.00% 0 222 75.00%

57.50 MSFT141220C00057500

0.01 0.00 0.01 0.00 0.00% 264 574 71.88%

51.50 MSFT141220C00051500

0.01 0.00 0.04 0.00 0.00% 32 381 54.69%

55.00 MSFT141220C00055000

0.02 0.00 0.02 0.00 0.00% 20 8126 64.06%

49.00 MSFT141220C00049000

0.02 0.01 0.02 0.00 0.00% 19 21806 30.86%

49.50 MSFT141220C00049500

0.02 0.00 0.04 0.00 0.00% 2 3045 39.06%

54.00 MSFT141220C00054000

0.02 0.00 0.02 0.00 0.00% 80 212 59.38%

51.00 MSFT141220C00051000

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, 2014                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   

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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.

References

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Stock Price Quote of Microsoft Corp (MSFT)

Stock Price Quote as of Mon, Dec 15th, 2014High 47.67 Low 46.5550.0452Wk High (-6.73%)Since 11/14/14

34.6352Wk Low (+34.77%)Since 01/14/14

Open 47.20 Prev Close 46.95Volume 29,266,466 Avg Volume 28,458,510EPS 2.58 Div & Yield $1.24 (2.63%)P/E Ratio 18.28 Market Cap $384.69 BWeighted Alpha

+25.16 Standard Dev -0.63

Sectors Computer - Software, All MSFT Sectors