A stock option gives a person the right, but not an obligation, to buy common stock in a company at a specified price on a particular ‘expiry’ or ‘maturity’ date. If the option is not exercised on this date it expires and is worthless – no money and no stock changes hands. However there is a legal obligation on the seller to provide the stock at the option price to the option holder if it is claimed (see below for definitions).

The underlying theory of stock option valuation is simple, though the mathematics is complex.

Let us consider an example: – the current price of one share of common stock of company ABC on the New York Stock Exchange is $25. You possess an option to buy one share of ABC for $25 today. As anyone can buy ABC common stock on the NYSE for $25, the option is worthless – you don’t need it. However if the option is to purchase one share for $20, the option is worth $5. It would be financially neutral if you bought the option for $5 and then used it to buy one share for $20 (for a total of $25), compared to buying one share for $25. If you had paid $5 for the option you would break even. However, if you had paid $2 for the option the option would be ‘in the money’ and you would benefit by $3 (total cost to you = $2 + $20 = $22 for one share of value $25).

The example is more complex if you have an option to buy the share in 4 weeks at a price of $20. The valuation of the option now depends on a number of time based factors. If you pay $5 for the option and the share prices fall to $15, you don’t need the option. It would be cheaper to buy the share without it. However if you pay $5 for the option and the share price rises to $30, probably it is valuable – you will have paid $5 for the option and in 4 weeks another $20 for a share worth $30.

So how do you value Stock Options offered by your company?

Your company ABC wishes to retain your excellent executive services, and has today offered you – free – an option to buy 500,000 shares of common stock at $12 per share in two years. What is the value of this option to you now? ABC is not listed on any stock exchange so there is no public information about options for your company.

There are three items of information you need in order to determine a value to you: the cost of the option now, the expected share price in two years, and the current interest rate. As there is no trade in your stock on a public exchange, you would need a current share price to estimate the expected price in two years.

One way to estimate the assumed share price of the company according to market standards is to assume a PE (price: earnings) ratio. For this you need to know the company earnings (revenue less cost of sales, and operating expenses), and the number of shares currently held in the company. Your company ABC earned $10million in the last year and the owners have 10 million shares. An ‘average’ PE ratio in the open market is 15, currently valuing the company at ($10million*15) = $150million, or ($150million / 10million shares) = $15 per share.

You estimate the company will grow 5% per annum. In two years, therefore, the company will be valued at ($150million * 1.05 * 1.05) = $165million. As there are 10million shares, the value will be $16.5 per share. However, when a company offers stock options, it normally creates new shares on redemption of the options, so there will be 10,500,000 shares, valuing the company at $15.7 per share in two years.

Our option is to buy shares at $12.00. In two years, therefore, the value of an option will be ($15.7-$12.00) = $3.70. The value of 500,000 options will be $1.85million. Finally we need to convert $1.85million in two years into cash today. To do this we discount by the interest rate, currently 1% per annum. We then find the value today of our stock option is ($1.85million/ (1.01*1.01)) = $1.81million. This then is our estimate of the value of our stock option.

In this calculation we have made a number of assumptions about your company and the future:-

– We have assumed that a fair value for your company is 15 times earnings (there are other methods);

– We have assumed the company will grow 5% per annum;

– We have assumed the discount rate (risk free interest rate) will remain at 1%;

– We have assumed you will be able to sell your stock for at least the purchase price once you have it – else it has less value!

– We also have not taken any tax effects into account.

How do you value Options purchased on the Open Market?

Stock options are also bought and sold on the open market. The pricing mechanism is different though. These options are bought as investments or speculation and the calculation of the prices includes probabilities of the stock rising or falling.

The terminology of options, and the main variables determining the price of an option.

A stock option is called a ‘derivative’ instrument, as the value of the option derives from the value of an underlying instrument, in this case one share of common stock. The current price of the stock is called the ‘spot price’, while the price at which the option allows you to buy common stock is called the ‘strike price’. An option may be on a single share, but may also be used to purchase fractions or multiples of shares in a company. The number of options required to buy 1 share is called the ‘cover. The volatility is the normalized square root of the variance of the share price around its mean.

There are essentially two types of options: – a ‘call’ option allowing you to buy stock at a specified price (as discussed above and the subject of this article), and a ‘put’ option allowing you to sell stock at a specified price (these will not be explicitly discussed in this article). There are also two forms of options: – the ‘European’ option which can only be exercised on a specific date – the ‘expiration date’ and the ‘American’ option which can be exercised at any time before the ‘expiration date’. Always ensure which option you are talking about as both forms are available in many countries around the world.

There are two other variables determining the value of an option: – the risk free interest rate (e.g., Government bonds) and the share price volatility. Finally the valuation formulae quoted here are only relevant to ‘vanilla options’ – standardized options traded at most exchanges, and not ‘exotic options’, often designed for specific persons or situations and are not freely tradable.

There are a number of formulae for calculating the theoretical value of an option.

The most widely used model is the Black-Scholes valuation (for which Merton and Scholes received the Nobel Prize in Economics – Black had died in the interim) though there are also Binomial models and the Heston model. They are all based on probability theory and the mathematics is daunting, though achievable on a spreadsheet. I have not recreated the formulae here, though links have been provided to the source. Suffice to say the formulae involve partial differential calculus, natural logarithms, exponentials, and a sound understanding of the normal and binomial distributions.

The valuation theories involve the concept of ‘investor neutrality’. This means that an investor, who buys the option, uses it to buy the share on the expiry date and then sells the share should be in an identical cash value position to the investor who buys the share today and then sells it on expiration date. The complexity is caused by considering the interest rate, where the investor who buys the share today ‘loses’ interest in the cash, and by considering the share volatility where a share could easily rise and fall a couple of percentage points in a single day, thereby affecting the price.

The strict theoretical valuations also only work on the European type option, where the option can only be exercised on a particular date. The American option, which can be exercised on any date prior to expiration date is far more complex, but is valued at higher than the European option due to greater flexibility in volatile times.

In order to understand how the variables affect the option price let us consider the following scenarios:-

Assume a share in company ABC, with spot price (current price) of $100.00 (10000 cents).

– Assume the date today is 1 July 2010, and the expiry date is 30 September 2010 (92 days).

– Assume a strike price of $120.00

– Assume a risk free interest rate of 1% per annum.

– Assume a volatility of 45%.

– Assume a cover of 1.

The Black-Scholes model would give a valuation of 294 cents. In other words the value of an option to purchase 1 share in company ABC on the 30 September 2010 at $120.00 where the current price is $100.00 would be $2.94. This seems low, but the reason is that the share price is unlikely to reach $120, so statistically you should not pay a large amount. However if you lowered the strike price to $110 the value of the option would be $5.30, and at $100 the value of the option would be $9.05. In this case the volatility of the share and the saving of interest imply it would cost you the same to buy the share now at $100, or an option at $9 and the share at $100 at end September.

The immediate conclusion is that the changes in value are not linear but are logarithmic or exponential. It is therefore not as simple as increasing the value by a fixed percentage as any of the variables change.

Conclusion.

The valuation of stock options is complex. The easiest way for the layman is to determine whether there is a standard option trading on the securities exchange with the same strike price and expiration date. The market maker (the person who creates the option) will determine the correct theoretical value.

In any other circumstance it is advisable to consult an economic or financial advisor to determine the correct value.

Definition of stock : http://www.investorwords.com/4725/stock.html

Black-Scholes: http://en.wikipedia.org/wiki/Black%E2%80%93Scholes

Binomial: http://en.wikipedia.org/wiki/Binomial_options_pricing_model

Heston Model: http://en.wikipedia.org/wiki/Heston_model

Other options:

Fairmark simplified : http://www.fairmark.com/execcomp/valuation.htm

Wharton University : http://knowledge.wharton.upenn.edu/article.cfm?articleid=363

Dinkytown web calculator: http://www.dinkytown.net/java/StockOptions.html