Valuation of Options
Options are a type of financial derivative whose value depends on the underlying asset. The buyer of a call option has the right, but not the obligation, to purchase the stock, commodity, or other asset at a specified price within a specific period. Call options are commonly seen in the corporate world as companies typically grant stock options to employees as an incentive scheme. An employee stock option programme aligns the interests of the employees with the investors and create a culture of ownership.
For the valuation of options, Black-Scholes model and binomial model are both commonly used option pricing models.
Black-Scholes Model
The model was discovered in 1973 by Fischer Black and Myron Scholes, who then received the Nobel Memorial Prize in economics for their discovery. Robert Merton further extended the formula to account for dividends in the calculation.
Fair value of a call option price can be calculated using the following formula:
Where:
The formula requires the following assumptions:
S0 = Market price of the underlying asset at option grant date
X = Strike price to be paid by the holder if the option is exercised
σ = Volatility of the underlying asset
r = Risk free interest rate
q = dividend yield
t = Expected life of the option
By making the above inputs, a hypothetical at-the-money option with underlying stock priced at $100 at option grant date would worth around $36 per option. See financial model below.
By varying (i) stock price at grant date; and (ii) exercise price while holding other assumptions the same, one can perform sensitivity analysis to see how stock price and exercise price affect the fair value of an option. If plotted as a 3-dimensional graph, the option value (vertical axis) would look something like this:
A few points to note:
- For a given stock price, reducing the exercise price causes an option to be more in-the-money, thus more valuable.
- For a given exercise price, increasing the stock price causes an option be more in-the-money, thus a more valuable.
- For at-the-money option, the higher the stock price the higher the option value. However, option value as a percentage of the stock price stays the same.