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.
Binomial Model
The binomial model was first proposed by William Sharpe in 1978 and formalised by Cox, Ross and Rubinstein in 1979 and by Rendleman and Bartter in that same year.
While Black-Scholes model is essentially one formula that provides a numerical result based on inputs assumed, the binomial model considers the potential values of the underlying asset and the corresponding option over a range of time periods. It is thus more suitable in terms of valuing American options (ie, options that are exercisable not just at the expiration of option life, but any time prior to expiration).
Under this model, the time period from grant date to expiration is split into equal steps. The model then follows an iterative method to evaluate each period, considering either an up or down movement and the respective probabilities. Effectively, the model creates a binomial distribution of possible stock prices. The corresponding option values can then be derived, and the current value of an option equals to the present value of probability-weighted future payoffs.
Compared to the Black-Scholes model, the binomial model allows more flexibility in terms of catering characteristics of options in real life situation. For example, most options are not only exercisable at the end of the option life (European option), but are exercisable during a specific period within the option life (eg, exercisable at any time after a certain vesting period but before expiration). The binomial model can also reflect the assumption of stock price level at which the option holder would exercise the option prior to expiration (eg, when the stock price is higher than 2.5 times the initial stock price).