The Black-Scholes Option Valuation Model
Simply put, the model attempts to value options based upon the following assumptions:

    1. The underlying price distribution is lognormal over time.
    2. The risk-free rate is known and does not change.
    3. The volatility of the underlying is known and constant.
    4. Taxes and transaction costs do not exist.
    5. The underlying asset has no cash flows.
    6. The options are European.

Inputs to the model are:
  • underlying price - higher priced stocks will command a greater call premium as the option is that much more in the money and likely to be exercised.
  • exercise (strike) price - similarly, the lower the exercise or strike price, the greater the call premium, particularly if the current market price for the stock is high.
  • the risk-free rate - the higher the risk-free rate, the greater the call premium.
  • time to expiration - a longer time to expiration puts the call holder at an advantage, as there is more time for the stock price to rise above the exercise price.
  • volatility - volatility is the call buyer's friend. Because more volatile stocks will tend to fluctuate more in price, there is a greater potential for the stock to trade above its strike price.

All else being equal, call buyers will pay a high premium for an option with a low strike price and longer term to expiration with a high priced and volatile underlying stock. Black-Scholes represents a continuous-time form of option pricing.

Binomial Option Pricing
A model to value options, chiefly American ones, whereas Black-Scholes most often values European style options. The model attempts to price a European call option by valuing a separate portfolio with a similar payoff pattern as the call being priced. In its simplest form, the model is based upon two possible outcomes over a single period. In practice, it is often expanded to encompass more than two outcomes and a multi-period analysis. As distinct from the Black-Scholes Model, binomial option pricing represents a form of discrete-time option pricing.
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