The Black-Scholes Option Valuation Model
Simply put, the model attempts to value options based upon the following assumptions:
- The underlying price distribution is lognormal over time.
- The risk-free rate is known and does not change.
- The volatility of the underlying is known and constant.
- Taxes and transaction costs do not exist.
- The underlying asset has no cash flows.
- 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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