Black-Scholes Model Is Essentially a Formula Used

Black-Scholes model is essentially a formula used in the calculation of a theoretical call price for options. It is considered to be the fundamental model for pricing in the option market (Cretien, 2006). This model uses in its calculation the five main determinants of an option's price, which include stock price, strike price, volatility, time left until expiration, as well as risk-free, short-term interest rate (Hoadley, 2010). The computations executed by the Black-Scholes model result in prices that are close to actual market value as long as input variables are determined that are reasonably accurate (Cretien, 2006). A benefit resulting from the use of this model is that it provides traders with a means to compare market prices with alternative values while using different inputs (Cretien, 2006). The Black-Scholes model also assists in the prediction of movements in price for investments other than options by providing a way to compute implied variance for any assets that have options traded (Cretien, 2006). Moreover, the Black-Scholes model may be defined as a method for the theoretical pricing of options that is based primarily on risk-free arbitrage between options on the assets' prices and underlying assets The Black-Scholes model is considered as the most fundamental formula for pricing options (Crawford, 2003). It forms the basis for all option-pricing models (Cretien, 2006). Without the Black-Scholes model, the market for exchange-traded options would not exist as it currently does (Cretien, 2006). There have been subsequent models developed for price calculation of options, but it may be understood that these are merely variations on the Black-Scholes model (Crawford, 2003). The Black-Scholes model is considered by many to be irreplaceable, although it has some limitations as to how effective it is for the valuation of various types of options (Cretien, 2006).

The concept on which the Black-Scholes model is based is a normal distribution of asset returns, or that the underlying asset prices are in fact distributed lognormally (Hoadley, 2010). The main characteristic of a lognormal distribution in comparison with a normal bell curve is that the lognormal distribution exhibits a longer right tail (Hoadley, 2010). This type of distribution allows for any possible stock price between zero and infinity and does not allow for any negative prices (Hoadley, 2010). Furthermore, a lognormal distribution also exhibits an upward bias, which represents how a stock price can only drop 100% of its worth but can rise by more than 100% of its worth (Hoadley, 2010). However, distributions of underlying asset prices often significantly depart from the lognormal, and the pricing executed by the Black-Scholes model can be modified in order to effectively deal with non-lognormally distributed asset prices (Hoadley, 2010).

The most critical parameter that affects option pricing is volatility, which must be estimated as it cannot be observed directly (Hoadley, 2010). Implied volatility provides the price of an option, while historical volatility provides the value of an option. Furthermore, the value of an option is absolutely independent of the expected growth of an underlying asset, therefore rendering it risk-neutral (Hoadley, 2010). This constitutes the main reason why expected rate of return of a stock is not a variable in the Black-Scholes model (Hoadley, 2010). This functions to ensure objective agreement between different investors concerning the value of an option regardless of whether they agree or disagree about the future growth of the option (Hoadley, 2010). Therefore, the price determined by the Black-Scholes model is a risk-neutral valuation, and is essentially the compensation amount required by an option writer for writing a call and totally hedging the risk (Hoadley, 2010).

The main advantage to the use of the Black-Scholes model for the calculation of option price is that it is time-efficient, as it allows for the calculation of a large number of option prices within a short timeframe (Hoadley, 2010). The main limitation of the Black-Scholes model is that it only calculates the price of an option at one moment in time, at expiration (Hoadley, 2010). Therefore, this model cannot be effectively used to accurately calculate option price with an American-style exercise, which includes all exchange traded equity options (Hoadley, 2010). Also, empirical studies have determined that the Black-Scholes model produces and estimation bias, and that the underlying assumption of a lognormal distribution does not hold (McKenzie et al., 2007).

The formula for the calculation of a theoretical option price using the Black-Scholes model is as follows (Hoadley, 2010):

Where (Hoadley, 2010):

The variables are (Hoadley, 2010):

S = stock price

X = strike price t = time remaining until expiration, expressed as a percent of a year r = current continuously compounded risk-free interest rate v = annual volatility of stock price (the standard deviation of the short-term returns over one year).

ln = natural logarithm

N (x) = standard normal cumulative distribution function

e = the exponential function

The Black-Scholes model has been subject to empirical investigation in order to assess its accuracy in calculating theoretical option prices. McKenzie et al. (2007) sought to examine the statistical significance and accuracy of factors within the Black-Scholes model, including underlying probabilities. The researchers hypothesized that based on assumptions regarding the probability of an option, the Black-Scholes model would demonstrate levels that are statistically and economically significant (McKenzie et al., 2007). Evidence used in this study was from the Australian Stock Exchange.