Black Scholes (B&S) and Binomial Models are used for pricing options.
Black Scholes allows determining the theoretical call price, which uses different determinants of price including “stock price, strike price, time remaining, volatility, and the risk-free rate of interest” (Capiński and Kopp 61). B&S is based on the normal distribution of returns of the underlying asset. However, the returns are not normalized and there could be significant variations. The weakness of this model is that it ignores dividends paid. Furthermore, it is difficult to assess the volatility of stock prices, which changes over the period. Therefore, one of the limitations of B&S is that it is not suitable to determine the price of American options (Capiński and Kopp 61).
The binomial model addresses the limitation of B&S by breaking down the expiration time into smaller time intervals and a tree is developed that indicates stock prices for each interval identified. It allows determining possible stock prices at different time points till expiration. In the same way, stock prices are calculated backward from expiration to the current time. The model uses the same underlying assumptions about stock prices. The three main determinants used by the binomial model are stock price volatility, risk-free rate, and expiration time (Chance and Brooks 137). Therefore, the model can be used for pricing American options. However, it takes a long time to perform the binomial model.
The basic difference between American and European options is the time when these options can be exercised. European options are exercised only on the expiration date. The expiration date is specified in the option contract and it is fixed. On the other hand, the American option can be exercised at any time before the expiration date. Such options give more opportunities to investors as they can take advantage of the favorable price at any time before the expiration date (Kwok 11). For this reason, these options have more value. Therefore, it is difficult to price American options. Moreover, it could be noted that European options are often traded at a discount price as compared to American options.
There are other differences between American and European options. The underlying assets including stocks, funds traded on an exchange, and limited indices can have American-style options. On the other hand, European options are more common with broad-based indices. American options can be traded till the third Friday of the expiration month. European options can be traded till one day before the expiration date (Petters and Dong 357).
At-the-money (ATM) is a term used to refer to the “options that have a strike price almost equal to their current price of the underlying asset” (Bossu and Gottesman 37). Both put and call option has the same value. A strictly ATM option does not have an intrinsic value, and it only contains an extrinsic value. The delta value of ATM options is comparatively low and equal to or close to 0.5.
Out-of-the-money (OTM) call refers to options that have a strike price greater than the current price of the underlying asset” (Duarte 170). Similarly, Out-of-the-money (OTM) put refers to options that have a strike price lower than the current price of the underlying asset” (Duarte 170). An OTM option loses its value as the expiration date gets closer. Moreover, if the option is still OTM, then it becomes worthless. In both cases, there is no intrinsic value and the premium of the option is the extrinsic value.
Bossu, Sebastien and Aron Gottesman. Derivatives Essentials: An Introduction to Forwards, Futures, Options and Swaps. John Wiley & Sons, 2016.
Capiński, Marek and Ekkehard Kopp. The Black-Scholes Model. Cambridge University Press, 2012.
Chance, Don M. and Roberts Brooks. An Introduction to Derivatives and Risk Management. 8th ed., Cengage Learning, 2015.
Duarte, Joe. Trading Options For Dummies. Hoboken: John Wiley & Sons, 2015.
Kwok, Yue-Kuen. Mathematical Models of Financial Derivatives. Springer Science & Business Media, 2014.
Petters, Arlie O. and Xiaoying Dong. An Introduction to Mathematical Finance with Applications: Understanding and Building Financial Intuition. New York: Springer, 2016.