Here, we will derive formulas for European style Asian call and put options when we are taking geometric average of the underlying’s price. We will put expectation and variance of the geometric average into Black’s formula (generalized version), and will simplify it to obtain option formulas. The geometric average is defined by And, the continuously […]

The geometric average of the lognormally distributed random variables is a lognormal. And, Kemna and Vorst showed the expectation and variance of geometric average and derived the European style call and put options formula. The expectation and variance of geometric average is given by Where n(a,b) represents a normal distribution with mean a and variance […]

When we take the arithmetic average of the underlying, A(0,T) is calculated as Our Black-Schloes-Merton model and Black’s model rely on the assumption that the underlying price follows a lognormal distribution. And, the arithmetic average of the lognormally distributed random variables is not a lognormal. Turnbull and Wakeman (1991) valued arithmetic average Asian options using […]

To be updated.. Asian Option Asian options are options where the payoff depends on the average price of the underlying asset during the life of the option. The payoff from an average price call option is Where A(0,T) is the average value of the underlying from t=0 to t=T, inclusive. The popularly used Asian options […]

In addition to standard European and American call and put options, also called plain vanilla products, which are traded in exchanges, there are several exotic options, normally, traded in over-the-counter market. Financial engineers create them to make them more attractive and meet some specific need of a particular business. List of Exotic Options Asian Option

The generalized version of Black’s formula is given by This formula could be used to derive the European style option formulas for a derivative follows a lognormal distribution having the payoff Where Derivation of Black-Scholes formula using Black’s formula Now, we will derive Black-Scholes option pricing formula using this Black’s formula. As we know that […]

Futures price behaves in the same way as a stock that provides a dividend yield q equal to risk-free rate, r. Fisher Black [1976] provides the formula for valuing European futures call and put options which can be given by replacing So by Fo and setting q=r in Merton formula Where

There is no close-form solution for American-style option up to now. For applying Black-Schloes-Merton model to American options, let us consider non-dividend paying American call and put options, and dividend paying American call and put options separately. Non-Dividend Paying American Call Option Analysis shows in case of non-dividend paying American call option it is always […]