Black Scholes

The Black-Scholes function is used to price both American and European options on dividend and non-dividend securities. The Black-Scholes function can evaluate American and European options using the Black-Scholes, Modified Black-Scholes, Black (options on futures), Garman-Kohlhagen (options on spot foreign exchange), or Pseudo-American function. Options are priced with the Black-Scholes function that takes into account continuous dividends, a constant dividend yield, and discrete dividends. American options are evaluated with Fischer Black's Pseudo-American option function. The Pseudo-American function returns the maximum value of the option evaluated at each ex-dividend date and at expiration with the Modified Black-Scholes European function. At each ex-dividend date the time to expiration is the ex-dividend date, the AssetPrice is adjusted by all of the dividends, and the number of dividends expected to occur adjusts the StrikePrice. The Black-Scholes function was derived with the following assumptions:

The option can only be exercised on expiration date (European option only).

There are no transaction costs, margin costs or taxes.

The risk-free interest rate is constant over the life of the option.

The price changes of the underlying security are log-normally distributed.

Dividend payments are not discrete. The underlying security yields cash flows on a continuous basis.

The FinOptions function BlackScholes can be used to evaluate either American or European vanilla options.

Black Scholes French

The French Black-Scholes function, developed by D. French, can be used to price American and European options where the calendar days and trading days until expiration differ. The original Black-Scholes function was developed using calendar days in order to evaluate option prices. Although volatility is related to the number of trading days until expiration, interest is paid according to calendar days. D. French has suggested that two different time inputs should be used and applied to an adjusted Black-Scholes formula. The French Black-Scholes function takes into account these two time values with the variables, TimeTrading and TimeExpire. This allows for the evaluation of volatility according to trading days and the discounting of interest according to calendar days. Although this adjustment makes little difference in the evaluation of most options, it is important in the pricing of short life options. For American options, this function utilizes the Modified Black-Scholes American function with the French adjustment. Therefore, the theoretical value of an American option is the maximum of the intrinsic value or the calculated theoretical value of the option. The French Black-Scholes function can be used on European and American options that have a continuous dividend, a constant dividend yield, and discrete dividends.

The FinOptions function BlackScholesFrench can be used to evaluate either American or European vanilla options.

Whaley

The Quadratic Approximation developed by G. Barone-Adesi, R. E. Whaley and L. W. MacMillian, as an alternative to numerical methods for pricing American options. The Quadratic Approximation method is accurate and less computer intensive than Binomial, Compound-option or Finite-difference pricing methods. It can be used to value American call and put options on currencies, futures contracts, stocks, and stock indices. The technique involves estimating the difference between an American option and a European option.

American option = European option + American Surcharge

The method of determining the early exercise option is an iterative approach. Therefore, the Quadratic Approximation is computationally more intensive than the Generalized Black-Scholes function. The Quadratic Approximation function follows the same assumptions that apply to the Generalized Black-Scholes function. The Quadratic Approximation function can be used on American options that have a continuous dividend, a constant dividend yield, and discrete dividends.

The FinOptions function Whaley be used to evaluate American vanilla options.

EuroDollar

The Eurodollar function uses the Black-Scholes, Pseudo-American, Quadratic Approximation, Binomial, or the Bjerksund-Stensland function to price a Eurodollar option. The technique in straight forward and is as follows. The range for the AssetPrice is, 0 < AssetPrice £ 100, the range for the StrikePrice is, 0 < StrikePrice £ 100, YieldRate = InterestRate, and call options are evaluated as put options and put options are evaluated as call options. Once these conversions and parameters have been met, the variables are passed into the appropriate functions and the result is returned. There are no dividends to evaluate with this function. Although the ExerciseType flag can toggle between American and European, it is only used for the BlackScholes and Binomial functions. The Quadratic Approximation and Bjerksund-Stensland functions are American only option functions. The BlackScholes function uses the Pseudo-American function to price American Options.

The FinOptions function EuroDollar be used to evaluate either American or European vanilla options.

Binomial

The Cox, Ross, and Rubinstein binomial function developed in 1979 is a numerical technique to price both American and European options on stocks, futures and currencies. The construction of the binomial lattice attempts to dicretizes the geometric Brownian motion. At the limit, a binomial tree is equivalent to the continuous-time Black-Scholes formula when pricing European options. The binomial function works by dividing the time to expiration into a specified number of time steps or iterations. At each node for every iteration, the function calculates the up and down price movements using a predetermined probability, producing a binomial distribution for the price of the underlying asset. The distribution recombines at every node to generate a lattice or tree of underlying prices. To calculate the option value the binomial function uses each of these prices. By valuing the option at each node in the tree, determination of early exercise is evaluated. The size of the lattice or tree is determined by the "Iteration" term in the function. By increasing the number of iterations, the accuracy increases, as does the computational time. The binomial function has been extended to handle a continuous dividend, a constant dividend yield, and discrete dividends on both American and European options.

The FinOptions function Binomial be used to evaluate either American or European vanilla options.

Jump Diffusion

The Merton jump-diffusion function assumes that the underlying asset follows a jump-diffusion process. The valuation of the option follows a stochastic process other than a log-normally distributed function such as the Black-Scholes function. The function requires two additional parameters to be estimated; the expected number of jumps per year l and the percentage of the total volatility explained by the jumps g. The jump-diffusion function is used to price European options. These options can have a continuous dividend, a constant dividend yield, and discrete dividends.

The FinOptions function JumpDiffusion be used to evaluate European vanilla options.

Bjerksund Stensland

The Bjerksund-Stensland function developed in 1993 is an alternative closed-form solution for pricing American style options. The function is computationally very fast and analytically efficient. Numerical data indicates the Bjerksund-Stensland function may be more accurate in pricing long dated options than the Quadratic Approximation function. The Bjerksund-Stensland function can be used on American options that have a continuous dividend, a constant dividend yield, and discrete dividends.

The approximation is based on an exercise strategy corresponding to a trigger price. If the underlying asset price is greater than or equal to the trigger price it is optimal to exercise, and the value must equal S-X. If the cost of carry is greater than or equal to the risk free rate it will never be optimal to early exercise the American call option, and the value can be found using the Generalized Black-Scholes function.

The FinOptions function BjerksundStensland be used to evaluate American vanilla options.

Roll-Geske-Whaley

Roll (1977), Geske (1979a), and Whaley (1981) developed a formula for the valuation of an American call option on a stock paying a single dividend. If there is more than one dividend, usually the final dividend date is the only consideration. In this case the Roll-Geske-Whaley function can be used, but the underlying asset price is adjusted by the present value of all the remaining dividends. The Roll-Geske-Whaley function can only be used to value American Calls with zero or more dividends. The function can only handle discrete dividends. This technique is computationally fast closed-form approximation that executes quickly compared to numerical techniques.

The FinOptions function RollGeskeWhaleyCall be used to evaluate an American vanilla call option.