Numerical Searching Methods and Option Pricing Models Set (Set 3)
Newton-Ralphson method - or simply the Newton's method is one of the most commonly used numerical searching method for solving equations. Usually Newton's method converges well and quickly, but the convergence is not guaranteed. Newton's method requires an initial value. This values can determine the way the search is converged. The major challenge to using this method is that the first differential (first derivative) of the equation is required as an input for the search precedure.
Package Set 3 - Numerical Searching Methods and Option Pricing Models - 14 programs *
Numerical Searching Method - Newton-Ralphson | Numerical Searching Method - Secant Method | Implied Standard Deviation For Black/Scholes
Call - Newton Approach | Implied Standard Deviation For Black/Scholes Call - Secant Approach | Implied Standard Deviation For
Black/Scholes Call - Bisection Approach | Implied Standard Deviation For Black/Scholes Put - Newton Approach | Implied Standard Deviation
For Black/Scholes Put - Secant Approach | Implied Standard Deviation For Black/Scholes Put - Bisection Approach
Black-Scholes Option
Pricing Model - European Call and Put | Option Greeks Based on Black-Scholes Option Pricing Model | European Option Model on Asset
with Known Cash Payouts |
European Option Model on Asset with Continuous Cash Payouts (Index Option) | European Option Model on Currency| European Option Model on Futures
Secant method, unlike the Newton-Ralphson method, does not require the differentiation of the equation in question. Because of that, it can be used to solve complex equations without the difficulty that one might have to encounter in trying to differentiate the equations. Secant method requires two initial values. Test shows that this method converge a little bit slower than the Newton-Ralphson method.
The implied standard Deviation or implied volatility is the volatility value that would make the theoretical value (in this case the Black-Scholes Model) equals to the given market price. To use Newton-Ralphson method, the first differential of the standard deviation with respect to the price (Black/Scholes) is required. In this case, we can use Vega (Kappa) the sensitivity of the call price to the implied standard deviation.
Unlike Newton-Ralphson precedure, Secant method does not require the first differential of the of the standard deviation with respect to the price (Black/Scholes) as an input. This makes Secant method a more convenient tool to use. Nevertheless, it does require an initial value for the iteration just as any other numerical precedures. Secant method does not converge as fast as the Newton-Ralphson method.
Bisection searching method utilizes linear interpolation. It uses a minimum and a maximum starting numbers in the iteration process. The steps it takes to convert depends greatly on the starting numbers. In general, this method takes more iterations to convert compares to the Newton method.
In this example, we derived call and put option price based on the Black-Scholes model. The function procedures are used. The first function, SNorm(z), computes the probability from negative infinity to z under standard normal curve. This function provides results similar to those provided by NORMSDIST( ) on Excel. The second function and the third function compute call and put prices, respectively.
This program contains option sensitivities (delta, gamma, vega, theta, and rho) formulas and source code. Option sensitivities are also know as the Greeks. They measures how sensitive the option price is toward changes in its parameters. All Greeks are available in user-defined VBA functions with mathematical formulas.
When a stock issues dividend, cash is paid to the holder of the asset. The call holder does not receive any part of the payout. When the stock goes ex-dividend, its value will usually decreased by approximately the amount of the dividend distribution.
Some assets have numerous distribution of cash payouts. An example is a broad-based stock market index portfolio (say SP500), in which nearly everyday one component stock or another will pay a dividend. Merton (1973) has derived a variant of the Black-Scholes model for an asset that pays dividends continuously.
In 1983, Garman and Kohlhagen developed a model that computes European currency options. This program demostrates the computation of Currency option prices.
Black in 1976, developed a variant of his basic model that can be applied to compute options on futures and forward contracts. The following demostrates the computation of futures option prices.
* Two programs in this set also available in Set 1.
(Similiar for Put option)
(Similiar for Put option)
(Similiar for Put option)
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