Efficient algorithms are needed to predict the accuracy and increase the speed of the prices of the American and European put options. A put option is an agreement between two parties that allows the buyer the right to sell commodity at a certain time for a certain price, which can be also known as the stock price. The buyer is obligated to purchase the commodities at the strike price if the seller is willing to sell it. The seller is not obligated to sell his/her commodities, but has the right to do so. In order to gain this privilege, the seller has to pay a premium in order to get the put option. This premium cannot be recovered if the seller chooses to sell or not. Pricing the numerical valuation of the standard put option can have many effects on determining the put options value.

There are some algorithms that were used to determine the price of the American put option: binomial trees, finite difference approach, quasi-analytical solutions, randomization, etc. However, the standard algorithm used to determine the price of the American put was introduced in 1984 by Geske and Johnson—the Richardson’s extrapolation. Richardson’s extrapolation can be used to derive a price from a series of prices, which depend on the option maturity. Alfredo Ibanez introduced a more efficient algorithm using Richardson’s extrapolation and Newton’s root finding method. Option prices are smooth and convex functions of underlying price, so Newton’s method can compute the fixed point quickly, and Richardson’s extrapolation was then used to derive the correct order of the error term. The use of Richardson’s extrapolation allowed for the control of the error of the extrapolated prices.

http://goliath.ecnext.com/coms2/summary_0199-281818_ITM

http://en.wikipedia.org/wiki/Richardson_extrapolation

http://en.wikipedia.org/wiki/Put_option