Quantitative Finance Collector is a blog on Quantitative finance analysis, methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management.

One big issue of pricing barrier option with Binomial tree or other lattice method is its slow convergence rate, the barrier option value converges very slowly as the number of tree or lattice levels increase, often requiring unattainably large computing times for even a modest accuracy. A typical plot of barrier option binomial tree results against its analytic value looks like

source from paper Enhanced Numerical Methods for Options with Barriers
where the pricing performance is in a sawtooth fashion, with severe periodic spikes that move away from the correct result, which is a nightmare for a researcher because adding more steps doesn't necessarily mean to yield a more accurate answer.

How to value a stock option with discrete dividend was briefly introduced at http://www.mathfinance.cn/valuation-of-stock-option-with-discrete-dividend/, where the main goal is to compare the performance of different methods, namely, Escrowed dividend model, Chriss volatility adjustment model, Haug & Haug volatility adjustment model, Bos volatility adjustment model, and Haug, Haug and Lewis method. I didn't include lattice method for comparison because non-recombining binomial tree is computer intensive, especially when the number of dividends is large.

In the book Options, futures and other derivatives by John Hull, how to deal with discrete dividend with a binomial tree is explained in detail, see page 402, fifth version, where future discrete dividend is divided into two types:
1, known dividend yield. For instance, there will be a 3% dividend 3 months later (3% of the stock price), it is straightforward to handle it as the binomial tree is recombined when the nodes are multiplied by a percentage, so basically what we need to do is to construct a tree like usual before ex-dividend date, and then shift all the left tree nodes down by (1-dividend yield), that's it, the number of nodes are the same as for non-dividend binomial tree;

(source from Options, futures and other derivatives)

2, known dollar dividend. For instance, there will be a 2.5 dollar dividend 3 months later, so before ex-dividend date the binomial tree is constructed as usual but exactly at the date after ex-dividend, the whole nodes are shifted down by 2.5 dollar, and then a new binomial tree is constructed, because the nodes are shifted by an absolute amount number, the new binomial tree is not recombined any more, which means much more nodes than the non-dividend case.

Binary option is a one of the simple &  common type of derivative, where the payoff is either a certain amount of prescribed cash, called cash-or-nothing option, or shares, called asset-or-nothing option. Intuitively, cash-or-nothing option holders receive cash if the option finishes in the money, asset-or-nothing option holders receive shares of asset if in the money, thereafter binary options are often named as digital options.

The pricing of binary options is straightforward under GBM framework, the widely used Black Scholes formula can be easily adopted for binary option valuation. Once we understand the principle and know how to price it, the next step probably is to trade binary options. There are several online option trading platform for an individual investor to choose, the one I'd like to review is EZTrader, who has revolutionized the way binary options are traded on the internet today, by supplying its customers with a simple, exciting, dynamic and highly profitable trading platform, very different from traditional option trading. Due to the simplicity and speed of our binary options trading system and the low minimum investment amount, it is able to reach investors with different profiles all over the world. Ranging from sophisticated investors that are looking for ways to hedge their positions in the traditional market, to amateur day traders looking for some "action" without risking large amounts of money, EZTrader developed a system suitable to most of everyone's goals.

EZtrader have taken the fear and uncertainty out of Forex trading to focus on an existing new kind of trade. At EZtrader you can trade Binary Options. With binary options you simply choose whether the stock price will go up or down by the expiration time and place you call or put accordingly. With EZtrader your winning return is fixed, you don’t have to leverage millions of dollars with every trade or setup complicated stop loss strategy. With EZtrader binary options everything you need is right in front of you.

Why do I select this online trading service? well, there are at least the following advantages of EZtrader I am aware of:
- A member is able to trade Nasdaq, Dow Jones and Commodities based options;
- Hourly trades;
- Open an account is free, absolutely No Fees;
- Members can choose to withdraw fund as they want;
...

Besides simplified trading process, EZTrader members are provided with a complete set of tools to help them optimize their trading. Tools include live financial news, references to financial sites and a wide variety of tradable options, and more. Check EZtrader home page regularly for new promotions that will help you to get the most out of your trades, for example, one promotion is: If you deposit a total of $550.00 today, Monday, February 22nd, 2010, you will receive a bonus of $250.00 (%45)Registration is totally free and there are no commissions to pay ever.

To start trading, first go to trading area after sign in, you will find a pool of options to choose

Choose an option to trade from the list of available options, then select the type of trade, either CALL or PUT, enter the amount you would like to trade. you can change the trade type from CALL to PUT or vice-versa even after entering an amount, finally click 'Trade' to execute your trade. Simple & new binary options trading platform, start applying your derivative quantitative skills directly at EZTrader.

When asked how to value a stock option without dividend or with continuous dividend, many people would refer to Black Scholes formula, but how to price an option with discrete dividend then? certainly Black Scholes model can't be used directly since one of its assumptions is continuous payout. Paper Back to Basics: a new approach to the discrete dividend problem by Haug, Haug and Lewis summarizes the following ways:
1, Escrowed dividend model, which is the simplist and the least accurate way as a result. The basic idea of Escrowed dividend model is to adjust the current stock price by deducting the present value of future dividends, and plug in the replaced stock price to Black Scholes formula;
2, Chriss volatility adjustment model, besides replacing current stock price, this model adjusts volatility as well because the Escrowed dividend model alone decreases the absolute price standard deviation, hence underestimates an option's value. However, Chriss model yields too high volatility if the dividend is paid out early in the option’s lifetime, which generally overprices call options;
3, Haug & Haug volatility adjustment model; which is more sophisticated than Chriss model and takes into account the timing of the dividend, unfortunately, the authors show this method performs particularly poorly for multiple dividends stock option;
4, Bos volatility adjustment model, a even more sophiscated model than Haug & Haug, but still, it performs poorly for large dividends or long term options;
5, Lattice method, for example, non-recombining binomial tree introduced in the bible book Options, Futures, and Other Derivatives with Derivagem CD (7th Edition), we all know it is time-consuming;
6, Haug, Haug and Lewis method introduced in the above-mentioned paper, the basic idea is to calculate first the ex-dividend option price by Black Scholes model, then discount back the ex-dividend value under equivalent martingale measure. The authors demonstrate the high accuracy of their model with several examples afterwards.

Below is sample Matlab codes I wrote for comparision, a single dividend is used for simplicity, results similar to the table listed in the paper

Another derivative calculator shared with you, ATOM - Advanced Tool for Option Modelling is a C++ option calculator covers:

price, implied volatility and Greek letters;
Black-Scholes analytic formula;
binomial tree lattice;
Cox-Ross-Rubinstein parametrisation;
Jarrow-Rudd equal-probabilitiy parametrisation;
control variable technique;
Broadie-Detemple penultimate node analytic approximation;
Monte carlo simulation with the following variance reduction and normal sampling techniques:
antithetic variable;
moment matching, also known as quadratic re-sampling;
Mersenne Twister pseudo-random numbers;
Halton quasi-random numbers;
Box-Muller polar normal inversion;
Moro normal inversion;
unlimited maximum number of steps in binomial trees and unlimited maximum number of trials and time intervals in Monte carlo simulations;
exotic option support: Asian average price, binary cash-or-nothing and asset-or-nothing, chooser option;