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Monday, March 29, 2010

Quantitative Finance Collector

High Dimensional Sobol Sequences

Quasi-Monte Carlo methods are very efficient in solving low dimensional integration problems, for example, pricing a path-dependent exotic option, among those low discrepancy sequences, Sobol is undoubtedly one of the best and most widely used due to its high convergence speed, how to generate Sobol Sequences was shared at posts Sobol sequence generator, Primitive polynomials for Sobol sequences, halton and sobol sequences and Sobol and Generalised Faure sequences.

However, empirical research have shown Quasi-Monte Carlo Sobol sequences perform poorly for high dimensional integration problems, it can never be more efficient than the ordinary Monte Carlo simulation, some researchers suggest to use Sobol sequences with Brownian bridge for a better result. A recent working paper shared at Articles uses a small trick - randomized Sobol sequences to avoid the poor performance high dimensional problem, indeed it is straightforward to add this trick into your codes.  Below is a comparison graph demonstrating the performance, very impressive.

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Upcoming Webinars by MathWorks

Attending a free webinar is a good way to learn efficiently. Here is a list of upcoming one-hour live online webinars about MathWorks products which I think useful to math finance studying.

DateTitleSession Time
24 Mar 2010Data Acquisition with MATLAB10:00 a.m. (Central european Time)
6 Apr 2010Global Optimization with MATLAB Products9:00 a.m. (U.S. EDT)2:00 p.m. (U.S. EDT)
8 Apr 2010Digital Signal Processing Using MATLAB9:00 a.m. (U.S. EDT)2:00 p.m. (U.S. EDT)
13 Apr 2010Best Practices for Verification, Validation, and Test in Model-Based Design9:00 a.m. (U.S. EDT)
2:00 p.m. (U.S. EDT)
15 Apr 2010MATLAB for Excel Users in Computational Finance10:00 a.m. (United Kingdom-GMT)


Enjoy.
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Binomial Tree Option Pricing with Discrete Dividends

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;
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(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.
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Quant Project Outsourcing

I haven't updated my blog for several days as I have been very busy last week with my own research, courses. In the mean time I also did a few outsourcing projects, for example, some recent projects including a portfolio Value at Risk calculation excel with macro supporting advanced input & output form; an American option calculator considering discrete dividends in VBA; data clean, missing data imputation for a small hedge fund; probability of future stock price exceeding a barrier in Matlab, etc.

Some selected customers' review:
Very satisfied, with the level of work -  muzammmil
Fantastic effort abiao, I really appreciated you turning this job around for me so quickly. If I ever have the need for a quant to help with another custom function, you are my man. - ABCInvestor
Thanks for your efficiency, you are definately on my list if I need a quick quant. - John

with received feedback
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If you happen to have some math finance projects for outsourcing, you may consider to give my group and me chance. Although we can't guarantee we are capable of meeting all your requirements, we can try best to insure you a satisfied result as long as we promise to undertake the project, at a low cost. Please email me at abiao @ mathfinance.cn (remove space) for a quote and proposal if you want, cheers.
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The Big Short: Inside the Doomsday Machine

Most of us have read the book Liar's Poker: Rising Through the Wreckage on Wall Street, one of the books that define Wall Street during the 1980s and are highly recommened by many people in academia and industry, where the author Michael Lewis described his experiences as a bond salesman in a humon sense. I just got to know today from a friend of mine recommending a new book The Big Short: Inside the Doomsday Machine , as described:
Quotation
A brilliant account—character-rich and darkly humorous—of how the U.S. economy was driven over the cliff. Truth really is stranger than fiction. Who better than the author of the signature bestseller Liar’s Poker to explain how the event we were told was impossible—the free fall of the American economy—finally occurred; how the things that we wanted, like ridiculously easy money and greatly expanded home ownership, were vehicles for that crash; and how shareholder demand for profit forced investment executives to eat the forbidden fruit of toxic derivatives.


I don't know, considering the high quality of Liar's Pocker, it may be worth reading. I pre-ordered just now at Amazon, if you also want a bed reading book, order one at only $15.09 at Amazon to be released on March 15, 2010. The Big Short: Inside the Doomsday Machine.



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Calibrating Stochastic Volatility Models with Heuristic Techniques

Stochastic volatility models, specifically, Heston model, SABR model, are introduced before and become the widely used among academia and industry. However, the calibration process is difficult because generally the pricing requires numerical integration, and calibration requires to find five and eight parameters instead of only one for Black Scholes model.

Found a paper Calibrating Option Pricing Models with Heuristics, where the author look into the calibration of Heston (1993) and Bates (1996) models. Finding parameters that make the models consistent with market prices means solving a non-convex optimisation problem. Optimisation heuristics is suggested for this issue, more specifically they show that Differential Evolution and Particle Swarm Optimisation are both able to give good solutions to the problem.

Take a look if you are interested, in the Appendix the R and Matlab codes are given for a better understanding. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1566975
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Improve Your Trading With Objective Method

After the successful release of free ebook 14 Critical Lessons Every Trader Should Know, elliott wave international decides to present another new free report: Improve Your Trading With Objective Method.

You’ve heard the common trading advice: “Successful traders know how to control their emotions, instead of being controlled by their emotions.” I bet you're thinking easier said than done, huh? As a trader, you’re bombarded with countless possibilities that can make decisive action a stressful hire wire act.  It’s no wonder your emotions can get in the way.

That’s where Elliott Wave International’s free report can help. You’ll discover how to manage your positions objectively – plus control your emotions – so you make the most of each high-confidence trade set-up.

Learn more and download your free report.

There’s even a bonus lesson included on “Protective Stops,” so you can learn critical exit strategies.

If you’re a trader or considering trading, this report is a must-read. Rid yourself of emotional trading and learn to objectively identify high-confidence trade set-ups. Visit Elliott Wave International to download your free report.
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VaR Historical Simulation

Following Value at Risk xls and var backtesting, a third post about using historical simulation for Value at Risk calculation. We know one shortcoming of historical simulation is: the result highly depends on the choice of sample data length, VaR result does not vary often or changes suddenly. Despite this weakness, HS is still popular due to its obvious advantage: easy to implement, and no distribution assumption required, which is especially appealing if the estimate of distribution assumption is difficult. Several ways have been proposed to improve HS's performance, here are two selected methods with good results I personally use.

1, The Best of Both Worlds: A Hybrid Approach to Calculating Value at Risk by Jacob Boudoukh1, Matthew Richardson and Robert F. Whitelaw. By hybrid it means this approach is a combination of RiskMetrics's parametric method and Historical Simulation. The basic idea is: since we can allocate larger weight to recent data and smaller weight to remote data for exponential weighted moving average (EWMA) volatility calculation, hence improves the backtesting performance of parametric method, why can't we then apply a similar principle to historical simulation? make sense? so it estimates the VaR of a portfolio by applying exponentially declining weights to past returns and then finding the appropriate percentile of this time weighted empirical distribution. The following results are from the paper The Best of Both Worlds: A Hybrid Approach to Calculating Value at Risk, page 11. It does improve compared with the vanilla historical simulation and EWMA parametric method, nice.
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Friday reading list of this week

Friday again, just a final kind remind, since Change of Friday Reading List Setting, I have been updating Friday reading list on page articles, for example, the list of this week includes:
1, Testing for Asymmetric Dependence, http://www.bepress.com/snde/vol14/iss2/art2/;
2, Index-Exciting CAViaR: A New Empirical Time-Varying Risk Model, http://www.bepress.com/snde/vol14/iss2/art1/;
3, Improving Portfolio Selection Using Option-Implied Volatility and Skewness , http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1559642;
4, Trading Activity and Bid-Ask Spreads of Individual Equity Options, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1553222;
5, The Method of Simulated Quantiles, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1561185

Keep an eye on page articles.
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Inverse Graphing Calculator

An interesting application of Inverse Graphing Calculator, where you enter any word from A to Z into your calculator and then get a graph of the curve.

For instance, if you write an equation:
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you would get a graph below:
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Creat your own at http://www.xamuel.com/inverse-graphing-calculator.php
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