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Download Terreneuve: A lightweight C++ library for quantitative finance applications.

**[may 12, 2007]**

Nassim Taleb podcast about the Black Swan. Another recent appearance. Update: Taleb's site has links to lots of reviews as well as the Colbert clip. It's weird to see one of your profs on TV, let alone on a show that people actually watch. The only review I've read so far (NYTimes) is apparently Taleb's least favourite judging from its position on the page and the extensive rebuttal he's prepared (a portion of which appeared in the Times' letters section). Taleb's crazy, but *good* crazy.

**[february 8, 2007]**

My final thesis: "Stock Option Pricing with Cash Takeovers". MATLAB code here.

Also of interest, final Computing Methods project, a MATLAB implementation of Joshi and Stacey's Intensity Gamma model with Christelle Ho Hio Hen, Aaron Ipsa and Dharmanshu Shah:

**[january 27, 2007]**

Asset allocation is always a tricky question. I somewhat dogmatically invest in index funds but when it comes to deciding the allocation between stocks and bonds I roughly follow Benjamin Graham's prescription "The investor may vary his holding of common stocks between the 25% minimum and the 75% maximum." That is I hold about 25% bonds... ;-) When it comes to dividing up the equity piece, I'd be lying if I said I approached it rigorously. Although I'm relatively certain about which indices I wanted to invest in, I always felt like the final allocation was a bit...uh...heuristic. It was a lot easier when the Canadian government forced you to invest 80% of your retirement savings in Canadian investments.

Here are some useful links about asset allocation:

- https://retirementplans.vanguard.com/VGApp/pe/PubQuizActivity?Step=start - Vanguard questionnaire to help you divide between stocks and bonds
- http://www.tdcanadatrust.com/mutualfunds/pdf/mut_app.pdf - more involved TD questionnaire, includes several sample portfolios

**[january 22, 2007]**

(Probably only useful for NYU students) NYU's Virtual Business Library, a compendium of information sources on companies, industries, countries, etc.

**[january 21, 2007]**

(via sam) Article from BW "Outsmarting the Market" on Barclays Global Investors quantitative approach to investing. Some interesting bits:

*I came here because this is where the leading edge in my area of research is now.* (One of my profs also claimed this. I hope it isn't just a euphemism for "I wanted to get paid".)

*The global macro group came up with a notable example of the latter by devising a set of signals that can pinpoint the timing of an economy's pivot from recession to expansion. By buying a country's stocks and shorting its bonds before its recovery was generally recognized, BGI's Global Ascent fundwas able to generate total alpha of 4% in 2005 and 2006.*

*Even as individual signals have come and gone, earnings quality has been BGI's single richest source of alpha over the last decade.*

*Whatever a new idea's provenance, as a rule, BGI will not deem it portfolio-worthy unless it first passes four tests. In BGI speak, they are collectively known as SPCA, for Sensible, Predictive, Consistent, and Additive.*

*Recently, Duke University's David A. Hsieh, a leading hedge fund scholar, theorized that only $30 billion in alpha is realizable annually from the $30 trillion market value of all stock and bond markets worldwide.* (thank goodness for derivatives...)

**[january 20, 2007]**

Paper by Leif Andersen on MC discretization of Heston. Mentions the Lord, et al. paper which we used to discretize Heston in Case Studies but includes this diss: "While this conclusion overall appears sound, the resulting discretization scheme is largely heuristic and uses essentially none of known analytical results for the Heston model." Andersen proposes two methods based on approximating the CDF of variance using a truncated Gaussian distribution and a combination of distributions respectively.

This was presented this past week at a new monthly seminar series organized jointly by NYU and Columbia: http://www.ieor.columbia.edu/seminars/NY_Quantitative_Finance/

Peter Carr will be speaking on Monday (Jan 22nd) at the Columbia seminar on "Derivatives Replication under Laplace Dynamics" http://www.ieor.columbia.edu/seminars/financialengineering/2006-2007/spring/index.html

Updated MathFinSeminars.

**[january 8, 2007]**

Happy New Year! I just wanted to note for the record that I am the OISS featured student of the week, weighing in with my pithy remarks on life in the Big Apple. Read at your own risk.

Just to give this post some math finance content...congratulations to Jim Simons on his selection as Financial Engineer of the Year. To tell the truth I find the choice a bit puzzling. Simons is well known as the head of Renaissance Technologies which has a truly incredible track record, but unlike past winners I don't think he has made many public contributions to the field.

**[december 21, 2006]**

What a ridiculous two years... (doing the flashback squiggle from Wayne's World)...

Scene I: A deserted office park, its a freezing cold December Ottawa day, early in the morning. Coffee in hand our intrepid hero enters the testing center to write the GRE. Things are going well, until he hits it, the question of death. Not especially hard, not especially easy, but somehow the one question that our hero can't solve quickly and can't bring himself to move past. Five minutes pass. Ten. There's still time to finish, the other questions will be easy right, one minute per question should be enough. Got it...oh oh, ten questions left and five minutes? Disaster! Less than 750 on the quant section? Give up on NYU and the rest of them buddy...

Scene II: One month later. The basement of an apartment building in Toronto, another testing center. A bunch of Chinese and Indian kids (including our hero). Ten minutes to test time, feeling good, there's still a chance. Suddenly, the room fills with the sound of...jackhammering...? Everyone starts to chatter nervously, its got to be temporary right? The testing drone emerges, "Everyone is going to have to be rescheduled." Rescheduled? I'm going to miss all the deadlines! "Well, if you choose to write the test you're going to have to deal with the noise. We can't let you cancel your results. We've got noise blocking headphones." Noise modulating headphones is more like it... Everyone else leaves, some in tears. Well, let's try it, some of the music I listen to is worse than this. 800 quant, 780 verbal, 6 on the essay. And no hardship bonus. Couldn't get an inch from ETS either...bastards.

Who knows if it made a difference, but a few months later I got in to NYU and the rest is well, the rest is this blog. I still can't believe it.

**[december 6, 2006]**

Two links about the Large Homogeneous Pool approximation for CDO pricing:

- http://ucs.kuleuven.be/seminars_events/other/files/3afmd/Schloegl.PDF "STOCHASTIC METHODS FOR PORTFOLIO CREDIT DERIVATIVES" Lutz Schloegl (How do you think that's pronounced? Shloy-gul? Schloegl! Come on say it, you'll like it!)
- http://www.mathematik.uni-ulm.de/finmath/ss_05/fe/LHPlus.pdf "LH+: A Fast Analytical Model for CDO Hedging and Risk Management" Lehman research note

And one nice overview paper with some interesting anecdotes:

**[december 3, 2006]**

Roundup of recent hedge fund coverage via mahalanobis. One theme running through recent stories is the possibility of passively replicating hedge fund strategies. Is Merrill planning something in this space? (Is it even a space yet?) See this piece by ML analyst Benjamin Bowler who also shows up in this Economist article "Send in the clones":

*Alternatively, Benjamin Bowler of Merrill Lynch suggests investing in hedge-fund niches, such as merger arbitrage. Instead of spending time and money investigating which takeovers to back, a mechanical strategy could back all deals that met certain parameters. This, too, could be done at low cost.*

**[november 21, 2006]**

Speech given today at the NYSE by one of the Fed governors Kevin Warsh "Financial Markets and the Federal Reserve". Some interesting financial engineering related points about how the Fed uses asset prices to imply thinking of market participants and the challenges it faces trying to do so.

*Why can't market prices be more assuredly relied upon? Asset prices contain term premiums, credit risk premiums, and liquidity premiums that vary over time and are themselves related to market expectations and uncertainty. Consequently, it can be difficult to determine whether movements in asset prices reflect a change in expectations, in uncertainty, or in some combination of premiums.*

**[november 14, 2006]**

Article from today's WSJ about demand for graduates from math finance programs "Wall Street Warms To Finance Degree With Focus on Math". NYU's Steve Allen makes a cameo but Berkeley's Linda Kreitzman gets top billing! Sounds like there will be even more programs this year with UCLA, Rutgers and Minnesota throwing their hats into the ring.

*At the Sloan School of Management at the Massachusetts Institute of Technology, a master's in finance is in the works, targeted at MIT undergraduates in engineering, math and science. "Hedge funds are already recruiting MIT undergraduates, but they would rather hire them with an additional year of finance," says Paul Osterman, deputy dean at the Sloan School.*

UPenn's Management and Technology program also competes in this area.

**[october 31, 2006]**

IAFE endorsed event next Wednesday at the New York Academy of Sciences featuring some interesting speakers: Robert Engle, Doyne Farmer, David Shaw and Eugene Stanley. It's free!

I've seen Robert Engle speak before and it's always interesting and it's always about vol. Doyne Farmer is at the Santa Fe institute but is also associated with The Prediction Company, now a UBS subsidiary, which builds automated systems for trading financial instruments. David Shaw is well known as the founder of his eponymous firm which among other things is attempting to build a supercomputer for drug design. Eugene Stanley has published many papers often classified as "econophysics" and has been mentioned here earlier as co-author of a paper which discovered an inverse cubic law for for the distribution of stock price variations. Should be fun.

Some links:

- http://www.kk.org/outofcontrol/ch22-b.html - Chapter 22 "Prediction Machinery" talks about chaos theory and Doyne Farmer, from Kevin Kelly's book "Out of Control"
- http://www.bearcave.com/bookrev/predictors.html - review of "The Predictors" by Thomas Bass which tells the story of The Prediction Company
- http://polymer.bu.edu/hes/articles/ - Eugene Stanley papers
- inverse cubic law paper

**[october 21, 2006]**

We're pricing American options in Computing Methods...

- http://www.dma.unive.it/105-02.pdf - put boundary
- http://www.iam.fmph.uniba.sk/institute/sevcovic/papers/cl20.pdf - call boundary
- not related but interesting - a patent by two Citadel guys that talks about toxic trades

**smart / get things done** | http://www.joelonsoftware.com/articles/fog0000000073.html

A guide to *conducting* interviews but certainly of value to interviewees as well. I recognized many of the techniques from interviews I've been on. Luckily, it appears that I've avoided most (I won't say all!) of the pitfalls mentioned in the article.

**more questions** |

in-place reverse string 20 red and green balls, two buckets, what's the best way to place them to pick a red ball after shuffling the buckets, what is the probability of picking a red ball how would you use m-c to simulate vasicek, what is variance of the short-rate in vasicek what is price of option that pays ST/St? what is the delta of a binary, how about gamma, how would you hedge generate fibonacci sequence into an array what effect does an increase in vol have on delta, what about decrease in time to maturity call with strike 100 costs 30 and strike 120 costs 10, is there an arbitrage, if so what is it is there an option that cannot be priced on a binomial tree can you price an american with m-c inputs to a black-scholes pricer for equity options write an algorithm to count number of ways to ascend n-steps by one or two steps at a time try recursive, try iterative one out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased whats the diff between a java interface and abstract class (yes java has abstract classes) what trade would you put on if you knew a dividend was going to be announced what keyword is associated with allocation from the heap how would you price an option which allows you to choose between having one strike 100 call or two strike 120 calls is a european call option path dependent how does liquidity risk affect puts and calls differently what inputs are required to price a convertible how can you get survival probabilities from cds spreads what does the command netstat -ap do if a user reported a java out of memory exception what would you tell him to do (to diagnose the problem wiseass) what does the sql having clause do, what about group by what is the difference between a pointer and a reference (you think you know but you don't...) int *x = null;x += 7; - what is the value of x after these statements what are problems with the mean variance approach how does barra make such nice covariance matrices

**[october 19, 2006]**

Presentations from a 2005 conference at Cambridge "Developments in Quantitative Finance". An amazing collection of experts on a wide variety of topics and all the talks are in MP3 form.

Now playing: http://www.newton.cam.ac.uk/webseminars/pg+ws/2005/dqf/dqfw05/0318/hunter/all.mp3, Chris Hunter, a former instructor here at NYU (see http://math.nyu.edu/~chunter/) on "Applications of financial mathematics to trading".

Also, of relevance to Nassim Taleb's class today (and to my thesis project) - when does a mixture of distributions produce thin tails: http://www.wilmott.com/blogs/kurtosis/index.cfm/2006/6? As Peter pointed out to me today, Heston or other stochastic volatility models cannot produce a frown (with volvol you can increase the tails and with correlation you can change the skew but you can't thin the tails). A mixture of distributions or our proposed takeover model can achieve that.

**[october 17, 2006]**

Ahhh, the return of the link dump! Links related to our Computing Methods project - an implementation of Joshi and Stacey's Intensity Gamma model.

- http://www.quarchome.org/igini.pdf - Joshi presentation
- http://perswww.kuleuven.be/~u0009713/creditrisk/Stacey.ppt - Stacey presentation
- http://www.nomura.com/resources/europe/pdfs/dynamiccdomodelling2.pdf - Baxter (of Baxter and Rennie fame) proposes another model - in a nutshell: "Existing continuous models have no jump terms, and existing jump models have no continuous terms. The new model has both. It has both a continuous Brownian-motion term and a discontinuous Variance-Gamma jump term."
- http://www.fmpm.ch/docs/9th/papers_2006_web/9112b.pdf - Moosbrucker Variance Gamma copula model

**[october 7, 2006]**

Interesting paper from data provider Tick Data about issues and approaches to filtering tick data. Our current econometrics assignment has us looking at volume data which has some "anomalous" data corresponding to days before holidays and option expiries. Although this paper is about filtering price data, a lot of the points it makes seem general to the filtering of any type of financial data.

*The filtering of marginal errors involves a tradeoff. Filter data too loosely and you still have unusable data for testing. Filter data too tightly and you increase the possibility that you overscrub it, thereby taking reality out of the data and changing its statistical properties. Overscrubbing data is a serious form of risk. Models that have been developed on overscrubbed data are likely to find real-time trading a chaotic experience.*

**questions** |

derive the formula for beta write a program to calculate beta and alpha given x and y in C (assume that you have an immense amount of data) what is the variance of a spread (given individual vols and corr) given x and y uniform variables what is the probability that x+y <= .5 does expected return of a stock influence the value of an option (in b-s? in the real world?) what are the assumptions underlying black-scholes how do you test the significance of a factor what is a good null hypothesis for beta of a stock wrt market how would you correct anomalies in a time series pick five integers a b c d e, imagine a five cube, you start a ball at the origin and send it in the direction (1/a 1/b 1/c 1/d 1/e), what distance will it travel before reaching the origin again give me an example of something you have thought about quantitatively explain your thesis project given a large timeseries for a stock and for the market what things would you do to provide a high-level summary of the data in a short timeframe given four years worth of daily observations of return, a mean return of $1 and a std.dev of $15 what is the annual sharpe ratio? what is the std. error of this ratio? given a daily volatility what is the yearly volatility, what assumptions underly this calculation, if the series exhibited positive autocorrelation would the yearly volatilty be higher or lower a doctor tells you a test is 95% accurate, you get a positive result (you have the condition) and ask how likely is it that you have the condition, what is the answer? what information do you need to answer this? how would you optimize a portfolio of 20 S&P components to track the entire S&P 500 how would you use monte-carlo to find pi what option strategy would you use if you thought volatility would increase, what are the break-even points of the strategy have you done empirical work write a program in matlab to calculate the dependent variable given series of independent variables and factors why did you decide to move to finance give an example of an option that has negative vega what is the delta of an atm call what is gamma, is gamma higher for a shorter or longer maturity atm option value a two year lease on a gold mine given the current and forward price of gold and the cost of mining gold how can you tell whether a regression parameter is stable given this libor curve draw a line indicating the swap rate how would you bootstrap a yield curve quickly tell me how much a par 4% coupon four-year maturity bond would be worth if rates rose to 4.05% why do people issue cdos two red, two green socks in a drawer what is the chance of picking two socks that are a pair derive the b-s pde, explain a painter is known for the oblong figures in his paintings, one explanation is that he perceives everything as oblong, is this a possible explanation? why did scholes and merton win the nobel prize what would cost more: a portfolio of call options on individual stocks or a call option on a basket of stocks how would you price an american style option on a binomial tree why do real-world probabilities not matter on a binomial tree which courses did you like best what kind of job are you looking for if you had two triple-B bonds (same notional) where would the attachment point of the mezz tranche be, what about 100?

**[september 26, 2006]**

Robert Ferstenberg from MS presents joint work with Robert Engle on execution risk this Thursday, September 28th at 5.00 PM at Cornell's facilities on Broad Street. Registration required, click on the link to register and for info on future seminars by NYU profs Gatheral ("the wiliest practitioner in finance") and Carr!

**[september 14, 2006]**

Carr, Madan paper discusses using information from stock indices to price options on companies. One key insight is to use the CAPM model to divide the company's risk into an index (market) specific component and a firm specific component. They then suggest that the first component should be priced consistently with index options (e.g. risk-neutral measure) while the second can be priced using statistical probabilities.

An interesting paper because it brings together some of the topics we've covered so far in:

- Computational Methods - pricing options using characteristic functions and the FFT (the paper includes some sample matlab code for this), the Variance Gamma model
- FE Methods for Corp Finance / Econometrics - time-series methods, factor models, modeling residuals using non-normal distributions

Also the variance gamma model is the basis of the "intensity gamma" approach to CDO pricing proposed by Mark Joshi and linked below.

**[september 4, 2006]**

Courant Fall '06 class schedule. First day of classes tomorrow, plus I'll be speaking briefly at the orientation for the incoming class of Math Finance students. Everyone's experience is different but hopefully some of my advice will be useful for someone. If not, well, at least there's the free food.

**[august 30, 2006]**

Interesting program of talks on...what else...credit derivatives at Fordham: New York City's Jesuit University. Speakers include MC guru Paul Glasserman on computational aspects and Baruch's Liuren Wu speaking "about evidence and theory linking stock option implied volatilities and CDS spreads underlying the same reference company". Friday, September 29th from 11 to 5.30, free for full-time students! Added it to MathFinSeminars.

Speaking of computational aspects of credit derivatives, take a look at Mark Joshi's recent paper Intensity Gamma: A new approach to pricing portfolio credit derivatives.

**[august 28, 2006]**

(via infoproc) Infoproc's Steve Hsu and his student Brian Murray "On the volatility of volatility".

Abstract: The Chicago Board Options Exchange (CBOE) Volatility Index, VIX, is calculated based on prices of out-of-the-money put and call options on the S&P 500 index (SPX). Sometimes called the "investor fear gauge," the VIX is a measure of the implied volatility of the SPX, and is observed to be correlated with the 30-day realized volatility of the SPX. Changes in the VIX are observed to be negatively correlated with changes in the SPX. **However, no significant correlation between changes in the VIX and changes in the 30-day realized volatility of the SPX are observed.** We investigate whether this indicates a mispricing of options following large VIX moves, and examine the relation to excess returns from variance swaps.

Lots of other good stuff at infoproc - Shiller on the incredible magnitude of this housing boom, the predictive value of house prices (watch out below?), comments on the recent quant article in Businessweek, put/call ratio for statistical arbitrage...browse the archives!

**[august 18, 2006]**

Thin posting schedule over the past two months...gone fishing effect anybody? Well, I've not got much to add except that I wrapped up my internship on August 4th and have been taking some R&R here in lovely Niagara Falls for about a week-and-a-half now. We're planning to finish up our gruelling schedule of barbecuing and afternoon napping in time to get back to school for September 2nd. Looking for some MF content...um...how about this question from Heard on the Street: find the formula for an at the money call-option assuming zero interest rates and a stock that follows a Bachelier process (arithmetic Brownian motion).

**[july 28, 2006]**

Fall semester NYU professors include Robert Almgren (Econometrics), Attilio Meucci (Capital Markets) and David Shimko (a new course called Financial Engineering Models for Corporate Finance). Almgren is a pioneer in optimal execution, Meucci is the author of a recent book "Risk and Asset Allocation" and Shimko is a long-time contributor to Risk Magazine and a founder of the firm Risk Capital.

**[july 7, 2006]**

Electronic resources from NYU libraries (requires NYU id to access). Ebrary is a recently announced collection that has good material on many subjects including math finance: *Ebrary is a full text collection of over 30,000 current books from the world's leading scientific, academic and reference publishers.*

**[july 6, 2006]**

Nice calendar of upcoming announcements which may move the market. Tomorrow is the employment report.

**[july 5, 2006]**

(via freakonomics) "Testing the Efficiency of Markets in the 2002 World Cup" by Ricard Gil and Steven Levitt.

**get flexible** | http://online.wsj.com/article/SB115206391458398060.html

Article about FLEX options, exchange-traded equity options which allow customers to specify strike price, expiration date and other parameters.

*For example, some investors hedge bond positions with put options on a stock, and FLEX options allow them to ensure the expiration of the put coincides with the maturity of the bond, Mr. Kolanovic said. This has been common among investors in General Motors Corp. credit. The put is meant only as disaster insurance -- in case of bankruptcy.*

**[june 30, 2006]**

Site for the recent credit risk conference hosted by NYU includes a ton of (hopefully) cutting edge research from luminaries in the field.

**[june 26, 2006]**

Paul Wilmott on "Volatility forecasting, option trading and Crash Metrics". Play it in Windows Media Player to get psychedelic accompanying graphics. From Paul Wilmott's blog, one of a few quantblogs (quogs?) hosted at http://www.wilmott.com/blogs.cfm (cold fusion!). Discovered all this good stuff via quantjock who is coming to NYC!

Another link from quantjock about structured products being marketed to individual investors: http://online.wsj.com/article/SB115085644419185995.html. Covers some of the same ground as the recent IAFE conference and includes a quote from Phelim Boyle who presented some work on structured products. The bottom line is that as cool as they may sound (TD had one called "S&P Bear Notes") structured products are often difficult to understand and loaded with fees.

At the same conference, one of the members on the panel on hedge funds (Andrew Weisman from ML) noted how mutual fund prospectuses currently include information about standard deviation of returns. He made a crack that as they expand into more hedge-fund like products they'll have to start explaining skewness and kurtosis as well. I thought it was funny but...I guess maybe you had to be there.

**[june 22, 2006]**

Illuminating thread on Wilmott. How would you BS delta hedge a stock that happens to move up one percent every day? A related question that's been making the rounds at work: what happens if the stock price path is flat? The answer to the first question is in the thread. It also illuminates the answer to the second question which is that in the Black-Scholes world, delta hedging works only when the integral of log changes in the stock price from zero to T is sigma^2 * T (i.e. conforms to the model). A flat stock price does not satisfy this and so the hedge does not work! To use terminology borrowed from Mike Lipkin, it falls outside of the "phase space" of the Black-Scholes model. The probability is not just of measure zero, it is actually impossible. Other examples of stock price paths outside of the phase space: hitting zero, going negative, jumping.

**[june 14, 2006]**

Discussion of today's WSJ piece on insider trading including some insightful comments and links to research papers on the topic. The consensus seemed to be that allowing insider trading is a bad idea, leading to higher bid/ask spreads, higher volatility and reduced liquidity.

**[june 11, 2006]**

Entry from economist Brad Setser's blog about the use of financial technologies to "drive faster". Lots of stuff on credit derivatives and some interesting links (Has Financial Development Made the World Riskier?, PIMCO's Mark Kiesel on hedge fund strategies in credit markets).

This idea that derivatives are a great service to mankind is kind of funny. Certainly, they help to distribute risk, but the mechanism of that is enabling investors to speculate. The incentives for those who try to sell new types of derivatives is not to "complete markets" and enable optimal risk redistribution, it is to make money by finding bets that investors would like to take (preferrably with a comfortable margin!).

**[june 9, 2006]**

Interesting article on a new way of finding signal in noise.

*One fundamental observation enabled this vast improvement: They were able to visualize the areas in which there was no sound at all. The two researchers used white noise — hissing similar to what you might hear on an un-tuned FM radio — because it’s the most complex sound available, with exactly the same amount of energy at all frequency levels. When they plugged their algorithm into a computer, it reassigned each tone and plotted the data points on a graph in which the x-axis was time and the y-axis was frequency.*

**[may 28, 2006]**

Books and recent papers by Ecole Polytechnique Professor Rama Cont. I was lucky to attend his seminar (abstract) this past Thursday at NYU. His talk addressed the problem of model calibration: finding model parameters which reproduce the prices of a set of observed option prices. More specifically he addressed the fact that most such techniques produce *point estimates* without a clear idea of the *parameter uncertainty* associated with the estimates.

Roughly, the approach consists of sampling the "model-space" (e.g. combinations of parameter values for a given model) and then creating a weighted combination of these models. The key innovation is that these (scalar) weights are assumed to be IID random variables from some distribution. A prior is assumed for this distribution and then refined by finding the distribution which optimizes the fit to the observed prices, this is similar to the weighted Monte Carlo approach but with weightings applied to the weightings (!) rather than sample paths.

Finally, to price, take draws from the model space and price the option in each. Take draws from the weight space to blend the models. This will produce a distribution of possible prices which, although providing less comfort than a single price, has value: "However, the non-uniqueness of the solution of the original calibration problem is not simply a mathematical nuisance: the multiplicity of solutions contains interesting information on model uncertainty, which is lost through the process of regularization." This last quote is taken from a paper by Cont and Ben Hamida Recovering volatility from option prices by evolutionary optimization which covers some of the same ground and is well worth the read.

**[may 24, 2006]**

(via quantrecruiter) Interesting article about applying news analysis to algorithmic trading. As the article points out, the analysis is only half the battle. Using it as an input to an algorithm seems much more challenging.

SUNY SB's Steven Skiena has done some cool stuff on the analysis side at TextMap which keeps track of news appearances for a vast array of people, places and things. Here is Textmap's page for Google which includes a popularity time series and other automatically generated info. Unfortunately it's not especially current (news articles from May 14th) and some of the info is kind of wonky. It brings to mind some of the strange results produced by Google's Trends service. And since we're on the subject of Google, the charts at Google Finance also attempt to allow correlating news events with price changes.

A kind of related idea I had was sparked by a seminar given by David Mordecai at NYU. He specializes in systemic risk and really emphasized the importance of understanding contracts and tax law to mitigate such risks. I wondered why the text of contracts couldn't be analysed algorithmically for anomalies or for alarming passages. Another application could be looking for warning signs in financial statements.

**implied copulas** | http://www.iafe.org/upload/HullSlides.pdf

Slides from John Hull's IAFE presentation on May 2nd. Lots of snazzy graphs.

**[may 11, 2006]**

There was a bug in the CDO code! I was calculating the break even spread for each path, but this should be computed using the expected value of the fixed and floating leg. I've updated the code, executables and tarball. For those few who have downloaded it, please get this latest version. It has at least one less bug!

This highlights the fact that expectations of nonlinear functions are not equal to functions of expectations (Jensen's Inequality). It also made me realize how important it is to have an idea of what types of answers are reasonable. This could come from market knowledge, from other models, or from mathematical intuition about your model's bounds and behaviour in different scenarios. I guess the difficult thing is that these things may lead you astray as well: the results could match your intuition and yet be completely incorrect (for example scaled by two).

Thanks to Aaron Ipsa for helping me find the issue.

On the bright side I get to post a nifty new graph which shows that the mezzanine tranche is far less affected by changes in correlation than the other tranches:

**[may 10, 2006]**

Project for Interest Rate and Credit Models: CDO Pricing in Gaussian Copula. Monte Carlo simulation to price a CDO and calculate the break even spread subject to a ton of simplifiying assumptions: flat correlation, flat forward default intensity, constant interest rate, constant recovery per firm, constant notional per firm. Still, those assumptions make it extremely tractable and the results still give some intuition about the effect of correlation on the valuation of different tranches.

CDOs are essentially a gamble on when or whether firms will default on their debt. The CDO is divided up into "tranches", that can be thought of as layers which are eaten into as defaults occur. The equity tranche is at the "top" and begins to lose money as soon as a single firm defaults. More senior tranches are protected since the upper layers have to be eaten away before they are exposed.

So why does the equity tranche price drop as correlation increases? Leif Andersen gave a nice analogy in class. Think of a minefield where the mines symbolize defaults. If correlation is low the mines are scattered about willy nilly. If correlation is high, there may be many mines in one place (a supermine!). The equity tranche is like a unsuspecting villager walking across the field. If he hits any mine, he's toast. If the mines are widely scattered, the chances of our poor villager making it across in one piece are low since many paths across the field will cross a mine. What if they were concentrated in just a few places? Well, to the villager, a mine is a mine is a mine, so a field with just one or two supermines amounts to a field with less mines. His chances of getting across in one piece are much better! Now substitute an armour plated humvee for our villager, and you can soon see why supermines are bad news for senior tranches.

**[may 8, 2006]**

Project for Interest Rate and Credit Models: Vasicek (interest rate model) binomial tree implementation. Vasicek's model is invariably used to introduce interest rate models because of its simplicity and the analytical expressions that can be derived for the short rate, bond prices and bond options. Too bad about those negative interest rates!

I just finished studying Vasicek's loan loss distribution approximation for the exam tonight. More on Vasicek: 2004 FEOY award winner One-on-one.

**[may 6, 2006]**

Online calculator for bond option pricing using the Vasicek model. Includes a very nice explanation of the formulas along with detailed calculations. I'm using this to test my project for Interest Rate and Credit Models. There are many more calculators and other useful articles on Jan Roman's site including lecture notes for a class he teaches called "Analytical Finance" at Malardalens University in Sweden.

**puffy** |

I just read through the $$ article on GS from the April 29 Economist, and just like the cover story it is relatively fact free. Interesting bits:

- Today, non-American revenues are growing particularly fast and should exceed 50% of the total before long. More than a third of the firm's profits come from activities that either did not exist at the time of the public offering or were too small to matter.
- Goldman through its various trading entities touches about one-third of all the share trades in America's financial markets
- Indeed it has become harder to distinguish between who is a Goldman client and who is a Goldman competitor...Examples abound. Would General Motors be better off if Goldman had merely sought out a buyer for the property arm of its financing operation, instead of itself joining the buy-out group, as it recently did? The bank cites numerous times when it advised on a deal and then provided a hedge of some sort that immunised the buyer from a risk.
- This aggressive approach is thought to have contributed to a minor regulatory complaint that, in turn, led to the cancellation of Goldman's contract as the sole non-Japanese firm distributing funds through the country's massive postal savings bank.
- Mutual funds: Goldman's results, as evaluated by Morningstar, which analyses funds, are distinctly mediocre
- Equity research: In an annual poll by Institutional Investor, Goldman ranks as mediocre. Many investors contend it is no more compelling than the research put out by any of the big banks and is not as good as that of more specialised firms, notably Bernstein. Company executives, however, who have privileged access to Goldman's analysts and bankers, say they have high regard for their research.

**[april 28, 2006]**

Exhaustive discussion of bounds for European and American style options. Would've come in handy on Wednesday's Continuous Time Finance exam ;-) Ok, without peeking, which is worth more: the right to sell a stock for K at time T or the right to decide *today* to sell the stock at time T? Why?

**[april 24, 2006]**

Inspired by Suman I've adopted del.icio.us as a quick way to keep track of links of interest. Since Suman already was tagging relevant content with financialengineering, I've decided to stick with that standard. Check the above link or subscribe to its RSS feed to get a hopefully steady diet of linkage. I've also added it to MathFinLinks.

- http://www.nytimes.com/2006/04/20/business/20swap.html - Cautionary tale of local governments shooting themselves in the foot with derivatives.
- http://www.marginalrevolution.com/marginalrevolution/2006/04/why_dont_we_hav.html -
*Why Don't We Have Tax Futures*- Does everyone want to hedge against higher marginal rates? It is suggested that holders of tax-free bonds (e.g. municipal bonds) are hurt by lower marginal rates. I interviewed with a municipal bond trader and he mentioned that people also factor in a probability of losing tax-free status. - http://www.dbresearch.com/servlet/reweb2.ReWEB?rwkey=u1562380 - (via aleablog) Emerging market default probabilities from CDS spreads
- http://infoproc.blogspot.com/2006/04/alpha-geeks.html - "Non-linear returns to brainpower" - exhibits 1 and 2: Kai-Fu Lee and the quants at Goldman's Global Alpha hedge fund (I've added Info Processing to MathFinLinks, math finance is a recurring topic, check the archives for many interesting thoughts and links)
- another Goldman link via quantjock: http://www.smartmoney.com/barrons/index.cfm?story=20060410&src=fb&nav=RSS20
- http://www.maths.ox.ac.uk/~shaww/hestonstovol.pdf - good introduction to the Heston stochastic volatility model

**calendar** | http://www.iafe.org/events.php?event_id=1231943847

May 2nd, 5.30 PM - IAFE presents "Valuing Correlation-Dependent Credit Derivatives: Implying Copulas from Market Data" Presentation by John Hull. Probably based on the Hull-White paper The Perfect Copula. Apropos:

*We show how a one-factor copula model can be implied from iTraxx or CDX tranche quotes. The copula that is implied is “perfect” in that it fits the tranche quotes exactly. What we are doing in this paper is analogous to what Breeden and Litzenberger (1978) and Jackwerth and Rubinstein (1996) did when they implied a future stock price distribution from European option prices.*

**[april 20, 2006]**

- http://www.investmentactuarysymposium.org/pdf/handouts/I6_Payne.pdf - "Modeling Complex Derivatives" - from an insurance industry conference - lots of interesting stuff about Monte Carlo
- http://www.jstatsoft.org/v05/i08/ziggurat.pdf - mentioned in the above paper, describes the Ziggurat method, a fast way to generate random variables from a given decreasing density
- http://papers.ssrn.com/sol3/papers.cfm?abstract_id=103772 - A General Treatment of Barrier Options

**stupid calculus tricks** | http://mathworld.wolfram.com/Erf.html

The cumulative normal distribution function shows up *a lot* in math finance. It appears in Black-Scholes (twice!) and pops up in other places too. On a recent PDE in Finance assignment we were asked to show that the solution to the PDE for a barrier option can be expressed in terms of the normal CDF. The normal CDF can be expressed in terms of erf: the error function. I've put the Mathworld link for erf above, it has lots of useful information.

Anyways, we were then asked to show that the integral of the normal CDF is actually the solution of a PDE with final time condition max(x,0) - something that looked suspiciously like a call. The integral is not trivial (to me anyways) and we were not asked to calculate it, only to describe it qualitatively. I decided to plug it into Matlab's symbolic solver and this is what popped out:

(graphic generated by this awesome Online Equation Editor)

[Update: Actually the integral of the normal CDF is quite trivial, not sure why it didn't occur to me earlier. Anyways it can easily be evaluated using integration by parts and the fact that the normal CDF and PDF go to zero at negative infinity! This yields the following which is identical to the above but easier on the eyes:

If we change the "strike" from 0 to K in this formulation, it amounts to replacing the above function of x with a function of (x-K) and it starts to look more and more like Black-Scholes.]

If you plot it you will see that indeed it looks like the value of a call. This is another illustration of the relationship between call prices and barrier prices and between prices and probability densities. These properties are exploited to create static and semi-static hedges as well as in the study of local volatility.

An early study of these properties is Breeden, Litzenberger "Prices of State-Contingent Claims Implicit in Option Prices".

Another relevant link, my classmate Simon Leger's derivations of closed-form solutions via both discounted expectation and PDE methods: http://homepages.nyu.edu/~sl1544/KnownClosedForms.pdf.

**[april 18, 2006]**

Peter Forsyth's page at the University of Waterloo. Includes some introductory material about Computational Finance and lots of interesting papers (These looked interesting: Dynamic hedging under jump diffusion with transaction costs and Hedging with a correlated asset: solution of a nonlinear pricing PDE)

**[april 16, 2006]**

FT versus MR throwdown! The main event: Should you melt down your pennies? Apparently, with copper and zinc prices rising they may soon be worth more dead than alive. FT says melt. MR says wait. Why does MathFin care? MR justifies waiting by saying you should not exercise an option before expiration. As the comments point out, that only applies to a call option with carrying costs less than the risk-free rate. Lots of great comments:

*And there _is_ a good reason to exercise your penny options: pennies have large carrying costs. Do you have enough glass jars to hold _tons_ of pennies? Or do you have to rent mini-storage, or tell your wife not to park in the garage until the price of copper falls? These carrying costs are mathematically equivalent to a dividend on the underlying asset, if the dividend is a fixed amount instead of an assumed-constant yield.*

Some discussion on executive stock options in there as well.

**air goldman** | http://quote.bloomberg.com/apps/news?pid=10000006&sid=aJO3R8Wb8vWE

At first I thought Goldman was trying to buy British Airways...they're actually trying to buy the British Airports Authority which runs Heathrow and other airports in the UK, but I liked "air goldman" too much to give it up. Still raises the question of Goldman's motives. A clue:

*"There is a bit of desperation" among managers of buyout funds, said Stuart Fraser, who helps oversee $28 billion in investments at Brewin Dolphin Holdings in London, including BAA shares. "Money has flowed very aggressively to private equity and hedge types. There is a pressure because of the amount of money that they are sitting on. If you don't invest the money, there is no performance fee."*

Canadian content: Potential partners included "Canadian pension funds Borealis and the Ontario Teachers' Pension Plan.''

**[april 12, 2006]**

Course page for an elementary PDE course at the University of Minnesota.

**[april 10, 2006]**

(via bigpic) Report on the management of the Yale endowment portfolio which is "structured using a combination of academic theory and informed market judgement". Not much B-S (!), mostly mean-variance analysis and (in)efficient markets. In practice, that means diversifying away from very efficient markets (fixed income and domestic equity) and emphasizing those markets where active management can make a difference (emerging markets, event-driven, venture capital and...um...timberland).

I thought the best bits were the diatribe against fixed income managers (p.14), the sidebar on Liquidity (p.25 - *Investors should fear failure, not illiquidity.*) and the opaquely named Degree of Investment Opportunity (p.36) which proposes a simple way to evaluate the degree of efficiency in a market.

**film school** | http://math.nyu.edu/~atm262/files/spring06/scicomp/a5/crystal.avi (51 Mb)

After the reflection: crystal-2.png

One of the enduring attractions of math is the way that simple dynamics can generate quite complex phenomena. In the latest Scientific Computing homework we wrote an ODE solver. Plugging in a simple result of Einstein's for the crude modelling of vibrations of a crystal lattice (see question 6 at the end of these notes) we were able to generate these pretty amazing graphics. The code as usual is at http://math.nyu.edu/~atm262/files/spring06/scicomp/a5/.

**never say never** | http://www.iht.com/articles/2006/04/03/business/buffett.php

(via infoproc) IHT article about Warren Buffett's bet on world stock markets. Apparently BH has sold $14 billion in "long-duration equity index put contracts", effectively insuring buyers against drops in "four major equity indexes...three of which are outside the United States". The actual indices are not named. This part seemed excessively cautious...

*For Berkshire to lose the $14 billion that the company says is at risk, all four indexes covered by the puts would have to fall to zero, according to Gary Gastineau, managing director of ETF Consultants. Historic trends show that is unlikely to happen.*

I suppose another (bearish) take is that Buffett's stated philosophy is to buy undervalued stocks. Selling puts allows him to make some money while waiting for that to happen. Probably good money too since we're taught that out-of-the-money puts are usually more "expensive" in terms of (Black-Scholes) implied volatility. There are various explanations for this but one that seems persuasive, especially in light of the conventional wisdom about the smile becoming more pronounced after the October '87 crash, is that this is simply a reflection of increased demand by (rationally?) crash-fearing investors.

Risk Glossary has a good overview of this phenomenon. Don Chance's typically clear exposition of implied vol includes this funny remark: *It is fair to say that mathematicians have devoted excessive hours of human and machine time to researching the smile with little if any regard to the reasons why the smile exists.* (this leads to the footnote: *Perhaps if these reasons were found, the mathematicians would be out of work.*).

**[april 5, 2006]**

Links for the latest scientific computing assignment:

- http://developers.sun.com/prodtech/cc/documentation/ss11/mr/man3c++/slice.3.html
- http://gcc.gnu.org/onlinedocs/libstdc++/libstdc++-html-USERS-3.4/classstd_1_1gslice.html
- http://www.roguewave.com/support/docs/leif/sourcepro/html/stdlibref/gslice.html
- http://developers.sun.com/prodtech/cc/documentation/ss11/mr/man3c++/gslice.3.html

slice and gslice allow you to manipulate C++ valarrays as you would in Matlab (kind of).

update - 7 april - comet ODE links:

- http://wps.prenhall.com/wps/media/objects/884/905485/chapt4/proj4.3B/proj4-3B.pdf
- http://math.etsu.edu/Multicalc/Capstone/Cap2/cap2.pdf

**[april 3, 2006]**

Stephen Ross (of Cox-Ross-Rubinstein fame) will be trashing behavioural econ at Thursday's Courant seminar. Sounds like fun!

**[april 2, 2006]**

Lots of interesting stuff over at aleablog. Here's a bit on the impact of algorithmic trading from the Annual Volume Survey of the Futures Industry Association.

**quants wanted** | http://www.financetech.com/showArticle.jhtml?articleID=184417510

(via mahalanobis) Article from a magazine called "Finance Tech" about the demand for math, physics, CS and mathfin grads (whew) on Wall Street.

**[march 30, 2006]**

Ridiculously complete C reference.

**drunken birds** | http://mahalanobis.twoday.net/stories/228354/

Some historical insight into the random walk in two and three dimensions.

*The lesson of Lord Rayleigh's solution is that in open country the most probable place of finding a drunken man who is at all capable of keeping on his feet is somewhere near his starting point.*

**what can be a numeraire** | http://www.wilmott.com/messageview.cfm?catid=19&threadid=5838

Mark Joshi: "the numeraire has to be an asset of positive value. Definition of asset: something you can assign a dollar value to."

An example from our recent HomeWork in continuous time finance. Given assets S (UK stock), P (UK bond), D (US bond) and X ($/pound exchange rate), the dollar assets (a.k.a. tradeables) are XS, XP and D. Pound tradeables are P, S and B/X. X is not a tradeable in either pounds or dollars.

Every ratio of one tradeable to another (in the same currency) can be made a martingale under the same measure through changes in drift of the underlying Brownian motions. If you choose XS as the numeraire you have the ratios: XS/XS (this is always 1 and a trivial martingale), XP/XS = P/S and D/XS.

Some examples: The numeraire B/X can be used to price the option that pays off (ST - K)+ in dollars. (see http://www.math.nyu.edu/faculty/kohn/cont-time-finance/section3.pdf). The numeraire XS can be used to price the option that pays off XT(ST - K)+. How? Using the "Fundamental Theorem of Asset Pricing". See: http://www.math.nyu.edu/~cousot/Teaching/IRCM06/FTA.pdf, http://www.math.nyu.edu/~pender/teaching/ctstimefin/Notes7.pdf, Baxter & Rennie ch. 4 (see MathFinBooks).

Some more cool examples in this paper by Benninga et al. On the Use of Numeraires in Option Pricing.

**database** | http://www.finance-and-physics.org/Library/articleh2.html

Huge list of math finance papers and lecture notes.

**[march 28, 2006]**

Aleablog on the problem with housing index based futures and options (mentioned below on 3/23): *Main reason :time basis risk ,the proposed index has a lag of 2 months on release and reflects transactions as old as 5 months => this makes it useless for hedgers and even more so for speculators unless you think than speculating on past events will make you rich.*

Damn, I was looking forward to buying myself a virtual house. If only I could buy a real house at the five months ago price.

**linkdump** |

- http://www.math.uu.se/research/telecom/software/ - stochastic simulation using MATLAB
- http://en.wikipedia.org/wiki/Exponential_distribution -
*The exponential distribution is used to model Poisson processes, which are situations in which an object initially in state A can change to state B with constant probability per unit time lambda. The time at which the state actually changes is described by an exponential random variable with parameter lambda. Therefore, the integral from 0 to T over f is the probability that the object is in state B at time T.*- used in math finance to model time to default

**[march 27, 2006]**

Ito's product rule: d(XY) = XdY + YdX + dXdY (if either of dX or dY has no dW term then dXdY = 0 and this is the standard product rule) From this it is straightforward to derive Ito's division rule: d(X/Y) = (YdX-XdY-dXdY)/Y^2 + (XdYdY)/Y^3 Note that in this case that there is an asymmetry. If dY has no dW term then this becomes the standard division rule. However, if only dX has no dW term, the dYdY term remains. Using the change of measure technique with the stock numeraire, this term turns "d2" into "d1" in the Black-Scholes formula.

Credittrade provides transaction, data and information services for the credit market. The product education section looks interesting. Saw it mentioned in a paper entitled Insider Trading in Credit Derivatives from proceedings of 2005 conference on credit risk. The 2006 conference is coming up at NYU Stern (see MathFinSeminars and click on NASDAQ DRP seminars) but it costs mad ducats.

**new financial order inc.** | http://online.wsj.com/article/SB114307310436605753.html free alternate

"S&P Will Launch Indexes to Track Housing Prices...The indexes, which plan to launch in April, will serve as the basis for futures and options contracts that will trade on the Chicago Mercantile Exchange." Sounds like this is the fruit of research by economists Robert Shiller and Karl Case into creating ways for investors to acquire a diversified real estate portfolio as well as allowing better hedging of the associated risks. Of course, its timing suggests that it will be used for less lofty purposes, but without speculation it won't be of much use so lets hope it gets a healthy start.

See Daniel Gross' Housing Bubble Insurance for more background on the product and on Shiller and Case's contributions. Gross has these cautionary words: *Individuals have shown a great propensity not to buy insurance on financial assets, even when such insurance is readily available. Most individual investors don't hedge their large stock holdings: Did you buy puts on the Nasdaq-100 Trust to guard against the drop in your tech stock portfolio?* (Which reminds me that I should probably keep an eye on these...)

Some interesting related links:

- http://cowles.econ.yale.edu/P/cd/d10a/d1006.pdf - Readable discussion paper from '91 motivating "Index-based Futures and Options Markets in Real Estate". It also explains how the existence of such a product could in turn spur the development of home equity insurance: protection against a large drop in price between purchase and sale. The risks of offering such a product could then be hedged using the corresponding real-estate derivatives. Other publications related to Shiller's book "The New Financial Order": http://newfinancialorder.com/nfo.htm

- http://www.econ.yale.edu/~shiller/risk.htm - 2001 article about an earlier abortive attempt: In 1990, Shiller joined economist Karl Case, and Weiss, a former MBA student of Shiller's at Yale, to develop a real estate futures market. Their efforts got as far as a Chicago Board of Trade press release in 1993 announcing the development launch of a home equity insurance market, but the venture never took off. "It's kind of bizarre," says Shiller, "We have futures markets for some very unimportant things, like pork bellies, but we don't have them for real estate, which is a major asset category." Recently, Shiller has been proposing global income risk-sharing markets in which citizens, already defacto long their own country, could hedge their income risk by going short their domestic national income. For diversification benefits, investors could go long in foreign global income-sharing securities. (GDP-linked securities have also been a recent subject on Marginal Revolution)

- http://www.cme.com/files/CmeCsiHousingIntroWhitePaper.pdf - Includes this description of how the underlying index (the so-called "Case-Shiller Index") is calculated: "CSIs are fundamentally based on observed changes in home values. In particular, CSW collects data regarding transactions on all residential properties during the time period in question. Next, CSW conducts a search of its accumulated database to find information regarding any previous sales for the same home... The sale pair is thereupon aggregated along with all other sales pairs found in a particular region to create the index." Shiller's paper on the technique is here: http://cowles.econ.yale.edu/P/cp/p07b/p0781.pdf. An Excel spreadsheet of historical index values: http://www.cme.com/files/HousingData.xls. More at CME: http://www.cme.com/trading/prd/env/housingover16250.html

**clamor** | http://online.wsj.com/article/SB114187184703793317.html?mod=interactive free alternate

Article about Nicole El Karoui, a professor at L'École Polytechnique in Paris and one of the big names in math finance teaching. This interview with a student (windows media) talks about a typical class in which they covered stochastic volatility models. (Bluffers guide to being a quant: "Stochastic vol models mean the market is not complete"... Just kidding ;-) There are a few students in our class who took her courses.

**[march 17, 2006]**

(via MR) Another blog for MathFinLinks with recent topics including the Dubai stock market crash (link), prediction markets, the stock-picking abilities of psychologists (link) and a link to an interestingreview of Fortune's Formula by Elwyn Berlekamp.

I turned up the phrase *jacta alea est* (*" the die is cast," uttered by Julius Caesar, immediately before he crossed the Rubicon*) at this interesting site: http://www.ancientlibrary.com/smith-dgra/0081.html.

**meanwhile at baruch** | http://faculty.baruch.cuny.edu/lwu/890/fin890spring2006.html

Outline and reading list for Professor Liuren Wu's class for doctoral students "Options Markets".

*This class discusses the frontiers of the option pricing literature. After a brief review of the options market, including market conventions and stylized facts, I'll go through old and new option pricing models from the perspective of modeling security returns with time-changed Lévy processes. It is a new framework that can encompass pretty much all existing models. It also provides an intuitive way to desigining new models.*

**[march 8, 2006]**

AXA Re writes a policy for the World Food Program that pays out $7 million "if the rainfall measured at 26 weather stations around Ethiopia falls below a certain level between March and October".

*The first gauge of success of the trial program was whether an insurance company would actually write a policy based on Ethiopian weather data. That has worked, partly because MDA Federal Inc., a company in Rockville, Md., that provides satellite-based weather forecasting information to agricultural clients, said it would ensure the accuracy of the rainfall numbers.*

*In all, the World Food Program, a United Nations agency, received five proposals from major international insurance companies. It accepted the offer from AXA Re, which already issues weather-related policies in other parts of the world.*

*Taking out insurance on the vagaries of Mother Nature is common practice. Natural gas suppliers do it, knowing that their profits will dip if the winter is mild. The same goes for farmers, who take out insurance in case of frost or excessive heat. Hurricane coverage and flood coverage are now standard offerings as well.*

Read the post at MR for more insight.

**be a drip** | http://www.qqqdirect.com

Saw an ad for this in the WSJ this morning. The pitch is "QQQ + Monthly Reinvestment - Broker Fees = No Excuses". I believe it puts ETFs on a more equal footing with index funds. ETFs have lower fees than index funds but since they trade like stocks you have to pay a commission to buy them. This has always made index funds more appealing to me because they allow small regular investments (dollar-cost averaging) without fees.

This service allows you to invest a monthly amount in the Nadaq-100 as small as $10 without *any* fees: no brokerage fees and no account fees either! The plan is run by a company called Mystockfund. They offer a few other plans but basically you can make very cheap planned purchases. The key is that the purchases are planned. They can offer low fees by following Henry Ford's old strategy: you can buy at any time you want as long as its a Thursday. By batching the orders up in their "execution window" I assume they take advantage of economies of scale. Market orders cost $13 which must further subsidize the planned purchases. Interestingly all sales are executed as market orders. That's a bit puzzling since there are economies of scale on that side as well, but I guess they are aiming to attract a clientele more interested in buying than selling.

**[march 7, 2006]**

Teach yourself econ using this handy compendium of MIT Opencourseware links covering topics from behavioural econ to time series analysis.

**reality not theory** | http://www.ieor.columbia.edu/feseminar/Lipkin.ppt

Interesting talk by Mike Lipkin on the importance of supply and demand (and the failure of Black-Scholes) in the determination of option prices. I heard a version of this talk at the Stern volatility seminar a few weeks back. Mike is an erstwhile chemistry Ph.D who is now a trader on the ASE. He can often be found sitting in on our classes as well as at the Thursday MathFinSeminars. Mike mentioned the inverse cubic law paper I cited on Feb. 22.

**[march 5, 2006]**

Interesting and exhaustive discussion of hitting times for Brownian motions. Hitting times are useful for answering questions like when will a gambler go bankrupt or when will a stock hit a certain value. Also includes some spreadsheets illustrating various properties of hitting times.

**[march 2, 2006]**

Good reference on interpolating yield curves. Lots of other interesting stuff at the RiskWorX site which is based in Johannesburg, South Africa. Rooibos for everyone!

**[february 28, 2006]**

Some links about stats and probability...

http://courses.ncssm.edu/math/Talks/PDFS/Taxis%20for%20MAA.pdf - Approaches to the taxicab problem: You see a taxicab numbered, say, 313. Assuming that the taxis in this city are numbered consecutively and they're all out and about what is the total number of taxicabs in the city. More: http://ask.metafilter.com/mefi/14383. True story, a fishy company recruiting at a recent job fair asked resume submitters to put a solution to the problem in the subject line. My guess before reading the above was 2*313 - 1 (e.g. assume the number you saw was the expected value - n*(n+1)/2n = 313). I never heard back from them...

http://www.freakonomics.com/blog/2006/02/25/is-it-harder-to-win-a-gold-medal-in-luge-or-to-win-the-nobel-prize-in-economics/#comments - I guess the link says it all. Lots of good comments including this: *In fact, that reminds me of my favorite unsolved (to me—maybe statisticians have addressed this) probability/statistics problem: you have a weatherman who predicts whether it will rain or not as an N% chance (N is multiples of 10 from 0 to 100). How do you evaluate his accuracy?*

http://mahalanobis.twoday.net/stories/1629071/ - Chicken soup as a statistics teaching aid. (Why not pho? In the analogy between tasting soup and polling what's a bo vien?)

**from soul sides to swaps** | http://steadyblogging.blogspot.com

Check out Suman Ganguli's blog, I've added it to MathFinLinks. Like QuantJock he's in the Haas MFE program and his blog is an eclectic collection of recent news (Indian mangoes are coming to America!), tech, travelogue, music (house, crunk, jazz, 313... sounds familiar but he's a lot more up-to-date than I am) and of course financial engineering.

**[february 27, 2006]**

By discretizing the points in an interval and judiciously applying Taylor expansion, a boundary value problem can be solved using matrix operations! We tried this method out in assignment 3 in Scientific Computing. The application to math finance is in finding numerical solutions to pricing problems that can be posed as BVPs. My MATLAB code for the assignment is here: http://math.nyu.edu/~atm262/spring06/scicomp/a3. See question 9 in Goodman's Linear Algebra Theory notes. The writeup for the last few parts of this question is here.

- ordacc.m allows you to construct a finite difference approximation to a given derivative and calculates the approximation's order of accuracy. It uses the method of undetermined coefficients as described in Goodman's Local Analysis notes. Handy for coming up with the elements to use in the matrix.

- bvpsolv2.m and bvpsolv4.m implement 2nd and 4th order accurate solutions to 1/2uxx = f given f and bvpcheck.m and bvpconv.m provide a way of verifying them. The second-order solution gives rise to a tridiagonal matrix. The above notes also describe a quick way to solve such a system. I used Baris Sümengen's MATLAB code for solving a tridiagonal system which implements this technique.

**[february 24, 2006]**

Just got back from the Stern symposium - some links related to things that I discussed with people after the seminar (so not volatility related...):

- http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:cond-mat/9803374 - "Inverse Cubic Law for the Probability Distribution of Stock Price Variations"
- http://www.publicaffairsbooks.com/publicaffairsbooks-cgi-bin/display?book=1586481819 - THE CHASTENING Inside the Crisis that Rocked the Global Financial System and Humbled the IMF - PAUL BLUSTEIN
- http://www.publicaffairsbooks.com/publicaffairsbooks-cgi-bin/display?book=1586482459 - AND THE MONEY KEPT ROLLING IN (AND OUT) Wall Street, the IMF, and the Bankrupting of Argentina - PAUL BLUSTEIN
- http://www.iirusa.com/CDO/files/IIR_U2052_Duggar,%20Madhur.pdf, http://www.cfainstitute.org/conferences/Event_1166/pdf/CDOOutlook2005.pdf, http://www.cfainstitute.org/conferences/Event_1166/pdf/StructuredCreditHandbook.pdf - some stuff on CDOs from Citi

**[february 21, 2006]**

Stern symposium on volatility trading this Friday.

**[february 17, 2006]**

- http://www.symmys.com/AttilioMeucci/Research/Talks/AttilioMeucci_BeyondBlackLitterman.pdf - subject of yesterday's Courant seminar, using MC simulation and copulas to move beyond the "Normal assumption" (other interesting stuff on Meucci's site as well, he is teaching the Capital Markets and Portfolio Thory class this semester)
- http://wsomfaculty.case.edu/ritchken/documents/Chap_1.pdf and http://wsomfaculty.case.edu/ritchken/documents/Chap_9.pdf - two chapters from a book on interest rate and credit derivatives
- http://www.dpmms.cam.ac.uk/~twk/Joshi.pdf (via WSP) - Mark Joshi on how to become a quant (he mentions integral of log x, I got asked what is the derivative of x^x... but nobody asked me to explain my Master's thesis ;-)

**[february 11, 2006]**

The Pignetti is back with two new posts, one on Wall Street "templates" (a non-coder might call them archetypes) and a link to this post on a blog called C++ Quant addressing what exactly quants do (added this to MathFinLinks). I found it pretty handy. Being new to the finance industry, its still not quite clear to me what types of jobs are out there or how day-to-day routines differ materially between buy / sell side, front / mid / back office, fixed income / equities, etc. Some of these distinctions are also addressed on C++ Quant's site so explore it a bit.

**linkdump** | http://math.nyu.edu/~atm262/files/spring06/scicomp/a2

Links related to Scientific Computing assignment 2...

- http://www.cs.utah.edu/dept/old/texinfo/glibc-manual-0.02/library_2.html - error codes defined in errno.h
- http://gcc.gnu.org/ml/gcc-help/2005-05/msg00146.html and http://www.devx.com/tips/Tip/22530 - why you should prefer function objects to function pointers for performance
- http://mathworld.wolfram.com/FresnelIntegrals.html - the integral of cos(tx^2) includes a Fresnel integral

**[february 10, 2006]**

Added information about the University of Waterloo's seminar series in New York City to the seminars page. They will host some interesting speakers including University of Waterloo alumnus David Li.

**[february 8, 2006]**

(via MR) Lots of interesting posts on this economics blog including the observation that "individual investors have a striking ability to do the wrong thing" (link) and volatility strategies in FX markets (link).

Incidentally, Prasanta Chandra Mahalanobis was the founder of the Indian Statistical Institute. He contributed much to statistics including the concept of Mahalanobis distance which "is a useful way of determining similarity of an unknown sample set to a known one." His biography from the Kolkata ISI site illustrates one application:

*Mahalanobis's interest in anthropometry remained strong and two large-scale anthropometric surveys were carried out under his direction in the United Provinces and Bengal. Based primarily on the D-square statistic, many of the important anthropological inferences drawn from the data collected in these surveys have stood the test of time. For example, the conclusion that Bengal Brahmins resemble other castes of Bengal more closely than they resemble Brahmins from elsewhere in India has been corroborated by many subsequent studies.*

I wonder what the Bengali Brahmins thought of that?

**[february 3, 2006]**

In return for helping set out nametags and checking names off the guestlist I was able to attend this year's IAFE Financial Engineer of the Year dinner. At $600 a plate, I think that's a pretty good deal. Who says there's no such thing as a free lunch...er...dinner? On top of that, it was a source of pride as a Canadian to see the award go to Dr. Phelim Boyle, the head of the University of Waterloo's MFinance program, for his many contributions to FE especially in the area of Monte Carlo simulation. Dr. Boyle's acceptance speech was truly inspiring, drawing a direct line between his childhood in rural Ireland and his current achievements.

I had the opportunity to speak with Dr. Boyle as well as his wife, Mary Hardy, who is a distinguished financial engineer in her own right and the current holder of the CIBC Chair in Financial Risk Management. From talking to them I gathered that admission to Waterloo's MFinance program is becoming more and more competitive. The program sounds similar to NYU's but with more emphasis on econometrics and less emphasis on computing since many of the candidates already have strong computing skills. Graduates find jobs on Bay Street, but since the degree is not yet widely recognized on Wall Street it can be difficult to come directly to New York. Hardy also made reference to another issue of concern, the patenting by Columbia of low discrepancy sequences. Apparently, the patent is predated by research conducted by Boyle and others, however the patent precludes anyone but licensees from publishing further work on the subject (even in Canada). In his speech Boyle made direct reference to this issue (he cited the patent number!) and tied it together with the recent patent-related trials of Waterloo's Research in Motion.

On a lighter note, I admit it was a bit of a nerdfest. One of the speeches featured an extended explanation of how many girls you should go out with in order to be sure to have chosen the best one. I'm not sure if I should apply this algorithm when interviewing for internships. I'm inclined not to. Also tucked into the program was a page of truly awful riddles. To their credit, the reading of the answers was accompanied by groans from the audience. For example: *What do you give a future financial engineer for Christmas?* ... *A Black-Derman-Toy*.

One regret: I didn't find the answer to the question posed in the previous entry...

**[february 1, 2006]**

A financial engineering approach to healthcare savings accounts? After a semi-rant against these vehicles (which shows that just because you're in finance doesn't mean you have to be a conservative) Dr. Joannas has some very interesting ideas:

*Instead of investing the proceeds of the health saving accounts in hazardous investments, why not invest them in something tangible which will ALWAYS go up in value?*

*Why not create a framework to invest these funds in buying treatments now for later use? Why not buy beds in hospitals? Doctor and surgeon’s fees? Blood, chemicals and the likes?*

*Actually, given patient history or genetic background, one may be more at risk to a given illness. Then if they contract another disease for instance, they could trade the treatment they bought for another one.*

What would Merton do?

**[january 29, 2006]**

Upcoming symposia hosted by Stern's NASDAQ Derivatives Research Project. The next one is on February 24, 2006 on the topic of trading volatility and will include a talk by Marco Avellaneda.

I was able to attend the symposium on Friday on Credit Derivatives and got to hear interesting presentations from David Lando, Arvind Rajan and Peter Carr. Stephen Figlewski gave the introductions and noted that the origin of the word symposium are the Greek words for drinking together. Apparently he has been on leave for the past two years working at Citigroup and has only recently returned to Stern. I've mentioned Figlewski's site here before - it has a lot of really interesting stuff on volatility.

Some assorted notes...

- Lando said that the data on credit spreads has become much better since CDS' are so liquid and there were now standard contracts for other types of credit derivatives, he also mentioned a paper (Blanco Brennan Marsh 2005 I think) which discovered that CDS spreads lead corporate bond spreads possibly as a consequence of greater liquidity
- using default intensity type models allows a lot of the existing term-structure models to be (re)used in modelling default
- One model / analogy for the effect of economic factors on default is the effect of increased pollution levels on asthma attacks. Increased pollution may lead to more asthma attacks but asthma is not contagious so it will only affect those who already have asthma.
- Price movements in equity and mezzanine tranches (the equity tranche is the first to be hit by default losses followed by mezzanine) during April / May last year (a bad time for the credit market apparently) indicated that maybe hedge funds had sold protection on the equity tranche and hedged by buying protection on the mezzanine tranche in some ratio. Once defaults started to happen, hedge funds tried to close out the equity positions causing the price movements to become inversely correlated.

- Rajan's talk was really interesting. Apparently he is starting a trading desk at Citigroup in this area (after having headed credit research and fixed income strategy there). His observations were that the arrival of "structured technology" in a market coincides with a rally. Why? Inefficiencies are removed and buyers and sellers are matched more efficiently. He also indicated that the popularity of CDOs has to do with changes in supply and demand - he referred to a barbell shaped demand for risk - e.g. central banks want AAA investments but other investors (e.g. petrodollars? Asian investors?) want to take on outsize risk positions. CDOs allow these two types of securities to be easily packaged. Moreover securitization can make dusty corners of the market user-friendly. Since they are broad-based they allow buyers to participate without havig to become experts in the area (he gave the example of middle market loans).
- Get long assets that are being securitized!
- Corporate bonds cluster around BBB. Not a coincidence since companies issue enough debt that the bonds become risky.
- The virtuous cycle of mean reversion: whenever credit spreads widen issuance accelerates driving down spreads. Play this cycle for new asset classes...
- But get out before something bad happens - e.g. April May 2005. A nice discussion in thisBank of England paper (see page 3).
*In effect, the structured credit market was reminded that credit risk has a major idiosyncratic element; that even the most sophisticated statistical models can be found wanting when they are detached from fundamentals and/or based on short runs of data; that hedging can hurt when a trade is ‘crowded’; and that, in such circumstances, volatility can suddenly spike and spreads move in unplanned-for ways.* - Base correlation is not a good model (check out this Wilmott thread for more on base correlation). Better models are comimg and will be implemented but may change the market. Right now there are a wide range of "deltas" (e.g. proportion required to hedge) but if larger deltas are required then the trades become less attractive. How long will it take for there to be a consensus? It took fifteen years for the interest rate derivatives market to converge!
- Some other directions: modelling the relationship between correlation and volatility. Hedging using deltas for individual names.

- Carr presented research on the relation between CDS spreads and equity options. He focused mainly on hedging techniques - firstly static replication of CDS legs with default-free and defaultable strips, and secondly on replicating these defaultable bonds (a security which pays $1 in the case of no default and nothing otherwise) by dynamic trading in calls and stocks. Pretty much the same material Carr presented earlier this month at Stanford: http://finmath.stanford.edu/seminars/documents/stanfordtalk1.pdf, http://finmath.stanford.edu/seminars/documents/BSjtd.pdf
- The key point was that using Black-Scholes with "jump to default" (developed by Merton in 1976 paper entitled "Option pricing when underlying returns are discontinuous") allows the defaultable bond to be replicated with only two securities (the call and stock) because they will all jump to zero ("a call on a worthless stock is worthless") when the company defaults.

**[january 27, 2006]**

Good explanation of principal component analysis starting from the basics.

**[january 26, 2006]**

Discussion on dividend swaps on Wilmott. Just learned about these yesterday in Continuous Time Finance. It allows the dividend stream to be stripped from an investment, usually an index like the S&P500, and sold separately. The floating leg can be replicated by shorting a forward contract, lending long bonds such that they pay off the forward price at expiry and going long one dividend paying stock. You collect dividends on the stock until expiry at which time the forward and bond cancel out the final value of the stock. (check out notes on "Model Free Dynamic Replication") Apparently the payment for the dividend stream is often used to buy a call option.

Uses? Apparently taxes are the main one - e.g. "Dividend swaps are employed when an investor cannot take advantage of tax benefits which may accrue to another, or cannot use investment opportunities (such as a scrip dividend alternative) more valuable to another investor, and decides to lend the shares to a borrower for whom the dividend is more valuable. Thus, both share in the benefit." (from http://www.icgn.org/organisation/documents/slc/code_final.pdf). However "they can also be a simpler way to hedge a large block of dividend risk for a relatively reasonable price."

More info on this Wilmott thread http://www.wilmott.com/messageview.cfm?catid=3&threadid=26937 with its somewhat dim view of corporate ethics: "Div swaps are typically done on major indexes like SPX and Eurostoxx which can make the contracts fairly standard and liquid, not so much on high-yield baskets, and definitely not on single stocks which should make every desk wary that I might be a chairman in desguise wanting to receive dividends on my shares right before going to the board to increase them."

**[january 25, 2006]**

The internship season is upon us and with it the need to worry about interviews and interview questions...

**[january 21, 2006]**

Funny blurb in Saturday's WSJ about a stock picking contest for Playboy playmates. Apparently Jenny Mc Carthy's sister Amy was up more than 20% with a portfolio of five stocks at the end of trading Thursday. This was the part that made me laugh...

*Some experts scoff at the models' early success. "In any large group of stock pickers, some will show great success over short periods on a purely chance basis," says Eugene Fama, finance professor at the University of Chicago Graduate School of Business. "The nature of journalism is to focus on such chance outcomes."*

Hm...guess Fama's not looking for an invitation to the mansion. Still, can a mutual fund be far behind? Can you imagine the prospectus?

**[january 16, 2006]**

Agam Brahmi is a CS major at UNC who's not quite sure what to do next... I'm not sure if he's decided yet but two of the avenues he's considering are math finance or computing in finance. He's posted some interesting links recently on those topics and its worth digging into the archives for more (and also for an entertaining digression about why he's 99.99% sure he's not doing a PhD?... systems/networks boring... biocomputation? oh how things have changed). Here are the recent links that caught my eye:

I've added this (and wallstreetprogrammer) to MathFinLinks. I think the SUNY link can go into MathFinNotes.

go to MathFinArchive