I was looking for a mathematical formula for the probability of a stock price touching a strike price of an option any time during it’s lifetime, assuming the stock price follows a traditional Black-Scholes model of Brownian Motion. I found some traction from from http://quant.stackexchange.com/questions/235/probability-of-touching, but unfortunately the equation did not seem to work according to freely available online calculators. My understanding is that ThinkOrSwim has this as a “proprietary” calculation on their options, however I don’t have that platform, and would like to calculate this “in house,” so to speak.

Anyway, one of the answers to the previous post linked to https://github.com/barrycarter/bcapps/blob/master/box-option-value.m, which is a Mathematica script for calculating the probability of a stock price hitting a “box” in a matter of x hours.

It follows the equation ..

Now, I am not a big math person, but this is saying this is the probability of a particle in Brownian Motion hitting “m” between time “t1″ and “t2″. The math here isn’t particularly important (well it actually is the most important, but for people like me who have to stand on the shoulders of someone who understands how to actually come to the above equation, it is just a piece of information in a puzzle).

So, after experimenting with the best way to actually use this script, I first converted it to Sage, an open source mathematics program that only works on Linux. This was a kind of cludgy solution that I did not like since my “toolbox” of finance programs that I build is run on .NET on Windows. Eventually I found that I could do the entire calculation in Python using the open source SciPy and NumPy. After a failed attempt at getting those to run on IronPython for .NET (which honestly should work, it is just in its infancy really and I think it is simply a bug), I decided to just use native Python on Windows. This works well for my purposes since the program can simply call the python script at the command line, pass the appropriate parameters, and then parse the result.

Here is the above script that was written in Mathematica, rewritten into Python using SciPy and NumPy.

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import sys
import scipy.stats
import scipy.integrate
import scipy.special
import math
from scipy import inf
 
if (len(sys.argv) < 7):
print "Must specify all variables"
 
p0 = float(sys.argv[1])
v = float(sys.argv[2])
p1 = float(sys.argv[3])
p2 = float(sys.argv[4])
t1 = float(sys.argv[5])
t2 = float(sys.argv[6])
 
def pdflp(x,p0,v,t1):
return scipy.stats.norm(math.log(p0), math.sqrt(t1)*v/math.sqrt(365.2425*24)).pdf(x)
 
def cdfmaxlp(x,v,t1,t2):
return (1-scipy.special.erf(x/(v*math.sqrt(t2-t1)/math.sqrt(24*365.2425/math.sqrt(2.0)))))
 
def upandin(p0,v,p1,p2,t1,t2):
integrand = lambda x: (pdflp(x,p0,v,t1) * cdfmaxlp(math.log(p1)-x,v,t1,t2))
return scipy.integrate.quad(integrand,-inf,math.log(p1))
 
def hitleftedge(p0,v,p1,p2,t1,t2):
integrand = lambda x: (pdflp(x,p0,v,t1)*1)
return scipy.integrate.quad(integrand, math.log(p1), math.log(p2))
 
def downandin(p0,v,p1,p2,t1,t2):
integrand = lambda x: (pdflp(x,p0,v,t1) * cdfmaxlp(x-math.log(p2),v,t1,t2))
return scipy.integrate.quad(integrand, math.log(p2), inf)
 
def probabilitytouch(p0,v,p1,p2,t1,t2):
return (upandin(p0,v,p1,p2,t1,t2) + hitleftedge(p0,v,p1,p2,t1,t2) + downandin(p0,v,p1,p2,t1,t2))
 
print probabilitytouch(p0,v,p1,p2,t1,t2)

Now you can just run this script from the commandline, like “python scriptname currentprice volatility strike1 strike2 time1 time2″. So, for example, let’s take SPY, the popular ETF for the S&P 500. Right now the price of the stock is $128.60, the volatility is 21.49%, and let’s look at the probability of hitting a strike of $130.00. Notice that we are not using multiple strikes to price a box like the original script intends, which is fine. And an expiration of November 19th, which is 21 days away (21*24 is 504 hours).

I would simply run the script in a command prompt (assuming python, SciPy, and NumPy are installed).

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c:/python27/python.exe "c:/myscript.py" 128.60 .2149 130.0 130.0 1 504

This gives the output ..

(0.8027563748048097, 8.583401667801735e-09, 0.0, 0.0, 1.1850503070997317e-06, 1.1480513621859987e-11)

There are 6 numbers here separated by commas. All together they represent the probability of the price hitting the “box” between strike1 and strik2 (which in our case is simply a line because they are the same value, $130). Each probability has a number after it (such as 8.583401667801735e-09) which represents the error of the calculation, so values 1, 3, and 5 are the ones we are looking at. We add the first value, 0.8027563748048097, to the third value, 0.0, to the 5th value, 1.1850503070997317e-06. Now since the 5th value is so close to 0 we count it as 0. The probability is therefore .8027 or 80.27% of the stock hitting $130 between now and 21 days from now.

Keep in mind this follows the idea of Brownian Motion of stocks and all the rules of the Black-Scholes formula and is not “law” for where the stock will be 21 days from now. That is to say, this is a mathematical approximation of an approximation of a process that cannot really be described mathematically. So why is it valuable?

Well, the way I am using it is not to predict where a price will be, but where it has a low percentage to be, or more simply to find where the price has a high probability of not being on a certain date. This helps in a strategy where you are selling options (more specifically, veritcal credit spreads). We want to know the probability of a short strike being hit, and typically we want a 20% or lower chance of a short strike being hit. This formulation gives us something to quantitatively calculate and shoot for. It is important that this not take on the only piece of the decision making process though, and you should use some kind of outlook or technical analysis to further analyze where you expect the stock to go.

There is a workaround for the silly way in which Vista checks to make sure that you are actually upgrading a previous version of Windows. I wrote about this previously, where I actually had to install Windows XP to use the 32 bit upgrade of Windows Vista after I had “mistakenly” formated over my XP installation to install a clean copy of Windows Vista.

There is a way around this that I mentioned in the previous article though.

Install Windows Vista 32 bit on a clean partition using the partition tools, but do not enter in a license key. Once Vista is completely installed you can boot into it for the first time. At this step you will figure out that even when entering in a valid key (although for an upgrade), you will be unable to actually use it because Vista still demands that it be upgraded from a previous version and not a clean install. To get around this, just install Vista on top of itself as an upgrade which circumvents the whole purpose of this silly routine … making sure you have Windows XP or Windows 2000. When you upgrade your unlicensed version of Windows Vista 32 bit you will have no ill effects, a “semi” clean install, and a useless “Windows.old” directory that you can just delete. With this method you do not have to touch your XP disk, although you will have to install Vista twice. What a pain.

Alright, I went to add notepad and wordpad to my Send To folder like I always have for each of the previous operating systems. Thanks to the “simplified” (and admittedly cleaner) users folder and the disappearance of settings, I could not find the way to customize the Send To command. A search in the Vista help produces basically nothing.

I finally found the solution. Just enter in the following in your power search box (Start -> “Start Search”).

%APPDATA%\Microsoft\Windows\SendTo

Once the folder is opened just drag shortcuts of any program you would like to appear there into that folder. Not as easy as WindowsXP is it?

An alternate title for this post could be “How I managed to memorize my Vista key in one day.”

In previous versions of Windows when you purchased the Upgrade version of the new operating system you could always just insert the CD of the previous version during the install to verify that you indeed had a previous version and were eligible for the upgrade version. In Vista, as far as I can tell, this is not possible. It actually checks for a previous operating system for an upgrade key. This makes for a huge inconvenience that in the end is completely retarded because there is a “workaround” that nullifies the whole process.

So I get my copy of Vista Ultimate the other day, mostly because I am technical enough to troubleshoot the problems, the first DirectX 10 games are coming out soon, and I like eye candy. I proceed to jump right into Vista x64 (the 64 bit version). I take my Windows XP partition, along with the 15 other partitions that I foolishly made (to keep partition sizes down, which I honestly do not care much about anymore), delete all of them, and create 3 new partitions from within the Vista install program. I format and install Vista onto the smallest of the three and everything installs great. It loads up no problem, all my drivers (except for my graphics card and ATI remote wonder) are pre-installed with the OS. I easily find my video card drivers on Nvida and install them easily. The ATI remote wonder, unfortunately has some problems, but by installing the manufacturer drivers (X10) and then the software I overcome everything. However, I my Disk Stakka does not seem to work in x64 and in looking in the benefits of x64 right now, and the drawbacks of device compatibility, I decide to switch back to x86 (32 bit).

Now comes the fun. What I should have tried (and since I prefer clean installs I did not try this) was to start the x86 disk inside of Windows itself and tried to “downgrade” through that. Unfortunately I cannot tell you if that works because before I could try that I had already started my standard clean install routine that I have done throughout all versions of windows.

So I am in the Vista install program after booting off the DVD and I decide to format the OS partition and do a clean install. It does not stop me from doing that. I go to install onto the clean partition and then it asks for the key. No problem. I put in the key and then Vista says, wait a second … you are an Upgrade version! I need to see an older version of Windows to allow you to do a 32 bit upgrade. This was not a problem with a 64 bit upgrade because they require you to do a clean install anyway.

So it wants me to install Windows XP to install Windows Vista 32 bit? I am not prepared to give up just yet though, so I choose not to enter in a key at all and just authenticate when Windows is up and running. So I go through the whole install process again and load up windows. I go to the windows authentication screen and enter in my key …. only to get an error about my key being for an Upgrade only. At this point I am pretty pissed off because sitting next to me is my shiny XP CD and I am literally begging the computer to just read it and say that its ok, I am not a pirate trying to steal software.

I end up biting the bullet and installing XP and then doing the stupid upgrade (I hate upgrades too). Luckily, since I just installed XP very fast, no service pack 2 or anything (I didn’t even activate my XP install, the first time it booted I inserted Vista and started the install process), it “forced” me to do a clean install. Unfortunately it did leave a “windows.old” directory on the partition, but that was easily deleted. I was worried that it would leave an older version of NTFS on there, but as far as I can tell all the Vista features (which consists of symlinks) work.

Later I found a nice workaround that negates the entire process and makes Microsoft look really stupid. Here I was jumping through hoops to make sure Vista knew that I had XP and was upgrading properly. To work around that all you have to do to do a “clean” install without XP is install Vista without a key. Once you boot up into Vista for the first time, just insert the CD/DVD again and then choose to “upgrade” to the same exact version. You don’t even have to let the software know that XP was ever installed. Using this process this could have been the first time you used Windows and you can use the upgrade option.

Way to go Microsoft. If securing your upgrade versions from pirates and users who are not really upgrading was your goal, you utterly failed. In the process you managed to make everyone else jump through hoops.

This post is obviously a spoiler, I would avoid reading it if you have not seen the show! Then again you can probably predict the ending if you have ever imagine up a semi-happy ending to some kind of hero like possible disaster.

Was it just me or was the Heroes big finale on Monday more of a big Fizzle? All season we have a build up to a supposedly spectacular ending that involves a bomb that is supposed to go off (from mysterious circumstances no less) in the middle of New York City. What we end up with is a cookie cutter ending that seems taken directly from any number of super hero comic books. That is all well and good, because obviously Heroes is a super hero show inspired by such works. On the other hand, the show had a kind of sophisticated story line (quasi-Lost like almost) that intrigued us all season. To end on such a predictable and comic book like ending was a let down for me, personally, because I expected some sophistication and surprise.

There were some surprises that we encountered. The little girl, Lilly, who can see anyone in the world and their exact location, mentioned that she knew of a person that was more evil than Silar and could see her when she saw him. Seeing as we are short some villains after the Finale and that the “League of Heroes” is together (corny and predictable), they might as well fight this new super villain (if the next season wasn’t so weird, more on that later).

Peter must live. He is the main character, so I expect him to live through that. I will be disappointed if he did not. I am already disappointed because the future that the showed a couple episodes before the finale, the bleak dark future where Silar is president and Peter is jaded but powerful, the world is splintered, and there is a kind of silent war on “heroes” … now that was interesting! I want a season about that future, not the warm fluffy future where the bomb does not go off and love between brothers conquers evil. Give me a break, I really hoped after seeing that bleak future that Heroes would finally surpass Lost as a sophisticated show (and yes I know Lost has its silly main stream unsophisticated moments). Unfortunately, only a mere couple of episodes later they disappoint me with the finale no less.

Then there is next season.

We could have new people and new storylines and new ideas and new threats and new bad guys and new heroes. So I would prepare the audience for that idea, that it’s not just a continuing serialized storyline about only these people. It’s a little more the 24 model than the Lost model.

I heard rumors that season 2 will be a prequel, however I have not confirmed that on the internet except for the fact that a twin mini-series will run while Heroes takes its break sometime next season called “Heroes:Origins” that explains the back stories of the characters. However, it seems here that we will see the some of the same characters but the writers are giving themselves the opportunity to trash, forget, or create characters at their discretion. You have to be cautious here though, since the story is character based. As Lost has found, introducing new characters can be a tricky affair, and many times people tend to just plain hate new characters because they appear so shallow to the old characters that have been well developed. However, unlike Lost, Heroes characters are not all that well shaped and developed. Instead of having to develop complex character personalities and back stories (ala Lost), Heroes simply adds a simple back story and personality and lets the audience concentrate on their super power. So shuffling characters in and out may work out for Heroes. Only time will tell.

I will say, it is a blessing and a curse, this final episode. One thing that I hate is an epic cliff hanger that I have to wait all summer to find the ending to. One thing that I love is an epic cliff hanger that I have to wait all summer to find the ending to. In the case of the Heroes finale, I won’t have to deal with either situation because it was not epic or much of a cliffhanger at all.

I am in the process of reviving this blog and finishing some kind of easy redesign. WordPress has grown a lot since I started using it, so I am incorporating many things quickly with themes and widgets, but will have to customize everything over time.

I am very busy lately since I graduated with a new job. It turns out that you have very little free time after college. Trying to fit everything you want to do and everything you have to do is hard in the few hours after work, and then the weekends you are busy doing even more stuff.

I would say stay tuned .. but chances are I am the only one reading this. Plus that weird guy Tom who keeps reading my blog.

I am in the process of redesigning once again to a more modern and functional theme based on K2. Have more time to work on this now that I have graduated from college.

Stay tuned.

It’s fairly easy, and a construct of existentialist philosophy.

There are two branches of explanation to the meaning of life. These are broken down into consequences and content. Consequences would be an expectation that life is building up to an ultimate goal, or that anything a human being does has an inevitable consequence and meaning. On the other side is content, where the meaning is within life itself.

Now under each of these categories there is a subjective and objective explanation as to the meaning of life.

Subjective Consequence

This is best described in religion. Instead of confronting the meaning of life, or looking for it in consequence that can be measured or observed, we look beyond our own world and explanation into the heavens. Thus the meaning of life is not discovered or attained until death, and then it is only subject to a deity or other religious belief. The problems with this are the same old problems with religion when discussed in a philosophical context. There is the problem of God and the status of God as evil or good, the notion that one must die to know, and the lack of any kind of reasonable evidence in our everyday experience.

Objective Consequence

Rocks. This is how my professor in existentialist philosophy put it. Objectively, that is using evidence in the real world, and observed logical conclusions, the consequences of our actions are just a different pattern of atoms. If I decide to throw my pencil across the room, then it is across the room. If not, then it is still here. The consequence of such actions, in context of the meaning of life, is that when all is said and done and billions of years have passed, rocks in outer space may be in a different place.

Subjective Content

The meaning of life is whatever makes me happy and whatever I see as valuable to me. This sounds like a reasonable claim. However, it is not. This is a leap away from the question, avoiding the meaning of life by assigning value to meaninglessness. It is the path of empty desire. Self fulfilling and is more of a distraction than anything else.

Objective Content

Meaning comes from everyone around us, and is evaluated objectively and logically. The ultimate truth of which comes with death. Everyone dies. Therefore the one objective and most reasonable answer for the “Meaning of Life” is death itself.

Makes you feel all warm and fuzzy, huh? Those existentialist are sure the life of the party.

This man has obviously never been laid

Our favorite asshole lawyer, Jack Thompson, has discovered *gasp* that people are actually naked under their clothes. This revelation came when Jack discovered that the game Oblivion accurately rendered women nude under their shirts.

Like most Republican prudes, Jack has never seen a naked woman and believes that people come from storks, at the ordering of God. He was also dropped on his head as a child.

I really do not understand the issue here. He does not even mention what I have already pointed out the most shocking aspect of this mod. How does plain nudity offend anyone, *unless* they are retarded prudish idiots? Maybe someone needs to talk to Jack about how babies are made.

In a game where you have missions to murder people and can kill people with swords and magic at will, the one thing the media and zealots pull out is a couple breasts. My solution? Get laid. I swear how messed up is this country? Isn’t this fairly analogous to art? What’s next, a condemnation of naked Barbie dolls? Perhaps you should rate them mature. People are naked, this is a fact. People have sex, this is a fact. There is no choice in the matter either, both of these things have to happen to continue living as a species. What does not have to happen is senseless violence. The fact that in American culture the thing that is natural is taboo and the thing that is not is totally integrated into our culture is just plain sad.

Don't cry Kobe, you Vindictive Bastard

Kobe Bryant likes to prove his points in the playoffs, and he does not care if it means that the team goes home for the summer. He is the most vindictive bastard in the NBA and will make his team suffer because he feels slighted by the press. Tonight Kobe took only 3 shots in the entire second half and only scored 1 point off a technical shot. What an ass. I am only mad because I was forced to pick him in my playoff fantasy game and I thought he would fire away because with the Suns finally reaching their normal tempo, the Lakers pretty much had little chance playing an uptempo game.

We have seen it before. The Lakers take a loss when Shaq and Kobe are on the same team, a loss where Kobe takes a bunch of shots, and the media criticizes him. The next game Kobe comes in and takes not only less shots, but so few it leaves everyone scratching their head. There are only a couple explanations for this.

1. Kobe was taken out by defense – This is the most improbable. Kobe himself recognizes he can get any shot he wants, he has said so. We all know he can.
2. Kobe was told by Phil Jackson to not take shots or he thought his team had a better chance to come back by passing off – No, again not likely. Phil Jackson has no problem with Kobe taking shots. Down by 15 at the half and watching it go up to 25 by the 4th should be a good reason to start taking the ball .. if nothing else to throw the crowd off.
3. Kobe was criticized for taking too many shots in Game 6, he “shows them” that they need him more than he needs them – Vindictive bastard raise your hand. I guess he made his point. You want me to take no shots? We lose by 30+.

Point taken Kobe. Thank you. Go home.