Exploring Brownian Motion and Coin Tossing in Relation to Time and Bet Size

[courtrigrad]

Let's say we restrict 6 coin tosses to a period t so that each toss will take $$\frac{t}{6}$$. The size of the bet is $$\sqrt{\frac{t}{6}}$$

Then why does $$\sum^n_{j=1} (S_{j}-S_{j-1})^{2} = 6 \times(\sqrt{\frac{t}{6}}) = t$$. Or more generally why does:

$$\sum^n_{j=1}(S_{j}-S_{j-1})^{2} = n\tiimes(\sqrt{\frac{t}{n}})^{2} = t$$

Also why does $$E[S(t)] = 0 , E[S(t)^{2}] = t$$?

Thanks

 

[courtrigrad]

any ideas or advice?

thanks

 

[polyb]

any ideas or advice?

thanks

Question: Is this for a high school or university level course work? Or are you just doing your own research? The three posts that you have started is usually geared towards the advanced upper level undergraduate work and/or graduate level work. Just curious!

BTW, here is a link to the mathworld write-up on these particual subjects. Though they are brief it may help!

http://mathworld.wolfram.com/WienerProcess.html

You should be able to link to the other questions from this page.

 

[courtrigrad]

I am a high school student doing a science fair project in this topic

Thanks

 

[polyb]

I am a high school student doing a science fair project in this topic

Thanks

Brownian motion can be likened to flipping a coin in a series. Even though there are two possibilities the statisitcs work out to describe very robustly the random process. Brownian motion is a great way to introduce yourself to random processes and the coin flipping is real easy. Einstein's lesser known work had everything to do with brownian motion and diffusion.

What exactly are you trying to focus your project on because there is a lot to this subject and it is easy to get lost in the thicket of theoretical underpinnings.

 

[courtrigrad]

I am focusing on Black-Scholes Equation but studying this ins advance.

 

[polyb]

I am focusing on Black-Scholes Equation but studying this ins advance.

So what are you trying to demonstrate, that even though markets are stochastic in nature that the risk function of a protfolio can be worked through and accounted for?

BTW, what do you know about Tsallis statisics and/or non-extensive systems?

 

[courtrigrad]

I am an extreme beginner. I was focusing on a specific problem in my quantitative finance bok involving the density of options using Black Scholes Equation.

 

[polyb]

I am an extreme beginner. I was focusing on a specific problem in my quantitative finance bok involving the density of options using Black Scholes Equation.

I'm guessing, at least with that problem, your are looking to show that as the density of options(or derivatives) increases the less the risk to the protfolio, which is the whole point of Black Scholes, risk minumization. WOW! I bet you feel like you got more than you bargained for!

Here is a small write up on black scholes from math world:
http://mathworld.wolfram.com/Black-ScholesTheory.html

There are some citations and links in there that you may find useful. Other than that the subject matter you are approaching gets pretty involved and there is a lot to it, as you may have noticed by now. If you are up to it, you may want to look into econophysics, which is all about what you seem to be investigating. I doubt if I could help with much else at the moment but if you could, tell me what your project is covering. There is a lot to your topic and it may help to slim it down to something that you would be more comfortable with!

 

[courtrigrad]

yea I am using Wilmott's book, and basically I actually need help narrowing down my topic.

thanks

 

[polyb]

yea I am using Wilmott's book, and basically I actually need help narrowing down my topic.

thanks

So what ideas are you working on to narrow this down? There is a lot and for a high school science project it would seem reasonable to include some science before jumping off into finance land.

For example you could easily talk about random processes, brownian motion, statistics, and/or stochastic processes and how they relate to finance plus how finance got the math/theoretical framing from physics. Of course it would seem that more thought would be put into the physics than finance but then again that would be 'econophysics'. Here are a couple of links:

http://www.econophysics.org/
http://www.unifr.ch/econophysics/

There is plenty of stuff in both of those sites, maybe it will help. How far along are you in your math? physics? finance?

 

[courtrigrad]

I am currently studying calculus from Courant's book and took an AP Statistics class. I was in a dilemma as to whether I should first finish all the problems in the calculus book, but I realized that the quantitative finance book teaches you for the most part the calculus. Then I plan to continue formally studying calculus. Currently I have not taken a physics course. Now in finance I have basically gone over put and call options, no arbitrage, long and short calls etc..

The main thing I need is a problem where I can actually develop a hypothesis. I looked at some problems at the book and considered doing those. But are those really science fair project topics?

Thanks

 

[courtrigrad]

because it doesn't make sense to limit yourself physchologically right?

 

[polyb]

Not a problem!

You may want to consult with your teacher on this because it is a pretty tricky territory. I don't know if it could be considered a science fair project unless you started stressing the econophyics stuff and even then it may be a little over your head at the moment. You could pull it off but you would have to draw a lot topics into your project to bring it all together. That would be the hard part and I don't know what kind of time constraints you are under, so go talk to a teacher that could advise you.

Oh yeah, your last question from the top of the thread, I think that is more about establishing the statistcal independence of the events in the process. I think the first statement is about the distribution, which for a Gaussian one averages out to 0, and the second one is the standard deviation of that distribution. I could be wrong though, I don't have any notes or books with me and it has been a while since I touched on this subject!